Download - Time-dependent inverse box-model for the estuarine circulation and primary productivity in the Strait of Georgia

Time-dependent Inverse

Box-model For The Estuarine

Circulation and Primary

Productivity in The Strait of

Georgiaby

Olivier Riche

Licence et Maıtrise, Universite du Havre, 1997D.E.S.S, Universite de Toulon et du Var, 1998M.Sc., Universite du Quebec a Rimouski, 2002

A THESIS SUBMITTED IN PARTIAL FULFILMENT OFTHE REQUIREMENTS FOR THE DEGREE OF

Doctor of Philosophy

in

The Faculty of Graduate Studies

(Oceanography)

The University Of British Columbia

(Vancouver)

September, 2011

c© Olivier Riche 2011

Abstract

During 2002–2006, a comprehensive set of observations covering physical, biological,

radiative and atmospheric parameters was obtained from the southern Strait of Geor-

gia (SoG), Western Canada by the STRATOGEM program. Monthly time series of

estuarine layer transports over 2002–2005 were estimated using a time-dependent

2-box model in a formal inverse approach. These transports are then consistent with

the temperature and salinity fields, as well as riverine freshwater inflow (R) and at-

mospheric heat fluxes. Uncertainty was analyzed by resampling observations using

bootstrap methods. The transport time series were then combined with observa-

tions of nutrient concentrations to construct monthly time series of nutrient uptake

for nitrate, phosphate, and silicic acid.

Analysis of these time series suggests that the SoG estuarine circulation is not

very sensitive to the seasonal changes of R. Comparison of the surface layer transport

(U1) and R yields the first observational relationship between the SoG estuarine

circulation and R. This relationship (U1=2.68 m2s−2/3 × 103 R1/3) is consistent with

estuarine theories. Although the flows change slightly with the freshet, a 5-fold

change in R results only in a 40% change in U1.

Based on the calculated sink of near-surface nutrients, net primary productivity

is estimated to be 212 gC m−2 yr−1, which is similar to values obtained differently

in similar estuaries. Comparison of the nitrate and phosphate uptake rates suggests

that the primary productivity (PP) is mainly new PP during spring and summer.

ii

Abstract

Thus, PP is mainly controlled by the upwelling supply of nutrients through deep

inflow and entrainment. The uptake of silicic acid (Si) is almost two times larger

than the uptake of nitrate during diatom spring blooms, while it is similar during the

summer blooms. Such a high Si uptake suggests that spring diatoms form heavier

frustules or that heterotrophic silicoflagellates compete with diatoms for Si.

Speculative considerations based on comparison of the estimated production rate

of near-surface oxygen and new PP also suggest that the regenerated PP is small.

In addition, the summer heterotrophic respiration might be in excess by as much as

2 gO m−2 d−1 relative to the net PP.

iii

Preface

The author’s contribution to this work was: a) the collection of some of the data in a

field program designed by others, and b) the analysis of the entire dataset described

here. The author will be the lead author in manuscripts to be submitted.

iv

Table of Contents

Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ii

Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iv

Table of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . v

List of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viii

List of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . x

Glossary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiii

Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xviii

Dedication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xix

1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.1 Physical Oceanography . . . . . . . . . . . . . . . . . . . . . . . . . 5

1.2 Biological Oceanography . . . . . . . . . . . . . . . . . . . . . . . . . 11

1.3 Two-layer Model and Governing Equations . . . . . . . . . . . . . . 14

1.4 Objectives, Approach, Thesis Contributions and Plan . . . . . . . . 17

2 Inverse Methods and Box Models . . . . . . . . . . . . . . . . . . . . 19

v

Table of Contents

2.1 Introduction: Mathematical Framework . . . . . . . . . . . . . . . . 19

2.2 Inverse Problem: Estimating the Circulation of the Strait of Georgia 29

2.3 Solution Uncertainty and Residuals . . . . . . . . . . . . . . . . . . . 46

2.4 Forward Problem: Estimating the Net Primary Productivity . . . . . 49

3 Observations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54

3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54

3.2 Data Sources . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54

3.2.1 STRATOGEM . . . . . . . . . . . . . . . . . . . . . . . . . . 54

3.2.2 JEMS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56

3.2.3 Freshwater Inflow, Surface Heat and Air-sea Fluxes . . . . . . 57

3.3 Box Model Inputs . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58

3.3.1 Separation Depth . . . . . . . . . . . . . . . . . . . . . . . . 59

3.3.2 Spatial Averaging and Hypsography . . . . . . . . . . . . . . 60

3.3.3 Time Dependence . . . . . . . . . . . . . . . . . . . . . . . . 61

3.3.4 Input Time Series . . . . . . . . . . . . . . . . . . . . . . . . 61

3.3.5 Air-Sea Oxygen Flux . . . . . . . . . . . . . . . . . . . . . . 71

3.3.6 Riverine Inputs . . . . . . . . . . . . . . . . . . . . . . . . . . 71

4 Circulation and Transports . . . . . . . . . . . . . . . . . . . . . . . . 79

4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79

4.2 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81

4.2.1 Estimates of Estuarine Transports and Mixing Exchange . . . 81

4.2.2 Sensitivity Analysis . . . . . . . . . . . . . . . . . . . . . . . 85

4.2.3 Residuals of the Conservation Equations . . . . . . . . . . . . 96

4.3 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101

vi

Table of Contents

4.3.1 General Circulation in the Box Model . . . . . . . . . . . . . 101

4.3.2 Seasonal Transports . . . . . . . . . . . . . . . . . . . . . . . 105

4.3.3 Comparison With Transports in the Strait of Juan de Fuca . 107

4.3.4 Net Outflow from the Strait of Georgia . . . . . . . . . . . . 113

4.3.5 Circulation Sensitivity to the Freshwater Inflow . . . . . . . . 115

4.3.6 Annual Variability of the Circulation . . . . . . . . . . . . . . 117

4.3.7 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121

5 Nutrients Uptake and Primary Productivity . . . . . . . . . . . . . 125

5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125

5.2 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129

5.2.1 Supply and Sink Rates of Nutrients . . . . . . . . . . . . . . 129

5.2.2 Supply and Sink Rates of Oxygen . . . . . . . . . . . . . . . 138

5.2.3 Uptake Ratios . . . . . . . . . . . . . . . . . . . . . . . . . . 144

5.2.4 Estimates of Net Primary Productivity . . . . . . . . . . . . 149

5.2.5 Estimates of Net Community Productivity . . . . . . . . . . . 157

5.3 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 160

6 Discussion and Conclusion . . . . . . . . . . . . . . . . . . . . . . . . 183

6.1 Thesis Contributions . . . . . . . . . . . . . . . . . . . . . . . . . . . 183

6.2 Seasonality and Variability of Water Transports in Estuaries . . . . . 185

6.3 Estimates and Variability of Net Primary Productivities in Estuaries 195

6.4 Recommendations for Future Studies . . . . . . . . . . . . . . . . . . 201

Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 210

vii

List of Tables

2.1 Total Depths and Volumes of the Model Boxes . . . . . . . . . . . . . 30

2.2 Inversion Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

2.3 Coefficients of Variation in the Parametric Bootstraps . . . . . . . . . 48

4.1 Analysis of Variance and F-tests for TD and QSS Transports . . . . . 82

4.2 A Priori and Estimated Values of the Equations Residuals . . . . . . 99

4.3 SoG Transports from Previous Studies . . . . . . . . . . . . . . . . . 111

5.1 Averages of Surface Supply and Sink Rates of Nitrate (mol s−1) in the

Euphotic Zone of the SoG. . . . . . . . . . . . . . . . . . . . . . . . . 139

5.2 Averages of Surface Supply and Sink Rates of Phosphate (mol s−1) in

the Euphotic Zone of the SoG . . . . . . . . . . . . . . . . . . . . . . 140

5.3 Averages of Surface Supply and Sink Rates of Silicic Acid (mol s−1)

in the Euphotic Zone of the SoG. . . . . . . . . . . . . . . . . . . . . 141

5.4 Averages of Surface Supply and Sink Rates of O2 (mol s−1) in the

Euphotic Zone of the SoG. . . . . . . . . . . . . . . . . . . . . . . . . 142

5.5 Estimates of the Average Excess of Heterotrophic Respiration . . . . 160

5.6 Comparison of Si-replete and Iron-replete Bloom in Coale et al. [2004]

and SoG Spring Bloom . . . . . . . . . . . . . . . . . . . . . . . . . . 177

viii

List of Tables

5.7 Comparison of Si-depleted and Iron-replete Bloom in Coale et al.

[2004] and SoG Summer Bloom . . . . . . . . . . . . . . . . . . . . . 177

6.1 Ranges of PP rates in Temperate Estuaries . . . . . . . . . . . . . . . 197

6.2 Macronutrient Limitation During Summer . . . . . . . . . . . . . . . 198

ix

List of Figures

1.1 Geography of the Strait of Juan de Fuca/Haro Strait/Strait of Georgia 4

1.2 Detailed Sampling Area of STRATOGEM and Other Sampling Im-

portant Locations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

1.3 Along-channel Cross-section of the Strait of Juan de Fuca/Haro Str-

ait/Strait of Georgia System . . . . . . . . . . . . . . . . . . . . . . . 8

1.4 Fraser River Discharge . . . . . . . . . . . . . . . . . . . . . . . . . . 11

2.1 Physical Fluxes and Processes in the Box Model . . . . . . . . . . . . 31

2.2 Biogeochemical Fluxes and Processes in the Box Model . . . . . . . . 32

2.3 Chart of the Inversion Procedure . . . . . . . . . . . . . . . . . . . . 48

3.1 Vertical Profiles of Salinity and Temperature . . . . . . . . . . . . . . 62

3.2 Hypsography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63

3.3 Salinity and Freshwater Time Series . . . . . . . . . . . . . . . . . . . 65

3.4 Temperature and Surface Net Heat Flux Time Series . . . . . . . . . 66

3.5 Heat Flux Observations and Estimates . . . . . . . . . . . . . . . . . 68

3.6 Surface Heat Budget . . . . . . . . . . . . . . . . . . . . . . . . . . . 69

3.7 Phosphate Concentration Time Series . . . . . . . . . . . . . . . . . . 72

3.8 Nitrite+Nitrate Concentration Time Series . . . . . . . . . . . . . . . 73

3.9 Silicic Acid Concentration Time Series . . . . . . . . . . . . . . . . . 74

x

List of Figures

3.10 Dissolved Oxygen Time Series . . . . . . . . . . . . . . . . . . . . . . 75

3.11 Air-sea Oxygen Flux Time Series . . . . . . . . . . . . . . . . . . . . 76

3.12 River Biogeochemical Inputs Time Series . . . . . . . . . . . . . . . . 78

4.1 Transport Estimates and Their Errors . . . . . . . . . . . . . . . . . 86

4.2 Transport Estimates and Errors in Quasi-steady State . . . . . . . . . 89

4.3 Circulation Sensitivity to Separation Depth . . . . . . . . . . . . . . 91

4.4 Circulation Sensitivity to the Trade-off Parameter α=γ s−1 . . . . . . 94

4.5 Circulation Sensitivity to the a Priori Parameter β . . . . . . . . . . 97

4.6 Residuals of the Mass, Salt and Heat Equations . . . . . . . . . . . . 102

4.7 Comparison Between Li et al. [1999]’s Exchange Flow and U1 . . . . . 108

4.8 Advective Transports in HS Box Model . . . . . . . . . . . . . . . . . 110

4.9 Surface Seaward Transport Plotted With Respect to Freshwater Inflow 118

4.10 Residuals of the Fit of U1 as a Power of R . . . . . . . . . . . . . . . 119

4.11 Annual Mean of SoG Transports and Freshwater Inflow . . . . . . . . 122

4.12 Annual Variability of SoG Transports and Freshwater Inflow . . . . . 123

5.1 Surface Box Nitrate Supply and Sink Rates . . . . . . . . . . . . . . . 130

5.2 Surface Box Phosphate Supply and Sink Rate . . . . . . . . . . . . . 131

5.3 Surface Box Silicic Acid Supply and Sink Rates . . . . . . . . . . . . 132

5.4 Surface Box O2 Supply and Sink Rates . . . . . . . . . . . . . . . . . 133

5.5 Surface Nitrate and Phosphate Uptake Rates . . . . . . . . . . . . . . 150

5.6 Surface silicic acid and Phosphate Uptake Rates . . . . . . . . . . . . 151

5.7 Surface Dissolved O2 Release and Phosphate Uptake Rates . . . . . . 152

5.8 NPP Rate Estimates based on Net Biological Uptake Rates of N and P158

5.9 Estimate of the Chl-a-normalized NPP Rate . . . . . . . . . . . . . . 161

xi

List of Figures

5.10 Comparison of the Chl-a Normalized NPP Rate With Averaged PAR

over the Mixing Layer . . . . . . . . . . . . . . . . . . . . . . . . . . 162

5.11 Net Community Production Rate Estimates Based on Net Biological

Production Rates of O2 . . . . . . . . . . . . . . . . . . . . . . . . . . 163

5.12 Estimates of the Excess of Regenerated NPP rate and Heterotrophic

Respiration Based on NCP and New NPP Rates . . . . . . . . . . . . 164

5.13 Estimates of the Diatom and Total Autotrophic Biomass . . . . . . . 169

5.14 SoG Nitrate Concentrations at 0 m, Average at 0 m and Over 0–30 m 171

5.15 SoG N and Si Average Concentrations at 0 m and Over 0–30 m . . . 173

5.16 Estimate of the Net Biological Uptake Rate of Si . . . . . . . . . . . 174

5.17 Estimates of the Si:N and Si:P Ratios Based on the Net Biological

Uptake Rates of Si, N and P . . . . . . . . . . . . . . . . . . . . . . . 175

6.1 Comparison of STRATOGEM and IOS SoG Temperature Box Averages189

6.2 Comparison of STRATOGEM and IOS SoG Salinity Box Averages . 190

6.3 Comparison of STRATOGEM and IOS SoG Nitrate Box Averages . . 191

6.4 Histogram of FW Input in the SoG over 1912-2008 . . . . . . . . . . 194

6.5 Mean Monthly FW Input in the SoG . . . . . . . . . . . . . . . . . . 196

xii

Glossary

α Trade-off parameter

A′ Matrix A scaled by W−1

A Data matrix of m rows and n columns

b Data vector of m rows and 1 column, containing internal and external sources

c Data vector b associated with z

Chl-a Chlorophyll-a pigments

D′

r Diagonal matrix with diagonal coefficients d′i(α2), r rows and columns

d Separation depth between SoG top and bottom boxes

d′i(α2) ith Diagonal coefficient of D′

r(α2), equals to λ′2

i

λ′2

i+α2

ε Estimated residuals associated with x

ε Residuals vector of m rows and 1 column

E Net upward entrainment transport

F Heat flux

FSW SW component of F

xiii

Fall Fall starts in August and ends in October

FW FreshWater

GPP Gross primary production

HS Haro Strait

Ik Square identity matrix, k rows and columns

J Objective function to minimize to find x

k Effective rank ≤r and λk ≪ λi for i > k

λi ith Singular value of the SVD

Λ′ Matrix of the singular values of A′, m rows and n columns

Λ−1 Inverse matrix associated with Λ, n rows and m columns

Λ Matrix of the singular values of A, m rows and n columns

Λ′−1 Inverse matrix associated with Λ′, n rows and m columns

λ′

i ith Singular values of A′

LLS Linear Least Squares method

LW LongWave

M Mixing exchange rate due to vertical turbulence

m Number of rows of a matrix

N Nitrogen based on nitrate and nitrite

xiv

n Number of columns of a matrix

NCP Net community production

NPP (Total) net primary production

NPPn New net primary production

NPPr Regenerated net primary production

ωi ith Diagonal element of W

O2 Dissolved dioxygen gas

P Phosphate

PP Primary production

QSS Quasi-Steady State

R FW input rate in the SoG

r Rank of the matrix A with r≤ min(m,n)

Ra Autotrophic (phytoplankton) respiration

Rh Heterotrophic (zooplankton and bacteria) respiration

S Colum-scaling diagonal matrix, n rows and n columns

s Value of each diagonal element of S

Si Silicic acid

SoG Strait of Georgia

xv

SoJdF Strait of Juan de Fuca

Spring Since the earliest spring blooms can occur in February, spring is defined as

February-April

Summer Summer starts in May and ends in July

SVD Singular Vector Decomposition

SVDM Singular Vector Decomposition Methodology

SW ShortWave

Tu Data resolution matrix, m rows and m columns

Tv Solution resolution matrix, n rows and n columns

T Operator of matrix transposition

TD Time Dependence

U′ Square matrix of the data singular vectors of A′, m rows and columns

ui ith Column of U

Uk Rectangular matrix of the k first data singular vectors, m rows and k columns

U Square matrix of the data singular vectors of A, m rows and columns

u Vector of the flow speed

U1 Seaward surface horizontal transport of the estuarine circulation

U2 Landward deep horizontal transport of the estuarine circulation

xvi

U01, U02, W01, and W02 A priori values of U1, U2, W1, and W2

V′ Square matrix of the solution singular vectors of A′, n rows and columns

vj jth Column of V

Vk Rectangular matrix of the k first solution singular vectors, n rows and k

columns

V Square matrix of the solution singular vectors of A, n rows and columns

V1 SoG top box volume

V2 SoG bottom box volume

VH HS box volume

W Row-scaling diagonal matrix, m rows and m columns

W1 Downward vertical transport of the estuarine circulation

Winter Winter starts in November and ends in January, before the earliest spring

blooms starts

x Estimated solution vector

xA A priori solution vector of n rows and 1 column

xU Vector of the estimated transports associated with the estuarine circulation

x True solution vector of n rows and 1 column

x Mean of x

z Solution vector centered around xA

xvii

Acknowledgments

I wish to first thank professor Rich Pawlowicz for giving me the opportunity to

do research in field oceanography in Beautiful British Columbia. I also wish to

thank him for his longstanding academic and financial support during both the field

sampling phase and the thesis writing phase of my PhD. I would also like to thank

the professors who have been or are still on my supervisory committee: professors

Susan Allen, Maite Maldonado, Doug Oldenburg, Philippe Tortell, and finally Grant

Ingram. They provided guidance during the long process of shaping my research.

Professors Allen and Maldonado also helped reviewing several versions of my thesis

and provided very valuable insights. Finally, I would like to thank the people, too

many to be named, who helped in the collection and analysis of the data used in my

research.

xviii

Chapter 1

Introduction

Estuaries are very productive marine ecosystems. Marine organisms living in an estu-

ary can either migrate in and out (e.g., salmon, large marine mammals) or reside there

for most of their life cycle (phytoplankton and zooplankton). Phytoplankton and zoo-

plankton are food sources for higher levels in the food web. An estuarine ecosystem

represents an important asset for local economy and for recreational tourism. In an

estuary, the economical and recreational exploitation of the ecological resources can

greatly contribute to the dynamics of the ecosystem when the exploitation becomes

excessive, e.g. sport fishing [Peterman and Steer 1981]. Monitoring the ecological

condition of an estuary (e.g. biomass of phytoplankton and growth rate, nutrient

concentrations and uptake rates as estimated in Chapter 5) is an important step in

understanding the dynamics. Better understanding of the ecosystem dynamics can

help develop a sustainable estuary in the long term. The motivation of this thesis

is to study the recent conditions of the Strait of Georgia (SoG) and to provide esti-

mates of the estuarine circulation of the SoG (chapter 4), the primary productivity

of the SoG ecosystem, and its variations over seasonal and interannual timescales

(chapters 5 and 6).

In the rest of this section, I will introduce the concepts of estuary, estuarine circu-

lation and box model. In the next sections, I will introduce the physics (section 1.1),

the biology (section 1.2) of the SoG and the mathematical equations (section 1.3),

and define the objectives, the approach and the significant contributions of the thesis

1

Chapter 1. Introduction

(section 1.4).

An estuary is “semi-enclosed body of water where freshwater and seawater masses

mix or overlap forming vertical and horizontal gradients along their main chan-

nel” [Cameron and Pritchard 1963]. Real-life estuaries that fit this definition can

greatly differ in geomorphology and hydrological regimes [Dyer 1973].

Studies typically focus on shallow, partly-mixed estuaries under a temperate cli-

mate, e.g. the Chesapeake Bay estuary [Harding et al. 2002]. Apart from partly-

mixed estuaries, the possible salinity structures of an estuary [Dyer 1973] can be

highly-stratified salt-wedge estuaries (e.g. Fraser River mouth [Halverson and Pawlow-

icz 2008]), fjord-like estuaries (e.g. SoG [Thomson 1994, Pawlowicz et al. 2007])

characterized by a deep basin separated from the shelf by a sill near the mouth, and

well-mixed estuaries (e.g. lower Delaware estuary [Moore et al. 2009], Gwangyang

Bay [Shaha et al. 2010]). The literature contains descriptions of estuaries over a

wide geographical range, in: arctic regions [Sorokin and Sorokin 1996], subtropical

regions [Wua and Chou 2003], and subpolar regions [Kristiansen et al. 2001].

An estuary tends to get narrower landwards and quickly widen seawards. This

geomorphological feature varies from estuary to estuary [Gay and O’Donnell 2009]

and plays a role in the amplification of the seaward flow, as discuss later in this

chapter.

An estuary is a dynamic physical system. Horizontal and vertical transports can

change the properties of the water column. Organisms can be transported vertically

to shallow or deep waters, and horizontally into and out of the estuary. The transports

can also alter factors necessary for growth and survival of organisms (nutrients, light,

food abundance, etc.). Thus, the physics and the biology of an estuary are coupled

processes. Since they vary over many time and space scales, these processes are

difficult to sample, analyze and understand.

2


The Strait of Georgia Ecosystem Monitoring (STRATOGEM) program was an

example of an extensive study of the Strait of Georgia (SoG), an important estuarine

system in the coastal waters of British Columbia [see Figs 1.1, 1.2, and Pawlowicz

et al. 2007]. One of the goals of STRATOGEM was to understand the variability of

estuarine circulation and primary productivity over the entire SoG using 3 years of

data [Pawlowicz et al. 2007]. This thesis fulfils this goal of STRATOGEM by pro-

viding time series and averages of the mass transports and the primary production

(PP) rates. Other important goals were to understand the physical factors that con-

trol the timing of the phytoplankton bloom [Collins et al. 2009], to estimate and

analyze secondary production and the associated secondary productivity [Sastri and

Dower 2009], and its interaction with primary production [El-Sabaawi et al. 2009,

2010].

An estuary is a complex and dynamic system, but it can be simplified for study. A

basic approach to study an estuary is to reduce it into a two-layer system [MacCready

and Geyer 2010] supplied with freshwater (FW) and seawater by external sources,

and exchanging heat and gas (e.g. O2 and CO2) with the atmosphere. Such a simple

approach, a so-called box model, can give one useful insights. A combination of boxes

and equations can be used to fit a particular system: e.g., Strait of Georgia [Pawlow-

icz et al. 2007], Salish Sea (formerly known as Strait of Georgia/Haro Strait/Juan de

Fuca system) [Li et al. 1999, Pawlowicz 2001], St. Lawrence system [Bugden 1981,

Savenkoff et al. 2001], Chesapeake Bay [Hagy et al. 2000, Austin 2002]). This ap-

proach usually works well even though estuaries greatly differ in freshwater (river

flow, evaporation/precipitation, ice sheet formation), tide and wind regimes all of

which may affect stratification and mixing differently. This approach also works well

even though estuaries differ in geomorphology (e.g. bathymetry, coastline and con-

nection to the ocean).

3


Figure 1.1: Geography of the Strait of Juan de Fuca/Haro Strait/Strait of Georgia.

Northern and Southern Entrances of the SoG are indicated: Johnstone Strait (JS)

and Boundary Pass (BP), respectively. The dashed squares indicates the sampling

area of the STRATOGEM and JEMS (Joint Effort to Monitor the Strait of Juan de

Fuca) programs described in Chapter 3.

4


Using a box model allows one to quantify aspects of the estuarine circulation

and the ecosystem productivity. Transports and tracer fluxes in, out and within an

estuary are basic parameters of interest as they help one to quantify whether fresh-

water, plankton and particles remain in, are exchanged within, or whether they leave

the estuary [Thomson 1994]. The estuarine circulation will be introduced in further

detail in section 1.1. The knowledge of the estuarine circulation allows one to make

estimates of biogeochemical and biological quantities by accounting for advection.

1.1 Physical Oceanography

The oceanography of an estuary is always influenced by its geomorphology. The SoG

is a deep basin isolated by sills from seawater sources. It is 220-km long and oriented

in a northwest-southeast direction. Its width ranges between 25 and 55 km [ Thomson

1981, 1994 and Fig. 1.1]. On average, it is about 155 m deep, but its central and

northern areas can reach 400 m and deeper (Fig. 1.3). Its horizontal area ranges

from 3.5×109m2, at 155 m, to 7×109m2, at the surface (see hypsography, Chapter 3,

Fig. 3.2). The total volume of the SoG is 1.1 ×1012m3 (Chapter 2, Table 2.1).

The main passage for seawater is through the Southern Entrance, mainly through

Boundary Pass. The Southern Entrance represents 93% of the mass transport into

the SoG versus 7% for the Northern Entrance [Thomson 1994]. Through Boundary

Pass, the connection is restricted by a sill that is located near 160 m depth [Davenne

and Masson 2001]. Thus, the SoG is like a fjord, isolated from open ocean sources.

The Fraser River plume is one of the characteristic oceanographic features of the

SoG. It is easily visible on satellite imagery during the peak of the Freshet when the

load of sediments is the highest of the year [Stronach 1981]. The plume can carry

freshwater into the SoG farther than the Fraser River mouth. The plume dynamics

5


Figure 1.2: Detailed Sampling Area of STRATOGEM and Other Important Sam-

pling Locations. STRATOGEM water sampling stations are the 9 open diamonds (S1

to S5). The important locations other than the STRATOGEM stations are the trian-

gles. The sampling at the STRATOGEM stations and at other important locations

is described in Chapter 3.

are complex to study because, among other aspects, they can vary over different

timescales: e.g. semidiurnal to annual timescales [Halverson and Pawlowicz 2008].

The surface of the SoG within and near the plume is fresher than the rest of the

6


SoG. Average plume practical salinity ranges between 15 and 26 [Halverson and

Pawlowicz 2008]. The plume can move across the SoG, and along the east coast of

the SoG northward and southward, but tends to remain in the southern SoG. On

average the rest of the SoG is strongly stratified in the upper 20 m (see section 3.3.1

and Pawlowicz et al. [2007]) because of freshwater input from the rivers. Below 20

m, the water column is weakly stratified because of mixing of surface and deep SoG

water and entrainment of deep water. I will discuss the vertical structure of salinity

and temperature in further detail later in this chapter and chapter 3.

On large timescales (&1 month), the overall circulation of the SoG is an estuarine

circulation driven by freshwater (FW) input, entrainment and tidal mixing (Fig. 1.3).

I will discuss the estuarine circulation and deep water advection in further detail in

the next paragraphs. Note that previous studies suggest that the deep flow in the

Central SoG is on average geostrophic and cyclonic: i.e northward flow on the eastern

side of the SoG, and southward on the western side [Stacey et al. 1991, Marinone

and Pond 1996]. In addition, the surface circulation can be affected by the wind

on small timescales (≪1 month) [St. John et al. 1993, D′Asaro and Dairiki 1997].

Winds are driven by the dominant pressure system (Aleutian low or Pacific high) on

larger scales (&1 month) [Thomson 1994, Marinone and Pond 1996]. The wind can

drive the surface circulation by drift, advection and mixing [Thomson 1994]. The

wind varies from overall strong winds blowing to the north and northwest in winter

(due to the Aleutian low) to overall weaker winds blowing to the south and southeast

in summer (due to the Pacific high). The wind is also affected by the local coastal

topography that channels the wind [Thomson 1981].

The hydrology of the SoG is also an important factor in the estuarine circulation.

Fig. 1.4 shows the Fraser River discharge over the 2000s. The river discharge has a

very marked seasonal cycle. However, the magnitude of the freshet peak varies from

7


Figure 1.3: Along-channel Cross-section of the Strait of Juan de Fuca/Haro Str-

ait/Strait of Georgia (SoJdF/HS/SoG) System. This is a schematic view of the So-

JdF/HS/SoG system. The straight arrows indicate the surface and deep transports

in the SoJdF and SoG. The round arrows indicate the vertical mixing in the SoJdF,

HS and SoG. The arrow size indicates the relative magnitude of the mixing (see text).

The light grey area shows the representative along-channel bathymetry with respect

to the distance from the western end of the SoJdF. It also shows the STRATOGEM

and JEMS (Joint Effort to Monitor the Strait of Juan de Fuca) sampling areas.

8


year to year. In particular, the STRATOGEM sampling period (2002–2005), captured

both the largest and the smallest flows of the 2000s: the 2002 freshet peak has the

largest discharge, and the 2004 freshet peak has the smallest. The Fraser river is the

largest single contributor to the FW discharge into the SoG, about 63% of the total

FW inflow according to linear regression coefficients from the literature [Pawlowicz

et al. 2007]. Usually the total discharge is of the order of 103m3s−1 in winter and

104m3s−1 in summer. However, in winter local large rainfalls could provide more

freshwater than predicted by estimates based on the regressions. The freshwater

mass from the Fraser River and other rivers is almost always mixed with salt within

the riverbed before it enters the SoG itself because of the intrusion of salt wedges

through the river mouth [Geyer and Farmer 1989, Halverson and Pawlowicz 2008].

The Fraser river can also supply heat and nutrients to the SoG, in particular nitrate

and silicic acid, both essential to siliceous phytoplankton like diatoms.

Fig. 1.3 shows a schematic view of the SoG general circulation. On a seasonal

time scale, the SoG estuarine circulation is mainly forced by the FW inflow and

the vertical mixing. Dense deep seawater enters the SoG from Haro Strait (HS) at

Boundary Pass, mixes with deep SoG seawater, and sinks to intermediate depth (50–

200 m) or near the bottom (200–400 m) [Waldichuk 1957, LeBlond et al. 1991, Masson

2002, Pawlowicz et al. 2007]. The depth to which it sinks is controlled by the density

of the oceanic seawater and the vertical turbulent mixing modulated by the neap-

spring cycle [LeBlond et al. 1991]. The magnitude of the vertical turbulent mixing in

the Strait of Juan de Fuca (SoJdF), HS and SoG are qualitatively represented by the

size of the curved arrows in Fig. 1.3. Compiled data on the vertical eddy diffusivity

suggests that HS has the largest values while the SoG has the smallest [Li et al. 1999,

Masson 2002]. The surface SoG water is a mixture of deep SoG water and freshwater

from the rivers.

9


Deep seawater entering the SoG originally comes from the Pacific Ocean. Seawa-

ter intrusions start entering into the SoJdF at depth of 100–200 m [Pawlowicz et al.

2007]. Seawater is transported through SoJdF with little change in its oceanic charac-

teristics [Pawlowicz 2001, see Figs 5 a-d]. However, when Pacific seawater is upwelled

or downwelled along the coast of Vancouver Island, the properties of the seawater

intrusions in the SoJdF can change. Upwelling and downwelling are seasonal coastal

processes with interannual variations. Thus, properties of the oceanic intrusions in

the SoG, like temperature, salinity, dissolved oxygen (O2) and nutrients, can change

from year to year [Masson 2002].

Previous studies have estimated the water transports at various locations inside

and close to the SoG with various approaches. All these studies provide reasonably

consistent average estimates of water transports. The approach used in this the-

sis enables one to estimate the seasonal changes of the water transports besides to

estimate their average values. Godin et al. [1981]’s estimates were based on the inte-

gration of current measurements across two channels: SoJdF and Johnstone Strait

(JS). England et al. [1996] estimated the transports indirectly by first estimating

flushing time with different techniques of mixing box-model. Marinone and Pond

[1996] used a sophisticated and complex 3D model of the SoG. Li et al. [1999] built a

prognostic time-dependent box-model of SoJdF/HS/SoG system with parametrized

mixing. Pawlowicz [2001] used inverse modelling and a box-model framework, and

applied it to SoJdF/HS/SoG system. Pawlowicz et al. [2007] used a hierarchy of

mixing box-models to determine both flushing times and transport magnitudes in

the SoG. Using a box-model approach enables one to compare estuaries. Later in

Chapters 4, 5 and 6, the SoG and other other estuaries (Chesapeake Bay and St.

Lawrence) will be compared.

10


Jan Jul 2001 Jul 02 Jul 03 Jul 04 Jul 05 Jul 06 Jul 07 Jul 08 Jul 0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1


0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1F

rase

r R

iver

dis

char

ge (×

104 m

3 s−

1 )


0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Figure 1.4: Fraser River Discharge. This is a time series of the Fraser River discharge

measured in 104 m3s−1, at Hope, about 150 km, east of Vancouver. The grey-shaded

time segment shows the STRATOGEM time frame.

1.2 Biological Oceanography

In the SoG, primary production (PP), the biomass of phytoplankton, is available

as food for zooplankton. Phytoplankton is the lowest link in the SoG food web and

thus, the survival of organisms higher in the food web may depend on PP. In the

SoG, PP varies seasonally and peaks in spring [Harrison et al. 1983]. Although the

11


spring bloom usually occurs in mid or late March, in 2005 it occurred in February.

During later blooms, in summer and in fall, PP is usually smaller than during spring

blooms. Note that hereafter we will define the spring season as the period which

starts in February and finishes in April. The other seasons are defined accordingly

(see Glossary, p. xiii). Summer and fall blooms produce less PP because of limitation

due to nutrients (summer) and light (fall) [Harrison et al. 1983]. Episodic (spring and

summer) grazing by zooplankton and sinking of, most likely, diatoms can reduce the

biomass, thus it can balance a large rate of phytoplankton growth during favourable

light and nutrient conditions. The spring maximum in PP coincides with an intense

uptake of nutrients and eventually leads to the surface minimum of nutrients in

summer [Masson 2006].

All phytoplankton species require dissolved nitrogen (e.g., nitrate and ammo-

nium; nitrite when in large concentration) and phosphate for growth and cellular

maintenance. However, diatoms, a dominant group of siliceous phytoplankton dur-

ing SoG spring bloom, require silicic acid to build frustules, their outer shells. Other

phytoplankton groups, like silicoflagellates (cosmopolitan marine phototrophic flag-

ellates able to form an external siliceous skeleton [Henriksen et al. 1993]), also require

silicic acid, and competition can happen when surface level runs low, for instance,

after a large spring bloom. Deeper, in the aphotic zone, nutrients remain at about

relatively constant level [Masson 2006] because of lack of PP. Through vertical tidal

mixing and discrete wind-induced mixing events, deep nutrients are transported to

the surface to replenish the surface nutrient pool [Yin et al. 1996, 1997]. Some stud-

ies suggest [Mackas and Harrison 1997] or assume [Li et al. 2000] that the nutrient

supplies may be a limiting factor during intensive uptake events (spring and summer

blooms).

Previous studies have analyzed the composition of SoG phytoplankton communi-

12


ties, and provided estimates of the SoG average PP, seasonality and average primary

productivity (PP rate). These estimates are broadly consistent with each other, al-

though they used different techniques of measurement or estimation that can lead to

different definitions of the PP and the corresponding PP rate. I will discuss this mat-

ter in detail in Chapter 5. Harrison et al. [1983] reviewed the SoG biology including

seasonal PP estimates and cycles. Mackas and Harrison [1997] provided a complete

budget of new PP based on the sources of nitrate in the SoG. Li et al. [2000] used

a biology-physics coupled box-model of the SoG to estimate PP and its variability

assuming nutrient limitation in summer. Pawlowicz et al. [2007] estimated new PP.

In the SoG, phytoplankton biomass is dominated by diatoms during spring blooms.

The most abundant diatom species is the chain-forming Skeletonema costatum [Har-

rison et al. 1983]. Generally in spring and summer there is a succession of diatom

species:Thalassiossira sp. are the most abundant diatoms in early spring, Skele-

tonema costatum blooms overlap with Thalassiossira sp. blooms or closely follow

them, and finally blooms of Chaetoceros sp. and Nitzschia sp. occur in late spring

and summer [Harrison et al. 1983]. Diatoms are known to have a resting stage [Mc-

Quoid and Hobson 1996]. Diatoms move into their resting stage when the availability

of nutrients and light become unfavourable to growth. The resting cells of diatoms

could provide the seed population for the following year blooms [Harrison et al.

1983, McQuoid and Hobson 1996]. Diatoms have to compete for nutrients and light

with other phytoplankton groups. In particular, in spring and summer they compete

for nutrients with silicoflagellates (e.g. Ebria tripartita), autotrophic flagellates (e.g.

Gymnodinium sp.), and autotrophic ciliates (e.g.Myrionecta rubra, previously known

as Mesodinium rubrum). Gymnodinium sp. are more abundant during summer, but

can also bloom during spring [Harrison et al. 1983].

In the SoG, mesozooplankton graze on phytoplankton. Mesozooplankton include

13


copepods, dominated by Neocalanus plumchrus, and euphausiids, dominated by Eu-

phausia pacifica. Some copepods can shift their diet like Metridia pacifica, an omniv-

orous copepod species, while other zooplankton species, like Neocalanus plumchrus

can not do this [El-Sabaawi et al. 2009, 2010]. Copepods and euphausiids coexist

throughout the year because their abundance is strongly influenced by temperature:

copepods are more abundant during spring and summer, euphausiids during win-

ter [Harrison et al. 1983]. Neocalanus plumchrus overwinter in a dormant stage and

dominate the deep zooplankton community during winter [Campbell et al. 2004].

All higher levels in the foodweb of the SoG are not well-known yet. However, links

between mesozooplankton and their predators, e.g. salmon sp. [Garay and Soberanis

2008] and herring [Therriault et al. 2009], and between fish and their predators, e.g.

seabirds, have been investigated [Therriault et al. 2009].

1.3 Two-layer Model and Governing Equations

In order to write a two-layer model of the SoG, it is necessary to make several

assumptions about the circulation. First, many physical processes, e.g. wind events

like storms, the flux of correlated variations due to turbulent mixing (see Eq. 2.57

and section 2.2 for further detail), diurnal, semi-diurnal and spring-neap tides are all

lumped together as vertical transports in the box model and only their contribution

over long timescales affect the vertical transports. This is a useful assumption since

the STRATOGEM dataset generally contained monthly cruises only. Second, the

observations of temperature and salinity, and the associated box averages used in

the two-layer model are assumed to be in a mass, heat and salinity balance over long

timescales. The conservation equations of three physical tracers (mass, heat and

salinity) are written to estimate the SoG circulation which satisfies the dynamics of

14


the SoG over long timescales:

Mass conservation:

− div(ρu) =∂ρ

∂t(1.1)

Heat conservation:

− div (ρCpTu−K ∇(ρCpT ))=∂ρCpT

∂t(1.2)

Salinity conservation:

− div (ρSu−K ∇(ρS)) =∂ρS

∂t(1.3)

where ∇() represents the gradient of the bracketed quantity and div() its divergence:

∇() = [∂

∂x(),

∂

∂y(),

∂

∂z()]T (1.4)

div() =∂

∂x() +

∂

∂y() +

∂

∂z() (1.5)

where x, y and z represent displacements along the horizontal and vertical directions

(denoted by x-, y- and z-direction), ρ the density, T the temperature, S the salinity,

u the flow speed, Cp the water specific heat capacity, and K represents the diffusivity

either due to molecular diffusion (KM) or due to eddy diffusion (KE). Note that Cp

the specific heat capacity varies less than 5% over the range of temperature and

salinity of seawater [IOC, SCOR and IAPSO 2010], so that it can be assumed to be

constant in Eq. 1.2 and the equations derived from Eq. 1.2. Note that the Practical

Salinity is assumed to be proportional to the Absolute Salinity, which is reasonable

15


in estuarine waters [IOC, SCOR and IAPSO 2010]. Derivation of the box-model

equations from Eqs 1.1–1.3 are detailed in Chapter 2.

Using the two-layer paradigm, the SoG model is defined with two boxes and

the conservation of the mass, heat, salinity (Eqs 1.1–1.3), nutrients and dissolved

oxygen (collectively represented as q) are written in a two-layer system. These tracer

equations can be written as:

Vj∂qj∂t

= (uj)q + (φj)q + εj (1.6)

where qj is the average of q over the box j (j =1, top box, and 2, bottom box),

(uj)q the sum of all the advective transports of tracer q into and out of the box j,

(φj)q the sum of the internal sources and sinks of tracer q in box j, and (Vj∂qj/∂t)

the time rate of change of the total amount of tracer q in box j. The rightmost

term εj represents the estimated error due to measurement error, assumptions and

neglected physical and biological processes. Storage occurs when the time derivative

is positive. On the other hand, drawdown occurs when the time derivative is negative.

Any net biological uptake of nutrients (e.g., nitrate, phosphate and silicic acid) and

net biological production of dissolved O2 are represented by φ terms.

The net uptake of nutrients represented by (φ1)N for nitrate and (φ1)P for phos-

phate will be used as an indirect estimate of the primary productivity. Their scal-

ing into organic carbon currency using either an estimated Redfield ratio (see sec-

tion 5.2.3) or the expected Redfield ratio [Redfield et al. 1963] will provide an esti-

mate of the net primary productivity in various forms: total, new, and regenerated

(defined in section 5.1).

16


1.4 Objectives, Approach, Thesis Contributions

and Plan

The primary objective of this thesis is to quantify the estuarine circulation and the

nutrient supplies and sinks over monthly to interannual time scales over 2002–2005.

Then, I use this quantitative knowledge to estimate the nutrient uptake rates, and,

by carefully scaling, net primary productivity (NPP rate).

My approach is to infer the monthly estuarine transports using inverse modelling

in a 2-box model. Then, one can infer the nutrient supplies and sinks using a forward

box model in the same 2-box model.

I now summarize the contributions of this thesis on the understanding of the

SoG circulation and its primary productivity. In chapter 2, a formal mathematical

framework and a time-dependent inverse two-box model of the SoG are constructed in

order to infer the monthly variability of the SoG circulation. This is a more rigorous

approach than it has been used previously in the literature. A careful analysis of

the approximations made in the derivation of the budget equations of physical and

biogeochemical tracers is carried out in sections 2.2 and 2.3.

Chapter 3 contains a detailed description of the oceanography of the SoG as

observed during STRATOGEM. In chapter 4, transport time series of the SoG estu-

arine circulation are estimated over three years with a monthly time resolution, for

the first time. This inference is based on consistency of the transports with obser-

vations of salinity and FW input, temperature and surface heat fluxes. Using these

estimated SoG transports, I provide an observational study (one of the few obser-

vational studies based on a rigorous mathematical framework) of the relationship

between FW input (R) and surface seaward transport (U1) in the SoG and possibly

17


for any estuary.

The analysis of the transport time series provides some novel findings. It suggests

that the seasonality of the total upward transport (W2) is very small, even when

the seasonality of the riverine inflow (R) is large. Based on 2002–2005 data, the

seasonality of surface seaward transport (U1) is weakly linked to (R), while the

seasonality of other transports is not linked to R. The transport analysis is discussed

in more detail in section 6.2.

In chapter 5, I estimate the seasonal and annual average rates, with error estima-

tion, of the net primary production (NPP rates) based on SoG nutrient budgets over

three years. When the estimated NPP rates are compared with previous estimates

in the SoG and in other estuaries, I show that the estimated NPP rates are reason-

able and typical of NPP rates in a temperate estuary (comparison in section 6.3).

The analysis of the nutrient budgets suggests that, as proposed first by Mackas and

Harrison [1997] for N, the estuarine entrainment is the largest supply of N, P and

Si. The comparison of the estimated NPP rates based on N and P budgets suggests

that the average f-ratio is large (average f=0.88, see section 6.3). These results are

discussed in more detail in section 6.3. Recommendations for future work are given

in section 6.4.

18

Chapter 2

Inverse Methods and Box Models

2.1 Introduction: Mathematical Framework

The thesis objectives lead to two mathematical problems.

First, I define a mathematical problem associated with the estuarine circulation

of the matrix form:

Ax = b. (2.1)

In chapter 1, the equations of the conservation of the mass, heat and salt in the

top and bottom boxes of the form of Eq. 1.6, can be written into Eq. 2.1, as will

be done in section 2.2. I have a direct knowledge of salinity and temperature whose

values are used in the definition of A and whose time-derivatives are used in the

definition of b in Eq 2.1. I also have a direct knowledge of freshwater input and

heat fluxes (also used in b). But I have no knowledge of the advective transports

(x, unknown term) associated with the estuarine circulation. The problem is then

to derive an estimate of the advective transports x from the observations A and b

where A and b both have errors associated with them.

The second mathematical problem is more straightforward mathematically, but

involves more assumptions. It is associated with the primary productivity. It has the

matrix form:

AxU − b = x. (2.2)

19

Chapter 2. Inverse Methods and Box Models

In chapter 1, the equations of the conservation of the nutrients and dissolved O2

in the top and bottom boxes of the form of Eq. 1.6, can be written into Eq. 2.2,

as it will be done in section 2.4. I have a direct knowledge of the concentration of

nutrients and dissolved oxygen (represented by A in Eq. 2.2), the associated influxes,

and the time rate of their concentration changes (b). I also have an estimation of

the transports from the estuarine circulation problem (xU, the transports associated

with the circulation which are assumed to be known in this second problem). From

this knowledge, it is possible to quantify the sink terms (x, unknown term) which

are assumed to be associated with primary productivity, and after further analy-

sis estimate the new primary productivity associated with some of the nutrients. I

can also qualify the net community productivity associated with the dissolved oxy-

gen balance and analyze the variability of the estimated net biological uptake rate

associated with the silicic-acid concentration.

To be able to apply the appropriate methodology to solve these mathematical

problems, I first need to define the notions of forward and inverse problems. These

notions are connected to the notion of well-posed and ill-posed problems. According

to Wunsch [1996] a well-posed problem is a mathematical problem that has a unique

“well-behaved” solution. Note that “well-behaved” means that this solution is stable

to perturbations on the boundaries, on the initial conditions and on the sources.

By convention, such problems correspond to textbook cases or classic problems in

mathematics and physics [Wunsch 1996]. An example of such a classic problem is

Dirichlet’s problem of Laplace’s equation ∇2φ=ρ where φ is the unknown field [Wun-

sch 1996]. Such well-posed problems are often called “forward” problems. However,

many real-life problems are ill-posed. An ill-posed problem can have multiple, irreg-

ular solutions which may be sensitive to the data. An irregular solution can be for

instance a fast-changing solution with at least a jump between two values. An ill-

20


posed problem can also be created from a well-posed problem by interchanging the

unknown quantities of the classic problem with some of the known quantities. These

problems can often be described as inverse problems because they involve taking the

inverse of a matrix.

In this chapter and the following chapters, I will define the forward problem as

the mathematical problem that consists of finding the unknown sink terms in the

conservation equations by addition and subtraction of the advection terms. This

problem is similar to Eq. 2.2. Its solution is unique by construction. In the particular

case of the nutrients in the SoG, the sink term represents a net biological uptake

rate or net sink. I will define an inverse problem as finding the unknown transports

when no simple way is available to compute the solution by addition or subtraction

of the known terms. This problem is similar to Eq. 2.1.

In the case of the estuarine circulation in the SoG, these problems can be written

more precisely in matrix form as:

Ax = b+ ε. (2.3)

where A is a matrix of m rows and n columns (m-by-n) that contains known coeffi-

cients (including spatially-averaged temperature and salinity) of the mass, heat and

salt equations, b is an m-by-1 vector of all internal and external sources, and x is

an n-by-1 vector that contains the unknown advective transports. The net advective

flux of mass, heat and salt is the vector Ax. In the particular case of a unique and

exact solution of Eq. 2.3, the residuals ε must equal 0. Finding a unique and exact

solution x is not always possible. This becomes a general rule with real-life problems.

Since solutions are not exact in real-life problems, there are residuals ε( 6=0) in

Eq. 2.3:

ε = Ax− b. (2.4)

21


There is generally some uncertainty introduced in Eq. 2.3 by sampling errors in the

observations because the sampling errors can have propagated to A and b. If any

important process is missing from the model defined by Eq. 2.3 a misfit error is

also added to the uncertainty ε. As it is not clear whether the solution x is stable

to perturbations in the inverse problem (and whether the solution in the forward

problem is stable by propagation of the errors of the transports) it is necessary

to carry out an error estimation (detailed in section 2.3) and a sensitivity analysis

(discussed in chapter 4) for any solution.

The inverse problem can be solved by a Singular Value Decomposition (SVD)

Methodology (SVDM) [Wunsch 2006]. The SVD of any m-by-n matrix A is a de-

composition:

A = UΛVT (2.5)

The matrix Λ is a m-by-n diagonal matrix [Wunsch 2006]. The r non-zero coefficients

on the diagonal are called the singular values (λi) of the SVD. The singular values

are arranged in decreasing order (λ1 > λ2 >...> λr). The rank of A is also equal to r.

By definition, the rank of a matrix is the maximum number of linearly independent

columns (or equivalently rows) of the matrix. Thus, the rank r is bounded by:

r ≤ min(m, n) (2.6)

The rest of Λ is padded with zeros. The columns of matrices U and V form two sets

of orthonormal vectors: m vectors ui and n vectors vi. The vectors ui and vi are

called singular vectors of A. They can be used to decompose the elements of b and

x, respectively.

The SVDM is based on this generalized diagonalization (Eq. 2.5) of m-by-n ma-

trices (square or non-square). The SVDM can be used to find a solution of any type

of linear system of equations: just-determined (when m=n), overdetermined (when

22


m>n) and underdetermined (when m<n). Unlike the Linear Least-Squares approach

(LLS), the SVDM can work for a linear system with rank deficiency. Rank deficiency

happens when r< min(m,n). With a rank-deficient system of linear equations, the

LLS cannot distinguish between certain equations in Eq. 2.3 or elements of vector

x [Wunsch 2006]. The SVDM helps to make appropriate choices. It still leads to a

reasonable estimated solution x. The SVDM also provides additional tools to analyze

the system and the solution.

The inverse problem to be solved is formally overdetermined because 6 conserva-

tion equations will be used to determine the 4 unknown transports. With both LLS

and SVDM, the “best” solution of this inverse problem, x is found by minimizing

the norm of the residual ε. This norm is the objective function J defined by:

J = εTε. (2.7)

The LLS solution, the so-called “pseudo-inverse”, x (in Eq. 2.3) exists if the

system is full-rank. It minimizes Eq. 2.7. It is defined by:

x = (ATA)−1ATb (2.8)

However, the SVD solution:

x = VΛ−1UTb, (2.9)

which can be rewritten as

x =r∑

i=1

(uTi b)

λi

vi, (2.10)

exists even if the system is rank-deficient. The matrix Λ−1 is the inverse of Λ defined

as a n-by-m diagonal matrix with λ−1i on the diagonal. The rest of the matrix is

padded with zeros. Eq. 2.8 (when it exists) and Eq. 2.9 are equivalent and this

follows from the decomposition of A in Eq. 2.5. With the estimated solution x the

23


corresponding residual ε is:

ε = −

m∑

i=r+1

(uTi b)ui (2.11)

which is the part of b unresolved by the range vectors and is spanned by the nullspace

vectors ui(i=r+1...n). The range vectors are the vectors ui and vi (i = 1, ..., r) that

satisfy:

ATui = λivi and Avi = λiui (2.12)

while any nullspace vectors satisfy:

ATui = 0 and Avi = 0. (2.13)

To separate the range vectors from the nullspace vectors, it is sometimes useful to

define the m-by-r matrix Ur and the n-by-r matrix Vr, the matrices formed by the r

first singular vectors ui and vi. The range vectors ui and vi are associated with the

singular value λi. By construction, any vector ui or vi that is not a range vector has

to be a nullspace vector. Note that in Eq. 2.11 if the system is full-rank and m < n,

ε = 0. The true solution x is spanned by the range vectors vi and can be written as:

x =

r∑

i=1

(uTi b)

λivi +

n∑

i=r+1

γivi (2.14)

where the second right-hand term is the part of x unresolved by the range vectors vi

(i=1, ..., r). For this reason γi’s for j=r+1, ..., n are unknown because the remaining

vi (i=r+1, ..., n) are nullspace vectors (i.e. Avi=0).

When an objective function J, more sophisticated than Eq. 2.7, is chosen to

improve the solution x, the SVD enables one to understand the effects of such a

function [Wunsch 2006]. For instance,

J = εTε+ α2zTz (2.15)

24


where the solution z is the solution x centered around an a priori average xA so that:

z = x− xA and c = b−AxA. (2.16)

The coefficient α2 is a coefficient that quantifies the compromise made between min-

imizing the residuals ε and being close to an a priori estimate xA. The corresponding

LLS solution is:

z = (ATA+ α2In)−1ATc (2.17)

which can be written using the SVD as

z = V(ΛTΛ+ α2In)−1ΛTUTc. (2.18)

The SVD solution in turn can be rewritten as

z =

r∑

i=1

λi(uTi c)

λi2 + α2

vi. (2.19)

By comparing Eq. 2.10 and Eq. 2.19 (setting xA = 0 without loss of generality),

it clearly appears that the scaling has “tapered” the coefficients of vi, and thus

the norm of the solution x. The term tapering refers to the weighting down of the

coefficients of the vector vi by α2 [Wunsch 2006]. The coefficients of the vector vi

(in Eq. 2.19) are different from the values they would have in the simple SVD form

(Eq. 2.10). This ensures that there is always an inverse solution even if some of the

λi’s ≪ α (in this case λi2 + α2 ≃ α2.). However, this procedure also adds a bias to

the equation residuals ε. A bias is a difference between the estimated value and the

expected value of any quantity. Using the SVD, the new residual ε is:

ε = −

m∑

i=r+1

(uTi c)ui −

r∑

i=1

α2(uiTc)

λi2 + α2

ui. (2.20)

25


The true solution z(= x−xA) is spanned by the range vectors vi and can be written

as

z =

r∑

i=1

λi(uTi c)

λi2 + α2

vi +

n∑

i=r+1

(vTi z)vi (2.21)

where the second right-hand term is the part of x unresolved by the range vectors

vi(j=1...r).

When I applied Eq. 2.10 to get the “pseudo-inverse” solution of the system Eq. 2.3,

the inverse procedure gave large month-to-month uncertainty. Instead, it was more

useful to apply Eq. 2.19 to get the so-called “optimal” inverse solution x. The inverse

optimal solution x minimizes an objective function J:

J = εTW−2

ε+ zTS−2z. (2.22)

whereW and S are weighting matrices [Wunsch 2006]. This generalization of Eq. 2.15

represents a compromise between the observations and the a priori knowledge. The

inverse optimal solution of Eq. 2.22, x is of the form:

x = xA + (ATW−2A+ S−2)−1(ATW−2)(b−AxA) (2.23)

where z and c have been expanded using Eqs 2.16, and using the SVD:

x = xA +V′(Λ′TΛ′ + S−2)−1(Λ′TU′TW−1)(b−AxA) (2.24)

where Λ′, U′ and V′ are the matrices of the singular values, data and solution

singular vectors when the SVD is applied on the scaled matrix A′ (=W−1A).

This inverse procedure has been devised to improve and build on the “pseudo-

inverse” procedure. It tapers the coefficients of vi when there are very small singular

values (associated with unstable solutions) of the matrix A. It also constrains the

optimal solution to keep values closer to the positive a priori solution xA. In Eq. 2.22,

26


several new matrices and vector have to be defined: the matrices W and S (6-by-

6 matrix and 4-by-4 matrix, respectively) and the a priori solution vector xA. The

matrices W and S are applied to scale the system equations and the solution, respec-

tively. W normalizes the conservation equations so that all conservation equations

have the same weight in the solution (so-called row-scaling) and it has the form:

W =

ω1 0 0 0 0 0

0 ω2 0 0 0 0

0 0 ω3 0 0 0

0 0 0 ω4 0 0

0 0 0 0 ω5 0

0 0 0 0 0 ω6

(2.25)

In Eq. 2.22, the matrix S normalizes the solution x so that the scaled version of

x has components with a magnitude of O(1) (so-called column-scaling). S is a square

diagonal matrix of identical non-zero coefficients:

S =

s 0 0 0

0 s 0 0

0 0 s 0

0 0 0 s

(2.26)

The actual values of the inversion parameters (s, ωi and xA) are shown later in

this chapter (see Table 2.2).

The solution (data) range vectors can be used to resolve the solution (data).

Their resolution capacity can be measured by the solution (data) resolution matrix

27


Tv (Tu). In the case of the simple SVD, these matrices are defined by

Tv = VrVTr (2.27)

and

Tu = UrUTr (2.28)

For instance, one can use the matrix Tv (Tu) to find the relationship between the

estimated solution x (data vector b) with respect to the true solution x (b):

x = VrVTr x (2.29)

and

b = UrUTr b (2.30)

and the condition for x = x (or b = b) is A full rank: r=n (or r=m). In the case of

the tapered SVD of the row- and column-scaled matrix A′, the resolution matrices

are defined by:

Tv = SV′

rD′

r(α2)V′

rTS−1 (2.31)

and

Tu = WU′

rD′

r(α2)U′

rTW−1 (2.32)

where the matrix D′

r(α2) is a r-by-r diagonal matrix which non-zero coefficients

d′i(α2) are of the form:

d′i(α2) =

λ′

i2

λ′

i2 + α2

(2.33)

The matrices V′

r and U′

r are the matrices V′ and U′ with r columns. For α = 0,

D′

r(α2) = Ir and Eqs 2.31–2.32 turn back into Eqs 2.27–2.28. Note that when λi ≪ α,

d′i(α2) ≃0. If the matrices Tv and Tu are different from the identity matrices, some

of the r solution (data) range vectors cannot help to completely resolve the solution

28


(the data). They can be considered as nullspace vectors because they are associated

with such small λi’s that they cannot be useful as range vectors. When the data

resolution matrix is not the identity matrix, one could reason that there is not enough

information to distinguish some of the equations from each other. An alternative

explanation could be that some of the equations have more weight than the others

despite the a priori weights ωi’s. This is similar to a situation where some of the

equations are linearly dependent.

The data resolution matrix can help build a “data ranking” assessment and de-

termine which observations are the most important [Wunsch 2006]. The diagonal

coefficients help to compare the relative weight of each equation. The non-diagonal

coefficients show the strength of the linear dependence between each pair of equa-

tions.

2.2 Inverse Problem: Estimating the SoG

Circulation

The introduction (chapter 1) suggested that the dynamics of the SoG system set up

an estuarine circulation. The SoG estuarine circulation allows one to idealize the SoG

as a two-box system. Table 2.1 gives the volume and depth of each SoG box. It is also

necessary to define the other systems which communicate with the SoG. In this SoG

idealization, the only connection to the open ocean is through the Southern Entrance.

Haro Strait (HS) is directly connected to the SoG by the Southern Entrance and it is

the most important neighbouring system (as shown in chapter 1). Because of strong

vertical mixing in HS, the water column is usually quite uniform relative to the

SoG, although this does not mean it is perfectly uniform. Table 2.1 also defines HS

29


Domain Name Total Depth (m) Volume ×1011 (m3)

Upper SoG (V1) 30 1.9

Lower SoG (V2) 370 9.1

Haro Strait (VH) 200 1.6

Table 2.1: Total Depths and Volumes of the Model Boxes. Note that the SoG sea

surface is about ∼7×109 m2

volume and depth and illustrates why it is represented by a smaller box than the

SoG. Fig. 2.1 shows the different processes and fluxes in, out and between the SoG

boxes and the different tracers affected by them (Ti, Si, with i=1 representing the

surface box, 2 the deep box, H the HS box, R the river).

At this point in the description of the box model, it is important to define the

separation depth (d) between the SoG boxes. I define d as the depth above which

the flow in the SoG is seaward and below which the flow in the SoG is landward.

The surface outflow carries freshwater from the rivers and seawater from the SoG,

while the deep inflow brings dense seawater from HS. Analyzing the vertical profiles

of salinity and temperature in the SoG can help locating these two types of water

masses and determining their depth ranges. In section 3.3.1, I will explain how the

separation depth d is chosen. Later in chapter 4, I will set d to 30 m.

The two-box model is a useful paradigm to study an estuary. This idealization of

the SoG into two boxes will enable me to study both the physics (Fig. 2.1) and the

biology (Fig. 2.2) in the same domains and carry the estimates of the circulation to

the advective transports of nutrients and dissolved O2. It will also help when it is

necessary to make comparison with other systems. Previous studies have used this

approach successfully [Li et al. 1999, 2000, Pawlowicz 2001, Johannessen et al. 2003,

30


Figure 2.1: Physical Fluxes and Processes in the Box Model. The left-hand boxes

represent the SoG, and the right-hand box the Haro Strait. The thick arrows rep-

resent the transports in and out the SoG (U1, U2, W1 and W2). In the model, the

surface SoG water enters the surface of HS (U1) while the deep HS water enters the

deep SoG (U2). The arrow thickness approximates the relative magnitude of trans-

ports. The upper left arrow represents the freshwater inflow (R). The thick wavy

arrow represents the turbulent and radiative heat fluxes (F ). FSW is the shortwave

component of F (thin wavy arrow) that penetrates deeper into the SoG than the

longwave component.

31


Figure 2.2: Biogeochemical fluxes and processes in the Box Model. The same con-

vention for the boxes and the arrows in Fig. 2.1 apply to this figure. The sink terms

are denoted by φ1 and φ2 (top and bottom, respectively) and are inside small boxes

with inward arrows. Air-sea exchange fluxes are represented by a double wavy arrow.

32


Pawlowicz et al. 2007] to investigate various aspects of the SoG oceanography.

In a two-layered system, a classical approach consists in applying Knudsen’s

hydrographic theorem [Dyer 1973]. However, there are several issues when one studies

a real system which is necessarily more complex than the simplified estuary used in

Knudsen’s theorem. These issues involve the use of multiple tracers, and the degree

to which a quasi-steady approximation is valid.

Knudsen’s approach can only combine two sets of equations to close the prob-

lem: mass and salt conservation equations. The salt conservation equation is the

reasonable choice of tracer to estimate the estuarine circulation. However, inferring

the water transports based on only the salinity can be an ill-posed problem if there

is any inaccuracy or inconsistency in the data. Inaccuracy and inconsistency can

lead, in the salt equations, to coefficients indistinguishable from zero within the er-

ror bars [Smith and Hollibaugh 1997, Dale et al. 2004]. It can also lead to inaccurate

advective transports for other tracers (in particular biogeochemical tracers) that are

important in multidisciplinary studies. Using a formal inverse approach enables one

to add other tracers, for instance temperature, and improves the reliability of the es-

timated transports [Roson et al. 1997, Pawlowicz and Farmer 1998, Pawlowicz 2001].

One more tracer adds two equations to solve: one for the surface box, and another

for the bottom box. The formal inverse approach enables one to handle a system

with more equations than unknowns.

Time dependence may be an important element of the SoG modelling, especially

at the surface where changes can have a short timescale in spring and fall. To compute

the time derivatives, the numerical scheme described later in section 3.3.3 is used.

The salinity can change quickly during the early freshet (spring). Heat and salt can

be stored in the SoG and released later. Thus, the salt and heat inflows and outflows

33


need not to be balanced all the time. On the other hand, the time rate of volume

change can be shown to be negligible over periods of a couple of months. For instance,

Godin et al. [1981] estimated a net inward transport of 350 m3s−1 based on long-

term sea-level change (as high as 19 cm) measured from tide gauges during April to

June 1973. Assuming that this time rate of change remains reasonable for the smaller

STRATOGEM study area and different analysis period, this net inward transport

can be considered to be negligible because it is smaller than the largest surface mass

inflow (greater than 1165 m3s−1 according to a linear regression from the literature,

section 3.2.3), the freshwater discharge. The net inward transport is smaller than

the freshwater discharge by one order of magnitude on average, and by two orders

of magnitude in summer. Note that there could be also a net barotropic transport

around the Vancouver Island (e.g. SoG inflow at Johnstone Strait and outflow at

Juan de Fuca Strait), but sea level observations cannot be used to study this. This

is because an inflow and an outflow of the same magnitude produce no change in sea

level.

The inverse problem in Eq. 2.3 is a linear form of the differential equations of

conservation of mass, heat or salt, which were introduced in Chapter 1 as Eqs 1.1–1.3.

There is no internal source of heat or salt. The sources of freshwater, seawater

and heat have been identified only at the boundaries of the SoG, for this reason

they will be applied later as boundary conditions. Eqs 1.1–1.3 can be simplified by

assuming that the water is incompressible. Thus, Eqs 1.1–1.3 can be rewritten in the

34


form:

Continuity equation:

− div(u) = 0 (2.34)

Heat conservation:

− div (Tu−K ∇T )=∂T

∂t(2.35)

Salt conservation:

− div (Su−K ∇S) =∂S

∂t(2.36)

where ∇() represents the gradient of the bracketed quantity and div() its diver-

gence:

∇() = [∂

∂x(),

∂

∂y(),

∂

∂z()]T (2.37)

div() =∂

∂x() +

∂

∂y() +

∂

∂z() (2.38)

where x, y and z represent displacements along the horizontal and vertical directions

(denoted by x-, y- and z-direction), ρ the density, T the in-on-site temperature, S

the salinity (measured as practical salinity), u the flow speed, and K represents

the diffusivity either due to molecular diffusion (KM) or due to eddy diffusion (KE).

When discussing the salt and heat budget equations in this section, we will show that

KM ≪ KE and thus K ≃ KE. The diffusivity K represents diffusion in all directions

and it depends on the gradient of T (or S). Note that Cp the specific heat capacity

varies less than 5% over the range of temperature and salinity of seawater [IOC,

35


SCOR and IAPSO 2010], so that it can be assumed to be constant in Eq. 2.35 and

the equations derived from Eq. 2.35.

The quantities from Eqs 2.34–2.36 are now integrated over the SoG boxes. The

continuity equation (Eq. 2.34) can be integrated over the top and bottom boxes of

volumes V1 and V2, respectively (see Table 2.1) to obtain the first two equations of the

inverse problem. According to the divergence theorem, if the surrounding boundaries

of the boxes are called a1 and a2, the integrals of the divergence over the volumes V1

and V2 in Eq. 2.34 can be replaced by the fluxes through the surfaces a1 and a2.

Continuity equation:∫

a1

u · da1 = 0 (2.39)

∫

a2

u · da2 = 0 (2.40)

The inner product of the vector u and dai is represented by a dot inside the integrals.

The vectors da1 and da2 represent elements of the surfaces a1 and a2, respectively, and

are oriented outward from the volumes. This orientation is chosen so that outward

flow is positive. Noting that at the surface there is a river inflow R and at depth water

intrusions U2, a negligible inflow at the northern connection (see chapter 1), and no

flow through the bottom and through the walls of the SoG boxes, Eqs 2.39–2.40

36


become:

Mass budget:

− U1 −W1 +W2=− R + ε1 (2.41)

U2 +W1 −W2 = ε2 (2.42)

The residuals ε1 and ε2 in Eqs 2.41–2.42 represent the estimation errors on u. The

transports in the top and bottom boxes U1, U2, W1 and W2 in Eqs 2.41–2.42 can be

closely represented by integrals:

surface outward transport:

∫

SC1

u · da1= U1 (2.43)

deep inward transport :

∫

SC2

u · da2=− U2 (2.44)

vertical transport :

∫

SB

u · da1 = W1 −W2 (2.45)

=−

∫

SB

u · da2

where SC1 and SC2 defined the cross-section of the southern connection (see chap-

ter 1) in the top and bottom boxes while SB is the separation boundary (at depth

d) between the top and bottom boxes. Although the vertical advective transport

(Eq. 2.45) is on average over the year a net upward transport, its magnitude or di-

rection can change seasonally and over the pycnocline. Eq. 2.45 summarizes these

cases by introducing the average upward and downward advectives transports W2

and W1.

W1 =

∫

SB′

u · da1 (2.46)

37


W1 is the downward transport and SB′ represents the locations of SB where the

speed is outward (with respect to the boundaries of the top box), u · da1 >0 and

W2 = −

∫

SB′′

u · da1 (2.47)

W2 is the upward transport and SB′′ represents the locations of SB where the speed

is inward, u · da1 <0. This decomposition of the vertical transport is important later

when the vertical advective transports of heat and salt have to be estimated.

Then, the next step is to obtain the heat and salt budget equations by estimating

the advective transports of heat and salt, and integrating the conservation equations

of heat and salt, Eqs 2.35–2.36, over the top and bottom boxes. The molecular diffu-

sion (diffusivity KM) for heat and salt can be neglected in Eqs 2.35–2.36 after scaling

it against eddy diffusion (diffusivity KE) following Pond and Pickard [1978]. A typi-

cal molecular diffusivity for heat and salt is 10−7 and 10−9m2 s −1, respectively [Pond

and Pickard 1978]. The eddy diffusivity depends on the properties of the mean flow

and it has the scale of a speed times a length:

− q′v′ = KE∂q

∂y(2.48)

where q′ and v′ represent fluctuations about quantity q and horizontal speed v along

y-direction, respectively. The overlined quantities q′v′ and q are time-averages and

represent fluxes of correlated variations over a timescale shorter than the circula-

tion timescale. So, to scale the eddy diffusion, it is necessary to estimate the speed

associated with the SoG circulation and the characteristic lengthscales.

The characteristic horizontal and vertical lengthscales in the SoG are about 104

and 102m, respectively. The order of magnitude of the horizontal speed ranges from

0.1 to 1 m s−1 [Thomson 1994, 1981]. Based on these scales and the continuity

equation (Eq. 2.34), the scale of the vertical speed is at least two orders of magnitude

38


smaller than the scale of the horizontal speed. These numbers lead to horizontal and

vertical eddy diffusivities of 0.1–1×104 and 0.1–1 m2s−1, respectively. These values

are consistent with the maxima given by Pond and Pickard [1978], 105 and 10−1

m2s−1, respectively, suggesting weaker horizontal mixing in the SoG and vertical

mixing of similar magnitude in the SoG. However, these values are large enough to

prevail over molecular diffusivity.

The flow speed and the tracer quantity (q represents either T or S) are written

in the form of a mean and fluctuations about that mean at any location in the SoG:

u = u+ u′ (2.49)

q = q+ q′ (2.50)

where the mean is defined over a timescale characteristic of the circulation timescale.

Thus, the advective fluxes in the conservation equations represent estuarine trans-

ports. Since the molecular diffusion is neglected, K ≃ KE . Using Eqs 2.48, 2.49

and 2.50, the conservation of the quantity q (either T or S) can be written:

− div(q u+ q′u′) =∂q

∂t+ e (2.51)

where e is the error due to the assumption on the molecular diffusion and the simpli-

fication by the terms qu′, q′u, q′u′, and ∂q′

∂tover the circulation timescale. The term

q′u′ represents any flux of correlated variations. Similarly to obtain the equations of

the mass budget, Eqs 2.41–2.42 are integrated over the volumes V1 and V2 and yields

the advective transports using the divergence theorem:

−

∫

a1

(q u+ q′u′) · da1=

∫

V1

∂q

∂tdV1 + e1 (2.52)

−

∫

a2

(q u+ q′u′) · da2=

∫

V2

∂q

∂tdV2 + e2 (2.53)

39


The time-derivative integrals 1

Vi

∫Vi∂q/∂t dVi are exactly equal to the time deriva-

tive of the box averages q1 (corresponding to T1 or S1) and q2 (T2 or S2) of the

quantity q since the volumes V1 and V2 do not change significantly with time.

−

∫

a1

(q u+ q′u′) · da1= V1

∂q1∂t

+ e1 (2.54)

−

∫

a2

(q u+ q′u′) · da2= V2

∂q2∂t

+ e2 (2.55)

The means u and q can be further approximated. If the spatial variations of the

speed are neglected, the speed mean can be replaced by the mean over the surface,

ua. Thus, the surface integral of the mean speed leads to one of the transports defined

by Eqs 2.43–2.45 and 2.46–2.47, depending on the surface considered (U1, U2, W1 or

W2). The tracer mean over the surface, q can be replaced by the tracer mean over the

volume qv, further simplifying the form of the conservation equations. Once these

approximations are made, the integral advective flux becomes:∫

a

(q u+ q′u′) · da =

∫

a

qvua · da+

∫

a

q′u′ · da+ e (2.56)

where e takes into account the abovementioned approximations on q and on u. In

Eq. 2.56, since only correlated fluctuations of q′ and u′ will contribute to the second

right-hand integral (∫aq′u′ · da), this integral is likely to describe the turbulent fluxes

due to entrainment and mixing exchange. However, the timescale of these fluctuations

are smaller than the changes that the observations can resolve. Thus, this term is

unknown in this problem although only its monthly average would be required. If

the fluctuations occur on a larger timescale than expected, they will contribute to

the error term e. If this surface integral is neglected Eq. 2.56 becomes:∫

a

(q u+ q′u′) · da =

∫

a

qvua · da+ equ (2.57)

40


where equ is the total error due to the approximations on qu made up to this point.

The surface integral is applied to the model boxes and simplified by inspecting the

transports through the surface, the bottom, the separation boundary, the northern

and southern connections, and the walls:

∫

SC1

(q u+ q′u′) · da1= q1U1 + eSC1(2.58)

∫

SC2

(q u+ q′u′) · da2=− qHU2 + eSC2(2.59)

∫

SB

(q u+ q′u′) · da1 = q1W1 − q2W2 + eSB (2.60)

=−

∫

SB

(q u+ q′u′) · da2

(2.61)

where qH is the average in HS box, the other advective and turbulent fluxes over

the surface, the bottom, the walls are zero and the fluxes through the northern

connection are assumed to be negligible compared to the ones through the southern

connection.

Similarly to the continuity equations, the equations of the advective and turbulent

transports (Eqs 2.58–2.60) and conservation equations (Eqs 2.54–2.55) lead to the

41


budget equations of heat and salt in the top and bottom boxes:

Heat budget:

− T1 U1 − T1 W1 + T2 W2= V1

∂T1

∂t− TR R (2.62)

−a

ρ0CpFSW

∫ d

0

k e−kzdz −a

ρ0Cp(F − FSW ) + ε3

TH U2 + T1 W1 − T2 W2= V2

∂T2

∂t(2.63)

−a

ρ0Cp

FSW

∫∞

d

k e−kzdz + ε4

Salt budget:

− S1 U1 − S1 W1 + S2 W2= V1

∂S1

∂t+ ε5 (2.64)

SH U2 + S1 W1 − S2 W2= V2

∂S2

∂t+ ε6 (2.65)

The quantities Ti, Si, Ui, Wi, εj , R, F , FSW and k variables are time-dependent

scalars (where i is defined as in Fig. 2.1, j as in Eqs 2.41–2.42 and 2.63–2.65).

The quantities εj, R, F , FSW and k represent the equation residuals (as defined in

the introduction), the freshwater inflow, the net surface heat flux and its shortwave

(hereafter SW) component and a light attenuation coefficient (in m−1). In Eqs 2.62

and 2.63, F and FSW fluxes are converted into ◦C m3 s−1 by the factor a/(ρ0Cp)

where a is the SoG surface area (Table 2.1), ρ0 a reference density and Cp the water

specific heat capacity. I assume a single-band light attenuation of FSW , the SW

component of F, or “blue” component [Kara et al. 2005]. It penetrates deeper than

the longwave component and some fraction can potentially enter the lower box. It

decays according to k, the light attenuation (over the characteristic distance k−1,

42


average 3.6 m), based on an estimated 1% photosynthetically available radiation

(PAR) level in the SoG (at an average depth of 15 m), using STRATOGEM data.

I will refer to Eqs 2.41–2.42 and 2.62–2.65 as the “flux form” of the conservation

equations by opposition to the following “tracer difference form” equations:

Mass budget (unchanged):

− U1 −W1 +W2 =−R + ε1 (2.66)

U2 +W1 −W2 = ε2 (2.67)

Heat budget:

(T2 − T1)W2 = V1

∂T1

∂t+ (T1 − TR)R (2.68)

−a

ρ0CpFSW

∫ d

0

k e−kzdz −a

ρ0Cp(F − FSW ) + ε3

(TH − T1)U2 + (T1 − T2)W2= V2

∂T2

∂t(2.69)

−a

ρ0Cp

FSW

∫∞

d

k e−kzdz + ε4

Salt budget:

(S2 − S1)W2 = V1

∂S1

∂t+ S1 R + ε5 (2.70)

(SH − S1)U2 + (S1 − S2)W2= V2

∂S2

∂t+ ε6 (2.71)

There is an advantage to using Eqs 2.66–2.71 instead of Eqs 2.41–2.42 and 2.62–

2.65. A numerical issue that becomes relevant when inversion is attempted is that

the advective terms (lefthand side of Eqs 2.62–2.65) are usually significantly larger

than the other terms (righthand terms). Thus, the advective terms can dominate

43


these smaller terms, during the inversion procedure. The inversion then tends to

show only mass conservation which is a weak constraint on the circulation. However,

the differences of the advective terms, in Eqs 2.66–2.71, are of the same order of

magnitude as the other sink terms and the forcing terms. This makes the inversion

procedure more likely to produce tracer conservation, as well as mass conservation.

The “tracer difference form” is obtained by combining Eqs 2.62–2.65 with Eqs 2.41–

2.42. Adding Eqs 2.41–2.42 to Eqs 2.62–2.65 does not change the formal validity of

Eqs 2.62–2.65. In Eqs 2.68–2.71, I chose to express all the conservation equations in

terms of U2 and W2, while W1 and U1 only appear in Eqs 2.41–2.42. The terms U1

and W1 mathematically result from the addition of U2 and R, the freshwater flow

in the case of U1, and from the difference between vertical upwelling W2 and deep

landward flow U2 in the case of W1. The “tracer difference form” of the conservation

equations shows that the heat content and the salt content of the SoG boxes are

controlled by forcings, sources, vertical upwelling and water intrusions.

Upward and downward transports (W2 and W1) are convenient mathematical

parametrizations of the circulation. But, in physical terms, it is more appropriate

(and efficient) to separate entrainment processes associated with a net unidirectional

mass flux from turbulent mixing processes, in which no net mass flux occurs but a

tracer flux does occur. The parameter E (=W2-W1) can be defined as the entrainment

while M (± W1) as the rate of turbulent mixing (mixing exchange for short). On

average W2 > W1 in positive estuaries since upwelling should occur more often

than downwelling. In this case, M vanishes completely from Eqs 2.41–2.42, which

is consistent with the properties of turbulent mixing. Thus, the quantities E and

M enable one to interpret the mathematical transports U1, U2, W1, W2 in physical

terms.

The next step of the conservation equations consists of writing these constraints

44


in a matrix equation of the form of Eq. 2.3. This form of the conservation equations

will be convenient for applying the LLS or the SVDM. The matrix A and the vectors

x and b are defined as

A =

−1 0 −1 1

0 1 1 −1

0 0 0 T2 − T1

0 TH − T1 0 T1 − T2

0 0 0 S2 − S1

0 SH − S1 0 S1 − S2

(2.72)

x =[U1, U2, W1, W2]

T (2.73)

b =

−R

0

V1∂T1

∂t+ (T1 − TR)R− a

ρ0Cp(H1 FSW + F − FSW )

V2∂T2

∂t− a

ρ0CpH2 FSW

V1∂S1

∂t+ S1 R

V2∂S2

∂t

(2.74)

where H1 (=0.99 for d=30 m) and H2 (=0.01 for d=30 m) are the integral coefficients

appearing in Eqs 2.68–2.69, respectively.

Table 2.2 shows the actual values of the inversion parameters (s, ωi and xA).

Horizontal and vertical transports in the vector x were scaled by s, 5×104m3s−1, as

this is known to be a reasonable scale for magnitude [Godin et al. 1981, Li et al. 1999,

Pawlowicz et al. 2001, Masson and Cummins 2004, Pawlowicz et al. 2007]. The scaling

45


Parameter W row-scaling S Column- xA a Priori Transports

scaling

Diagonal/Vector ω1, ω2 ω3, ω4 ω5, ω6 s U01, U02, W01, W02

Coefficients

Scale 1.6, 1 2.6, 1.5 4.7, 0.4 5 4.5, 4, 2.1, 6.2

×104 ×105 ×105 ×104 ×104

Unit m3 s−1 ◦C m3 s−1 psu m3 s−1 m3 s−1 m3 s−1

Table 2.2: Inversion Parameters

coefficients ωi are obtained by estimating the residuals of the conservation equations

assuming that the magnitude of the transports were about 5×104m3s−1, and the

sources and forcings at their absolute maximum value. The a priori estimates of xA

elements was taken as the approximate average of the “pseudo-inverse” (Eq. 2.9) for

the whole time series, but is also consistent with previous estimates (see chapter 1

and Table 4.3). The xA components reflect the pattern of the estuarine circulation

in the SoG where vertical turbulent mixing M(= W01) is usually smaller than the

other transports. The entrainment E(= W02 −W01) and the landward transport U02

are close and the seaward transport U01 is greater because of the contribution of the

river inflow R (maximum at about 104m3s−1).

2.3 Solution Uncertainty and Residuals

Eq. 2.3 is known with an error ε. The solution x is known with an uncertainty εx.

In the standard inverse approach, these uncertainties can be estimated as a function

of uncertainties in the observations b. However, there are also uncertainties in the

46


matrix A, as its coefficients are derived from observations of T and S.

In the standard inverse method (described in the introduction), only small changes

in b are taken into account to estimate the error in the inverse solution. More sophis-

ticated procedures known as the total least-squares or the total inversion [Wunsch

2006] can be used to take into account the small changes in both b and A. How-

ever, these procedures are non-linear. The use of non-parametric and parametric

bootstraps [Efron and Tibshirani 1993] enables one to use a linear procedure.

Fig. 2.3 shows the different steps that lead to the inverse solution x. When boot-

strap procedures are applied to the input data of the top and bottom boxes, the

coefficients of the replicates of the vector b and the matrix A are changed by an

amount ∆b and ∆A, respectively. This implies that the replicates of the solution x

and the residuals εi’s, through the inverse procedure, are also changed by an amount

∆x and ∆εj. This shows that the bootstrap can provide one with an estimate of

the error of all the knowns (b and A) and unknowns (x and εj) of the problem.

The latter is based on small changes not only of the elements of b but also of the

coefficients of A.

The method of non-parametric and parametric bootstraps is based on Efron and

Tibshirani [1993]. In the case of the observations from the different hydrographic

stations, it consists of resampling the vertical profiles in a given cruise with repetition

allowed and generating replicates of the box averages with these new profile sets. In

the case of observations obtained at one particular location (forcings and sources), a

parametric bootstrap was applied assuming a reasonable coefficient of variation (ratio

of standard error over average, see Table 2.3). The non-parametric and parametric

bootstraps are combined to produce enough bootstrap replicates (200 replicates) for

all the known quantities. It is then possible to estimate the residuals εj of Eqs 2.66–

2.71, the error in x the solution of the inverse problem, and the errors ∆A and ∆b on

47


Figure 2.3: Chart of the Inversion Procedure. The data are input on the lefthand

side. After averaging and time derivation, they yield the solution, on the righthand

side.

Variables Coefficient of Variation (%)

N, P and Si concentrations 15

TH and SH 2.5

F and FSW 15

TR and R 15 and 10, respectively

Table 2.3: Coefficients of Variation Used in the Parametric Bootstraps. They are

applied to the riverine inputs, the biogeochemical tracer concentrations, the heat

fluxes and HS salinity and temperature. N is nitrate+nitrite, P phosphate and Si

Silicic acid.

48


equations of the biogeochemical tracers (see next section).

2.4 Forward Problem: Estimating the Net

Primary Productivity

To solve the forward problem, I will keep the same idealization of the SoG into a

two-box model (Fig. 2.2) with time dependence and the same separation depth d.

The solution is, however, more simple than in the case of the inverse problem, as it

does not require the inverse procedure. Fig. 2.2 lays out the boxes, the biogeochemical

fluxes and processes and the biogeochemical tracers monitored in the SoG. Phosphate

(P), nitrate (N), silicic acid (Si) and dissolved oxygen (O) exchanges in the two SoG

boxes are represented by the following Eqs 2.77–2.84 of conservation in the “tracer

difference form” (for consistency with the physical box model). These equations result

from integrations similar to the integration of Eq. 2.51 where q can be the tracer N,

Si, P or O. For instance, the “Flux form” of the conservation equations for P are

written as:

− P1U1 − P1W1 + P2W2 = V1

∂P1

∂t− PRR− (φ1)P + ε′1 (2.75)

PHU2 + P1W1 − P2W2= V2

∂P2

∂t− (φ2)P + ε′2 (2.76)

where surface processes, for instance for P, are net upwelling (P2W2−P1W1), advec-

tive export (−P1U1), river inflow (PRR), the net biological uptake (φ1)P and the sur-

face storage/drawdown rate term (V1∂P1/∂t), and deep processes are net upwelling

(P2W2 − P1W1), deep intrusions (PHU2), the net biological uptake (φ2)P and the

deep storage term (V2∂P2/∂t). Note that we will use the words “storage rate term”

49


or “storage rate” instead of “storage/drawdown rate term”. Storage occurs when the

time derivative is positive, so that the storage rate is positive too. In the other hand,

drawdown occurs when the time derivative is negative, so that the storage rate is

negative in this case. Any net biological uptake of nutrient and net biological pro-

duction of dissolved O2 are represented by φ terms. The general equations rewritten

50


in the “tracer difference form” are:

P budget:

(P2 − P1)W2 = V1

∂P1

∂t+ (P1 − PR)R− (φ1)P + ε′1

(2.77)

(PH − P1)U2 + (P1 − P2)W2 = V2

∂P2

∂t− (φ2)P + ε′2 (2.78)

similarly for the other nutrients and O2

N budget:

(N2 −N1)W2 = V1

∂N1

∂t+ (N1 −NR)R− (φ1)N + ε′3

(2.79)

(NH −N1)U2 + (N1 −N2)W2 = V2

∂N2

∂t− (φ2)N + ε′4 (2.80)

Si budget:

(Si2 − Si1)W2 = V1

∂Si1∂t

+ (Si1 − SiR)R − (φ1)Si + ε′5

(2.81)

(SiH − Si1)U2 + (Si1 − Si2)W2 = V2

∂Si2∂t

− (φ2)Si + ε′6 (2.82)

O2 budget:

(O2 − O1)W2 = V1

∂O1

∂t+ (O1 − OR)R− (φ1)O2

+ ε′7

(2.83)

− kO2

(Osaturation − Osurface

)

(OH − O1)U2 + (O1 −O2)W2 = V2

∂O2

∂t− (φ2)O2

+ ε′8 (2.84)

51


One difference between these equations and the inverse problem equations is that

each equation has a sink term (φ1 or φ2) of unknown size. In the case of O2, φ1 is a net

biological production term, that is a source term. The forward solution for φ1 or φ2

by any of the 4 biogeochemical tracers can readily be obtained by addition of all the

known quantities. At this stage, the transports (in particular U2 and W2) have been

obtained through the resolution of the inverse problem. Thus, the contribution of the

advective transports to the biogeochemical conservation equations can be estimated.

For instance, the sink term (φ1)N from the surface N budget is determined by the

equation:

(φ1)N = V1

∂N1

∂t− (N2 −N1)W2 + (N1 −NR)R. (2.85)

The air-sea exchange flux of O2 can be estimated once the piston velocity kO2,

and the difference between saturation level and surface level are known (see further

detail in section 3.2.3). Based on the budget made by Mackas and Harrison [1997], the

atmospheric deposition of N as well as the anthropogenic sources have been neglected

with respect to the other terms in the box model. The atmospheric deposition and

anthropogenic sources of P and Si have been assumed to be negligible. Turbulent

diffusion terms are combined mathematically to advective transports (for instance

(N2 − N1)W2 and (NH − N1)U2 for Eqs 2.79–2.80). This could be an additional

source of errors in the estimation of the sink terms (φ1)N , (φ1)P and (φ1)Si because

the turbulent component of the circulation, u′, and the corresponding correlated

variations of the biogeochemical tracers, q′, are unknown. For instance, the transports

of N in the top box V1 has the form:∫

SC1

(N u+N ′u′) · da1 = N1U1 + eSC1(2.86)

and ∫

SB

(N u+N ′u′) · da1 = N1W1 −N2W2 + eSB (2.87)

52


Eq. 2.85 is known with an equation error of ε′1. I can estimate the ε′j’s by applying

the estimation method in section 2.3. Further analysis is necessary on the sink terms

φ1 or φ2 of Eqs 2.77–2.84 to be able to estimate the primary productivity. This is

the goal of sections 5.2.3 and 5.2.4 and in Chapter 5.

53

Chapter 3

Observations

3.1 Introduction

This chapter describes the time series of the observed variables needed to find the

solution of the inverse and forward problems in chapter 2. First, I introduce the

sampling method used to collect the observations, and the data averaging used to

compute the time series. Time Series of freshwater, salt, heat and temperature are

used to estimate the parameters of the physical dynamics, and time series of nitrate,

phosphate, silicic acid and dissolved O2 are used to estimate the parameters of the

biogeochemical cycles. Then, I discuss the information from the time series that

reflect surface and deep characteristics of the SoG.

3.2 Data Sources

3.2.1 STRATOGEM

Using a hovercraft (CCGH Siyay), the STRATOGEM program was able to sample

9 stations (Fig. 1.2) over a 240-km long track in about 9 hours (including both

sampling and steaming times). The stations were located in the central and southern

regions of the SoG, between 48◦55.0’N and 49◦21.5’N. The surveys started early

in the morning (8-9 am) near the beginning of every month from April 2002 to

54

Chapter 3. Observations

June 2005, with more frequent sampling around the spring bloom. This amounts

to 47 cruises. In addition, another 9 cruises with a field sampling design different

from the standard STRATOGEM were carried out. This different sampling design

helped to investigate the Fraser River plume and stations different from the standard

STRATOGEM stations.

CTD (conductivity-temperature-depth) casts provided continuous vertical pro-

files of physical (pressure, temperature and conductivity) and biogeochemical tracers.

The CTD instrument was equipped with additional sensors to measure chlorophyll-a

fluorescence, dissolved O2 (oxygen), photosynthetically available radiation (PAR),

and beam transmissivity. Profiles were made at the front of the hovercraft (limiting

ship mixing) to sample at high vertical resolution the entire water column down to

within 15 m of the bottom. The data were binned (over 1 m) to obtain the continuous

profiles from the sensors and combined to obtain additional variables: e.g. salinity

and density. Note that salinity measurements are reported on the Practical Salinity

Scale PSS-78 [UNESCO 1981 a,b, 1983]. The vertical profiles of the variables of inter-

est for this study (temperature, salinity and dissolved O2) are accurate to ±0.003◦C,

±0.01 psu and ±0.2 mL L−1, respectively.

Water samples were also collected to measure macronutrients (nitrate/nitrite,

phosphate and silicic acid) in the water column, to identify and count phytoplankton

cells, and to provide a baseline for the correction of and comparison with CTD

measurements (dissolved O2, chlorophyll-a concentration and salinity). They were

taken at depths of 0, 5, 10 and 30 meters, from the front of the hovercraft, using

5 and 8 L Niskin bottles. In addition, in order to estimate the SoG deep profile of

macronutrients, samples were collected at 250 m at station 2-2 and 50, 100, 200, 300

and 390 m at station 4-1.

Water samples for macronutrients were sub-sampled on deck from the Niskin

55


bottles into acid-washed cups, filtered through a 0.7 µm GF/F, and then stored

and frozen in acid-washed vials for lab analysis (within 2 months). Water samples

(for the other biogeochemical measurements) were sub-sampled on deck from the

Niskin bottles into containers pre-rinsed with seawater. The error on the macronu-

trient concentration was estimated to be equal or less than ± 15% or 0.5 µM for

nitrate (short for both nitrate and nitrite), ± 8% for phosphate and silicic acid.

Macronutrient measurements on water samples that had not been frozen (analyzed

within a few days after sampling) showed that the accuracy could be significantly

improved. But logistical issues prevented this from being done on a regular basis.

Water samples for measurement of dissolved O2 and chlorophyll-a were kept fresh

and analyzed within 24 hours. Dissolved oxygen was estimated from water samples

using Winkler titration [Parsons et al. 1984, Culberson 1991] with an accuracy close

to or larger than 5%. Chlorophyll-a was estimated from smaller water sub-samples

(100–200 mL). The samples were vacuum filtered through polycarbonate membranes

of 0.2, 2 and 20 µm. Biomass of chlorophyll-a pigments was measured using a Turner

Designs 10AU fluorometer [Parsons et al. 1984]. Replicates suggest an uncertainty of

± 10%. Water samples (100 mL) for taxonomy were preserved using Lugols solution

in dark glass bottles and analyzed quantitatively for microplankton following the

Utermohl method [Hasle 1978]. Estimates of carbon biomass per cell were based on

determinations of cell biovolume [Haigh et al. 1992] and subsequent conversion of

biovolume into carbon biomass [Montagnes and Franklin 2001, Strathman 1967].

3.2.2 JEMS

The Joint Effort to Monitor the Strait of Juan de Fuca (JEMS) program collected

similar continuous profiles and water samples at three hydrographic stations in HS

56


(Fig. 1.1). Based on the available JEMS information [Newton et al. 2002], the meth-

ods of sampling and the accuracy of the measurements are similar to that of STRA-

TOGEM.

3.2.3 Freshwater Inflow, Surface Heat and Air-sea Fluxes

This section contains a description of the various sources of freshwater, heat, and

air-sea exchanges that contribute to surface changes of temperature, salinity and dis-

solved O2, respectively. Freshwater (FW) inflow estimates (see Fig. 1.4) are based on

the Fraser River discharge measured at Hope by Environment Canada (water station

08MF005). A linear regression (y-intercept 1165 m3 s−1 and slope 1.66) is used to

estimate the total freshwater inflow from the Fraser River discharge [Pawlowicz et al.

2007]. We assume other measurements (temperature, macronutrients, and dissolved

O2), necessary for the computation of the river fluxes into the whole SoG, to be

similar to those measured in the Fraser River at Hope. Thus, the observations used

to measure the temperature, the macronutrients and the dissolved O2 in the rivers

are based on the observations from the Fraser River at Hope.

The surface heat budget is composed of both a measured heat flux (ShortWave

radiation or SW) and estimated fluxes (LongWave radiation or LW, sensible and

latent heats). The SW heat flux is measured by a SW-band downwelling-radiation

sensor located 10 m above the ground at Totem Field station on the University of

British Columbia (UBC) campus (maintained by the Bio-Met program). The other

heat fluxes (LW, sensible and latent heats) were estimated using weather observations

at either Vancouver International Airport station (LW radiation) or Halibut Bank

buoy (sensible and latent heats), and the methods of Pawlowicz et al. [2001].

The air-sea flux of dissolved O2 is estimated using a Fickian formulation of the

57


flux [Woolf 2005, flux proportional to saturation level], an empirical formulation of

the gas piston velocity [Wanninkhof and McGillis 1999], and the observed surface and

estimated saturation levels of dissolved O2. The wind observations used for the gas

piston velocity come from the Halibut Bank buoy, while the dissolved O2 observations

come from STRATOGEM surface observations (between 1–2 m).

The sampling intervals of all these fluxes are different from the sampling period of

the observations from STRATOGEM. In general, these time series have a 30-minute

(SW heat flux) or 60-minute (all the others) intervals. This high time resolution was

kept to estimate the different surface source terms in the conservation equations.

Just before being substituted into the conservation equations, they were averaged

over a one-month moving window and interpolated to the STRATOGEM sampling

dates. This is a good practise to ensure that all the input data have approximately

the same characteristic timescale.

3.3 Box Model Inputs

Observations provide the necessary information to compute the unknown terms oc-

curring in the conservation equations in the physical and biogeochemical box models

(Eqs 2.66–2.71 and Eqs 2.77–2.84). The equation coefficients are the box-averaged

observed temperature, salinity, nutrients and dissolved O2 concentrations near the

surface (0–30 m) and at depth (30–400 m) in the SoG and in the entire water col-

umn of HS. The terms in the equations are the various surface sources of mass, heat,

nutrients or dissolved O2.

58


3.3.1 Separation Depth

The separation depth d defines the vertical separation between the surface outflow

and the deep inflow in the SoG box model. The surface outflow carries freshwater

from the rivers and seawater from the SoG, while the deep inflow brings dense seawa-

ter from HS. Analyzing the vertical profiles of salinity and temperature in the SoG

enables one to identify these two types of water masses and to determine their depth

ranges. The separation depth should roughly correspond to the lower boundary of

the pycnocline, below which the water column contains mainly seawater from HS.

In estuaries, salinity mainly controls the density, and determining the depth of the

halocline is equivalent to determining the pycnocline. Additional information on the

vertical structure of the water column can be obtain from looking at the tempera-

ture profile and determining the depth of the thermocline. A detailed analysis of the

vertical profiles (not shown) of temperature and salinity suggested an appropriate

separation depth would be below a level between 15 m and 30 m, based on the depth

of the halocline and the thermocline. These variations of the halocline and thermo-

cline are seasonal variations. In winter, both wind-induced mixing and convection

of cooling water tend to deepen the mixed layer close to 30 m or down to 50 m. In

summer, stratification due to both salinity (river freshwater) and temperature (sur-

face heating) tends to shallow the mixed layer above 30 m. Further analysis of the

vertical profiles at station S4-1, traditionally used to represent the SoG, suggested a

deeper separation depth, below the 50-m level [Pawlowicz et al. 2007].

The separation depth has to define not only the top box of the circulation box

model (section 2.2, Fig. 2.1), but also the top box of the primary production box

model (section 2.2, Fig.2.2). A shallow separation depth would underestimate the PP

and the associated productivity, while a deeper value would not affect the estimated

59


PP and its productivity. A detailed analysis was carried out using vertical profiles

of chlorophyll-a fluorescence (not shown), a proxy for primary production (PP), and

PAR (not shown), a variable to determine the depth range where photosynthesis

occurs. SoG PP was found to be located at or above 30 m where 1% and more of

PAR reaches the water column. The 30-m level corresponds to the deeper separation

depth suggested by the physical analysis in the previous paragraph. During Summer

however, the 1% level can be significantly shallower, and even <1 m in the Fraser

Plume. Thus, the analysis of SoG vertical profiles suggested that the separation

depth should be between 30 m (this analysis) and 50 m (Pawlowicz et al. [2007]).

To keep the box-model simple, a constant separation depth was chosen to be 30 m

(see also discussion in section 4.3.1). Note that the change of flow should happen in

a gradual transition layer between the inflowing and the outflowing water masses.

The two-layer box model ignores the thickness of the transition layer and assumes a

discrete change from a surface seaward flow to a deep landward flow at the separation

depth.

3.3.2 Spatial Averaging and Hypsography

To determine the best average of the observations from STRATOGEM over the

surface and bottom boxes, it is necessary to respect the SoG geomorphology and to

use all available data. In order to get the box averages for various tracers, a mean

vertical profile is first found by averaging all station data at a particular depth. Box

averages are then hypsographically weighted averages of the mean profiles over the

depth range represented by each box.

The hypsography (Fig. 3.2) shows that the area of horizontal sections of the SoG

decreases roughly proportionally to the depth. Thus, assuming a separation depth of

60


30 m, the volume of the surface box is smaller than, but not negligible compared to,

the volume of the bottom box: the surface box volume is about 21% of the bottom

box volume, and 17% of SoG volume.

In the case of HS, the box average is a simple average of the tracer over the

water column. Since the JEMS and STRATOGEM sampling dates are different it is

necessary to interpolate HS box averages to the STRATOGEM sampling dates.

3.3.3 Time Dependence

The transports and the primary productivity are expected to be time-dependent

(section 2.2). Thus, we need to calculate the time-derivative terms of physical and

biogeochemical inputs. In my box model, the time-derivative terms are computed

by estimating the derivative of the observation averages, in each box and for each

tracer. The derivative scheme is based on a 5-point parabolic fit which reasonably

represents the actual time derivative of the tracers. The reason for this choice is to

remove the sampling noise that appears from survey to survey in the temperature

and salinity time-derivatives. Time derivatives at the beginning and the end of the

tracer time series are handled by applying a 2- or 3-point time-difference scheme.

The time derivatives are part of the “known terms” like the surface sources and the

forcing terms.

3.3.4 Input Time Series

Salinity

In summer, average surface salinity rapidly decreased (Fig. 3.3a). This is due to

the addition of freshwater into the SoG surface water by the local rivers, and the

61


5 10 15 20 25 30100

90

80

70

60

50

40

30

20

10

0

salinity (psu)

dept

h (m

)

5 10 15 20100

90

80

70

60

50

40

30

20

10

0

dept

h (m

)

temperature (°C)5 10 15 20

100

90

80

70

60

50

40

30

20

10

0

dept

h (m

)

5 10 15 20 25 30100

90

80

70

60

50

40

30

20

10

0

dept

h (m

)

default d

maximum d

Figure 3.1: Vertical Profiles of Salinity and Temperature during STRATOGEM.

The envelope of temperature and salinity vertical profiles is shown at the 9 regular

locations. Each square is an average value of data binned over a 1 m. The dashed

lines show d–D, the range of the separation depth.

mixing of freshwater and SoG surface water during the freshet (a Glossary with the

definitions of the seasons can be found on p. xiii). The freshwater addition and water

62


0 1 2 3 4 5 6 7 8

0

50

100

150

200

250

300

350

400

section area (×109 m2)

dept

h (m

) surface area: ∼ 7 × 109 m2

surface volume: 1.9 × 1011 m3

bottom volume: 9 × 1011 m3

Figure 3.2: Hypsography of the Whole SoG. The 0-m section area is extrapolated

from the 1-m and 2-m values. In addition, the volumes of the top and bottom boxes

are indicated in m3.

63


mixing are fast processes with respect to the sampling period (1-2 weeks during the

early freshet, otherwise 1 month).

HS water enters and mixes with deep SoG water as seawater intrusions (Fig. 3.3b).

From spring until summer (or fall), HS salinity increases as it mixes with oceanic

seawater coming from the Strait of Juan de Fuca (SoJdF). At the same time, SoG

deep water mixes with denser HS water and it is entrained upward bringing more

salt to the SoG surface. The SoG deep salinity maximum was reached later than HS

salinity maximum because it takes time for the water to be transported into the SoG

and then mixed with the SoG deep water [Pawlowicz et al. 2007].

Temperature

The average surface temperature follows the surface net heat flux with a delay of 1 to

2 months (Fig. 3.4a). The surface net heat flux is the net budget of SW, LW, sensible

and latent heat fluxes. Eq. 2.68 shows the different processes that can contribute

to the changes of surface temperature: e.g., the summer heat influx, the outflow

transport of SoG water and the vertical exchange of surface and deep SoG waters.

The yearly maximum temperature is reached in summer.

In spring and summer, the well-mixed HS water enters and brings heat into deep

SoG (Fig. 3.4b). HS heat input peaks in late-summer when HS temperature is close

to the summer maximum temperature (similarly with the surface of the SoG). During

this same period, the SoG bottom water receives heat from HS and SoG surface. Since

heat transport and diffusion in SoG bottom occur through vertical water mixing, SoG

bottom average increases steadily until late August -early September after the yearly

temperature maximum occurs in HS (a delay of 1 to 2 months). In 2003 and 2004,

the temperature average of the SoG bottom box levelled off after September.

64


Apr Jul Oct 2003 Apr Jul Oct 04 Apr Jul Oct 05 Apr24.5

25

25.5

26

26.5

27

27.5

28

28.5

29

29.5

Sal

inity

(ps

u)

Apr Jul Oct 2003 Apr Jul Oct 04 Apr Jul Oct 05 Apr

0

2

4

6

8

10

12

14

16

18

20

Fre

shw

ater

× 1

03 m3 s−

1

Apr Jul Oct 2003 Apr Jul Oct 04 Apr Jul Oct 05 Apr30

30.5

31

31.5

32

32.5

Sal

inity

(ps

u)

SoG 0−30 m

FW

input rate

SoG 30−400 mHS 0−200 m

Figure 3.3: Salinity and Freshwater Time Series. In both panels, the left axis indi-

cates the Box averages of the Salinity (psu). Each marker is a box average from a

single cruise. In the upper panel, the right axis indicates the surface FW input rate

(m3s−1).

Observations, Estimations and Budget of Heat Fluxes

The observed SW radiation (Fig. 3.5a) and the estimated LW radiation, latent and

sensible heats (Figs 3.5b,c and d, respectively) are monthly averaged as explained in

65



7

8

9

10

11

12

13

14

Tem

pera

ture

( ° C

)

Apr Jul Oct 2003 Apr Jul Oct 04 Apr Jul Oct 05 Apr−10

−5

0

5

10

15

Flu

x (×

10

Wm

−2 )

SoG 0−30 m

Net Heat Flux in SoG


8

8.5

9

9.5

10

10.5

Tem

pera

ture

( ° C

)

SoG 30−400 m

HS 0−200 mb)

a)

Figure 3.4: Temperature and Surface Net Heat Flux Time Series. In both panels,

the left axis indicates the Box Averages of the Temperature (◦C). Each marker is

a box average from a single cruise. In the upper panel, the right axis indicates the

surface net heat flux (× 10 W m−2). The study of phase lags of between the SoG

and HS deep temperatures have been studied by Pawlowicz et al. [2007].

66


section 3.2.3. During summer, the maximum of the monthly averaged SW heat flux

(about 260 W m−2) is about 25% of the observed maximum (about 1010 W m−2)

because of the alternation of day and night. At the same time, both the estimated LW

heat flux and latent heat reach a minimum: about -80 and -60 W m−2, respectively.

The minimum of averaged sensible heat is very small (about -20 W m−2) during this

period. The estimated sensible heat flux in the SoG suggests that the heat exported

during winter and the heat imported during the summer were about the same. It

also suggests that the sensible heat flux is small compared to the other heat fluxes.

The observed and estimated surface heat fluxes were combined to determine the

net surface heat flux into the SoG (Fig. 3.6). The net heat flux shows heating (130

W m−2) in summer, and cooling (-60 W m−2) in fall-winter. A minimum net heat

flux of -90 W m−2 was observed in Jan 2004 and 2005, but winter 2003 was warmer

(closer to 0 with a January minimum of -50 W m−2).

Nutrients

The surface box averages of the nutrient concentration exhibit a typical seasonal

cycle of depletion during the spring-summer and replenishment during winter with

occasional fall depletion (Fig. 3.7: phosphate, Fig. 3.8: nitrate and Fig. 3.9: sili-

cic acid). Depletion periods were associated with the uptake of nutrients by phy-

toplankton during spring (between 50–75% of nutrients depleted), summer and fall

blooms [Harrison et al. 1983]. During depletion periods, the box average nutrient con-

centration never gets close to zero, but examination of the raw observations shows

that near surface nutrients can be completely depleted. It is generally thought that

replenishment corresponds to nutrient upwelling and low primary production.

Although nitrate and phosphate levels are at a minimum in summer, silicic acid

67



500

1000

W m

−2

a) SW flux

b) LW flux

c) Latent heat

d) Sensible heat


−100

−50

0

W m

−2


−200

−100

0

100

W m

−2


−50

0

50

W m

−2

Figure 3.5: Heat Flux Observations and Estimates Into the SoG. SW stands for

shortwave radiation, while LW stands for longwave radiation. SW flux is observed,

while LW flux, sensible and latent heats are estimated. The gray lines are the heat

flux data. The dashed black lines are the monthly averages. See section 3.2.3 for

further detail.

68



−50

0

50

100

150

200

250

300

W m

−2

SW

Sensible

Latent

LW

net heat fllux

Figure 3.6: Surface Heat Budget. The net heat flux is the sum of the shortwave

(SW), longwave (LW), latent and sensible heat fluxes.

69


level is often at a minimum in spring, recovering somewhat in summer. A possible con-

tribution to this replenishment is the river input of silicic acid (see section 3.3.6) [Har-

rison et al. 1991]. This occurs after the spring bloom unlike the replenishment of

nitrate and phosphate.

SoG bottom average and HS average time series show that nutrients are abundant

and continuously available at depth. Note that the SoG surface average of nutrients

also reached these deeper values during the end of the replenishment. The SoG bot-

tom average of phosphate has a slight increasing trend over the whole time series that

is not due to any error in the analysis of water samples. Apart from phytoplankton

uptake and nutrient upwelling, the phosphate concentration can be changed by the

adsorption of phosphate on sinking particles and its aggregation on organic matter.

However, our data set did not allow us to investigate the sinking of particles or or-

ganic matter. As previously observed in the Western North Pacific deep water from

1968 to 1998 [Ono et al. 2001], another explanation to the increase of the SoG phos-

phate concentration could be the non-local increase of the phosphate concentration

of the source water coming from the Pacific Ocean and entering in the SoG. Bottom

averages of silicic acid have a definite seasonal cycle with a maximum timing that

coincides with surface replenishment.

Dissolved Oxygen

The surface box average of the dissolved O2 (Fig. 3.10) increases with the increased

solubility level during winter and spring (Fig. 3.11) and with the high primary pro-

duction during spring blooms, as phytoplankton photosynthesized O2 in excess of

respiration needs (spring maximum about 7 mL L−1). The surface and bottom box

averages followed closely the same seasonal cycle, but the bottom range was markedly

70


lower than surface range: between 1.5 and 3 mL L−1 less than the top box average.

SoG bottom box average reached a minimum of about 3 mL L−1 in fall. In 2003,

it reached the minimum 1–2 months after the HS minimum. HS box average shows

that large amounts of dissolved O2 were exported during the fall-winter into the deep

SoG (HS maximum around 5 mL L−1).

3.3.5 Air-sea Oxygen Flux

The air-sea O2 flux was estimated as explained in section 3.2.3, using the surface

observed O2 and the estimated saturation level (Fig. 3.11). The flux was negative

(exporting from SoG) when the observed level exceeded the saturation level, and the

flux was positive in the other case (importing from the atmosphere). The maximum

flux of dissolved O2 exported from the SoG always occurred during the spring bloom,

with a large amount of oxygen produced. For instance, the highest average flux

occurred in spring 2004, reached around 1.2×104mol s−1, and arose from an oxygen

excess that ranges between 2.7 and 3.1 mL L−1 (39–47% above surface saturation).

3.3.6 Riverine Inputs

Inputs of nutrients and dissolved O2 were estimated as explained in section 3.2.3,

from observations at Hope in the Fraser River (section 3.2.3). The freshwater input

used to estimate the inputs of nutrients has been described in section 3.2.3 using

Figs 1.4 and 3.3a. The silicic-acid input was by far the most significant nutrient input

from the river by comparison with the amount of upwelled nutrients. Assuming a net

upwelling transport of 5×104m3s−1 (from the a priori estimates in chapter 2) and

using the bottom box averages, one can show that the river flux of P and N nutrients

(maximum 2 and 75 mol s−1, respectively) are much smaller than the estimated

71


Jul Oct 2003 Apr Jul Oct 04 Apr Jul Oct 05 Apr0

0.5

1

1.5

2

2.5

Pho

spha

te µ

M

SoG 0−30 m

HS 0−200 m

SoG 30−400 m

Figure 3.7: Phosphate Concentration Time Series. Each marker is a box average of

the phosphate concentration (µM) in the SoG or in the HS from a single cruise.

72



5

10

15

20

25

30

35

Nitr

ite+N

itrat

e µM

SoG 0−30 m

HS 0−200 m

SoG 30−400 m

Figure 3.8: Nitrite+Nitrate Concentration Time Series. Each marker is a box average

of the nitrite+nitrate concentration (µM) from a single cruise.

73



10

20

30

40

50

60

Sili

cic

acid

µM

SoG 0−30 m

HS 0−200 m

SoG 30−400 m

Figure 3.9: Silicic Acid Concentration Time Series. Each marker is a box average of

the silicic-acid concentration (µM) from a single cruise.

74



1

2

3

4

5

6

7

8

9

diss

olve

d O

2 mL

L−1

SoG 0−30 m

HS 0−200 m

SoG 30−400 m

Figure 3.10: Dissolved Oxygen Time Series. Each marker is a box average of the

dissolved O2 concentration (mL L−1) from a single cruise.

75


Jul Oct 2003 Apr Jul Oct 04 Apr Jul Oct 05 Apr−1.5

−1

−0.5

0

0.5

1

1.5

DO

flux

into

SoG

(×1

04 mol

×s−

1 )

5

5.5

6

6.5

7

7.5

8

8.5

9

9.5

10

diss

olve

d O

2 (m

L L−

1 )

observed O2 (1−2 m)

O2 saturation level (1−2 m)

air−sea O2 flux (1−month moving average)

Figure 3.11: Air-sea Oxygen Flux Time Series

76


upwelling input (100 and 1250 mol s−1, respectively), while it is close in the case of

the silicic acid: 700-1000 mol s−1 from the river is smaller but not zero compared to

2500 mol s−1 from the upwelling transport.

In the case of oxygen, the river input (1000-4000 mol s−1) was of similar order

of magnitude to the upwelled dissolved O2 (8900 mol s−1, still assuming a transport

of 5×104m3 s−1) or the amount of oxygen due to air-sea exchange (seasonal average

from -4000 to 2000 mol s−1, see Table 5.4, but maximum can reach 1.2×104mol s−1

from previous section).

77


Apr Jul Oct 2003 Apr Jul Oct 04 Apr Jul Oct 05 Apr 0

1

2

3

4

5

6

7

8

9

10

Nu

trie

nt R

ive

r F

lux

(× 1

02 m

ol s

−1)


1

1.5

2

2.5

3

3.5

4

4.5

5

5.5

O2 R

ive

r F

lux

(× 1

03 m

ol s

−1)

PNSiO

2

Figure 3.12: River Biogeochemical Inputs Time Series. Each curve indicates the

estimated river influx (mol s−1). Phosphate (P), nitrate (N), and silicic-acid (Si)

rates are indicated on the left axis, while dissolved-O2 rate is indicated on the right

axis.

78

Chapter 4

Circulation and Transports

4.1 Introduction

In this chapter, the objective is to estimate the advective transports in the SoG. The

estimation procedure extensively uses the SVD mathematical framework and the

SoG idealization as a two-box model described in chapter 2. Under this idealization

of the SoG, temperature and salinity are averaged over each domain represented by

the boxes. Water, heat and salt enter the deep box, are exchanged between the two

boxes and circulate in and out of the surface box. In the box model, these exchanges

are due to the estuarine circulation and the vertical diffusion.

I will apply the SVD method to compute an inverse solution for the transports

from the input data defined in chapter 3, solving, as closely as possible, the conser-

vation equations Eqs 2.66–2.71. The inverse solution (similar to Eq. 2.24) provides

an estimate of each transport (U1, U2, W1, W2) and the derived physical parameters

E(=W2 − W1), net upward entrainment, and M (=W1), vertical rate of turbulent

exchange of water parcels (mixing exchange for short). Each transport is associated

with an estimated error ∆xi and each conservation equation is solved with an esti-

mated uncertainty εj.

The transport estimates can be affected by the parametrization of the box model

and the inversion procedure. Time dependence alters the seasonal variability of these

estimates, and it increases the strength of the relationship between their seasonality

79

Chapter 4. Circulation and Transports

and the freshwater inflow seasonality. This aspect will be investigated and the alter-

native results (without the time dependence) are presented for later comparison and

discussion (chapter 6). I will also investigate the sensitivity of the estimates to the

depth of separation (d) between the boxes and the sensitivity to the parametrization

of the inversion (trade-off parameter α, a priori transports xA). I will determine the

robustness of the estimates to realistic changes in the parametrizations. Finally, the

residuals εj’s are analyzed and discussed to investigate their consistency with the a

priori error estimates of the conservation equations.

In the discussion section, I consider the general circulation of the SoG described by

the transport estimates and make comparisons with previous studies. The freshwater

inflow is an important forcing of the estuarine circulation. I will discuss the fresh-

water influence on the circulation seasonality and the sensitivity of the circulation

seasonality to the freshwater (FW) inflow. Finally, I will investigate the interannual

variability of the circulation over the three years of observations.

Here, I will highlight the main contributions of this chapter:

1. first inference of transport time series of the SoG estuarine circulation with a

monthly time resolution and over three years.

2. inference of SoG transports based on consistency with observations of salinity

and freshwater input, temperature and surface heat fluxes.

3. one of the few observational studies of the link between freshwater input (R)

and surface seaward transport (U1).

80


4. the finding that only small changes of U1 occur even with large changes of R.

5. the finding that there are only very small changes ofW2 even with large changes

of R.

6. the finding that, based on 2002-2005 data, annual changes of U1 are linked to

annual changes of R.

4.2 Results

4.2.1 Estimates of Estuarine Transports and Mixing

Exchange

Fig. 4.1 shows the seaward (U1) and net upward entrainment (E) transports, and the

mixing exchange (M) as defined in Fig. 2.1. The estimates of U2 (landward transport)

and W2 (total upwelling), and the corresponding errors are not shown because U2 is

not significantly different from E and there is little change in W2 time series.

Each band around the estimates represents the estimated standard error σbi using

bootstrap replicate statistics. Since the distribution of the transport replicates for a

single cruise was not always symmetric, lower and upper errors have been computed

using the difference between the 16th, 50th and 84th percentiles of the 200 replicates

(see Fig. 4.1) with σbi the average of lower and upper errors. TD columns of Table 4.1

show the values of σb, the mean of the σbi ’s, taken for x=U1, E, M and W2. In

81


x TD QSS

σe σb σ f p-value σe σb σ f p-value

U1 5.8 4.9 1.1 1.40 0.06 4.4 4.6 0.9 0.91 0.64

E 5.2 5.3 1.1 0.96 0.55 2.9 5.1 0.9 0.32 1

M 8.4 10 1.9 0.70 0.92 5.5 9.3 1.6 0.35 1

W2 4 4.8 0.9 0.69 0.93 3.2 4.6 0.8 0.48 1

Table 4.1: Analysis of Variance and F-tests for Time Dependence (TD) and quasi-

steady state (QSS) Transports. Columns σb, σe, and σ contain estimated standard

errors and variability of the transports U1, E, M and W2. Columns f and p-value

contain test statistics f and probability levels of the F-test on the estimated transports

U1, E, M and W2 . All values of σ’s are ×103m3s−1. The QSS columns will be used

later in section 4.2.2

addition, an overall mean value x and its corresponding error σ can be calculated:

x =1

l

l∑

i=1

xi. (4.1)

The error σ depends on σb the sampling and estimation error and σe the seasonal

variability of x contain in the time series xi:

σ =

√σb

2 + σe2

l(4.2)

where σe is:

σe =

√√√√ 1

l − 1

l∑

i=1

(xi − x

)2(4.3)

with l=47 the number of cruises: one cruise per month, except during March and

April 2003 and 2004, 2 or 3 cruises per month. The values of σe and σ are given in

82


Table 4.1, TD columns. Is there a seasonal cycle? A simple test can be derived by

taking a statistic f=σe2/σb

2 (f values of TD in Table 4.1).

This f statistic has an approximate F distribution with 46 and 199 degrees of

freedom [Harris 2001]. The larger the statistic, the more likely it is that the variability

in the time series is greater than its noise level. This can be quantified with the

associated one-sided p-value. The p-values go to zero as the seasonal variability gets

larger than the noise, and they approach 1 as time series variation becomes smaller

than the noise. These p-values are quite large for almost all of the parameters (xi’s

in Table 4.1), suggesting that, if there is any seasonal variability, it is too small to

be determined by the F-test. However, U1 shows a seasonal cycle (p=0.06, a 94% or

less confidence level).

Note that the test is somewhat conservative because it does not take into account

any autocorrelation in the time series. If autocorrelation were taken into account the

degrees of freedom associated with the time series and σe would be smaller than

46, thus σe would be larger than it is in Table 4.1 and f and p would be larger.

Table 4.1 shows that U1 is not constant with a reasonable small p-value (=0.06),

although the variance of the estimates for E, M and W2 is large enough that it hides

any likely seasonality. Note that p<0.05 is more commonly used as a significant test.

In Fig. 4.1, the general shape of U1 and E indicates that there is seasonality but

its magnitude is relatively small (σe about 5×103m3s−1) compared to the average

value (4–4.5×104m3s−1). The F-test suggests that the variability of U1 is statistically

significant, while one cannot make any conclusion about E. The average values are

(4.5±0.1)×104m3s−1 (1 σ) for U1 and (4.0±0.1)×104m3s−1 (1 σ) for E, while the

individual transports estimates range between 3.4×104m3s−1 and 6.2 ×104m3s−1 for

U1, and 2.8×104m3s−1 and 5×104m3s−1 for E (both representing a ratio of about

1.8). On average, the estimated error (1 σb) in U1 and E is around 5×103m3s−1.

83


The magnitude of the seasonality of the mixing exchange M is as large as its aver-

age value. The magnitude of M is on average lower than the other transports, about

(1.9±0.3)×104m3s−1 (1σ), while it ranges between 0.8×104m3s−1 and 4.4×104m3s−1

(a factor of 5). On average, the estimated error in M is about 1.0×104m3s−1 (1 σb).

This is consistent with the result of the F-test that suggests that the variability of

M is hidden by its sampling and estimation error.

Note that the magnitude of the seasonality of the total upward transport (W2 =

E +M , not shown) is very small (σe=4×103m3s−1) compared to the average magni-

tude of W2 (6.2×104m3s−1) and similar to the sampling and estimation error of U1

and E. Again, the F-test suggests that any seasonal variability of W2 is hidden by

the variance.

Fig. 4.1 also shows the freshwater inflow R (defined in chapter 3). There is a close

similarity between the circulation variability (both U1 and E) and the freshwater

variability. The circulation transports increase at the same time as the freshwater

input increases and vice versa. The relationship is not a simple scaling. Typically,

the freshwater inflow R varies by a factor of 5, but U1 by a factor of only 1.4. The

sensitivity of the circulation transport U1 relative to change in R is further discussed

in section 4.3.5. Similar relationships between R and the transports can be seen in

the interannual variability (see section 4.3.6).

On the other hand, there is a mirror-symmetry between the variability of M and

the variability of R. M decreases when R increases and vice versa. However, the

F-test indicates that the variance of M could hide the seasonality magnitude of M

and thus, the relationship between M and R could be hidden by the variance of M

as well.

Oscillations, of large magnitude and of approximate two-month period, are vis-

ible around March-April 2003 and 2004 (and possibly around May 2005) when the

84


sampling period usually is a week or two weeks. The model could be aliasing the

fortnightly tidal signal in the transports and the box averages. However, an analysis

of the phase between these oscillations and the tidal current (estimated at Active

Pass, east side of Vancouver Island) or the sea surface level (at Sand Head) in March-

April 2003 and 2004 show no clear relationship with the fortnightly tides. Instead,

these oscillations could also be associated with the model misfit defined in chapter 2,

section 2.2, Eqs 2.56–2.57 when the temperature and salinity averages at the sepa-

ration depth are approximated by the box averages. Before the freshet period, the

approximation error is the largest and it could propagate to the estimated transports.

4.2.2 Sensitivity Analysis

Time Dependency or Quasi-steady State

The purpose of this section is to determine the influence of time dependence on the

transport estimates in the budget equations (Eqs 2.66–2.71). Fig. 4.2 shows estimates

of the transports when one assumes quasi-steady state, while time dependence was

assumed in the previous Fig. 4.1. There is no major change in the general appearance

of the transports between Time Dependence (TD) and Quasi-Steady State (QSS).

In both cases, the mean values of the transports are very similar to each other, as

are the average errors. Fig. 4.2 shows the mean values x and the average error (1σ)

with TD and in QSS while Table 4.1 shows the errors σe, σb and σ with TD and

in QSS. Fig. 4.2 shows that the mean value of each QSS transport is close to the

corresponding TD transport. Table 4.1 also shows that the seasonality magnitude

σe of the QSS transports is smaller than of the TD transports. The F-test applied

to the QSS transports suggests that their variance would hide any seasonality. This

indicates that even if there is some seasonality in the QSS transports it is difficult

85



1

2

3

4

5

6

7x 10

4m

3 s

−1

mean E

mean M

mean U1

Freshwater

M

U1

E

Figure 4.1: Transport Estimates and Their Estimated Errors (σbi) . Transports are

the lines inside the strips, error ranges (σbi) are represented by the strips (see legend).

The markers and the bars, at the right hand side, represent the overall means (x)

and the errors (σ), respectively (U1 diamond, E triangle, and M circle) over the 47

surveys. 86


to distinguish it from the error in the time series.

A detailed analysis shows that there are minor differences between the TD and

QSS transports each year from November to January. QSS transports U1 and E

are slightly higher than TD transports during that period: on average 3% of x for

both U1 and E. QSS transports M and W2 are lower than TD transports during the

same period: on average 12% of x for W2 and 45% for M . As a consequence, the

magnitudes of U1, E and M with TD overlap with each other between October and

February. The magnitudes of U1, E andM in QSS are distinct between December and

February but overlap with each other earlier in fall between October and December.

During March-April 2003 and 2004, the two-month oscillations are larger with

TD than with QSS. However, when they occur in the case of TD the magnitude

of U1 overlaps with the magnitude of M within the uncertainties. This is also true

between E and M . This suggests that mixing exchange in the TD case could be as

large as the net upward entrainment and the horizontal transports. On the other

hand, the QSS estimates have smaller oscillations and only the magnitude of U1

tends to overlap with E. In both cases, during March-April 2003 and 2004, more

cruises were carried out per month. Thus, possible short timescale processes may

have been sampled and more probably undersampled. This may have introduced

noise in the transports estimates. In the TD case, short timescale variations are

introduced in the transport estimates. Such a difference between transports over

circulation timescales and transports over shorter timescales has been previously

observed in Chesapeake Bay: oceanic exchange rate has been estimated to be on

average (8±2.3)×103m3s−1 [Austin 2002], while mass transports have been estimated

between -2×104m3s−1 and 4×104m3s−1 when they were induced by short-timescale

(2–3 days) local and remote wind events [Wong and Valle-Levinson 2002]. Note that,

87


in Wong and Valle-Levinson [2002], local winds could force net mass transports to

be landward temporarily.

This analysis shows that the seasonality of the transports increases when the time

dependence is included in the budget equations. The difference between TD and QSS

is likely significant in the case of the surface outflow U1 according to the results of

the F-test (section 4.2.1).

Separation Depth

In this section, the influence of the separation depth (d) is investigated by using

average transports x and the corresponding error σ over a range of d. A realistic

range of d is between 15 and 50 m. Within this range, the choice of a particular d

has no significant effect on the transports U1, U2, E and M . The values taken by the

transports are similar within the uncertainties (Fig. 4.3).

The circulation transports take values around 4.5×104m3s−1 for U1 and 4×104m3s−1

for U2 and E. These values x change by less than σ with any change of d within 15–

50 m (Fig. 4.3): U1, U2 and E change by about or less than 3% of x, while M changes

by 10%. The circulation transports (U1, U2 and E) reach a weak maximum when d

is in the range 20–30 m (close to the default depth), but the difference between min-

imum and maximum values in that range of d is of the same magnitude as the error

(σ). Analysis and comparison of the actual transport time series indicate only small

changes in the seasonal variability of the circulation transports when d is changed

from 15 m to 30 m. As already observed in the case of d=30 m (section 4.2.1), U2 and

E (=W2 −W1) remain significantly close to each other for any other chosen value of

d.

The average x and the associated error σ of the mixing exchange M and the total

88



1

2

3

4

5

6

7x 10

4m

3 s

−1

U1

E

Freshwater

M

mean E

mean U1

mean M

Figure 4.2: Transport Estimates and Their Estimated Errors in QSS. Overall means

(x) and overall errors (σ) are the vertical lines and markers, at the right hand side,

(U1 diamond, E triangle and M circle). QSS overall means and overall average errors

are represented by the large markers and the thick lines.

89


upwelling W2 (=E +M) increase proportionally to the increase of d. However, the

changes are small. A linear approximation of the sensitivity suggests a 1 m change in

d causes M to change by only 102m3s−1. Doubling (halving) d from 30 m on average

increases M by only +16% (or -8%). The linear trend indicates that even with a small

d (a few meters) there is still mixing exchange occurring: about 1.3×104m3s−1 for

d=2 m. The slope of the other transports drops faster than the slope of M between

2 and 5 m, but it seems unlikely that the actual d is this shallow (Fig. 3.1). It is

more realistic to expect the shallowest reasonable value of d to be about 15 m.

Overall, this analysis indicates that the transports are relatively insensitive to

changes of d in the range 15–50 m. The default choice of d=30 m leads to only

small departure between the estimated transports and the transports for the true d,

whatever it may be. Therefore, even if the true value of d differs from the default

there should be little effect on the analysis done here.

Inversion Parameters

In chapter 2, the concept of a trade-off parameter (α) was introduced (Eq. 2.15) and

generalized (Eq. 2.22). In this section, the influence of this parameter on the trans-

ports is investigated. Fig. 4.4 shows that bias and variability are linked. However,

as shown next, the sensitivity of the results to changes in α are small. The trade-off

between bias and variability is relatively balanced. This is the reason why the default

value of α turns out to be a reasonable choice.

In the default solution, the value of α was set to 2×10−5m−3s, that is s−1 (Ta-

ble 2.2, Chapter 2). Now we rewrite the trade-off parameter as α=γ × s−1 where γ

is a scaling factor of the default parameter (s−1). To show this scaling factor in the

90


2 5 10 15 20 25 30 35 40 45 500

1

2

3

4

5

6

7x 10

4vo

lum

e tr

ansp

ort (

m3 s

−1 )

separation depth (m)

Default depth30 m

M

E, U2

U1

W2

Figure 4.3: Circulation Sensitivity to Separation Depth. Overall averages and errors

of U1, E, U2, M and W2 over 47 surveys are the vertical lines and markers. Averages

and errors of U2 and E have been shifted right and left, respectively. This prevents

vertical bars showing errors from overlapping. 91


equations, Eq. 2.22 is rewritten as:

J = εTW−2

ε+ γ2zTS−2z = εTW−2

ε+ zT(γ−1S)−2z. (4.4)

Eq. 2.22 can be equivalently rewritten as:

J′ = γ−2εTW−2

ε+ zTS−2z = εT(γW)−2

ε+ zTS−2z. (4.5)

These equations show that it is mathematically equivalent to scale down the matrix S

by γ−1 and scale up the matrix W by γ. Note that the scale difference between J and

J′ is not important when finding the minimum. In turn, the scaling of the inversion

parameters changes the size of the estimated transports. In Fig. 4.4, transports over

the whole time series have been averaged at a given value of γ. Fig. 4.4 shows that

the transports increase with decreasing γ and they approach the values found for

the simple SVD, case γ →0. The variability (vertical bars) tends to increase more

markedly than the average.

The transports can be mathematically represented by a vector form of Eq. 2.24

while introducing the scaling factor γ:

x = xA +r∑

i=1

λ′

i u′Ti W

−1(b−AxA)

λ′i2 + γ2 s−2

v′

i. (4.6)

Eq. 4.6 shows the behaviour of the transports with decreasing γ. When γ decreases

the coefficients in front of the v′

i’s (range vectors of the solution) increase, as well

as the norm of x− xA (the deviation of the transports about the a priori averages).

When γ is small enough (i.e. when γ ≪ λ′

i × s) Eq. 4.6 approaches Eq. 2.9 (the

simple SVD equation) because the cost function becomes:

J = εTW−2

ε (4.7)

When γ is large enough (i.e. when γ ≫ λ′

i× s), the second right-hand term of Eq. 4.6

is down-weighted by γ−2 and the first right-hand term, the a priori solution xA,

92


prevails, leading to transports very close to the a priori averages. This last behaviour

is also clearly shown in Fig. 4.4 in the magnitude of both transport averages and

overall variations.

This shows that choosing the scaling factor is equivalent to trading between bias

and variability. When γ becomes large (γ >1), the transports are more biased toward

the a priori averages and have less variability. Reciprocally, when γ becomes small

(γ <1), the transports are less biased toward the a priori averages and have more

variability.

However, note that the curves in Fig 4.4 are roughly horizontal. This suggest that

the a priori transports are close to the true transports, but might also imply that the

data and the inversion procedure provide little information about transports. Thus,

the optimal solution reverts to the a priori transports. In order to determine whether

this is the case, we will now deliberately vary the a priori transports to see the effects

on the solution.

A Priori Transport Averages

This section investigates the influence of the a priori averages (xA with components

xAi ’s equal to U01, U02, W01 or W02 in the default case, Table 2.2) on the transports.

The analysis of Fig. 4.5 shows that when the magnitude of the a priori averages

decreases, the transports tend to have large variability. They also tend to have a

larger magnitude than the a priori averages. Reciprocally, when the magnitude of

the a priori averages increases, the transports tend to have a small variability. In

addition, they tend to take values close to the a priori averages but smaller than the

a priori averages. The chosen a priori averages lead to transports with which the two

aforementioned effects are relatively well-balanced. Thus, the data has an effect on

93


0.1 0.2 0.5 1 2 5 101

2

3

4

5

6

7

8x 10

4

γ

Tra

nspo

rt (

m3 s

−1)

Default caseγ=1, α=s−1

Simple SVD caseγ=0

W2

U1

E, U2

M

Figure 4.4: Circulation Sensitivity to the Trade-off Parameter α=γ s−1. Overall

averages and errors of U1, E, U2, M and W2 over 47 surveys are the vertical lines

and markers.

94


the solution, confirming the a priori estimates.

In more detail, the change of xA is introduced by the scale parameter β such that

the new a priori averages are β xA instead of xA. Eq. 4.6 (with default α) can thus

be written:

x = β xA +r∑

i=1

λ′

i u′Ti W

−1(b− β AxA)

λ′i2 + s−2

v′

i (4.8)

where β appears in both right-hand terms. Fig. 4.5 shows the ratio between the

transports and the a priori averages (i.e. the ratios xi/(β xAi ) with xi equals to U1,

E, M , and W2) as a function of β. Both transport average and variability magnitude

(vertical bars) of the transports change with β. Eq. 4.8 suggests that the transports

take values somewhere between the true and the a priori estimates. If the transport

estimates were only based on the data, the ratios xi/(β xAi ) would only vary as β−1.

The ratios at β=2 would be close to half the ratios at β=1. If the transport estimates

were only based on the a priori information, the ratios xi/(β xAi ) would be close to

1. Observing the curves in Fig. 4.5, neither of these situations occurs. The ratio

xi/(β xAi ) is not proportional to β−1 nor always close to 1.

In Fig. 4.5, when β ≫1 the ratios xi/(β xAi ) tend to be smaller than 1 within

error bars. It is subtle for both U2 and E, but very marked for U1 and M . Since

the estimates x lie between the true estimates and the a priori estimates, when the

a priori values becomes greater than the true values the ratios xi/(β xAi ) have to

become smaller than 1. When β ≪1 the ratios xi/(β xAi ) tends to be greater than

1. When β decreases, all the ratios xi/(β xAi ) increase. It is not as clear as when the

situation β ≫ 1 (especially M) because the error bars are larger when β ≪ 1. Since

the estimates x lie between the true estimates and the a priori estimates, when the

a priori values become less than the true values the ratios xi/(β xAi ) have to become

greater than 1.

95


Thus as suggested by this analysis of the sensitivity to β, in the inversion pro-

cedure where the default a priori averages are used (β=1, Fig. 4.5), the transports

estimates are close to both the a priori estimates and the true estimates.

The analysis above suggests that the data information and the inversion procedure

provide enough information to approach the true estimates. This indicates that the

curves in Fig. 4.4 are roughly horizontal because the a priori estimates are similar to

the true values.

4.2.3 Residuals of the Conservation Equations

Figs. 4.6a–c show the residuals ε1, ..., ε6 of the budget equations (Eqs 2.66–2.71) and

the associated bootstrap variation. In the inverse procedure, the weighting scales,

ω1, ..., ω6, are the a priori values of ε1, ..., ε6. The weighting scales have been defined

in section 2.1 and Table 2.2 so that all the equations could be ranked against each

other’s absolute magnitude and be scaled to have the same weight. In addition, the

ωi’s give an idea of the sizes of the residuals when all the source and forcing terms in

the budget equations take on large values. If the actual residuals (when x = x) are

smaller than the ωi’s, one can expect that the inversion procedure worked reasonably

well.

As suggested above, at the end of the sensitivity analysis (section 4.2.2), the

estimated transports (x) are close to both the a priori solutions (xA) and the true

solutions (x). This should be also true of the actual residuals. The mathematical

framework in chapter 2 enables one to verify this criterion and to write the residuals

(ε) with respect to the residuals associated with the a priori solution (εA).

96


0.1 0.2 0.5 1 2 5 100.5

1

1.5

2

2.5

3

β (log scale)

Rat

io b

etw

een

tran

spor

t and

a p

riori

aver

age

U1

U2 E M

M

U1

E, U2

Figure 4.5: Circulation Sensitivity to the A Priori Parameter β . β is the ratio

between the new a priori average β xAi and the corresponding default value xA

i ,

(Table 2.2, U01, U02, W01, and W02). Overall averages and errors of U1, E, U2, M

and W2 over 47 surveys are the vertical lines and markers. 97


The actual residuals, using Eq. 2.20 and Eq. 2.32, can be written as follows:

ε =m∑

i=r+1

(u′T

i W−1εA)Wu′

i +r∑

i=1

s−2(u′Ti W

−1εA)

λ′

i2 + s−2

Wu′

i (4.9)

where εA = AxA-b, u′

i the range vectors of the SVD of W−1A, W the weighting

matrix (see Eq. 2.25), the coefficient s that scales the solution x. Note that the a

priori residuals (εAi ’s) are easily rewritten:

εA =

m∑

i=1

(u′T

i W−1εA)Wu′

i (4.10)

Using Eq. 2.11, the residuals associated with the “pseudo-inverse” can be rewritten:

ε =m∑

i=r+1

(u′T

i W−1εA)Wu′

i (4.11)

Thus, Eq. 4.9 suggests that the residuals (εi’s) cannot be as small as one would

require. They are smaller than the a priori residuals (εAi ’s, Eq. 4.10), and bigger than

the residuals associated with the “pseudo-inverse” (Eq. 4.11).

An analysis of the sum of the squared scaled residuals (last row in Table 4.2), the

quantity that is minimized in the inverse procedure (see chapter 2 and Eq. 2.15), indi-

cates that the information in the observations is useful and leads to an improvement

of the estimated transports relative to the a priori transports. The overall appear-

ance of the residuals (Fig. 4.6a–c and Table 4.2) indicates that the actual residuals

(ε′js) are smaller than or close to the a priori residuals (ωi’s). This suggests that the

residuals (εj’s) are consistent. As suggested earlier in this chapter (section 4.2.1), the

useful information provided by the observations is the seasonality of the transports.

In the mass equations (Fig. 4.6a), ε1 (top box) is one order of magnitude smaller

than ω1 (on average 3.1% of ω1 and maximum 11%), while ε2 (bottom box) is one

to two orders of magnitude smaller than ω2 (on average 1.1% of ω2 and maximum

98


Equation A priori Residuals Residuals with Residuals with

x = x (in % of ωj) x = xA (in % of ωj)

average maximum average maximum

Top Mass ω1=1.6×104 3.1 11 14 69

Bottom Mass ω2=1×104 1.1 4.4 10 10

Top Heat ω3=2.7×105 10.5 28 11 31.5

Bottom Heat ω4=1.5×105 30 72 27.5 71

Top Salt ω5=4.7×105 10 25 9 29

Bottom Salt ω6=3.5×104 8 33 73 251

Summed scaled squared εi’s∑m

i=1(εi/ωi)2 0.16 0.54 1.1 6.9

Table 4.2: A Priori and Estimated Values of the Residuals of the Conservation

Equations in the SoG Box Model . The last line provides the average and maximum

sums of the squared scaled residuals with x = x and x = xA.

4.4%). The residual ε1 is usually a lot larger than ε2 (on average 3 times larger). The

residual ε2 is usually close to zero because the residual curve lies within the error

bars (Fig. 4.6).

Note that the residuals ε1 and ε2 are smaller when x = x than when x = xA

(Table 4.2, last column). This means that even with a constant solution x = xA, the

residuals are smaller than the weighting scales ω1 and ω2. In section 2.2, Eqs 2.41–

2.42, associated with the residuals ε1 and ε2, are exact because the transports U1,

99


U2, W1 and W2 are the surface integrals of horizontal and vertical flow speeds as

defined by Eqs 2.43–2.47. As a consequence, only the estimation error associated

with the transports can affect the residuals. The error associated with the external

sources is either very small (in the case of the freshwater) or zero (in the bottom

box). Further analysis showed that ε2 is more sensitive to change of the inversion

parameters than ε1, but they both remain one order of magnitude smaller than the

expected residuals. The difference between ε1 and ε2 is more strongly marked than

what the order between ω1 and ω2 suggested: ε1 ≃3 ε2 instead of ω1 ≃2 ω2.

The residuals of the heat equations (Fig. 4.6b) are also smaller than or close to the

weighting scales ω3 and ω4. ε3 (top box) is on average 10.5% of ω3 and maximum 28%,

while ε2 (bottom box) is on average 30% of ω4 and maximum 72%. The bottom heat

residual has a clear seasonal variability with maximum in summer and minimum in

winter. These minimum and maximum are smaller but close to the a priori residuals.

All these values are close to the residuals found with the a priori transports. This

could suggest either a larger error than expected in estimation (section 2.2) or in

model misfit (Eq. 2.4).

The residuals of the salt equations (Fig. 4.6c) are smaller than or close to the

weighting scales ω5 and ω6. ε5 (top box) is on average 10% of ω5 and maximum 25%,

while ε6 (bottom box) is on average 8% of ω6 and maximum 33%. The bottom salt

residual takes values close to zero within the error bars, but these error bars are

large. These minimum and maximum are smaller but close to the a priori residuals.

As suggested above this could be due to estimation error or model misfit.

Assuming a model misfit ∆A, a solution error ∆x and a source error ∆b, the

residuals in Eq 2.4 become:

ε = (A+∆A)(x+∆ x)− (b+∆b) (4.12)

100


or

ε = (A+∆A)∆ x +∆A x−∆b (4.13)

with Ax = b for an exact solution x. In the case of residuals ε1 and ε2, ∆A is

zero because the equations are exact. Since the external sources are well-known (ε1)

or there is none (ε2), ∆b is negligible. Eq. 4.13 shows that the residuals ε1 and ε2

depend only on the estimation error ∆x. Since the solutions have been found by an

inverse procedure where the tapered solutions are biased, the bias can propagate to

the residuals (Eq. 2.20). This bias is a linear combination of the components of the

vector b, which contains the external sources.

Summarizing the analysis of the residuals, the optimal solution is consistent with

the a priori solution. Overall, it minimizes the residuals similarly to the a priori

solution. However, the optimal solution provides an improvement over the a priori

solution as suggested by Eq. 4.9. It minimizes the residuals of the mass budgets and

the bottom salt budget further more than the a priori solution (Table 4.2).

4.3 Discussion

4.3.1 General Circulation in the Box Model

In the previous section, we obtained 4 time series of the circulation (Fig. 4.1) in a

box model representation of the SoG system. In this representation, surface outflow

has a significant seasonal variation according to a conservative analysis of variance

(Table 4.1). The sensitivity analysis showed that the estimated transports were rel-

atively insensitive to the separation depth between the model boxes, but that time

dependence was a necessary assumption to conserve a significant seasonal signal in

the time series. The analysis of sensitivity to the inversion parameters and the a

101


Jul Oct 2003 Apr Jul Oct 04 Apr Jul Oct 05 Apr−2

−1

0

1

2

3

m3s−

1

Top mass Eq., a priori ε1= 1.6×104

Bottom mass Eq., a priori ε2= 104

Jul Oct 2003 Apr Jul Oct 04 Apr Jul Oct 05 Apr−1

−0.5

0

0.5

1

1.5x 10

5

°Cm

3s−

1

Top heat Eq., a priori ε3= 2.7×105

Bottom heat Eq., a priori ε4= 1.5×105

Jul Oct 2003 Apr Jul Oct 04 Apr Jul Oct 05 Apr−1.5

−1

−0.5

0

0.5

1

1.5x 10

5

m3s−

1

Top salt Eq., a priori ε5= 4.7×105

Bottom salt Eq., a priori ε6= 3.5×104

×103

(a)

(b)

(c)

Figure 4.6: Residuals of the Mass, Salt and Heat Equations of the SoG Box Model .

Although there is a seasonal variability in b), its magnitude is well below the a priori

scale of 1.5×105.

102


priori averages showed that the estimated transports were likely close to the true

transports. Finally, the residuals analysis showed that the residuals in the budget

equations were, overall, consistent with the a priori residuals.

Although transport estimates of the SoG circulation have been previously made

in various ad-hoc ways [England et al. 1996, Pawlowicz 2001, Pawlowicz et al. 2007],

this is the first set of estimates based on a rigorous inverse procedure. The monthly

timescale of these estimates can help to understand the net effect of all processes,

including tides on the estuarine circulation. In addition, physical analysis of the

estuarine circulation and historical freshwater data (later in section 4.3.5) suggest

that, one can, from an estimate of the freshwater discharge, readily obtain estimates

of the estuarine circulation for any period outside the 2002–2005 window. The time

series in section 4.2.1 provide monthly information (biweekly during spring 2003

and 2004) on the circulation based on the 3 years of observations. However, the

use of a timescale limited to a monthly resolution prevents one from analyzing the

detailed influence of fortnightly tides contributing to the vertical mixing (unlike Li

et al. [1999]). This would require a smaller time resolution, for instance a daily time

resolution.

The inverse box model is constrained by, not only the conservation of mass and

salt, but also conservation of heat [Pawlowicz and Farmer 1998], in contrast to Li

et al. [1999] and Pawlowicz et al. [2007]. Adding heat conservation is useful because

the estimated transports have to be consistent with an additional set of constrains,

which should improve accuracy. The heat budget may also be important when evap-

oration or temperature stratification are important (e.g., Roson et al. [1997]), al-

though it is not the case here.

Previous studies of the SoG [England et al. 1996, Li et al. 1999, Pawlowicz 2001,

Pawlowicz et al. 2007] have also estimated the surface outflow and deep inflow trans-

103


ports (see Table 4.3 for detail about the averages and ranges of the SoG transport

estimates and the approach used). In Table 4.3, overall the average transport es-

timates, from 3.6±1.1 to 5.5±2 m3s−1, and the winter-summer outflow range are

consistent with each other across these studies. However, the estimates of Pawlow-

icz et al. [2007] are slightly higher (5–5.5 m3s−1) because their approach allows a

greater seasonal variability by introducing deep water renewal inflow (0.4–4 m3s−1)

that combines with the deep inflow (4 m3s−1). Note that a more detailed comparison

of my estimates with those of Li et al. [1999] is carried out in section 4.3.2 and a

discussion about the use of additional boxes is in section 6.2.

From the outflow and inflow transports one can calculate a difference. The dif-

ference can be used to estimate a net rate of export of a tracer outside of the SoG.

It indicates a net outflow if the difference is significantly larger than 0, and an ex-

change rate if the difference is not. Thus, it is useful to keep the outflowing and

inflowing transports independently constrained by the observations, as in Pawlowicz

[2001], Pawlowicz et al. [2007], and unlike, for instance, Li et al. [1999] who assumes

that the outflow equals the inflow. However, the estimation of the net outflow is a

difficult task to carry out because it usually involves subtracting two large and sim-

ilar layer transports together [Godin et al. 1981, LeBlond 1983] and the estimated

net outflow has the same size as the estimated error. The net outflow from the SoG

is discussed later in section 4.3.4.

Although I have studied the sensitivity of the the inverse solution to various model

parameters, the box model (Fig. 2.1) is itself only an approximation. Two aspects of

the real SoG, not reflected in this sensitivity analysis, are the effect of the Northern

Entrance and the variability in the separation depth.

The layer transports through the Northern Entrance are generally assumed to

be smaller than the ones through the Southern Entrance because the cross-sectional

104


area of the channels is markedly smaller. It is not clear from the STRATOGEM and

JEMS observations alone if the layer transports through the Northern Entrance are

negligible. If there is no significant difference of water properties between Johnstone

Strait (JS) and Haro Strait (HS), the layer transports in the SoG box model would

be the sum of the layer transports at the Northern and Southern Entrance. Thus,

the layer transports through the Northern Entrance could contribute to the net

horizontal transport (U1-U2) in the model as much as the layer transports through

the Southern Entrance, if they were significantly different from each other.

In the SoG box model, the separation depth d between the top and the bottom

boxes is the separation depth between the surface seaward transport (U1) and the

deep landward transport (U2). Observations in the SoG suggest that d is likely to

change over a year from deep values in winter to shallow values, due to wind-induced

mixing and convection of cooling water, in summer, due to stratification by freshwater

addition and surface heating, (section 3.3.1, Chapter 3). However, in section 4.2.2,

the analysis showed that the transports are relatively insensitive to changes of d,

above and below 30 m.

4.3.2 Seasonal Transports

A detailed comparison of my results with those of Li et al. [1999] (Figs 4.7a–d) shows

a good general agreement between the models despite a difference in the approach and

the model setup. Both transports have the same seasonal pattern with low winter and

high summer magnitudes: in winter: 4×104m3s−1 for my estimate versus 3×104m3s−1

for Li et al. [1999]’s; in summer: 6×104m3s−1 for my estimate versus 7×104m3s−1

for for Li et al. [1999]’s. The large summer and small winter magnitudes in the case

of Li et al. [1999]’s model are due to the additional fortnightly-tide modulation of

105


the estuarine circulation which is not captured in our model (section 4.2.1). The

magnitude of the seasonal change is significantly smaller (about ±1×104m3s−1 for

my estimate and ±2×104m3s−1 for [Li et al. 1999]) than the average magnitude of

the transports (4.5×104m3s−1 for my estimate and 5×104m3s−1 for Li et al. [1999]).

The good agreement occurs generally during the freshet maximum although there

are some differences: a difference in the timing of the freshet maximum of about a

month (exactly the end of June in Li et al. [1999]’s and June in ours), secondary

peaks are present before and after the freshet maximum in my estimated transport

but do not appear in Li et al. [1999]’s. These differences are due to the two different

approaches. Li et al. [1999] use an idealized freshwater inflow and idealized deep

salinity to force their model while I used observed conditions and forcings in my box

model.

The relatively small variations of W2 (section 4.2.1), the total upwelling trans-

port (overall seasonality magnitude σe of 0.4×104m3s−1 and average transport of

6.2×104m3s−1), suggests a relatively constant upward transport. This particular

property of W2 is useful to estimate the overall residence time in the euphotic and

aphotic zones of the SoG. The analysis of Eqs 2.68–2.71 show that the time rate of

heat and salt outward advection leads to a residence time τ=volume/outflow: in the

top box (T1 and S1) τ1=V1/(W2 + R)=32±2 days, in the bottom box (T2 and S2)

τ2=V2/W2=169±8 days. Given the relative invariance of W2 and its large magnitude

with respect to R, these residence times are also relatively constant for the SoG box

model. The residence times τ1 and τ2 represent characteristic e-folding timescales in

the top and the bottom boxes. The salinity and the temperature adjust to 90 %

in about 2 × τ1 in the top box (2 × τ2 in the bottom box). For instance, a change

of the external source of surface (bottom) salinity from S1 (S2) to S0 would take

about two τ1 or two months (about two τ2 or a year). In this example, I assumed

106


that the external source remained constant for the sake of simplicity, but in reality

the external source would change. The resident times then indicate the lag in the

observation of the SoG properties relative to those of the source. This property was

used in Pawlowicz et al. [2007].

2002F M A M J J A S O N D0

2

4

6

8

10

12x 10

4

Li et al. 1999′s exchange flowU

1

(a)


2

4

6

8

10

12x 10

4

(b)

Figure 4.7: Comparison Between Li et al. [1999]’s Horizontal Exchange Transport

and Estimated U1 Transport: (a) U1 2002 estimate, (b) U1 2003 estimate

4.3.3 Comparison with Transports in SoJdF

One further step can be taken to validate my estimates with other studies considering

the larger system of SoG/HS/SoJdF [Godin et al. 1981, England et al. 1996, Pawlow-

icz 2001, Masson and Cummins 2004, Thomson et al. 2007, Sutherland et al. 2011].

107



2

4

6

8

10

12x 10

4

(c)

2005F M A M0

2

4

6

8

10

12x 10

4

(d)

Figure 4.7: Comparison Between Li et al. [1999]’s Horizontal Exchange Transport

and Estimated U1 Transport: (c) U1 2004 estimate, and (d) U1 2005 estimate

An estimate of the inflowing and outflowing transports from HS can be determined

by analyzing the recirculation fraction of surface water from the SoG [Pawlowicz

et al. 2007].

The recirculation fraction of surface water defines the fraction of the surface

water that leaves the SoG surface, mixes with water from the HS and finally reenters

into the SoG at depth. The mixed water of HS contains both surface water from

the SoG and deep water coming from the Strait of Juan de Fuca (SoJdF). The

deep water that comes from the SoJdF has retained the oceanic characteristics of

the Pacific Intermediate Water (PIW). The PIW leaves the deep ocean, enters the

108


SoJdF through the Juan de Fuca Canyon, and is advected with little mixing until it

enters HS [Pawlowicz 2001, see Figs 5 a-d]. Similarly to Pawlowicz et al. [2007], the

recirculation fraction δ can be used to express the mixing of PIW and surface SoG

water in HS:

qH = δ q1 + (1− δ) qPIW (4.14)

where q is the tracer (salinity, temperature, nutrients or O2) from either the SoG

surface (q1), HS (qH) or PIW (qPIW ). Eq. 4.14 simply expresses that HS water is a

mixture of SoG surface water and PIW. The recirculation fraction gives the fraction

of q1 that reenters the SoG by seawater intrusions (U2). In my box model, the PIW

properties are calculated from a climatology of offshore data at depths between 100-

200 m using CTD and bottle vertical profiles collected by IOS (Institute of Ocean

Sciences, Sydney, Canada). Eq. 4.14 is based on the assumption that the dominant

processes that change the characteristics of HS water are advection and mixing of

water entering and leaving HS.

One can apply a simple one-box model to HS (see Fig. 4.8), and write the

equations of conservation of mass and the tracer q as follows:

U1 − U2 = U3 − U4 (4.15)

q1 U1 − qH U2 = qH U3 − qPIW U4 (4.16)

where U3 is the surface outflow to the SoJdF and U4 the deep inflow from the SoJdF.

When rearranged Eqs 4.15–4.16 lead to Eq.4.14 and

δ =U1

U1 + U4

. (4.17)

In theory, only positive values δ ≤1 correspond to the assumption that advection

and mixing dominate the processes affecting any tracer in HS. Otherwise, Eq. 4.14

109


Figure 4.8: Advective Transports in HS Box Model. The advective transports U1 and

U2 are the seaward and landward transports to the SoG from Fig. 2.1, respectively.

U3 and U4 are the seaward and landward transports to the SoJdF, respectively.

is not true. However, in practice, there are several issues that can lead to δ values

outside the range 0≤ δ ≤ 1 and δ values different from the true values. We are using

box averages and a climatology to approximate the properties of the water masses

from the SoG, HS and the SoJdF. Therefore, one can expect that differences between

the estimated and true water properties can occur and yield a few wrong values of

δ out of the 47 that we had to estimate. Since we assumed that the advection and

110


Reference Average Transports Comment

(×104m3s−1)

This study Formal inverse 2-box model 2002-2005

4.5±0.5(1σb) outflow

4±0.5(1σb) inflow and net upward flow

3.4–6.2 winter-summer outflow range

Li 1999 Flows in a forward box model: from numerical simulations

4.5 mean value

3–7 winter-summer outflow range

Pawlowicz 2001 Inverse 2-box model: SoG/HS/SoJdF system

4.6±1.1 outflow (July 1998)

3.6±1.1 inflow (July 1998)

Pawlowicz 2007 3-Box mixing model: 2002-2005

5–5.5 (±2) annual inflow and outflow

4.4–8 winter-summer inflow range

4.7–9 winter-summer outflow range

4±2 summer deep renewal inflow

0.4±0.02 winter deep water inflow

4±2 constant deep water inflow

Note:

Marinone 1996 3D numerical model:

flow speed

no transport estimates

Table 4.3: SoG Transports from Previous Studies

111


mixing processes dominate the processes that affect a tracer, a tracer that necessarily

satisfies Eq. 4.14 is a conservative tracer, e.g. salinity. If there are other processes

affecting a tracer apart from advection and mixing and if the contribution is close to

or larger than the contribution of advection and mixing, larger errors on δ are likely

to occur.

As a consequence of these issues, the conservative tracers that rely on the most

accurate box averages lead to the most reliable estimates of δ. Thus, one would

expect that the salinity is the most reliable tracer to use to estimate δ with values

within the range 0≤ δ ≤1 and small errors. On the other hand, dissolved O2 is likely

to be less reliable than salinity since O2 can be added by photosynthesis near the

surface, removed by respiration at any depth by various organisms, and removed at

depth by biogeochemical reactions. In the case of temperature, apart from advection

and mixing only surface heat fluxes can affect temperature near the surface.

When I estimated δ using salinity, δ always satisfied 0≤ δ ≤1. The median of δ

is about 0.35±0.1 based on salinity, 0.55±0.35 based on temperature, and 0.7±0.5

based on O2. The medians of temperature and salinity suggest that δ average is

about 0.45, while the medians of the three tracers suggest that δ average is about

0.53. All these estimates of δ are reasonable because they suggest that the HS and

SoG outflows have similar order of magnitude. Li et al. [1999] did not estimate this

parameter, but their results imply exchange flows between SoG and HS very similar

to the exchange flows between HS and SoJdF. Thus, applying Eq. 4.17 with U1=U4,

an estimated value for Li et al. [1999]’s δ is also about 0.5. Using values at 100 m

depth in HS, Pawlowicz et al. [2007] obtained 0.6 based on three different tracers

and smaller estimated errors, but their most reliable estimate of δ, based on salinity,

is 0.52±0.02. Their fractions obtained when using water-column averages in HS are

0.1 lower and get closer to our values: 0.5, based on the three different tracers; 0.42,

112


based solely on salinity. Pawlowicz et al. [2007]’s smaller estimate errors can be

explained by the use of spot measurements for SoG tracers at either station S4-1

or station S5 (see Fig. 1.1). By rearranging Eq. 4.17, one can estimate the order of

magnitude of the deep inflow into HS (U4):

U4 = (δ−1 − 1) U1. (4.18)

The range of my estimates of recirculation fraction is 0.35–0.7. The average values

around 0.5 suggests that the layer transports at both ends of HS are similar. This is a

situation similar to Li et al. [1999]’s where surface outflow and deep inflow are close.

The range of the estimated HS layer transports is (1.9–8.4)×104m3s−1. These values

are smaller than previous estimated layer transports in HS or in the SoJdF [Godin

et al. 1981, Pawlowicz 2001, Masson and Cummins 2004, Thomson et al. 2007]:

values in HS or the SoJdF are of the order of 10×104 [Thomson et al. 2007, Table 3,

column 3, Qin is ∼8.7×104] while values on the Pacific end of the SoJdF are higher,

about 20×104. It is reasonable to find layer transports in SoG or in the northern end

of HS smaller than or equal to the averaged layer transports in HS or SoJdF. In an

estuarine system the magnitude of the layer transports tends to increase towards the

mouth of the system because of entrainment. In the case of the SoG/HS/SoJdF the

mouth is the Pacific end of the SoJdF.

4.3.4 Net Outflow from the SoG

Data assessment based on the resolution matrix of the inverse problem (Eq. 2.32 in

section 2.1) and the small residuals of Eqs 2.66–2.67 (see section 4.2.3) suggest that

the relationship between the SoG layer transports (U1 and U2) and the freshwater

discharge (R) is consistent with:

U1 − U2 = R (4.19)

113


Eq. 4.19 represents the net outflow from the SoG.

My results from the SoG box model provide a rough estimate of the net outflow

from the SoG, regardless of whether the contribution of the Northern Entrance to

the transports is neglected or included in the box-model transports. The estimated

net outflow from the SoG is consistent with the net outflow from the SoJdF. The

assumptions in Eq. 4.15 are that the net outflow out of SoG and the net outflow out

of HS are likely correlated and are both associated with the SoG freshwater forcing.

Eqs 4.15 and 4.19 are combined into:

U3 − U4 = R (4.20)

Eq. 4.20 approximates the net outflow from the HS. The next step would be to deter-

mined the SoJdF net outflow by estimation the layer transports in the SoJdF. Since

the SoG/HS/SoJdF works as a large estuarine system [Thomson 1994, Pawlowicz

2001, Sutherland et al. 2011], due to the upward entrainment of deep water, one

expects that the layer transports of HS and SoG would be smaller than those in

the SoJdF. Previous studies show that, generally, there is a net outflow in the So-

JdF [Godin et al. 1981, Thomson et al. 2007] with a mean magnitude of 2×104m3s−1

(about ∼2× R at R freshet peak).

Fig. 4.1 shows that the freshwater discharge is likely to be significantly larger

than the estimated error of the transports only during the freshet maximum (e.g.

6.2×104m3s−1 for U1, 4.8×104m3s−1 for U2 with error bars smaller than their dif-

ference in June 2002). Eq. 4.19 implies that one will only observe a net outflow

from SoG significantly different from zero during the same interval. The estimated

transports in Fig. 4.1 used to estimate the net outflow in Eq. 4.15 are always larger

(around 4×104m3s−1) than the freshwater inflow and they have average errors (±0.5

×104m3s−1) of the same size as the average freshwater inflow. Godin et al. [1981] al-

114


ready suggested that surface outflow and deep inflow were very close: their estimate

of the net transport was not significantly different from zero within their estimated

error bars. LeBlond [1983]’s justification of this result relies on the argument that

net transport comes from the difference of large but close transports. All the other

studies either assume exchange transports [Li et al. 1999] or assume that the dif-

ference is of the order of the freshwater outflow from the Fraser River [Pawlowicz

2001, Pawlowicz et al. 2007]. Results in Fig. 4.1 indicate that net outflow occurs

intermittently. A net outflow is significantly different from 0 only during particularly

large freshet discharges compared to the transports errors, e.g. in June 2002.

4.3.5 Circulation Sensitivity to the Freshwater Inflow

In section 4.2.1, it was noted that the circulation transports (U1 and E) and the mix-

ing exchange (M) are correlated with the freshwater inflow (R). There is an overall

trend for U1 to increase with increasing R, and for M to decrease with increasing

R. But, the trend of U1 is easier to show that the trend of E and M because the

larger variance in E and M hides any seasonality (see Table 4.1). An analysis of the

correlation between the seasonal variability of U1 and R, and between the seasonal

variability of M and R shows a strong and significant correlation for U1 (0.6±0.16

within a 90% confidence interval) and a very weak but significant anti-correlation for

M (-0.25±0.23 within a 90% confidence interval).

Fig. 4.9 shows the relationship between U1 and R. The overall relationship be-

tween U1 and R can be approximated by U1=(1.1±0.5)R+(4±0.3)×104m3s−1. This

linear approximation is close to the the mass equation between the horizontal trans-

ports (Eq. 4.19):

U1 = E +R (4.21)

115


with constant E. The seasonality of E (σe) can be larger than 0.5×104m3s−1, but the

F-test shows that most of it is hidden by the uncertainty in the estimates. However,

the physical meaning of such a relationship is unclear. It would imply an estuarine

circulation in the absence of runoff (i.e. R=0).

The linear approximation given above only works because the range of the fresh-

water R does not cover values lower than 2×104m3s−1. Since the estuarine circulation

is driven by the freshwater inflow, one would expect that U1 and E tend toward 0 as

R tends toward 0. Previous studies using either analytical models [Chatwin 1976] or

numerical models [Hetland and Geyer 2004] of estuaries showed that in theory U1 (or

E) varies as a small fractional power of R, e.g. R1/3. Baker and Pond [1995] were

studying the estuarine circulation in Knight Inlet, but found no linear correlation

between the surface dewinded transport and the river discharge. Fig. 4.9 shows the

data and the curve aR1/3 with a a proportionality constant (a=2.68×103 m2s−2/3).

Other curves of the form aR1/n, with n taking various fractional and integer values,

have been tried to quantify how tight the fit of U1 to R1/3 is by comparison to other

possible powers. Fig. 4.10 shows the plot of the residuals of aR1/n, that is the sum

of the squared misfits between the curve aR1/n and the data. Residuals quickly get

close to a minimum value when n is 3 or larger. Although it is difficult to determine

the exact fractional power, it is clear that the relationship is not consistent with large

powers.

The mass conservation equation (Eq. 2.66) and the very small variations of W2

(with W2) suggest that:

U1 = −M + constant (4.22)

in other words that U1 and M could be anti-correlated.

Similar analysis of the relationship between M and R and fits with power func-

116


tions of the form R1/n were attempted. However, the analysis was not conclusive and

no satisfying fit could be found. For instance, the analysis showed that even a con-

stant value for M could represent the relationship between M and R as accurately

as the functions R1/n with n≥ 1.

Given the role of freshwater in controlling the estuarine circulation, one may

expect to detect a rapid decrease of U1 as R becomes small in winter. Within the

range of observed R, the analysis of U1 shows no such general and rapid decrease of

U1. However, this is not incompatible with the theory. Within the range of observed

R, the curve aR1/3 is approximately linear and only starts to steeply curve down

when R < 2×103m3s−1. The value of 2×103m3s−1 is the smallest value for R found

in our data. The analysis of the relationship between U1 and R and the correlation

between U1 and M also indirectly suggests that M could depend on R although the

variance in M was too large to allow a direct analysis (see F-test in Table 4.1).

The basic conclusion is that the estuarine transport is in practise not very sensi-

tive to changes in freshwater input. However, this conclusion is consistent with results

from analytical and numerical models [Chatwin 1976, Hetland and Geyer 2004], as

well as some observations in an inlet [Baker and Pond 1995].

4.3.6 Interannual Variability of the Circulation

Fig. 4.11 shows, for each year and each transport, the median value x, the average

variability σb and the annual range (range between seasonal minimum and maxi-

mum). The transport averages do not change a lot from year to year. The mean of R

does not have a clear pattern while the mean of the transports reaches an extremum

in 2004. Overall, Fig. 4.11 suggests that annual range of all the transports changes

from year to year, but these changes are small and it is not clear that they are

117


0 2 4 6 8 10 12 14 160

1

2

3

4

5

6

7x 10

4

R ×103m3s−1

U1 m

3 s−1

Figure 4.9: Surface Seaward Transport Plotted With Respect to Freshwater Inflow .

The dashed line represents the theoretical curve aR1/3 (a=2.68×103), the thick line

the empirical fit E +R with E=(4±0.3) ×104 m3s−1 (Eq. 4.21).

118


0 1 2 3 4 0

0.5

1

1.5

2

2.5

3

3.5

4

n

sum

of s

quar

es s

cale

d in

to a

n av

erag

e er

ror

(×10

4 )

1

16

1

8

1

3

1

2

1

4

2

3

3

4

Figure 4.10: Residuals, or Sum of Squares εi2, of the fit of U1 as a power of R. The

sums of the squares have been scaled into an average error using

√∑47

i=1εi2

47

119


correlated to the interannual variability of R. Such a correlation of the transports

was expected at least in the case of U1 since the previous section suggests a clear

relationship between U1 and R. Note that the data in 2002 and 2005 are limited to

the last 8 months and the first 4 months, respectively. The annual averages of 2003

and 2004 estimated with all the data, the last 8 months of the year, and the first 4

months of the year indicate that no large difference occur in the 3 ways to estimate

the annual averages. This suggests that the estimated annual averages of 2002 and

2005 are likely close to the true annual averages.

Fig. 4.12 shows, for each year and each transport, σe, the seasonal variability.

Since the data over 2002 and 2005 do not cover a complete year period, the effect of

removing data from the time series has been also investigated. The seasonal variabil-

ity σe has been recalculated for the years 2003 and 2004 for each layer transport by

omitting the first 4 months or the last 8 months. The analysis shows that σe varies by

14% maximum (W1) and 8% on average, except for σe of W1 over the last 8 months

of 2004 that varies by about 50%. This indicates that the trend found for the annual

statistics of the transports are not significantly affected by using shorter annual time

series.

There is a common pattern between U2, W1 and W2 suggesting that in 2003 σe

is the largest of all the observed years. In the last two years, σe of all the transports

has decreased compared to their σe in 2002 and 2003. On the other hand, the σe

of R reaches the largest value in 2002, and it is decreasing over 2003–2004. This is

consistent with the time series of R in Fig. 4.1: R reached a peak of 1.5×104m3s−1

in June 2002. Fig. 4.12 could suggest a delay of about a year in the response of

the SoG to change in the freshwater inflow, but this is not realistic. Li et al. [1999]

carefully modelled interannual variability of the freshwater inflow and PIW salinity

to study the adjustment time of the circulation. They varied the freshet maximum,

120


the salinity maximum, the freshet timing, and the freshet duration time. Li et al.

[1999]’s analysis showed that the circulation adjusted within a few months. This is

consistent with estimates of residences time for the surface SoG which are a few

months (see section 4.3.2). During deep water renewals, residence time of deep water

can be two months although deep water renewals occur once a year [Pawlowicz et al.

2007].

Such residence times cannot affect annual averages in Fig. 4.12 and the 2003

maximum of U2, W1 and W2 is more likely the result of the error σb (Table 4.1), i.e.

the sampling and estimation error tends to hide the seasonality of the transports U2,

W1 and W2. This suggests that only the curve of U1 represents likely interannual

variability. However, over 2004–2005, the annual trend differs between U1 (decrease)

and R (increase). This suggests that the error σb is still too large in the estimated

transport U1 and prevents one from clearly estimating the interannual variability of

U1.

4.3.7 Conclusions

Here, I will summarize the main points of this chapter. First, the statistical analysis

of the layer transports of the SoG circulation suggests that the small seasonality of

U1 (Fig. 4.1), the surface outflow, is significant, and the seasonality magnitude is

sensitive to taking into account time dependence in the equations. Secondly, despite

a clear interannual variability of the freshwater input (Fig. 1.4), there is no clear

indication of interannual variability of the estuarine circulation (Figs 4.11 and 4.12).

These characteristics of the SoG estuarine circulation can be summarized by an

observational relationship found between U1, the surface outflow and R, the fresh-

water input. The observational relationship is consistent with the theory of estuarine

121


2002 2003 2004 20050

1

2

3

4

5

6

7

8x 10

4

volu

me

tra

nsp

ort

(1

04 m

3 s

−1)

16th and 84th percentiles

median

whiskers: min and max

U1, △ U2 and E, o W1 and M, x W2, and ⋆ R

R

W1 (and M)

U1

U2 (and E)

W2

Figure 4.11: Annual Mean of SoG Transports and Freshwater Inflow

122


2002 2003 2004 2005 1

2

3

4

5

6

7

8

9

σ e

(103 m

3 s−1 )

U1

U2 and E

W1 and M

W2

R

U1

W1 (and M)

U2 (and E)

R

W2

Figure 4.12: Annual Variability of SoG Transports and Freshwater Inflow

123


physics [Chatwin 1976, Hetland and Geyer 2004] and can be approximated by aR1/n

with n≤3 and a=2.68×103 m2s−2/3 on a physical basis or empirically by E +R with

E=(4±0.3) ×104 m3s−1(Eq. 4.21).

Further examination of the inverse procedure suggests that the layer transports

are not sensitive to the separation depth of the box model, the inversion parameters,

and the a priori transports (see section 4.2.2). In addition, the residuals of the mass,

heat and salt conservation equations (Eq. 2.22) are consistent (Table 4.2). The inverse

procedure provides an improvement since the sum of the squared residual of the

estimated solution are smaller than the sum of the squared residuals of the a priori

solution.

124

Chapter 5

Nutrients Uptake and Primary

Productivity

5.1 Introduction

Primary production (PP) is the amount of carbon (C) fixed by primary producers.

In the ocean, primary producers are phytoplankton. Phytoplankton use the energy

from the sun to fuel biogeochemical reactions that transform inorganic C (CO2) into

organic C (photosynthesis). A by-product of photosynthesis is O2. Photosynthesis

also requires macronutrients containing inorganic nitrogen (N) and phosphorus (P).

Diatoms, a particularly abundant type of coastal phytoplankton also require silicic

acid (containing Si and thereafter referred as Si) to build their frustules (siliceous

outer structure). In our study, the biological processes associated with the uptake of

macronutrients and the production of dissolved O2 are inferred from budget equations

of N, P, Si and O2 and from knowledge of the non-biological advective processes

(described in Chapter 2 and estimated in sections 5.2.1–5.2.2).

The rate at which PP occurs is called primary productivity (PP rate). PP rate can

be determined directly by measuring the rate of organic C production, O2 production,

or the uptake of nutrients. PP and PP rate are not simple concepts because they can

be defined and determined in different ways. Gross PP (GPP) is defined as the total

amount of organic C fixed by phytoplankton. From GPP, net PP (NPP) is obtained

125

Chapter 5. Nutrients Uptake and Primary Productivity

after subtracting autotrophic respiration (denoted Ra, respiration by phytoplankton

only). When the PP rate is estimated over a long period of time, for instance a day,

the PP rate of interest is often the NPP rate. In our study, the PP rate is estimated

over a month (or a few weeks during freshet). For this reason, the estimated PP

rate is an estimated NPP rate. The net community production (NCP) takes into

account not only the autotrophic respiration but also the heterotrophic respiration

(by bacteria and zooplankton). The NCP can be estimated with the net biological

production of O2.

The NPP rate can vary from season to season (section 5.2.4). NPP is usually

low in winter low light levels. The first bloom of the year usually corresponds to the

largest amount of PP and is usually called the “spring bloom”. Thus, spring is defined

here as the period when the spring bloom can occur. In the SoG, spring is therefore

the period from early February until late April, as suggested by chlorophyll-a (chla)

measurement (shown later in section 5.2.4, Fig. 5.8). The other seasons are defined

relative to the SoG spring bloom. Summer is from early May until late July, fall from

early August until late October, and winter from November until late January.

Nitrogen chemical species like nitrite, nitrate (both hereafter jointly referred as

nitrate) or ammonium are taken up by phytoplankton to fix organic C. Depending

on which chemical form of nitrogen is used by phytoplankton, NPP is differently

identified. The NPP associated with nitrate use is called new NPP since this form of

nitrogen is generally replenished by upwelling or input from outside of the euphotic

zone, although it is possible to remineralize nitrate in the euphotic zone under low

light conditions [Sarmiento and Gruber 2006]. In this study, the new NPP rate can

be estimated from the disappearance of near-surface nitrate. This disappearance

is assumed to occur because of net biological uptake. The NPP associated with

ammonium is called regenerated NPP and it depends on the efficiency of nitrogen

126


regeneration and the amount of sinking of organic nitrogen. The sum of the new and

regenerated NPP is called total NPP or simply NPP. One can approximate NPP

with new NPP if most of the nitrogen uptake relies on nitrate.

In this chapter, I will

1. derive three-year observational estimates of seasonal and annual averages of

primary productivity based on nutrient budgets in the SoG (with uncertainty

estimation).

2. show that, based on the data, primary production in the SoG is mainly due to

new primary production (chapter 5 and further detail in chapter 6).

3. show that the estimated primary production rate over 2002-2005 is typical of

NPP rates in temperate estuaries.

4. find that, as suggested first by Mackas and Harrison [1997] for N, estuarine

entrainment is also the largest supply of P and Si (as suggested by our analysis

of nutrient budgets).

5. find that the annual averages of Si:N and Si:P uptake ratios ( (24.7±2.2):16

and (24.7±2.2):1, respectively ) are larger than the expected ratios [Brzezinski

1985].

In the biological box model (section 2.4), the surface uptake rate of nutrients is

represented as a sink rate term in the 3 conservation equations of surface nutrients

127


(negative terms). The surface production rate of O2 is represented as a source rate

term (positive term) in the conservation equations of surface O2. I will collectively

refer to these four biological terms as either sink terms or net biological uptake rates

because a source rate is merely a negative sink rate. In addition to the biological sink

terms, there are advective sink terms in the conservation equations. In the biological

box model, I will discuss all the terms appearing in the budget equations to determine

which terms are important. In the budget equations, the supply rate terms are the

contribution of the net upwelling (advective term), and the riverine inflow (source

term), while the sink rate terms are the net biological uptake rate and the advective

export (advective term). Since most of these rate terms vary seasonally, an analysis

has to be made for each season.

Once the terms in the budget equations are determined (using the methods of

Chapter 2 and the input data of Chapters 3 and 4), it is possible to estimate primary

productivity by scaling the sink rate terms as sources of organic C. The relationships

between C, nutrients and O2 have to be determined. The linearity and non-linearity

of these relationships are explained in terms of error estimates (N and P) and sea-

sonal variability (Si and O2). The average slope of each relationship is compared to

the average molar ratios of the elemental composition (C, N, P, Si and O2) of phyto-

plankton in seawater found by Redfield et al. [1963], Brzezinski [1985] and Anderson

[1995]. The best estimate of the NPP rate is then chosen. The validity of the estimate

is investigated by analysis of the error estimates, comparison with the biomass, and

consistency between NPP rate estimates for different nutrients. For the rest of this

chapter, we will assume that the new NPP is almost completely equivalent to NPP.

In the discussion section 5.3, we will show that this is likely the case on seasonal and

annual timescales. Finally, these results are discussed at the end of this chapter.

128


5.2 Results

5.2.1 Supply and Sink Rates of Nutrients

Figs 5.1-5.3 show time series of the surface supply and sink rates of various processes

included in the budget equations (section 2.4, Eqs 2.77, 2.79, and 2.81). These include

the computed biological sink rate terms and storage rate terms (time derivative

terms). In particular, the biological sink rate terms are the terms (φ1)P , (φ1)N , and

(φ1)Si defined by:

(φ1)P = V1

∂P1

∂t− PRR + P1U1 + P1W1 − P2W2 + ε′1 (5.1)

(φ1)N = V1

∂N1

∂t−NRR +N1U1 +N1W1 −N2W2 + ε′3 (5.2)

(φ1)Si =V1

∂Si1∂t

− SiRR + Si1U1 + Si1W1 − Si2W2 + ε′5 (5.3)

where processes, for instance for P, are net upwelling (P2W2 − P1W1), advective

export (−P1U1), river inflow (PRR), net biological uptake (φ1)P and the surface

storage/drawdown rate term (V1∂P1/∂t). Note that we will use the words “storage

rate term” or “storage rate” instead of “storage/drawdown rate term”. The air-sea

flux of nutrients is small compared to other processes (based on assumptions made

in section 2.4), thus it is neglected with respect to the other terms in the equations.

For each of these processes, the general shape of the curves for P, N, and Si is

very similar: the maxima and minima occur at the same time. The seasonality of

each process is analyzed in further detail later in this section. The net upwelling

term (positive term) tends to have the largest magnitude with the smallest variabil-

ity. This reflects the near-constant upwelling and near-constant levels of nutrients in

129



−2000

−1500

−1000

−500

0

500

1000

1500

2000m

ol s

−1 (

nitr

ate)

Net Upwelling

Rivers

Zero Line

Net Biological Uptake

Advective Export

Storage Rate

Figure 5.1: Surface Box Nitrate Supply and Sink Rate

130



−100

−50

0

50

100

150m

ol s

−1 (

phos

phat

e)

Net Upwelling

Rivers

Zero Line


Advective Export

Storage Rate

Figure 5.2: Surface Box Phosphate Supply and Sink Rates

131



−4000

−3000

−2000

−1000

0

1000

2000

3000

4000m

ol s

−1 (

silic

ic a

cid)

Net Upwelling

Rivers

Zero Line


Advective Export

Storage Rate

Figure 5.3: Surface Box Silicic Acid Supply and Sink Rates

132



−1.5

−1

−0.5

0

0.5

1

1.5

2x 10

4m

ol s

−1 (

O2)

Net Upwelling

RiversAir−sea O

2 Flux

Net Biological Production

Advective Export

Storage Rate

Figure 5.4: Surface Box O2 Supply and Sink Rates

133


the deep water. The P net upwelling increases slightly over 2002–2005 (Fig. 5.2) due

to an increase of the P levels in deep waters (about 0.5 µM over 3 years, Fig 3.7)

while this was not the case for N and Si (Figs 3.8 and 3.9). There is no satisfactory

explanation based on our analysis of the SoG data for the increase of deep P con-

centration. The explanation may be associated with processes specific to P that we

have neglected. In particular, P concentration has been found to change because of

adsorption/desorption on particles and sediments [Lebo 1991], possibly modulated

by salinity as observed in saline lakes [Clavero et al. 1993] (adsorption rate increases

with salinity), on organic matter, iron oxides, aluminium oxides and apatite [Lucotte

and D’Anglejan 1983, Lebo 1991, McDowell and Sharpley 2003] (possibly modulated

by salinity like in saline lakes), and even on phytoplankton cells as suggested by a

recent study [Fu et al. 2005]. Another possible explanation is the large-scale increase

of the deep P concentration in the North Pacific. Such a long-term increase of deep

P has been previously observed in the Western subarctic Pacific [Ono et al. 2001].

However the increase was only 0.015 µM over 3 years [Ono et al. 2001]. All these

processes are beyond the scope of this study.

The other supply rate term, the river input rate (positive term), had the smallest

magnitude compared to all the other terms. The river terms are especially small for

N and P. In the case of Si, the river input rate is larger and reached a significant rate

of 1000 mol s−1 in summer (maximum of June 2002, Fig. 5.3), but this is still 50%

smaller than the net upwelling term.

The advective export rate (negative term) has the largest magnitude of all the

sink rate terms. It has a seasonal cycle with high magnitude in late winter-early

spring and low magnitude the rest of the year. In contrast with N and P, the Si

term had a more complex seasonal cycle of advective export. Fig. 5.3 shows that the

magnitude of summer Si advective export can be as high as the following maximum

134


magnitude of late winter-early spring, in some years (for instance summer 2002 and

winter 2003).

The net biological uptake rate (mostly negative term) has the largest change in

magnitude in spring, a magnitude comparable to advective export in summer, and

a low magnitude the rest of the year. The net biological uptake rate is expected to

remain purely negative. Positive values occur but are generally indistinguishable from

zero within the uncertainties. In addition, the positive values of the net biological

uptake rate were all of small magnitude: .100 mol N s−1, .10 mol P s−1 and .500

mol Si s−1. After the net biological uptakes, the magnitude of storage rate terms

(positive except in spring) is higher than the other sink rate terms during spring,

but it is smaller the rest of the year.

The budget terms and their relative importance in the budget equations can be

made more quantitative by analyzing annual and seasonal averages (Tables 5.1-5.3).

The largest annual rate term was, on average, the upwelling supply of nutrients: 1273

mol N s−1, 104 mol P s−1, and 2168 mol Si s−1, with average error bars representing

5% of the upwelling supply rates (Tables. 5.1- 5.3, column 2 row 5). These rates have

a very small seasonality (highest around summer, lowest in winter) with seasonal

and annual means overlapping within 1 or 2 times the average error bars (±57 mol N

s−1, ±4.5 mol P s−1 and ±94 mol Si s−1, Tables 5.1-5.3, last row, column 2). Net

upwelling was always the largest source of nutrients to the surface SoG.

This supply rate of nutrients was almost completely balanced by net biological

uptake and advective export. Over the annual timescale, storage rate is zero and

river inputs although positive, are relatively small. On average, about a third (P and

Si) to a half (N) of the net upwelling supply rate was used for the net biological

uptake: -543 mol N s−1, -38 mol P s−1, and -765 mol Si s−1 (Table 5.3, column 3

row 5). In winter, the net biological uptake was not significantly different from zero

135


within the uncertainties.

The seasonal analysis shows that, in summer, the net biological uptake rates of

N and Si were similar (N: -844 mol s−1, Si: -840 mol s−1 very close to 1:1) and the

rates of N and P were close to the Redfield ratio (N: -844 mol s−1, P: -54 mol s−1,

15.6:1) [Redfield et al. 1963]. Since these average rates are close to the expected

ratios [Brzezinski 1985, Redfield et al. 1963], this may suggest that phytoplankton,

in particular, diatoms, a group of siliceous phytoplankton, were blooming during

summer although surface nutrient concentrations in summer were at the lowest level

of the year (Figs 3.7–3.9). On the other hand, this may imply competition between

diatoms and the other phytoplankton groups that are better adapted to low nutrient

concentrations as suggested later in Fig. 5.13, section 5.3 [Miller 2004, pp 12–14]. For

instance, in summer, C biomass estimates (not shown here) based on phytoplankton

taxonomy of water samples suggests that the biomass of silicoflagellates was larger

than the biomass of diatoms. Silicoflagellates are nanoplankton equipped with a

hollow siliceous skeleton that are competing with diatoms for Si [Takahashi 1987]

The seasonal analysis shows that, in spring, the net biological uptake rate of

Si is larger than the rate of N (N: -796 mol s−1, Si: -1460 mol s−1, about 1.8:1)

and the rates of N and P were again close to the Redfield ratio (N: -796 mol s−1,

P: -57 mol s−1, 14:1). The Si:N ratio of uptake rates (about 1.8:1) and taxonomy

data (Fig. 5.13) suggest that spring diatoms were dominant and heavily silicified,

compared to summer diatoms. Although Si:N ratio is an average and variations

around the average are possible [Brzezinski 1985, Brzezinski et al. 2003a], ratios

close or larger than 2:1 are usually unlikely. However, ratios of 2:1 and larger have

been recently observed during iron- and Si-enrichment experiments in mesocosms

containing Central Equatorial Pacific water [Marchetti et al. 2010]. The spring Si:N

and Si:P ratios will be discussed in further detail in section 5.3.

136


The overall importance of the different rate terms in the seasonal analysis was

very similar to that of the annual average rates. The largest rate term was usually

net upwelling, followed by advective export and net biological uptake. There were a

few exceptions to this order because of the different seasonality of different terms.

In particular, the rate terms associated with the net biological uptake (N, P and Si)

decrease from spring/summer (largest magnitude) to winter (smallest magnitude or

negligible). The timing of the seasonal maximum and minimum of N uptake rate

was very similar to that of P and Si. Spring and summer rates were similar within

the uncertainties for P and N (but not for Si, spring rate about 2 times that of the

summer rate). The advective export of N, P and Si had a small seasonality from

spring to fall. During winter, advective export of P, N and Si was usually larger

(larger negative values).

The storage rate terms also had a seasonal pattern. The spring average rate took

the largest negative values (P -20 mol s−1, N -299 mol s−1, and Si -530 mol s−1),

indicating a high drawdown during spring blooms while the largest positive value

could occur either in fall (P 14 mol s−1 and N 229 mol s−1) or winter (Si 172 mol s−1).

A large positive storage term and the high levels of fall and winter surface nutrients

(Figs 3.7–3.9) suggests a net gain of nutrients in the surface layer due to the estuarine

entrainment. The N and P storage rates were negligible in summer.

137


5.2.2 Supply and Sink Rates of Oxygen

Fig. 5.4 shows time series of the terms appearing in the budget equation (section 2.4,

Eq. 2.83). In particular, the source rate term is the term (φ1)O2defined by:

(φ1)O2=V1

∂O1

∂t− ORR +O1U1 +O1W1 −O2W2 (5.4)

−kO2

(Osaturation −Osurface

)+ ε′7

(5.5)

where surface box processes are the net upwelling (O2W2 −O1W1), advective export

(−O1U1), river inflow (ORR), net biological production (φ1)O2, the air-sea exchange

flux kO2

(Osaturation −Osurface

), and the surface box storage/drawdown rate term

(V1∂O1/∂t). Note that, like in the previous section 5.2.1, we will use the words

“storage rate term” or “storage rate” instead of “storage/drawdown rate term”.

The O2 time series provide a different picture from nutrient time series (Fig. 5.4).

Unlike in the nutrient budgets, in the O2 budget the advective export (negative

term) was the largest sink rate terms. The advective export had also one of the

largest magnitudes with some seasonality. The largest magnitude occurs during late

spring-early summer. There is a slower export during the rest of the year, but its

magnitude is never smaller than 0.75×104 mol s−1. The net biological production of

O2 (positive term) was the next largest rate term with large peaks in spring (largest

in April 2004, about 2×104 mol s−1). The net upwelling (a positive term) has a large

magnitude (0.6×104 mol s−1 on average) with a seasonal cycle in opposite phase

with advective export. That is, periods of greater upwelling input are also periods

of greater advective loss. The storage term (a mostly negative term) and river input

(positive term) have similar magnitude but opposite sign and nearly out-of-phase

seasonal cycles with values closest to zero near winter. That is, periods of greater O2

138


N Net River Net Biological Advective Storage

Upwelling Uptake Export

Spring 1266 33 -796 -802 -299

Summer 1303 51 -844 -552 -42 (*)

Fall 1337 17 -446 -678 229

Winter 1183 26 -34 (*) -1075 99

Mean 1273 33 -543 -770 -7 (*)

Apr’02-Apr’05

Average ±57 ±4 ± 93 ±56 ±59

uncertainty

Table 5.1: Averages of Surface Supply and Sink Rates of Nitrate (mol s−1) in the

Euphotic Zone of the SoG. A positive term represents a supply while a negative

term represents a sink rate. Average uncertainties are based on the average standard

error of the seasonal and annual values. The terms have been daily interpolated to

make it possible to average the terms over a few complete years. The row “Mean

Apr’02-Apr’05” provides averages of the term over 3 years starting on and finishing

on Apr 1st. (*) These values are not significantly different from zero.

river input are also periods of greater O2 negative storage rate (drawdown rate). The

magnitude of the air-sea O2 flux was also large. The largest air-sea outflux (negative

139


P Net River Net Biological Advective Storage


Spring 104 1 -57 -68 -20

Summer 108 2 -54 -54 2 (*)

Fall 107 1 -31 -63 14

Winter 96 1 -6 (*) -84 7

Mean 104 1 -38 -67 1 (*)

Apr’02-Apr’05

Average ±4.5 ±0.1 ± 6 ±4 ±4

uncertainty

Table 5.2: Averages of Surface Supply and Sink Rates of Phosphate (mol s−1) in the

Euphotic Zone of the SoG. A positive term represents a supply rate while a negative






term) usually occurs in spring (Fig. 3.11, surface water oversaturated by biological

production), while the largest influx (positive term) usually occurs in fall (Fig. 3.11,

140


Si Net River Net Biological Advective Storage


Spring 2255 265 -1460 -1590 -530

Summer 2119 596 -840 -1723 153

Fall 2243 230 -657 -1682 133 (*)

Winter 2057 185 -62 (*) -2007 172

Mean 2168 327 -765 -1748 -18 (*)

Apr’02-Apr’05

Average ±94 ±38 ± 182 ±122 ±118

uncertainty

Table 5.3: Averages of Surface Supply and Sink Rates of Silicic Acid (mol s−1) in the

Euphotic Zone of the SoG. A positive term represents a supply rate while a negative






141


O2 Net River Net Biological Advective Air-sea Storage

Upwelling productivity Export Flux

Spring 6260 1599 8965 -12562 -4121 141 (*)

Summer 7124 3386 4619 -13467 -3563 -1901

Fall 5674 1707 3543 -10319 -729 (*) -125 (*)

Winter 4680 1434 3478 -10726 2224 1091

Mean 5976 2070 5174 -11831 -1639 -249 (*)

Apr’02-Apr’05

Average ±589 ±220 ± 1190 ±534 ±1042 ±239

uncertainty

Table 5.4: Averages of Surface Supply and Sink Rates of Dissolved O2 (mol s−1) in the

Euphotic Zone of the SoG. A positive term represents a source rate while a negative




Apr’02-Apr’05” provides averages of the term over 3 years starting and finishing on

Apr 1st. (*) These values are not significantly different from zero.

142


spring 2004 and about 0.75×104 mol s−1 in October 2003.

The budget terms and their relative importance in the budget equations can

be made more quantitative by analyzing annual and seasonal averages (Table 5.4).

In Table 5.4, on average, the largest rate term was the advective export (-11831

mol s−1, column 5). The net upwelling (5976 mol s−1, column 2) and the net biological

production rate (5174 mol s−1, column 4) were of similar magnitude. Least important

were the river input (2070 mol s−1, column 3) and the air-sea exchange flux (-1639

mol s−1, column 6). On average, over a few years the storage term was, as expected,

negligible within error bars (Table 5.4, Column 7). The overall importance of the

different rate terms in the seasonal analysis was very similar to that of the annual

analysis. The largest rate term was advective export, followed by net upwelling and

net biological production rate, air-sea exchange flux and river input. There were a

few exceptions to this order because of the different seasonality of these terms. In

particular, the net biological productivity decreases from spring (8965 mol s−1) to

fall and winter (3478 mol s−1 in winter). The magnitude of the air-sea O2 outflux

(negative term) had also a similar seasonality, decreasing from a spring maximum

(-4121 mol s−1) to a fall minimum (-728 mol s−1). In winter, the conditions switched

from oversaturation to undersaturation and the air-sea O2 flux became an influx

(2224 mol s−1). The storage terms also had a seasonal pattern. The largest magnitude

occurred in summer (O2 -1901 mol s−1). The O2 storage term indicates a maximum

drawdown in summer.

143


5.2.3 Property Uptake Ratios

In order to obtain an estimate of the net primary production rate (NPP rate), the

following equation can be used:

NPP = GPP− Ra (5.6)

or

NPP = NPPn +NPPr (5.7)

where Ra is the autotrophic respiration, and NPPn and NPPr are new NPP and

regenerated NPP, respectively. New NPP is a net amount of C fuelled by external

sources of nitrogen (N): e.g., nitrate upwelling, N2 fixation. New NPP rate can be

computed by using the f-ratio, the ratio of new N uptake over the new and recycled

N uptake. This f-ratio can only be estimated when the N cycle is accurately known,

i.e. ammonium and nitrate uptake, nitrogen fixation and nitrification [Sarmiento and

Gruber 2006]. In upwelling coastal regions, most of the new N is upwelled nitrate, thus

the f-ratio can be estimated as the ratio of nitrate uptake over the ammonium+nitrate

uptake. Observed f-ratios from different coastal areas were as high as 0.8 [Eppley

and Peterson 1979]. Legendre et al. [1999] suggest that f-ratio can be higher than

0.7 when nutrients are mainly supplied by circulation. In addition, the equivalence

between NPP and new NPP rate only occurs when NPP is mainly driven by nitrate

uptake. Later in this chapter and chapter 6, we will discuss the difference between

the estimated new NPP rate and total NPP rate in the SoG. In section 5.2.1, it was

noted that siliceous phytoplankton, mainly diatoms, contribute to the NPP. Their

contribution to the new NPP rate can be analyzed further by looking at the uptake

of silicic acid relative to the other nutrients.

144


The elements N, P and Si can be used to estimate the NPP rate or a particular

contribution to the NPP rate. In the N budget (Eq. 2.79), if N represents only

upwelled N (Fig. 3.8), the estimated NPP rate based on N will give an estimate of

the new NPP rate. On the other hand, P is used in both new and regenerated primary

production. Thus, the estimated NPP rate based on P will give an estimate of the

(total) NPP rate. The Si removal is associated with the diatom fraction [Miller 2004,

Sarmiento and Gruber 2006] of the total NPP rate because diatoms can use both

upwelled and regenerated forms of N. The Si removal is due to the growth of siliceous

phytoplankton, mainly diatoms. Before estimating the NPP rate, it is necessary to

determine the relationship between the sink rate terms of the four sampled elements

(P, N, Si and O2) and the unknown element (organic C). It is important to note that

the ratios C:O2:Si are more variable than the C:N:P ratios, and hence are treated

differently.

Using the notation introduced earlier the NCP rate is defined by:

NCP = NPP− Rh (5.8)

or

NCP = GPP− Ra − Rh (5.9)

where Rh is the heterotrophic respiration. O2 is not only respired by phytoplankton

(Ra) but also by zooplankton, heterotrophic bacteria and other organisms (Rh) that

are not part of the NPP rate (and not included in the box model). Only if zooplankton

and bacterial respiration is negligible compared to phytoplankton respiration, will the

C:O2 ratio reflect the NPP rate and be equal to the empirical mole ratio associated

with the phytoplankton composition.

Before analyzing the relationships between the sink rates of the sampled elements,

145


it is necessary to choose one of them as a common element. C has not been mea-

sured, although it is the currency of NPP rate. I decided to use P as the common

element for the analysis. Since the C:Si and C:O2 are complex and very variable in

time, the choice of a common element for comparison can only be made between

N and P. In upwelling coastal regions, new NPP is usually associated with nitrate

because recycling of N, as ammonium, is inhibited by light and usually occurs out-

side the euphotic zone [Sarmiento and Gruber 2006]. Contribution of N2 fixation by

cyanobacteria (e.g., Synechococcus and Prochlorococcus) to the surface N cycle is

also unlikely in upwelling coastal regions [Miller 2004]. But the cycle of N can be

complex and the observations collected for this study are limited to the observed

concentration of nitrate and nitrite. On the other hand, the cycle of P is simpler

being mainly controlled by external sources and used by both new and regenerated

NPP [Sarmiento and Gruber 2006].

The observed mole ratios for C:N:P:Si have been shown to have a global average

over the ocean and coastal areas. Average C:P:N:Si ratios are 106:1:16:16 [Brzezinski

1985, Redfield et al. 1963], while the C:O2 had different values according to different

authors : C:O2= 106:138 Redfield et al. [1963] and 106:150 Anderson [1995], Fraga

et al. [1998]) as this depends on (among other things) the amount of O in nitrogen

species (NH+4 , NO

−

2 and NO−

3 ). In this study the C:O2 ratio when considering only

autotrophic respiration is chosen to be 106:150. Following the argument of Fraga et al.

[1998], a ratio of 106:150 is preferable to a ratio of 106:138 because the primary source

of nitrogen, in the SoG box model, is nitrate (NO−

3 ), not ammonium (NH+4 ), and

organic C can be used to build complex molecules like lipids, not just carbohydrates

as assumed by Redfield et al. [1963] in the open ocean. The C:P molar ratio will be

146


assumed to be equal to 106:1.

C : P = 106 : 1 (5.10)

If the productivity estimates based on P and N are very similar, then NPP will

mostly be new NPP. In this case, there will be little regenerated productivity, so

that NPP will be primarily new NPP. The extent to which the C:P uptake ratio

approximation and the equivalence between new NPP and NPP rates, and the N

and P sink rate terms hold will be discussed later in this chapter and chapter 6.

Figs 5.5–5.7 show the surface N sink rate, surface Si sink rate and surface O2

source rate against the surface P sink rate. Each single-survey estimate is plotted

with the corresponding error bars based on bootstrap replicate statistics (defined

in Chapter 2) and each season is specified by a different marker. The average sink

term and the corresponding error range is also indicated for each figure (vertical and

horizontal dashed lines). All terms have been already converted into mol m−2 d−1

using the surface area given by the SoG hypsography (Fig. 3.2).

Fig. 5.5 shows the surface N sink rate plotted against the surface P sink rate

term. The surface N sink rate and the surface P sink rate are strongly correlated.

Most of the points are slightly below the line that represents the empirical Redfield

ratio N:P=16:1. A least-squares fit method yields a linear slope of (14.6±0.8):1. Most

of the points are aligned along a straight line with lowest values of P and N uptake

in winter, the highest values in spring, and intermediate values in fall and summer.

A smaller ratio than 16:1 might be expected since the surface N sink rate (nitrate

and nitrite) is associated with new NPP rate and the surface P sink rate with total

NPP rate. The net biological uptake of P is on average (5.8×10−4)± (6×10−5) mol

P m−2 d−1 with a range of 0 to 16 ×10−4 mol P m−2 d−1. Multiplying this P average

uptake, (5.8×10−4) ± (6×10−5) mol P m−2 d−1, by the average N:P ratio, 14.6:1,

147


leads to an average N uptake of (8.5×10−3)± (9×10−4) mol N m−2 d−1. In Fig. 5.5,

the negative values are generally not significantly different from 0.

Fig. 5.6 shows the surface Si sink rate plotted against the P sink rate. Most

of the points are above the Si:P=16:1 line and below the Si:P=32:1 line. The 32:1

line was arbitrarily chosen to provide an upper bound of the largest Si:P ratios

(average ratio, spring and winter ratios). The points in fall and summer are the

closest to the 16:1 line with the highest ratios of Si and P uptake in fall (19.3±4:1 in

fall, 14.4±3.8:1 in summer). Five fits constrained by the least-squares method yield

linear approximations of slope (24.7±2.2):1 over the year, (28.1±3.8):1 in spring,

(14.4±3.8):1 in summer, (19.3±4):1 in fall, and (28.7±14.4):1 in winter. Thus, high

Si:P occur in winter, spring and overall of the year. The net biological uptake from

the Si sink term was about (1.3×10−2) ± (2×10−3) mol Si m−2 d−1, with a range

from 0 to 6×10−2 mol m−2 d−1. The negative values were not significantly different

from 0.

Fig. 5.7 shows the surface dissolved O2 source rate plotted against the P sink

rate. The figure suggests that the O2:P ratio is highly variable, not only from season

to season, but also from survey to survey during the same season. Most of the points

are scattered away from the Redfield ratio O2:P=150:1 line. The highest values of

O2 production and P sink rates are in spring and the lowest values tend to be in

winter. Four fits constrained by the least-squares method yield linear approximations

of slope (128±21):1 over the year, (143±29):1 in spring, (78±25):1 in summer, and

(110±55):1 in fall. Annual average, spring and summer average ratios have similar

uncertainties. The winter values of the O2 net production and P net uptake rates were

small and noisy and led to winter uptake ratios with a large range. The best estimate

of the winter average ratio is (70±358):1. This average ratio is indistinguishable from

zero within the uncertainties. The net biological productivity inferred from the O2

148


source term is on average (8.3×10−2) ± (9×10−3) mol O2 m−2 d−1, with a range

from 0 to 23×10−2 mol O2 m−2 d−1. Most of the negative values were generally not

significantly different from zero.

The analysis of the average ratios suggests a good agreement for N:P between the

empirical Redfield ratio of 16:1 and the estimated value of (14.6±0.8):1 throughout

the year. It also suggests a very similar variability for the surface P and N sink rates.

In contrast, the Si:P and the O2:P average ratios agreed on the order of magnitude but

with less fidelity: Si:P =(24.7±2.2):1 instead of 16:1, O2:P over the year (128±21):1

instead of 150:1. In the case of Si:P, the uncertainty is about 10% of the average. The

spring and winter average ratios are markedly higher than 16:1. The overall annual

average, spring and winter averages are greater than expected (between +50% and

+100% on average). In the case of O2, the overall year average, spring and fall

averages are smaller than, but still close to, the 150:1 ratio. The spring ratio, 143:1,

is the closest ratio to the Redfield ratio. All the O2:P ratios are lower than 150:1

suggesting that other processes are using up the photosynthesized O2. The decrease of

the O2:P ratio from spring to summer suggest an increasing heterotrophic respiration,

in particular grazers respiration. Further discussion of these Si and O2 results will

be deferred until section 5.3.

5.2.4 Estimates of Net Primary Productivity

The previous analysis based on P (being the common element for comparison) pro-

vides an estimate of P:N:Si molar uptake ratios of 1:(14.6±0.8):(24.7±2.2). The N:P

estimate is very close to 16:1, the Redfield ratio. This suggests that the C:P:N molar

ratios can reasonably be approximated by 106:1:(14.6±0.8), found in section 5.2.3.

In Tables 5.1 and 5.2, the annual uptake averages of N and P are 543±93 molN s−1

149


0 5 10 15 20

0

5

10

15

20

25

30

35

40

Nitr

ate

10−

3 mol

N m

−2 d−

1

Phosphate 10−4 mol P m−2d−1

spring

summer

fall

winter

Average P (−−)and errorbars (...)

Average N (−−)and errorbars (...)

N:P=(14.6±0.8):1

16:1 line

Figure 5.5: Surface Nitrate and Phosphate Uptake Rates. The slope of the dotted

line is the overall N:P ratio 14.6:1, the slope of the dashed line, the Redfield ratio

16:1.

150


0 5 10 15 20−1

0

1

2

3

4

5


Sili

cic

Aci

d 10

−2 m

ol S

i m−

2 d−1

spring

summer

fall

winter

Average Si (−−)and errorbars (...)

Si:P=(24.7±2.2):1

spring(28.1±2.3):1summer(14.4±3.8):1fall(19.3±4):1

winter(28.7±14.6):1


16:1 line

32:1 line24.7:1 line

summer

fall

springwinter

Figure 5.6: Surface Silicic Acid and Phosphate Uptake Rates. The overall molar

ratio is 24.7:1 (slope of dash-dotted line) and the seasonal ratios (slope of dotted

lines) are 28.1:1 (spring), 14.4:1 (summer), 19.3:1 (fall) and 28.7:1 (winter). The

empirical ratio, 16:1 [Brzezinski 1985], and an arbitrary upper bound, 32:1, are the

slopes of the dashed lines. 151


0 5 10 15 20−10

−5

0

5

10

15

20

25


Dis

solv

ed O

2 10−

2 mol

O2 m

−2 d−

1

spring

summer

fall

winter

O2:P=(128±21):1

spring(143±29):1summer(78±25):1fall(110±55):1

winter(70±358):1


Average O2 (−−)

and errorbars (...)

150:1 line

fall

summer

winter

128:1 line

spring

Figure 5.7: Surface Dissolved O2 Release and Phosphate Uptake Rates. The slope

of the dash-dotted line is the overall molar ratio 128:1. The slope of the dotted lines

are the seasonal ratios 143:1 (spring), 78:1 (summer), 110:1 (fall), and 70:1 (winter).

The Redfield ratio 150:1 is the slope of the dashed line. 152


and 38±6 molP s−1, respectively. The N uptake is associated with external sources

of N and new net primary production (NPP) rate, while the P uptake is associated

with any sources (recycling and external supply) of N and total NPP rate. Applying

the empirical Redfield ratio N:P of 16:1 to the average uptake of P gives an estimate

of the total average uptake of N (nitrate, nitrite and ammonium included): 608±96

molN s−1. Given, the average accuracy of about 90 molN s−1, there is no marked

difference between the two uptake rates. This is consistent with the similarity between

the observed Redfield N:P ratio of (14.6±0.8):1 and the empirical Redfield ratio. As

we will see in further detail in section 5.3, the uptake of N is mostly supplied by

nitrate. As a result, new and total NPP rates are close.

Fig. 5.8 shows the plots of the two estimates of the total NPP rate based on

surface net biological uptake rates of P and N. The estimated total NPP rates were

computed from net biological uptake rates of P and N scaled as terms of fixed organic

C by using the average ratios C:P:N of 106:1:(14.6±0.8) and the atomic weight of

C (12 g mol−1) to convert from mole number to mass. The ratio P:N, 1:(14.6±0.8),

enables us to obtain a N-based total NPP rate scaled to estimate the total NPP rate

like the P-based total NPP rate. In this fashion, our estimates of the NPP rate based

on N and P are both estimates of the total NPP rate. Note that the cumulative

uncertainty from the transport estimates, the sink rates and the uptake ratios is

likely larger than the uncertainty on the NPP rates anticipated with the bootstrap.

Although a similarity between the SoG observed N:P uptake ratio and the em-

pirical Redfield ratio was found, it does not guarantee a similarity between the SoG

C:P ratio and the empirical Redfield ratio of 106:1. At this point in the analysis, it

is clear that without any data on the organic C net production rate in the SoG, it

is speculative to define the SoG C:P ratio as the Redfield ratio 106:1 (Eq. 5.10, sec-

tion 5.2.3). Previous studies suggest that, in the ocean, the surface C:N:P ratios are

153


sensitive to growth rate, nutrient concentrations and species composition [Sarmiento

and Gruber 2006]. For instance, C:N:P observed disappearance ratios in the South-

ern Ocean exhibit some variability between phytoplanktonic groups. For 1 mole of

P used by diatoms, 9–10 moles of N are used to fix between 63–94 moles of C [Ar-

rigo et al. 1999, 2000, Sweeney et al. 2000, Quigg et al. 2003], while Phaeocystis

sp., another dominant phytoplankton group, used between 18.6–19 moles of N to

fix between 133–147 moles of C. Experimental cultures of diatoms, dinoflagellates

and Phaeocistis sp groups also showed variability, 60:10:1, 120:15:1 and 80:10:1, re-

spectively [Ho et al. 2003]. In these studies, although the average ratios are close

to the empirical Redfield ratios (e.g. C:N:P= 124:16:1 in Quigg et al. [2003]), they

emphasize that the variability of the individual values of the ratios found at sea is

large as a result of the variability of the composition of the species sampled. In the

case of the SoG, the species composition is seasonal (Chapter 1, section 1.2), and

thus the C:N:P ratios could vary according to the composition of the bloom.

In Fig. 5.8, the two estimated NPP rates have very similar seasonality. Largest

differences occurred during 2003, when N was larger than P by 0.6 and 0.5 gC m−2

d−1 in spring and summer 2003, respectively. But, in 2005, the opposite occurred,

the P-based estimate was larger than the N-based estimate. On average, the differ-

ence between the P-based estimate and the N-based estimate was positive, about

0.12±0.08 gC m−2 d−1 (1 standard deviation). This difference is small compared to

the average NPP rate of the order of magnitude of 1 gC m−2 d−1. When averaged

over the period April 11th 2002– April 11th 2005 the two estimates yielded annual

NPP rates of 212±41 (for P) and 205±36 (for N) gC m−2 yr−1. On average, the

difference between the NPP rate (based on P) and the NPP rate (based on N) is

zero. The P-based and N-based NPP rate estimates are slightly different due to mi-

nor discrepancy in their variability during spring and summer (Fig. 5.8). Note that

154


although NPP peak was highest during the spring bloom (average 157 gC m−2 yr−1

based on P), the 3-month spring average equalled the average over summer.

An independent proxy for the organic C biomass, the chlorophyll-a (chl-a) average

biomass integrated over 30 m is plotted along with the NPP estimates to show the

spring bloom timing, and the consistency between NPP rates and phytoplankton

biomass. Fig. 5.8 shows that the NPP rate is roughly proportional to the biomass.

The maximum rates occur in spring when chl-a is the highest. In summer and early

fall the NPP rate can be high despite a low level of chl-a. It is important to note that,

since we are not using a direct measurement of C biomass, but instead the amount of

chl-a pigments, the total chl-a concentration can change because the concentration of

chl-a per phytoplankton cell might vary even though the number of phytoplankton

cells in the water column is kept constant. Since there is no direct estimate of C

biomass in the STRATOGEM dateset to determine the chl-a:C biomass ratio, the C

biomass has to be assumed to be proportional to the chl-a biomass even though it is

likely not to be the case.

The chl-a-normalized NPP rate depends not only on the cell C (QC , gC cell−1)

and chla quotas (Qchl−a, g chl-a cell−1), but also on the cell growth rate (µ, d−1)

which represents the number of cell division per day:

chla nNPP =QC

Qchl−aµ (5.11)

Thus,

chla nNPP =gC cell−1 d−1

chl−a cell−1= gC chl−a−1d−1 (5.12)

The growth rate is known to depend on temperature, light and nutrients among

other things [Eppley 1972, Laws and Bannister 1980, Geider et al. 1998].

The photosynthetic rate (the NPP rate normalized by phytoplankton biomass)

155


can be approximated by the ratio between NPP rate and chl-a biomass. It repre-

sents the ability of a phytoplanktonic cell (represented by the same amount of chl-a

biomass) to produce C. Fig. 5.9 shows that the largest chl-a-normalized NPP rates

occur in late summer (2002 and 2003) and in spring (2004 and 2005). The chl-a-

normalized NPP rate ranges between -0.8 and 47 gC gChl-a−1 d−1 with an average

of 11 gC gChl-a−1 d−1. Assuming no nutrient limitation, since phytoplankton receive

more light during the summer (with a seasonal maximum over June–July) than in

spring and fall, phytoplankton would need less chl-a pigments to produce the same

amount of C. Thus, the chl-a-normalized NPP rate should increase throughout the

period of Februrary-August. In 2003, chl-a-normalized NPP rate was low during

spring (around 10 gC gChl-a−1 d−1) and rose during summer (up to 24 gC gChl-a−1

d−1).

Fig. 5.10 shows the chl-a-normalized NPP rate and an estimate of the averaged

PAR over the mixing layer depth during 2002–2003. It is based on STRATOGEM

monthly-averaged SW radiation and PAR vertical profiles, and estimated albedo,

light attenuation coefficient and mixing layer depth by Collins et al. [2009]. The

averaged PAR represents the light received by phytoplankton when they are mixed

vertically through the water column. The seasonal changes of the primary produc-

tivity were significantly correlated with those of the averaged PAR: on average, there

was a positive linear correlation coefficient, about 0.6 associated with a p-value of

0.0024 and a confidence interval of [0.24–0.81]. The averaged PAR over the mixing

depth best correlates with the chl-a-normalized NPP rate among the variables de-

rived from the solar radiation observations. There was a smaller correlation with

PAR estimated at individual depth (at 5, 10 and 15 m, not shown). There was

no clear relationship between chl-a-normalized NPP rate and other environmental

variables: e.g., 0-m PAR level, average light extinction coefficient, surface SW heat

156


flux, and surface temperature. This is most likely because of the large scatter of the

chl-a-normalized NPP rate.

5.2.5 Estimates of Net Community Productivity

In this section, we will use the net biological production rate of O2 as a crude estimate

of the NCP rate. By comparing it with the new NPP rate, scaled in g O2 unit using

the empirical Redfield ratio O2:N 150:16 (see section 5.2.3), one can speculate on

the timing and the size of the regenerated NPP rate and heterotrophic respiration

Rh. Fig. 5.11 shows the estimate of the net biological production rate of O2, in gO2

based on the molecular weight of 32 g mol−1 and O2:N:P= 150:16:1. It is not possible

to directly use the O2:C ratio to find a NCP estimate in organic C since the O2:C

ratio is not known and the constant ratio that we assume, O2:C= 150:106, is only

associated with the net biological production rate of phytoplankton not the entire

community, i.e. phytoplankton, zooplankton, and bacteria. Note that the previous

analysis based on P (in section 5.2.3) provides an average estimate of P:O2 molar

uptake ratios of 1:(128±21). Thus, the net biological production in organic C unit,

based on our O2 net biological production rate and C:O2= 106:(128±21), leads to an

average estimate close to the NPP rates found in the previous section: 223±53 gC

m−2 yr−1 of over 2002-2005. This value is larger than, but still close to, the P-based

(212±41 gC m−2 yr−1) and N-based (205±36 gC m−2 yr−1) estimates of the NPP

rates as expected from our analysis of the O2:C ratio.

In the O2 surface budget, by definition, the NCP is the difference between the

total NPP rate and the heterotrophic respiration rate (section 5.2.3).

NCP = NPP− Rh (5.13)

157



0

0.5

1

1.5

2

2.5

3

3.5

4

× 1

0−

1 g

Ch

l−a

m−

2 o

ver

30

m


0

0.5

1

1.5

2

2.5

gC

m−

2 d

−1

N

P

Chl−a

Figure 5.8: NPP Rate Estimates Based on Net Biological Uptake Rates of N and P .

The NPP rates are obtained by multiplying the net biological uptake rates with the

C:N:P ratio defined in section 5.2.3. The chl-a biomass (gChla m−2) is based on the

surface-box average and integrated over 30 m. 158


Since NPP is the total of NPPn and NPPr

NCP = NPPn +NPPr − Rh (5.14)

Rearranging yields:

NCP− NPPn = NPPr − Rh (5.15)

In theory, the difference between the NCP and the new NPP rates (Eq. 5.15)

would then represent the difference between the regenerated NPP rate and Rh (het-

erotrophic respiration rate). Eq. 5.15 provides a way to estimate what the “excess”

of regenerated NPP (relative to Rh) is when NCP is larger than new NPP, and what

the “excess” of heterotrophic respiration (relative to regenerated NPP) is when new

NPP is larger than NCP.

The NCP rate in Fig. 5.11 (triangled solid line) corresponds to the net biological

production rate of O2 from Fig. 5.4. To be used in Eq. 5.15, the new NPP rate has

to be scaled in unit of mol O2 by assuming a 150:16 O2:N ratio (section 5.2.3) when

respiration is only autotrophic (Fig. 5.11, squared dashed line). It represents the net

production of O2 by autotrophs supplied by external source of N. A positive difference

between the NCP and new NPP rates in mol O2 would suggest that regenerated

NPP rate is significantly larger than heterotrophic respiration rate, while a negative

difference would suggest a larger heterotrophic respiration rate than the regenerated

NPP rate.

Fig. 5.12 provides the timing of possible excess of regenerated NPP (relative to

heterotrophic respiration) and excess of heterotrophic respiration (relative to regen-

erated NPP). In the case of excess of regenerated NPP, NCP is larger than new NPP

during October 2002-March 2003 (maximum 2.1 gO2 m−2 d−1), November 2003-April

159


Annual Spring Summer Fall Winter

-0.1±1.9 0.2±2.2 -1.8±0.8 -0.6±1.2 1.5±1.1

Table 5.5: Estimates of the Excess of Heterotrophic Respiration (<0) based on

Eq. 5.15 . Annual and seasonal averages (with 1 standard deviation) of the ex-

cess of heterotrophic respiration (<0) rates (gO2 m−2 d−1) based on Eq. 5.15 and

Fig. 5.12. Average error is ±1 gO2 m−2 d−1 (Fig. 5.7).

2004 (2.5 gO2 m−2 d−1 in 2003, 3.6 gO2 m−2 d−1 in 2004), in August 2004, Decem-

ber 2004 and March 2005. Thus, large excess of regenerated NPP during spring and

winter suggests that regenerated NPP is likely larger once a large amount of organic

C has been produced by phytoplankton (early spring and late fall).

An excess of heterotrophic respiration has to be consistent with an increase of

zooplankton and bacterial respiration. New NPP was larger than NCP in summer

(maximum excess of 3 gO2 m−2 d−1 in 2002 and 2.5 gO2 m

−2 d−1 in 2003, 2004, and

2005), spring 2003 (peak of 6.4 gO2 m−2 d−1). The rest of the time the difference

can be considered insignificant given the average error of ±1 gO2 m−2 d−1 (Fig. 5.7).

Averaging of the difference NCP- new NPP (Table 5.5) suggests that on average

only the summer difference (on average negative) is significant. This suggests that

heterotrophic respiration is large on average during summer and is likely associated

with an increasing biomass of herbivorous zooplankton feeding on phytoplankton.

5.3 Discussion

In the previous sections of this chapter, the NPP rate of the SoG and its variability

have been examined through estimates of NPP and NCP rates based on a budget of

160



0

10

20

30

40

50gC

gC

hl−a

−1 d

−1

Figure 5.9: Estimate of the Chl-a-normalized NPP Rate (gC gChl-a−1 d−1) , Using

NPP Based on P (gC m−2 d−1) and Chl-a (gChl-a m−2, over 0-30 m) in Fig. 5.8.

Shades indicate spring and summer.

161


May Sep 2003 May Sep 04 May Sep 05 May

0

50ch

la−

norm

aliz

ed N

PP

rat

e (g

C g

Chl

a−1 d

−1 )

May Sep 2003 May Sep 04 May Sep 05 May0

100

200

Inte

grat

ed P

AR

base

d on

mod

elle

d 0−

m P

AR

(W

m−

2 ) an

d ob

serv

ed P

AR

ver

tical

pro

files

Figure 5.10: Comparison of the Chl-a Normalized NPP Rate (thick line, circles,

left y-scale) With Averaged PAR over the Mixing Layer (thin line, squares, right

y-scale) . The averaged PAR is based on STRATOGEM PAR vertical profiles, and

estimated albedo, light attenuation coefficient and mixing layer depth by Collins

et al. [2009].

162



−1

−0.5

0

0.5

1

1.5

2

2.5

3

3.5

× 1

0−

1 g

Ch

l−a

m−

2 o

ver

30

m


−2

0

2

4

6

8

10g

O2 m

−2 d

−1

Net Community PP rate based on O2 (gO

2 m−2 d−1 and 32 g mol−1O

2)

New NPP rate (gO2 m−2 d−1 using N uptake and O

2:N=150:16)

Surface chlorophyll−a integrated over 30 m (g chl−a m−2)

Figure 5.11: Net Community Production Rate (gO2 m−2 d−1). The estimates are

based on net biological productivities of O2, new NPP rate (gO2 m−2 d−1, using N

uptake and O2:N=150:16 Redfield ratio), and integrated chl-a biomass (gchl-a m−2,

right y-scale) is also plotted. Average error is about ±1 g O2 m−2 d−1 (Fig. 5.7)

163



0

0.5

1

1.5

2

2.5

3

3.5

× 10

−1 g

Chl

−a

m−

2 ove

r 30

m


0

2

4

6

8

10

gO2 m

−2 d

−1


−6

−4

−2

0

2

4

gO2 m

−2 d

−1

NPPn

Depth−integrated Chl−a biomass

NCP−NPPn

Excess Recycling

Excess Respiration

Figure 5.12: Estimates of the Excess of Regenerated NPP Rate (NPPr) and Het-

erotrophic Respiration (Rh). Estimates are based on the difference between NCP and

new NPP (scaled with C:O2=106:150 and molar masses of O and C), in mol O2 m−2

d−1. Positive values indicate that NPPr >Rh, and negative values that Rh >NPPr.

Average error (dotted lines) is about ±1 mol O2 m−2 d−1 (Fig. 5.7). All rates are in

gO2 m−2 d−1

164


the sink rate terms of nutrients and production of dissolved O2. The analysis of the

sink rate terms shows that the nutrients near the surface were mainly supplied by

the net upwelling with a small seasonal variability while both the biological uptake

rate and the advective export were responsible for drawing them down. However,

in spring, the budget was unbalanced with a residual rate (the storage rate terms

in Tables 5.1 and 5.2) of -20±4 mol P s−1 and -299±59 mol N s−1 indicating that

the external sources of P and N were not sufficient in spring to sustain sinks (net

biological uptake rate and advective export). Nutrients were drawn (storage terms)

from the surface nutrient pool (initially in spring 3.8 ×108 P moles and 4.8×109

N moles over 30 m). This surface nutrient pools provide a sufficiently large storage

of nutrients to compensate for the unbalance between the spring sources and sinks

but are not completely depleted at the end of spring. The estimates of the spring

sink rates (net biological uptake and advective export) were similar within error bars

although the net biological uptake rate eventually became the largest of the spring

sink rates because of their trends (Fig. 5.1–5.3: biological uptake rate reaching a

maximum and advective export decreasing).

The estimates of NPP rates were obtained from two assumptions: first, the net

biological uptake rates of N and P in the SoG are associated with new NPP and total

NPP respectively, and second, the molar ratios C:P and C:N in the SoG are relatively

constant, and close to the empirical Redfield ratios. The first assumption is reasonable

given my analysis of the terms in the budget equations of P and N, although it still

requires validation by other studies. The net biological uptake rate of P and N had

very similar seasonality. The small difference between them suggests that overall N is

supplied to phytoplankton by external sources of N. NPP (average NPP, 212 gC m−2

yr−1) is mainly new production (average new NPP 187 gC m−2 yr−1). The overall

f-ratio is the ratio of average new NPP over average NPP, f=187/212=0.88. It is

165


larger but close to the maximum observed value in coastal systems [Eppley and

Peterson 1979, f=0.8, see new:total production p. 679] and larger than the minimum

for systems where the nutrients are mainly supplied by circulation [Legendre et al.

1999, f-ratio>0.7]. In coastal areas, high f-ratio (f>0.8) can be observed when spring

blooms are dominated by diatoms [Kristiansen et al. 2001](using estimated uptake

rates based on radioactive isotopes of nitrate and ammonium). These SoG NPP rate

estimates are consistent with other studies and this suggests that they are reasonable

estimates of the primary productivity.

The ratio of the net biological uptake rates shows a strongly linear relationship

of P:N ratio with a slope coefficient of 1:(14.6±0.8) very close to the Redfield ratio

of 1:16 (Fig. 5.5). The small difference between the 2 ratios could be due to N being

mainly supplied by external sources of N. On average, it is consistent with averages

of surface measurements of C:P:N ratios [Sarmiento and Gruber 2006], although

individual values vary within a larger range [Quigg et al. 2003]. For this reason

assuming a C:P:N ratio of 106:1:(14.6±0.8) seems a reasonable choice to estimate

the average NPP rate, but a more speculative choice to estimate time series of NPP

rate.

The second assumption made to estimate the NPP rates in C unit, that of con-

stant surface C:P and C:N ratios, is also speculative. There is no simultaneous mea-

surements or estimates of the SoG C net production rate, and SoG P and N uptake

rates to suggest such a relationship is consistent in the SoG. At the surface of the

ocean, it is traditionally assumed that the C:P and C:N ratios are sensitive to the

nutrient concentrations, the growth rate, and the species composition and are more

variable than the ratios found deeper [Arrigo et al. 2000, Sweeney et al. 2000, Quigg

et al. 2003, Ho et al. 2003]. However, this allows one to compare these estimated

NPP rates with those of other studies.

166


The annual average rate of NPP was estimated between 205±36 (new NPP rate

based on N) and 212±41 (NPP rate based P) gC m−2 yr−1 which is consistent with

recent estimates of NPP rates: 220 gC m−2 yr−1 using the same dataset [Pawlowicz

et al. 2007], 120–345 gC m−2 yr−1 [Harrison et al. 1983, on average 280 gC m−2 yr−1]).

The similarity between these estimates and my NPP rate estimates suggests that my

NPP rates are realistic rates. The estimates from Harrison et al. [1983]’s review

are based on C14 incubation uptake experiments over several hours. An estimated

280 gC m−2 yr−1 has to be regarded as a weighted average between NPP and GPP

rates [Marra and Barber 2004]. Their measurements are also spot measurements with

a large variance, while ours are time and space averages with smaller variance. The

classification of estimated PP rate into GPP or NPP rate depends on the period

of incubation of water samples among other things. A recent study [Williams and

Lefevre 2008] suggest that a better knowledge of the internal sinks and sources of

14C of the phytoplanktonic cells are necessary to be able to differentiate the type of

PP rate.

The annual estimate of the NPP rate over the spring-summer period alone was

on average 157 gC m−2 yr−1 (based on P) equally contributed by spring and summer

biomass. Spring biomass was characterized by higher chl-a level and higher nutri-

ent uptake and drawdown rate (negative storage rate), while summer biomass was

characterized by lower chl-a level but still large nutrient uptake rate. In Fig. 5.9, the

estimate of chl-a-normalized NPP rate (average 11 gC gChl-a−1 d−1) suggests that the

summer biomass is more productive that the spring biomass because of greater avail-

ability of light [Miller 2004]. The effect of warmer temperatures on diatom growth

rate can either follow an exponential law (doubling with a 10◦C increase) or a slower

linear relationship [Montagnes and Franklin 2001]. In either case, the surface temper-

ature in the SoG increases by 2.78◦C month−1 maximum over summer and between

167


depths of 0–30 m by 0.7◦C month−1 on average (Fig. 3.4, section 3). This suggests

a small effect of temperature on diatom growth on a monthly scale over the water

column (maximum 2.78◦C month−1). However there may be a significant effect on

the phytoplankton assemblage right at the surface over the spring-summer period

(10◦C and more over 6 months).

Using direct measurements of chl-a and C14 uptake rate (not shown) in the eu-

photic zone during 1988–1991 in the SoG, the chl-a-normalized NPP rate in the

SoG provide similar average and range to our estimates [Clifford et al. 1992, aver-

age 24.5 gC gChl-a−1 d−1, and range between 3 and 188.4 gC gChl-a−1 d−1]. The

range of estimated chl-a-normalized NPP rates found in the present work (from -8

to 47 gC gChl-a−1 d−1, Fig. 5.9) is smaller than maximum chl-a-normalized NPP

rates found in other studies in the SoG [Forbes et al. 1986]. My estimates are time

and space averages, while the other studies took spot measurements. Forbes et al.

[1986] parametrized the PI curve at locations along the coast of British Columbia,

in particular in the SoG. Based on 14C uptake and PAR measurements at the depth

of chlorophyll-a maximum they estimated the maximum chl-a-normalized NPP rate

to be on average 12.3±1 mgC mgChl-a−1 h−1 that is 295±24 when scaled to gC

gChl-a−1 d−1. But averaged over a day and over depth this becomes much smaller.

Fig. 5.13 shows the estimated C production rate of autotrophic phytoplankton

and diatoms based on species abundance at station 4-1 at 5 m (see section 3.2.1

in chapter 3. During summer blooms, the phytoplankton biomass was a mixed as-

semblage of phytoplankton species [Harrison et al. 1983], but it usually contained

a large amount of diatoms. As expected, on average summer nutrients were drawn

down in the N:P Redfield proportion (=15.6, Tables 5.1 and 5.2, -844 molN s−1 and

-54 molP s−1, respectively), and in the theoretical Si:N (=1) and Si:P (=15.6) pro-

portions (Table 5.3, -840 mol Si s−1) [Brzezinski 1985]. During the summer bloom,

168



5

10

15

20

25

30

35

40

45


5

10

15

20

25

30

35

40

45gC

m−

2 ave

rage

ove

r 30

mba

sed

on S

4−1,

5 m

Total AutotrophsTotal DiatomsS4−1, 5 m

Figure 5.13: Estimates of the Diatom and Total Autotrophic Biomass Over 30 m

Based on Cell Counts at Station S4-1 at 5 m. Shaded boxes indicate spring.

169


the concentrations of nitrate and phosphate were very low near the surface and low

throughout the water column (Fig. 5.14) as a result of large uptake of nutrients

during summer and, earlier, during the spring bloom. However the silicic acid con-

centration was either increasing or plateauing during summer blooms (Fig. 5.15),

possibly because of the additional riverine supply during the freshet [Harrison et al.

1991]: 596 molSi s−1 on average in summer (Table 5.3), and up to 1000 molSi s−1

(Fig. 5.3).

Under low nutrient concentrations, diatoms had to compete with other better

adapted phytoplankton groups [Miller 2004]: for instance, silicoflagellates (e.g.,Dictyocha

speculum) competed with diatoms for, in particular, silicic acid, while phototrophic

ciliates, abundant Myrionecta rubra [R Pawlowicz, A Sastri, S E Allen, D Cassis,

O Riche, M Halverson and J F Dower; unpublished data], for nitrate and phos-

phate only [Crawford and Tore 1997, Lagus et al. 2004]. Myrionecta rubra is a ciliate

species known to prey on phytoplankton, to retain them as endosymbionts, and to

sometimes retain them as permanent autotrophic organelles [Stoecker et al. 2009].

A recent study suggests that Myrionecta rubra is particularly well adapted to low

nutrient conditions because it can swim to reach deep nutrients [Lagus et al. 2004].

Spring bloom biomass was dominated by diatoms (Fig. 5.13) adapted to strive

in replenished nutrient conditions [Miller 2004, Sarthou et al. 2005]. Although, on

average, the nutrients were taken up in near N:P Redfield proportion (14.6, Tables 5.1

and 5.2, -796 molN s−1 and -57 molP s−1, respectively), the Si:N (=1.8) and Si:P

(=25) ratios (Table 5.3, -1460 molSi s−1) were higher than expected [Brzezinski 1985].

The surface concentration of silicic acid was steadily decreasing during the spring

bloom (Fig. 5.15). There was an excess drawdown of Si (on average, 1.7 higher than

in summer) and it peaked during spring with values 2 to 3 times larger than the spring

average (e.g. 11.3 gSi gChl-a−1 d−1 in spring 2003, Fig. 5.16). Seasonality was also

170


Apr Jul Oct 2003 Apr Jul Oct 04 Apr Jul Oct 05 Apr Jul

0

5

10

15

20

25

30

35

N

con

cent

ratio

nµM

individual station value at 0 m

stations average at 0 m

SoG 0−30 m average

Figure 5.14: SoG Nitrate Average Concentration at 0 m and Over 0–30 m. The plot

shows the average of SoG nitrate at 0 m and the top box average (0–30 m) for com-

parison. The averages depart from each other during spring and summer when near

surface nitrate tends to be very low (on average 3µM) compared to surface average

concentration (about 11µM). Note that the surface and depth-average phosphate

concentrations (not shown) exhibit similar trends.171


observed when investigating the Si:N and Si:P uptake ratios. In spring (and in fall as

well), the Si:N and Si:P uptake ratios (Fig. 5.17) were higher that the Brzezinski’s

Si:N (=16:16) and Si:P (=16:1) average ratios for healthy diatom cells. Since the N:P

uptake ratio (14.6±0.8):1, remained close to the Redfield ratio (Fig. 5.5), the reason

for higher than expected Si:N and Si:P ratios is not a nutrient limitation. There are

three ways to explain the excess drawdown of Si with respect to N and P during

the spring bloom: abundance of heterotrophic silicoflagellates, abundance of heavily

silicified diatoms, and a neglected abiotic and non-advective sink of dissolved Si.

Heterotrophic silicoflagellates could have been abundant and taken up Si, but not

N and P. Unpublished data from the STRATOGEM program suggests that, indeed,

at least one species of heterotrophic silicoflagellates, Ebria partita, was abundant in

the SoG during springs, e.g. 2004 and 2005 [R Pawlowicz, A Sastri, S E Allen, D

Cassis, O Riche, M Halverson and J F Dower; unpublished data at station S4-1, 5-m

depth]. Other silicoflagellates were present in the SoG, but they bloomed after the

spring bloom, in summer, and were autotrophic species (e.g. Dyctiocha speculum).

Heavily silicified and large diatoms tend to uptake Si faster than N and P. Spring

blooms mainly composed of heavily silicified diatoms could explain the higher than

expected SoG Si:N and Si:P uptake ratios. For instance, a study of Central Equatorial

Pacific phytoplankton community investigated the effects of iron- and Si-addition on

diatoms bloom using mesocosm experiments [Marchetti et al. 2010]. They showed

that Si:N and Si:P uptake ratios could vary manifold above the expected Si:N (16:16)

and Si:P (16:1) ratios [Brzezinski 1985] depending on the concentrations of dissolved

iron and Si. Note that the Si concentration in this mesocosm experiment varies within

5–14 µM, lower than in the SoG (Tables 5.6– 5.7). In turn, they show that this could

affect the species composition of the diatom assemblage and the cell division rate of

diatoms.

172


Apr Jul Oct 2003 Apr Jul Oct 04 Apr Jul Oct 05 Apr Jul0

10

20

30

40

50

60

70µM

Con

cent

ratio

n

Apr Jul Oct 2003 Apr Jul Oct 04 Apr Jul Oct 05 Apr Jul0

10

20

30

40

50

60

70µM

Con

cent

ratio

n

Si 0 m

Si 0−30 m

N 0 m

N 0−30 m

Figure 5.15: SoG N and Si Average Concentrations at 0 m and Over 0–30 m. The plot

show the averages of SoG N and Si concentration at 0 m and their top box averages

(0–30 m) for comparison. Only the Si averages at the surface (0 m) and throughout

the euphotic zone (0–30 m) remain high during summer (maximum about 40µM in

2002).

173



0.5

1

1.5

2

2.5

3

3.5

4

4.5

5S

i Net

Bio

logi

cal U

ptak

e R

ate

× 10

3 mol

s−

1

Figure 5.16: Estimate of the Net Biological Uptake Rate of Si Normalized by Chl-a

(gSi gChl-a−1 d−1). Shaded boxes indicate spring (dark) and summer (light).

174



0.5

1

1.5

2

2.5

3

3.5S

i:N U

pta

ke r

ate

ra

tio

Si:N

Si:P


7.3

14.6

21.9

29.2

36.5

43.8

51.1

Si:P

Up

take

ra

te r

atio

Si:N

Si:P

Figure 5.17: Estimate of the Si:N (lefthand) and Si:P (Righthand) Ratios Based

on the Net Biological Uptake Rates of Si, N and P . Shaded boxes indicate spring.

The dashed line indicates the average ratios (Si:P=24.7, Si:N=1.7). Some of the

fall and winter values are <0. But they are negligible within the uncertainties (see

Figs 5.5–5.6).

175


Other studies of iron- and Si-replete conditions for heavily silicified diatom blooms

have also been carried out at sea, in the Southern Ocean, with naturally-occurring Si-

replete and Si-depleted surface conditions [Coale et al. 2004, Brzezinski et al. 2005].

The experiments in iron-replete conditions are of interest to our study because they

shared some similarities with the surface Si and N concentrations, the Si:N uptake

rate and the abundance of diatoms in the phytoplankton during the SoG spring and

summer blooms. The Si-replete and iron-replete conditions (60–65 µM Si, Table 5.6)

correspond to early spring bloom in the SoG when Si level is the highest (16–57 µM

Si, Table 5.6). The associated N concentration in the experiment also corresponds to

the maximum concentration found in the SoG (26–29 µM and 9–26 µM, respectively,

Table 5.6). In the Si-replete conditions, the bloom was dominated by diatoms as it

happened during SoG spring blooms. The Si:N uptake rate was (2.1±0.5):1 [Coale

et al. 2004] (Table 5.6) and similar to the SoG spring rate of (1.9±0.14):1 (based

on spring rates of Si and N uptake, Figs 5.1–5.3). For both SoG estimate and Coale

et al. [2004]’s estimate, the Si:N uptake rate was larger than expected. Thus, Si:N

uptake ratio of 2 may be normal during spring condition in the SoG.

However, these Si-replete and iron-replete conditions are promoting less silicifica-

tion than the Si-replete and iron-depleted conditions. The Si:N uptake rate decreased

from (8.1±1.5):1 to (2.1±0.5):1 when iron was added. In Si-depleted conditions the

Si:N uptake rate remained around 1 regardless of iron-addition [Coale et al. 2004].

The concentrations in the Si-depleted conditions were somewhat different (0.5–4 µM,

Table 5.7) in Coale et al. [2004]’s experiment from the summer Si-concentrations in

the SoG (25–41 µM, Table 5.7), possibly because, on average, the SoG summer sup-

ply of Si was larger than its sink (Table 5.3). The associated N concentration in

the experiment somewhat corresponds to the maximum SoG concentration (18–22

µM and 6–14 µM, in Coale et al. [2004] and this study respectively, Table 5.7). In

176


addition, in the Si-depleted and iron-replete conditions, the phytoplankton was dom-

inated by non-diatoms in both Coale et al. [2004]’s experiment and the SoG during

summer, the Si:N uptake rate was (0.85±0.36):1 [Coale et al. 2004] (Table 5.7) and

similar to the SoG summer rate of (0.9±0.28):1 (based on summer rates of Si and N

uptake, Fig. 5.1–5.3).

Loss of dissolved Si out of the system might also explain the higher than expected

Si:N and Si:P uptake ratios in the SoG. If there is an another sink of dissolved Si in

Coale et al. [2004] this study

Si-replete bloom spring bloom

Si (µM) 60–65 16–57

N (µM) 26–29 9–26

Si:N uptake ratio (2.1±0.5):1 (1.9±0.2):1

Table 5.6: Comparison of Si-replete and Iron-replete Bloom in Coale et al. [2004] and

SoG Spring Bloom in This Study. Values come from Figs 3 E–H and pp 411–412

in Coale et al. [2004] and Figs 3.8,3.9,5.1 and 5.3 of this study.

Coale et al. [2004] this study

Si-depleted bloom summer bloom

Si (µM) 0–4 25–41

N (µM) 18–22 6–14

Si:N uptake ratio (0.85±0.36):1 (0.9±0.3):1

Table 5.7: Comparison of Si-depleted and Iron-replete Bloom in Coale et al. [2004]

and SoG Summer Bloom in This Study. Values come from Figs 3 E–H and pp

411–412 in Coale et al. [2004] and Figs 3.8,3.9,5.1 and 5.3 of this study.

177


the surface Si budget (Eq. 2.81), it has to be abiotic and non-advective. Since the net

biological and advective sinks have already been taken into account in Eq. 2.81 (see

section 2.4), the excess drawdown of Si would have to be a chemical uptake. If one

considers spring blooms when diatoms dominate phytoplankton biomass, the rate of

Si biological uptake has to be close to the rate of N biological uptake. The difference

between Si and N rates would be the rate of Si chemical uptake (1460-796=664

mol s−1), of the same size as the Si and N biological uptake rates (796 mol s−1) since

our estimated spring Si:N ratio is close to 2:1 (Fig. 5.6). In fall, diatom blooms also

occur with a Si:N ratio still larger than 1:1 but close. To account for the fall difference

by adding a chemical Si uptake, the fall chemical uptake would have to be about 211

mol s−1, that is about a third of the spring chemical uptake (664 mol s−1). Then, one

could assume that the chemical uptake rate should be very small the rest of the year,

in particular in summer when the rates of Si and N biological uptake are very close

to each other either we assume or not that the excess drawdown of Si is a chemical

sink. Thus, the chemical uptake of Si would have to be seasonal with a maximum in

spring as large as the biological uptake of Si. The only chemical sink of dissolved Si

that could be considered is the spontaneous precipitation of dissolved Si into silica,

particulate Si, in the water column. When Si precipitates, it is not accounted to as

dissolved Si anymore and can sink out of the euphotic zone. However, this process is

balanced by dissolution of silica back into dissolved Si. Note that precipitation also

occurs inside the diatom cells, but this has been accounted for as the biological uptake

rate, (φ1)Si in Eq. 2.81. In the water column, the balance between precipitation

and dissolution of Si depends on different factors, the first one is the difference

between the observed and saturation concentrations of Si [Sarmiento and Gruber

2006]. In the SoG, observed concentrations of Si range between 16–62 µM (Fig. 5.15)

on average, with a maximum of 73 µMmeasured on a March 2003 water sample, while

178


theoretical saturation concentration ranges from ∼900 µM up to 1500 µM [Sarmiento

and Gruber 2006]. There is a factor of one order of magnitude, at least, between

the observed and saturation concentrations. This means that spontaneous chemical

precipitation is very unlikely to happen in the SoG anytime, in particular during

spring. In a recent study in Monterey Bay, the estimated dissolution rate is smaller

than the estimated precipitation rate during spring bloom (close to 10%, Brzezinski

et al. [2003b]), because rate estimates include the phytoplanktonic production of

biogenic silica in the total precipitation rate [Brzezinski et al. 2003b]. This strongly

suggests that a large chemical uptake of Si apart from the net biological uptake by

diatoms is very unlikely.

Finally, there is a last explanation, but also very unlikely, for the higher than

expected Si:N and Si:P uptake ratios. Fast remineralization of N and P could occur

during spring bloom and result in low nitrate uptake and high and immediate ammo-

nium uptake by diatoms. Since the Si uptake rate is 1460 mol s−1, this implies that

the remineralization rate of N would be the total uptake rate of N (equal to Si uptake

rate) minus the uptake rate based on nitrate (our estimate): 1460-796=664 mol s−1,

and similarly, using the Brzezinski Si:P proportion (16:1), a remineralization rate of

P, 35 mol s−1. This would imply that the f-ratio is about 0.5, an unlikely value for

remineralization during diatom spring blooms that mainly rely on nitrate not am-

monium. In addition, remineralization of particulate organic N and P from diatoms

would occur once the organic coating on the diatom frustules [Sarmiento and Gruber

2006, Miller 2004] have been scavenged or broken down to allow the particulate or-

ganic N and P to be in contact with the water column. Note that frustules dissolution

can be enhanced by increasing salinity and bacterial scavenging [Roubeix et al. 2008]

as well as diffusion through the frustule pores [Leterme et al. 2010]. Recent estimates

of sinking biogenic Si flux suggest that most of the diatoms from the spring bloom do

179


not remain in the euphotic zone [Johannessen et al. 2005]: e.g., their daily average

of sinking biogenic Si particles varies between 3.6×10−3 mol Si m−2 d−1 and 2×10−2

mol Si m−2 d−1 based on yearly on-site trap measurements below 150 m in central

SoG. My average Si uptake rate have a similar size (Fig. 5.6). This suggests that,

during spring, some of the diatom organic matter, possibly most of it, sinks below

the euphotic and, thus, is not remineralized in the euphotic zone.

The above discussion strongly suggests that Si is decoupled from the other nutri-

ents (P and N) during spring and fall because another biogeochemical process other

than photosynthesis is taking place: e.g., higher rate of silicification, competition

of diatoms and silicoflagellates for Si, and higher rate of N and P remineraliza-

tion. Higher silicification rate is plausible based on previous observations and exper-

iments [Coale et al. 2004, Brzezinski et al. 2005, Marchetti et al. 2010]. Taxonomic

analysis of spring bloom water samples at one STRATOGEM location also suggests

the presence of heterotrophic silicoflagellates, Ebria tripartita. Since the analysis is

based on only one location, although a central station representative of the SoG,

one would require an analysis at additional locations to determine the impact of

heterotrophic silicoflagellates on spring Si uptake. Finally, the qualitative analysis

of the NCP rate estimates (Fig. 5.12) showed that regenerated NPP could be also

at work during spring blooms and winter after fall blooms (Table 5.5). However, it

did not occur every spring and fall. Thus, on average the regenerate NPP is small

(Table 5.5). This is consistent with the quantitative analysis of the NPP rate (based

on P uptake) and the new NPP (based on N uptake), and their comparison that

suggested that on average the regenerated NPP was low (section 5.2.5).

This can be further quantified by using the annual NPP rates based on P and

N (section 5.2.4) and assuming the P:N Redfield ratio of 1:16 instead of the ob-

served ratio of 1:14.6 (Fig. 5.5). Any positive difference between P and N would be

180


due to the unaccounted regenerated N. The NPP rate based on P, the total NPP

rate, is unchanged, 212 gC m−2 yr−1 (section 5.2.4), and the NPP rate based on

N, the new NPP rate, becomes 187 gC m−2 yr−1. This suggests an annual regener-

ated NPP rate of 25 gC m−2 yr−1 about 12% of the NPP rate. This yields to an

average f-ratio of f=0.88. In comparison, a maximum f-ratio of 0.8 was observed in

upwelling regions [Eppley and Peterson 1979] and a minimum f-ratio of 0.7 was sug-

gested in systems where nutrients are mainly supplied by circulation [Legendre et al.

1999]. F-ratios ≥0.9 were observed in a highly productive fjord during diatom spring

bloom [Kristiansen et al. 2001]. Thus, the result found here that SoG diatom bloom

is mainly fuelled by upwelled nutrients is consistent with previous observations in

temperate systems.

The analysis of the near surface O2 sink rate terms shows that the O2 was mainly

supplied by NPP and net upwelling and mainly removed by advective export. In

particular, in spring the system was in quasi-steady state within the uncertainties,

while in summer the O2 source was smaller than the sink leaving a residual (storage

term in Table 5.4) of about -1901±239 mol O2. This suggests a possible decrease

of the net biological production of O2 by a decrease of phytoplankton biomass (by

heterotrophic grazing) and an increase of heterotrophic respiration. The speculative

comparison of the NCP and new NPP rates (section 5.2.5) also suggested that on

average the summer difference NCP-NPPn was negative. Thus, there was an aver-

age summer excess of heterotrophic respiration (relative to regenerated NPP that

uses O2) in the euphotic zone, about 1.8 gO2 m−2 d−1 (Table 5.5), while the rest of

the year the average difference was zero. A comprehensive study of 28 US estuaries,

based on high time-resolution measurements of dissolved O2 over two years, looked

at the net ecosystem metabolic rate, the difference between the gross production

of oxygen by autotrophs and the respiration rate [Caffrey 2003]. They showed that

181


most of these estuaries were heterotrophic, producing less O2 that was respired. In

particular, their average annual rates of O2 respiration were based on night respira-

tion rates and varied from about 4 to 19 g O2 m−2 d−1 [Caffrey 2003]. Although my

O2 excess heterotrophic respiration (1.8 gO2 m−2 d−1) is smaller and the SoG total

heterotrophic respiration is unknown, my respiration estimate is consistent with the

rates found by Caffrey [2003].

182

Chapter 6

Discussion and Conclusion

6.1 Thesis contributions

In this section, I summarize the contributions of this research to the understanding

of the SoG circulation and its primary productivity. First, in chapter 2, a formal

mathematical framework and a time-dependent inverse two-box model of the Strait

of Georgia (SoG) were defined in order to infer the monthly variability of the SoG

circulation. In addition, a careful bootstrap of the input data of the box model [Efron

and Tibshirani 1993] is used to estimate and analyze the uncertainty of the estimated

transports. Although none of the mathematics used in the two-box model is new,

this is one of the few times that this combination of math has been applied to

an estuarine system and use to analyze the seasonal variability of the estuarine

circulation. A careful analysis of the approximations made in the derivation of the

equations of budgets was carried out (2.66–2.71, sections 2.2 and 2.3) and was later

applied to estimate and analyze the uncertainty associated with the data and the

solutions of the inverse problem (SoG circulation transports, Chapter 4) and the

forward problem (sink/source terms in nutrients and O2 budgets, Chapter 5). Note

that a resampling strategy was used to estimate the uncertainty in the physical and

biological estimates.

In chapter 4, transport time series of the SoG estuarine circulation were estimated

over three years with a monthly time resolution, for the first time. Uncertainties were

183

Chapter 6. Discussion and Conclusion

also carefully estimated. These estimates are based on consistency of the transports

with observations of salinity and freshwater (FW) input, temperature and surface

heat fluxes. Using these estimated SoG transports, I provide the first observational

analysis of the relationship between R (FW input) and surface seaward transport U1,

certainly in the SoG and perhaps for any large estuary. The analysis of the transport

time series provides additional novel findings. It suggests that the seasonality of the

total upward transport W2 is very small (on average ±11% of 6.2×104m3s−1, the

mean W2; maximum ±19.3%), even when the seasonality of R is large. Based on

2002–2005 data, the annual seasonality of U1 is weakly linked to R, while the annual

seasonality of other transports is possibly independent fromR. The transport analysis

will be discussed in more detail in section 6.2.

In chapter 5, the seasonal and annual average rates of the net primary production

(NPP rates) based on SoG nutrient budgets over three years are estimated. Errors

are also estimated. Later, in section 6.3, I will compare the estimated NPP rates with

previous estimates in the SoG and in other estuaries and find that the estimated NPP

rates estimated here are typical of NPP rates in temperate estuaries. The analysis

of the nutrient budgets (Chapter 5) showed that, as suggested first by Mackas and

Harrison [1997] for N, the estuarine entrainment is the largest supply of P and Si.

The comparison of estimated NPP rates based on N and P budgets suggests that

the average f-ratio is large (f=0.88, based on annual averages, see section 6.3). These

results will be discussed in more detail in section 6.3. Recommendations for future

work will be given in section 6.4.

184


6.2 Seasonality and Variability of Water

Transports in Estuaries

To my knowledge only a few studies have tried to analyze the seasonal transport

variability of estuarine circulation. One useful feature of the SoG system is the ex-

treme seasonality of the freshet. This is a rare feature among large rivers [WWF

( 2006)]. In the SoG, a few recent studies tried to determine the seasonal variability

of the circulation by estimating seasonal flushing times [England et al. 1996], or sum-

mer/winter water transports [Pawlowicz et al. 2007], or using sophisticated numerical

modelling combined with analysis of observed salinity and temperature [Masson and

Cummins 2004], and seasonal water mass analysis [Masson 2006]. Although rela-

tively simple estimates of estuarine circulation transports are a staple of the “gray

literature”, e.g. Burrard Inlet Environmental Program [1996], published analyses are

less common [Petrie and Yeats 1990]. A study of the exchange transport rate in the

Long Island Sound, a major urban estuary, was more successful at determining the

seasonal variability (winter/summer transport magnitude (1.8–3)×104m3s−1) of the

circulation, but this system is not dominated by the river inflow [Codiga and Aurin

2007].

Although some of the previous studies in the SoG and other estuaries have used

a large volume of data, they tended to focus on the average circulation and ignored

the time variability. Previous studies also relied on long time series of the trans-

ports [Pawlowicz et al. 2007, Austin 2002] and the assumption of steady state [Savenkoff

et al. 2001, Pawlowicz 2001]. Austin [2002] used a long multi-year time series (1985-

2001 and 1992-2001) of salinity of the Chesapeake Bay (CB) in a time-dependent box

model to infer an average exchange rate between the estuary and the ocean. Despite

185


the long multi-year time series used, only an average exchange rate was determined.

Interannual variability was only qualitatively analyzed. Determining the seasonality

of the water transports is yet to be done in CB [Austin 2002]. Savenkoff et al. [2001]

used a quasi-steady state inverse box model of the Gulf of St.Lawrence (GSL) and

both physical and biogeochemical budget equations to estimate the water transports

in and out of important regions of the GSL. In Savenkoff et al. [2001], eight trans-

ports over 4 different depth ranges over July to September were estimated in 32

boxes and required the use of biogeochemical tracers. However, only data from July

to September were used. Thus, these transport estimates are only representative of

late summer-early fall.

Using a 2 box model of the SoG circulation is a useful simplification of the vertical

structure of the circulation. But, this approach leads to several limitations due to the

number of boxes, the averaged properties in each box, and the shape of the boxes.

For instance, the transport in our bottom box combines the transports of intermedi-

ate water and bottom water. Deep water renewals contribute to deep inflow of dense

water into the SoG circulation [LeBlond et al. 1991, Masson 2002]. A recent study

combined 3 boxes, instead of 2 boxes, with a simple mixing-box approach to deter-

mine the transports within 0–50 m, 50–200 m and 200–400 m in the SoG [Pawlowicz

et al. 2007]. The transport within 200–400 m is associated with the deep water re-

newal and it has a summer/winter seasonality ranging from (0–4)×104m3s−1. In the

3-box model, the maximum magnitude of the deep water transport can be as large

as the average intermediate water transport (4×104m3s−1). In contrast, in our 2-box

model, the transports within 30–50 m, 50–200 m and 200–400 m are combined as

a weighted average, the bottom transport, U2. Given the size of our error bars and

the magnitude of seasonality of U2, the small seasonality of U2 does not provide any

useful information about the bottom water renewals. On the other hand, the rela-

186


tively constant magnitude of U2 suggests a very small variability of the combined

intermediate and deep water renewal.

S1 and S2 represent the box averages of salinity in the top and bottom boxes in

both the Knudsen’s relationship (Eq. 6.1) and in our equations (Eqs 2.66–2.71). In

both cases, the box averages appear instead of the averages just above or below the

separation boundary between the two boxes. This is a simplification of the equations

and the calculations of the transports (section 2.2). This is an important problem

when the top box represents two or more water masses, for instance in the river plume

during the Freshet: the high boundary salinity is approximated by the average of

the high boundary salinity and the low surface salinity. Regardless of the tracer

considered, the corresponding approximation error on the tracer is proportional to

the difference between the box average and the boundary average. In the case of

points close and below the separation boundary, the vertical gradient of the tracers

tend to be small, while in the case of points close and above the separation boundary

the vertical gradient is large (Fig. 3.1).

In addition, there are two issues with my representation of the SoG with two

domains within 0–30 m and 30–400 m. First, the data only cover the southern part

of the SoG, but it was assumed they were still representative of the whole. Other

recent sampling surveys over the whole SoG suggests that this is a reasonable as-

sumption [Masson 2006, Masson and Pena 2009]. Comparison of the STRATOGEM

temperature, salinity and nitrate box averages with box averages using their data

also suggests that this is a reasonable assumption (Figs 6.1–6.3). Fig. 6.1 shows that

average surface and bottom temperatures over the Southern SoG and over the whole

SoG are very similar. Fig. 6.2 suggests similar salinity seasonality but higher bottom

salinity in the STRATOGEM data. Fig. 6.3 suggests similar surface nitrate seasonal-

ity but more variable summer concentration during the freshet. Summer deep nitrate

187


levels are similar, but no obvious trend can be seen.

Secondly, we are averaging the properties of water at different locations. Thus,

the averaging of the surface properties ignore the spatial variability of the surface

water masses. There are three primary surface masses of water in the SoG that

have different properties: the freshwater plume, sinking dense water at the Southern

Entrance, and SoG surface mixed water everywhere else (see Chapter 1). This could

be further investigated by analyzing seasonal TS properties of these water masses.

For instance, it is not clear if the influence of the riverine input of freshwater and

silicic acid should be extended to the whole SoG, although the SoG top box averages

of the salinity and the silicic acid concentration are strongly influenced by the riverine

input (Fig. 3.3a and Fig. 3.9a) [Harrison et al. 1991].

Although we take into account the SoG bathymetry to determine the box aver-

ages (section 3.3.2), our estimates of the transports represent the overall inflow and

outflow. When comparing the SoG with another system, one can take into account

any narrowing or broadening of the cross-section along an estuary to improve the

accuracy of the estimated transports. For instance, with the more sophisticated ap-

proach of a tapered box model, one could estimate the transports upstream of the

mouth in a chosen segment of an estuary and compare them with estimated trans-

ports based on current meter measurements. Tapered boxes have been successfully

used in estuary box models to provide a better estimation of the estuarine circulation

and help compare different systems [Gay and O’Donnell 2009].

The freshwater (FW) input from rivers is the main forcing of estuarine circulation.

The FW input (R) in the SoG estuary has a large seasonal change compared to its

average value, on average 58% of mean R (Fig. 6.5). The main source of FW is

the Fraser River, one of the last large (longer than 1000 km) free-flowing rivers in

the world [WWF ( 2006), Figs 3–4 and Appendix 1]. One might then expect that

188



7

8

9

10

11

12

13

14

Tem

pera

ture

( ° C

)


8

8.5

9

9.5

10

10.5

Tem

pera

ture

( ° C

)

STRATOGEM 0−30 m

IOS 0−30 m

STRATOGEM 30−400 m

IOS 30−400 m

Figure 6.1: Comparison of STRATOGEM and IOS SoG Temperature Box Averages

189



25

25.5

26

26.5

27

27.5

28

28.5

29

29.5

30

Sal

inity

(ps

u)


30

30.5

31

Sal

inity

(ps

u)

STRATOGEM 0−30 mIOS 0−30 m


Figure 6.2: Comparison of STRATOGEM and IOS SoG Salinity Box Averages

190



5

10

15

20

25

30

35

Nitr

ite+N

itrat

e µM


5

10

15

20

25

30

35

Nitr

ite+N

itrat

e µM



Figure 6.3: Comparison of STRATOGEM and IOS SoG Nitrate Box Averages

191


this large seasonal change in FW input will lead to similar large seasonal change in

the water transports of the SoG. Knudsen’s relationship (a simplified box model)

suggests that in quasi-steady state the circulation magnitude is proportional to R

assuming that the salinity field is roughly constant:

U1 =S2

S2 − S1

R. (6.1)

However, in Fig. 4.1 the surface seaward transport U1 exhibits only a small seasonal

change compared to its mean over 38 months, on average 14% of U1, although high

outflow can be reached during certain years (for instance 2002, 6×104 m3 s−1). This

expectation is clearly incorrect. The seasonal change mostly affects the salinity field.

In section 4.3.5, it is shown that the relationship U1=f(R) resulting from the

inverse box model (Fig. 4.9) is compatible with theoretical relationships developed

by Hetland and Geyer [2004], MacCready and Geyer [2010]. In Fig. 4.9, the theoreti-

cal curve and the estimated U1(R) curve both suggest that sensitivity of the estuarine

circulation to changes of freshwater flow is low on a seasonal timescale. Recent ad-

vances in the theory of the exchange flow in partially mixed estuaries [MacCready

and Geyer 2010] suggest that on a seasonal timescale estuarine circulations are sensi-

tive to the river inflow as Ra (with a ≪ 1). The exponent a tends to be small for long

estuaries. Thus, the theory suggests that the sensitivity of circulation to freshwater

inflow is high only with very small values of R. The results of chapter 4 suggest that

this increased sensitivity could only happen in the SoG with R inflows smaller than

2×103 m3s−1. In contrast, all of our R values are in the range 2×103–104 m3s−1. In

the SoG, very low values are very unlikely. According to the 1912–2008 historical

data (Fig. 6.4), very few values are lower than 2×103m3s−1 (only 4 monthly means

out of 1162 range within 1.9×103–2×103m3s−1). This value is not significantly differ-

ent from the minimum observed over 2002–2005. That is, the observed winter inflows

192


R used in our box model were already typical minimal values (Fig. 6.4).

While U1 is sensitive to the seasonal changes of R according to Ra (with a ≪ 1),

the analysis of similar relationships between U2 and R, and W2 and R suggests that

either the sensitivity of U2 and W2 to R is very small or that there is no sensitivity at

all. This can readily be noticed in Fig. 4.1. The seasonality magnitude of U2 and W2

is equal or smaller than the seasonality magnitude of U1 despite the large seasonal

changes of the FW forcing.

How then does the estuary responds to changes in R? Eq. 6.1 suggests an increase

in R is nearly balanced by an increase in stratification near the surface. For instance,

Fig. 3.3 shows that S2 changes by less than 0.25 from April to August in 2002 and

2003, while S1 can vary by 3 in 2002 and by 1.5 in 2003, closely following the changes

of R during the freshet in 2002 and 2003.

In our box model, the estimated variability of the vertical mixing M is large,

on average 44% of the mean with maxima during the freshet and winter, but this

is partly due to large errors in the estimation (Table 4.1). The monthly sampling

period used in our box model is too large to capture the effect of spring-neap tidal

effect on M . So, the tidal effect on the vertical mixing over the sampling period

cannot be observed. The changes in M must be explained by another mechanism.

Vertical mixing can be reduced by an increase in stratification [MacCready and Geyer

2010]. In particular, during the freshet peak, one expects to see a maximal effect of

the stratification on the vertical mixing because the stratification increase with R. In

Eq. 2.57, the fluxes of correlated variations were neglected. If there is any vertical flux

of correlated variations over a monthly time period, neglecting this term increases

the error associated with the vertical transports W1 and W2 in Eq. 2.60.

Li et al. [1999]’s numerical box model of the SoG also showed a small seasonal

variability of the SoG estuarine circulation (see Chapter 4, section 4.3.1) with a

193


0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

x 104

0

1

2

3

4

5

6

7

8%

of 1

162

FW

mon

thly

ave

rage

s

Bins are centered and of size 200 m3s−1

Figure 6.4: Histogram of FW Input in the SoG over 1912-2008

parametrization of vertical mixing [Li et al. 1999, Equations (2) and (3)] depending

on stratification. The vertical mixing exchange was set to be high at spring tides, low

at neap tides and lowest during the freshet. The order of magnitude of the vertical

mixing in Li et al. [1999] (pp 10–11, MLi99 = ωgAg = 104m3s−1) is similar to mine

(on average 1.9×104m3s−1). In addition, the timing of this lowest vertical mixing at

194


the freshet in Li et al. [1999] is similar to the timing of the low values of M during

the freshet in our box model. In section 4.3.1 (comparison between our transport U1

with Li et al. [1999]’s) the similarity of the seasonal changes between the two sets

of results suggests that reduced vertical mixing [MacCready and Geyer 2010] during

the freshet may be diagnosed in our box model.

Despite seasonal adjustments to the FW forcing, relatively steady estuarine cir-

culation suggests a small sensitivity of the seasonal changes of estuarine circula-

tion to the FW inflow changes (Fig. 4.9): e.g. maximum increase of 33% above

mean U1 (4.6×104m3s−1) in summer 2002 for a peak of R of about 3 times mean R

(5.3×103m3s−1), maximum decrease of 11% below mean U1 in winter 2003 for the

lowest R, about 36% of mean R. The changes of R observed during 2002–2005 are

representative of the changes of R recorded between 1912 and 2008. So these limits

should hold in general. An important implication of this relatively steady estuarine

circulation is that nutrients are upwelled relatively constantly from depth to near

surface. This is shown by the analysis of section 5.2.1 where it was found that trans-

ports of upwelled nutrients and advected nutrients have a small seasonal variability

(Tables 5.1–5.3).

6.3 Estimates and Variability of NPP Rates in

Estuaries

Table 6.1 shows the range of PP rates for a large number of temperate estuaries in

North America and Europe [Wilson 2002, Heip et al. 1995, Therriault and Levasseur

1985], including the NPP rates from our study. Since there are many different studies,

the estimates correspond to a variety of types of measurement of PP rates. However,

195


Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec 0

0.2

0.4

0.6

0.8

1

1.2

1.4

FW

dis

char

ge (

×104 m

3 s−

1 )

Figure 6.5: Mean Monthly FW Input in the SoG from 97 years of data for Fraser

River scaled as described in section 3.2.3. Vertical bars are uncertainty of the means.

196


Reference Annual euphotic Time Period Location

PP rate gC m−2yr−1

This study net 212 2002-2005 SoG

(spring/summer 157)

Heip et al. [1995]7-560 1977-1995 12 estuaries in Eu-

rope

and North America

Therriault and Levasseur

[1985]

7-470 1974-1984 20 estuaries

in North America

Wilson [2002]0-192 1981-2000 5 UK and US

estuaries

Harding et al. [2002]mean 408 range 282-538 1982-1995 CB

Tian et al. [2000]160 1992-1994 GSL model

estimates range 100–212

Savenkoff et al. [2000]range 0-500, max in spring 1992-1994 GSL (Fig. 2 in ref.)

Boyer et al. [1993]mean 465 range 395-493 1985-1988 Neuse River

Pennock and Sharp

[1986]

mean 307 range 190-400 1981-1985 Delaware

90-820 1950s-80s 10 US estuaries

(820 Hudson) Table 1 in ref.

Testa and Kemp [2008]net 0-116 1985-2008 Patuxent River

assuming net uptake rates of P and N

Fig. 7 in ref. and Redfield ratios

Table 6.1: Ranges of PP Rates in Temperate Estuaries.

197


most of them are based on techniques of 14C uptake in incubation. In general, most

of these estimates would be somewhere between gross and net PP rates [Marra

and Barber 2004, Williams and Lefevre 2008]. In some cases, this is known more

precisely [Harding et al. 2002].

In Table 6.1, most estimates have numerical values of several hundreds of gC

m−2yr−1. The average NPP rate of the SoG is 212 gC m−2yr−1 (based on P uptake

rate). The biological analysis shows that the f-ratio could range between 0.5 and

0.88. The former comes from the analysis in discussion section 5.3, while the latter is

based on annual NPP rate of 212 gC m−2yr−1 and annual new PP of 187 gC m−2yr−1.

However, comparison of P and N uptake rates suggests that the f-ratio leans towards

values&0.8, and thus, the total NPP rate overall tends to be greater but close to 212

gC m−2yr−1.

The SoG total NPP rate overall of 212 gC m−2yr−1 sets the NPP rate of the SoG

as average and typical when compared to other NPP rates in Table 6.1. The most

productive estuary in Table 6.1, Hudson estuary [Pennock and Sharp 1986] has a

P N Si

Macronutrient Range (µM) 0.5–1.25 6–14 25–41

Redfield and Brzezinski factors 1 16 16

Normalized Range 0.5–1.3 0.4–0.9 1.6–2.6

Table 6.2: Macronutrient Limitation During Summer in the SoG. The first row

shows SoG surface range of the 3 macronutrient (Phosphate (P), nitrate (N), silicic

(Si) acid) concentrations during Summer (using Figs 3.7–3.9). Second row shows

the Redfield and Brzezinski ratios. Third row shows the ratio between the first and

second rows. The limiting nutrient has the lowest value on the third row.

198


primary productivity (820 gC m−2yr−1) that is 4 times larger than the primary pro-

ductivity of the SoG. The primary productivity of Hudson estuary is high because its

primary productivity is partly fuelled by anthropogenic nutrient waste inputs [Fisher

et al. 1988] and the estimate is based on GPP rate [Howarth et al. 2006, Hudson

estuary, 850 gC m−2yr−1 GPP rate over 1990s in the most productive area of the

estuary] (see Eq. 5.6). Since anthropogenic inputs to the SoG are negligable [Mackas

and Harrison 1997], during summer nutrient limitation is probably the most impor-

tant factor in controlling primary productivity in the SoG. Table 6.2 suggests that

nitrate limitration is probably the most critical of the 3 macronutrients.

Generally in temperate estuaries, a diatom bloom occurs in spring [Heip et al.

1995, Miller 2004] (seasons defined in Glossary p. xiii). In most of the estuaries of

Table 6.1, blooms can occur not only during spring but also during summer and

fall: e.g., GSL, CB, Delaware, Patuxent River estuaries [Heip et al. 1995, see Table

2]. Diatoms strive when light and nutrients are both plentiful [Sarthou et al. 2005,

Aiken et al. 2008]. Thus, in deep estuaries, spring conditions are ideal for diatoms

to grow. Surface nutrients have been replenished during winter and light level is

increasing. In shallow estuaries, the increase of turbidity, due to sediments or self-

shadowing of phytoplankton, can lower light level and delay the large bloom until

summer. In shallow estuaries, a large summer bloom can be fueled by an increase of

nutrients from rivers or regeneration of spring bloom nutrients [Harding et al. 2002,

Testa and Kemp 2008], combined with the increased summer light level. In the SoG,

conditions are favourable to diatom spring blooms: diatoms dominate the biomass

of autotrophic phytoplankton (Fig. 5.13), and nutrients are plentiful (Figs 3.7–3.9).

Fig. 5.8 shows that the largest chla biomass occurs in spring: 0.3 gChl-a m−2 in

April 2002, 0.16 gChl-a m−2 in April 2003, 0.37 gChl-a m−2 in March 2004, 0.3 gChl-

a m−2 in March 2005. They are all associated with the largest annual total NPP rate

199


(>1.2 gC m−2d−1 based on P uptake).

The NPP rate represents the production of organic carbon by photosynthesiz-

ers; a seasonal maximum NPP rate will probably be associated with a high biomass

(large number of photosynthesizers) while a low biomass (small number of photosyn-

thesizers) can still produce organic carbon but not necessarily at a high rate [Miller

2004]. Thus, there maybe a correlation between the NPP rate and the phytoplankton

biomass [Miller 2004, see Fig. 1.1]. Using our estimate of the NPP rate (Fig. 5.8),

the coefficient of correlation between new NPP rates and integrated chla biomass (a

proxy for C biomass, assuming C:Chl-a does not very greatly) is 0.65 with a 95%

confidence interval ranging between 0.44 and 0.79. Many mathematical models of

daily and annual PP rates also assume a proportionality relationship between pro-

duction and productivity [Boyer et al. 1993, Harding et al. 2002]. In agreement with

low winter biomass and in contrast with summer high productivity, winter produc-

tivity is low: winter productivity is 0.1 gC m−1d−1 at most (while summer average

is 1 gC m−1d−1), but the winter uncertainty is of the same order of magnitude

and the uncertainty suggests that there is no winter productivity on average (using

C:P:N=106:14.6:1 and 4th column in Tables 5.1–5.3).

The maximum magnitude of the NPP rate can vary from year to year. In other es-

tuaries, studies interested in interannual variability suggest that it is associated with

variability in the nutrients supply. In CB, the interannual variability is large and de-

pends on the river input of nutrients: 282–538 gC m−2yr−1 over 1989-1998 [Harding

et al. 2002]. On the other hand, in Patuxent River estuary the interannual variabil-

ity depends on atmospheric deposition of DIN (dissolved inorganic N) and wet years

tend to have higher NPP rates than dry years [Testa and Kemp 2008]. Since phyto-

plankton blooms depend on light and nutrient availability, the more variable of the

two, nutrient availability, is usually thought as the more likely to explain the inter-

200


annual variability of the NPP rate. According to our results, in the SoG there was no

large interannual change over 2002–2005. This is not surprising since we found that

the net entrainment of nutrients has small variability over seasons and over years.

Our estimates of the SoG NPP rate are based on new and total NPP rates. The vari-

ability of the regenerated NPP rate might have a non-negligible contribution to the

total NPP rate and its interannual variability. However, in the SoG the regenerated

NPP rate is not well quantified and the timing of the maximum regenerated NPP

rate suggests that regeneration might not be seasonal, but only episodic (Fig. 5.12).

For instance, our estimate of the f-ratio varies between 0.5-0.9, although our average

estimate suggests a high f-ratio (average f=0.88) based on estimates of the total and

new NPP rates (section 5.3). Such high f-ratios have been previously observed in

similar systems during diatom blooms using different measurement methods: e.g.,

high productive fjord (≥0.8 and up to 0.96) [Kristiansen et al. 2001] and in Monterey

Bay (average f-ratio=0.83) [Brzezinski et al. 2003b].

6.4 Recommendations for Future Studies

Review and discussion of the results in the previous chapters suggest that several

open questions in the analysis of the mass transports and primary productivity still

remain. These open questions need to be addressed in future studies. Here these

questions are discussed and recommendations for future work are made.

The physical analysis of the water transports failed to show the U1=f(R) rela-

tionship at low and high FW inflows (Fig. 4.9), where a match or mismatch with the

theoretical U1=aR1/m (1/m ≪ 1) might be more obvious. Unfortunately, obtaining

more data and studying further the estuarine circulation in the SoG will not pro-

vide the additional needed information. Indeed, a closer look at the history of Fraser

201


River daily discharge (Environment-Canada, see source detail in section 3.2.3) sug-

gests that R <2×103m3s−1 only 0.25% of the time over 1912-2005, with a minimum

of 1.73×103m3s−1, and R > 2×104 m3s−1 only 0.13% of the time, with a maximum

of 2.64×104m3s−1 (Fig. 6.4). Thus, the 2002-2005 data combined with the regular

Fraser River cycle has already provided enough information to fully determine the

variability of the FW forcing and, through our box model approach, the monthly

estuarine circulation.

Thus, it would be necessary to study other estuaries to gain further knowledge on

the U1=f(R) relationship. There are three conditions that estuaries have to satisfy if

one wants to be able to apply the theoretical U1=f(R) relationship, U1=aR1/m [Het-

land and Geyer 2004, p. 2689]:

1. Salt budget is dominated by the estuarine circulation.

2. Estuarine circulation is larger than FW flow: mass budget is dominated by

estuarine circulation.

3. Estuary domain lies within an approximately rectangular and prismatic chan-

nel.

Most estuaries are likely to satisfy the first two conditions on a monthly timescale.

If one investigates another system, it is recommended to choose an estuary in

which ranges of estimated transports would cover several orders of magnitude: e.g.

in the case of the SoG, 103 m3s−1, 104 m3s−1 and 105 m3s−1 would have been more

useful than only 104 m3s−1. A range with two observable orders of magnitude for

FW discharge is unusual, but perhaps not completely unreasonable, since the range

is at least one order of magnitude for many large rivers: for instance, the discharge

of the Fraser (0.2×104m3s−1–2×104m3s−1), Seine (0.03×104m3s−1–0.25×104m3s−1),

202


and Columbia (0.3×104m3s−1–2×104m3s−1) rivers [Pawlowicz et al. 2007, Huang

et al. 2009, Hughes and Rattray 1980].

In the physical analysis, the next open question concerns the average transports

due to deep water renewals (DWR) and their variability in the SoG. The physical

analysis (sections 4.2.1 and 6.2) emphasized that a 2-box model had provided little

information on the deepwater renewals apart from the average transport magnitude,

although this information was present in the data [Pawlowicz et al. 2007]. A 3-box

model could provide the information on the seasonal variability of the transports

associated with DWR. This would involve a reanalysis of the SoG data with a modi-

fied box-model layout. The SoG would be divided into top, deep, and bottom boxes.

New transports of water and tracers, and storage rates would be introduced in the

budget equations. As for the 2-box model, a critical choice is to decide the separation

depths between the boxes. There is no a priori reason to change the separation depth

between the top and deep boxes (d) beyond the choices made here in Chapter 4. On

the other hand, the separation between the deep and the bottom boxes (D) could

be based on the shallowest depth where the SoG water renewals can occur. Mass

transports between boxes may (or may not) be relatively insensitive to changes of

the separation depth between the deep and the bottom boxes. In the case that one or

more transports were to become markedly sensitive to one or both of the separation

depths (d and D), a remedy could be to use a discretization of the water column:

use a larger number of boxes (>3) limited by the size of the bins of the CTD vertical

profiles (section 3.2.1). Such a discretization would have a secondary benefit. In the

physical analysis (section 6.2), we discussed the approximation of the boundary av-

erage of the tracers by the volume average. In the limit of a large number of boxes,

the difference between the volume average and the boundary average would tend to

be negligible.

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Another unknown in the physical analysis is the relative size of the transports at

the Northern Passage compared to the ones at the Southern Passages. The only esti-

mates of the Northern Passage transports based on observations date back to Godin

et al. [1981] (cross-channel transect in Johnstone Strait, or JS). No recent estimates

based on observations were available to be compared with my estimates of the SoG

transports. All of the analysis is based on observations collected in the central and

southern parts of the SoG (Figs 1.1 and 1.2). My estimated transports cannot be

clearly associated with either the Southern Passage or the whole SoG. I chose to

assume that they were the transports at the Southern Passage based on the as-

sumption, found in the literature, that the transports at the Northern Passage are

about 7% of the transports at the Southern Passage. This assumption is based on

the comparison of the area of the cross-channel sections at the Southern and North-

ern Passages, assuming similar current speed (section 1.1). An examination of the

estimates from Godin et al. [1981] suggests that the ratio of transport magnitudes

between the Northern Passage (average transport about 3×104m3s−1) and the South-

ern Passage (average transport 105m3s−1) is 30%, but they considered transects in

the SoJdF (Southern Entrance) and in JS (Northern Entrance, farther from Sey-

mour Narrows) instead of exiting the SoG itself. The SoJdF is separated from the

SoG by the HS. Entrainment along the straits results in the amplification of the mass

transports in each layer in the downstrait direction. Thus, the layer transports at

the Southern Passage cannot be approximated by the SoJdF transports. Similarly,

transports at Seymour Narrows cannot be approximated by those in JS. Transport

magnitudes and their variability in both Northern and Southern Passages could be

estimated using our inverse box modelling approach, but it would require additional

data from the Northern end of Vancouver Island.

The last unknown in the physical analysis concerns the average magnitude of

204


transports and their variability over short timescales (< 1 month). In the physical

analysis (section 4.2.1), we noticed large oscillations with 2-month period (when

biweekly sampling was possible during March-April 2003 and 2004). The magnitude

of these oscillations is very large compared to the average magnitude of the transports

over spring, in particular for transport M . One possible explanation for these large

oscillations is that they could be due to shorter timescale (< 1 month) processes like

spring-neap tides [Li et al. 1999] and episodic wind-driven surface mixing [Collins

et al. 2009]. Thus, it would be useful to gather data on a shorter timescale than

the monthly timescale used in our study. Since a larger number of cruises would be

carried out, a smaller number of stations could be sampled.

The biological analysis of the Net Primary Productivity (NPP rate) suggests that

the spring and summer blooms have a stable seasonal pattern over a long timescale:

in spring NPP rate and biomass both reach their annual peak, in summer NPP rate

and biomass levels are lower than spring, but they remain high over a longer interval

than in spring. However both spring and summer NPP rates have a roughly equal

contribution to the annual NPP rate. How stable is this pattern? Collecting data

over a larger timescale (≫ 3 years) would help to answer this question. The NPP

rates could be estimated with a forward box model approach (section 2.4) based on

the estuarine circulation using the relationship U1=f(R) found here (Fig. 4.9) with

additional observations of the nutrient levels. Recently the Victoria Experimental

Network Under the Sea (VENUS) observatory has begun providing near real-time

and archived data at different discrete depths within the southern SoG [Dewey et al.

2010]. One of the objectives of the observatory is to provide long-term observations

of the SoG physical environment and ecosystem at various timescales. In the near

future, some of the VENUS sites may be equipped with automatic vertical profilers

to collect various physical and biogeochemical properties at mid-water. Ferries are

205


to be fitted with sensors to sample the surface of the SoG [Dewey et al. 2010], as has

been already done during STRATOGEM [Halverson and Pawlowicz 2008]. All these

data could be used to build long term vertical profiles of the biogeochemical tracers

necessary for a long timescale study of the SoG NPP rate and NPP.

The biological analysis suggested that there was a difference between spring and

summer in NPP rate, biomass (Fig 5.8), taxonomy (Fig. 5.13), Si:N and Si:P ratios of

uptake rates (Fig. 5.17). In particular, the Si:N and Si:P ratios were above the average

ratios (16:16 and 16:1) expected in coastal waters [Brzezinski 1985, Brzezinski et al.

2003a]. Could these differences be explained by different species of diatoms in the

PP: driven in spring by heavily silicified diatoms and in summer by less silicified

diatoms?

Recent studies of co-enrichment with silicic acid and iron, using central Pacific

Equatorial water in mesocosms and seeding of phytoplankton patch in the Southern

Ocean, suggested that different concentration levels of dissolved silicic acid could

heavily favour or disadvantage silicified diatoms depending on the Si levels and drive

the Si:N and Si:P ratios above the 16:16 and 16:1 expected ratios [Coale et al. 2004,

Marchetti et al. 2010]. Note that recent estimates of the biogenic silica content of

sinking particles in the SoG aphotic zone were consistent with high Si uptake during

spring blooms [Johannessen et al. 2005]. It is not clear if the high Si:P and Si:N in

the SoG is due to only one cause. First, the Si:P ratio is not as high as expected

when there is a high demand of Si by diatoms as suggested by the experiments of

co-enrichment with silicic acid and iron. Secondly, the possible spring competition

between diatoms and heterotrophic silicoflagellates for Si, suggested above, is based

on the taxonomy analysis of only one STRATOGEM station although a central loca-

tion and the representative of the SoG. At this point, the combined effect of these two

Si sinks cannot be ignored. Thus, I recommend a reanalysis of the STRATOGEM

206


taxonomy data looking for differences in diatoms composition and silicoflagellates

abundance, and a reanalysis of the size-fractioning data of extracted chlorophyll-a.

This reanalysis could provide a preliminary study. Secondly, an extensive study sim-

ilar in scope to Marchetti et al. [2010], but with SoG water could be carried out

to determine precise uptake rates, taxonomic and chemical compositions of phyto-

plankton.

In the biological analysis, using the estimates of the new and total NPP rates,

we attempted to estimate the f-ratio, the relative importance of external supply

of N (assuming negligible atmospheric N2 fixation) relatively to the total primary

productivity. Low f-ratios (.0.5) were found in previous studies of SoG biological

oceanography [Harrison et al. 1983] indicating that the internal supply, e.g. regen-

eration of surface organic N, primarily contributes to the supply of N. However, our

biological analysis suggested a higher average f-ratio, e.g. annual average of 0.88 over

3 years (section 5.2.4), with possible events of low f-ratios (section 5.2.5). The results

of these studies are not incompatible. High f-ratios (f>0.7) are expected in highly

productive ecosystems where the primary supply of nutrients is external [Eppley

and Peterson 1979, Legendre et al. 1999]. During coastal diatom blooms, succession

of high and low f-ratio events had been previously associated with the succession

of blooming diatoms and collapsing diatom blooms due to nutrient depletion [Kris-

tiansen et al. 2001]. Kristiansen et al. [2001] noted a clear transition between PP

based on external supply and PP based on nutrient regeneration occurring with the

depletion of available nutrients. Thus, we recommend that future studies in the SoG

include not only nitrate+nitrite and phosphate measurements, but also ammonium,

to directly estimate the regenerated NPP (see Chapter 5). Additional short-term

15N incubation experiments uptakes have been used to measure the nutrient uptakes

in previous studies [Kristiansen et al. 2001, Elskens et al. 2008] and these isotope

207


uptake rates could provide independent estimates of the nutrient uptakes rates.

In the biological analysis, we focused on the surface biogeochemical processes as

they provide information about spring and summer blooms, the NPP and the new

NPP rate. We did not analyze the rest of the SoG water column, below 30 m down to

400 m. Thus, one of the questions left open by our biological analysis is whether there

is a net biogeochemical production or consumption of nutrients and dissolved O2 in

the aphotic zone of the SoG, and if we can identify the corresponding biogeochemical

transformations. If there are deep biogeochemical processes, do they interact with

the surface biogeochemical processes? A similar box-model approach has been used

in a partially-stratified estuary to address this question [Testa and Kemp 2008].

Unfortunately, the quality of the nutrients data was not very high. The quality

of this data could be improved by updating the nutrients protocol and using small

batches of fresh nutrient samples for the analysis, instead of large batches of frozen

samples. However, this would make the sampling and analysis logistically more com-

plex. I estimated and compared the differences of nutrient concentrations between

replicates of fresh and frozen samples analyzed within a week, and frozen samples

analyzed within a few months. Nutrient concentrations tend to be more accurate

when samples were analyzed within a week, in agreement with Barwell-Clarke and

Whitney [1996].

In the biological analysis, I estimated the rate of the primary productivity which

is essential, as a food source, to the rest of the SoG ecosystem and adjacent systems.

One of the next open issues is now the export production due to SoG phytoplankton

and what are its downward (contribution to pelagic and benthic ecosystems) and

seaward (possible contribution to adjacent systems) components? Is the SoG estuary

a net autotrophic or a net heterotrophic system?

To carry out such an analysis of the transport of organic carbon, I recommend

208


collecting data on dissolved and particulate organic carbon (DOC and POC) in

addition to the nutrients and dissolved O2 in both throughout the water column in

the SoG and outside of the SoG (HS, SoJdF, JS). Collecting data on size-fractionated

biogenic silica could provide a way to separate the contribution of diatoms from

other phytoplankton species and estimate the contribution of the silica pump to the

exported production [Brzezinski et al. 2003b].

As introduced in the first chapter, the motivation of this research was to study

the SoG ecosystem condition at the lowest level of the foodweb, the primary pro-

duction, and the related parameters, the seasonal average of the NPP rate and its

seasonal variability. The next step would be to couple the NPP study with the study

of the next level in the foodweb, the Net Secondary Production (NSP), and the re-

lated parameters, the surface and deep net secondary productivity and their seasonal

variability. This would enable one to embrace the SoG ecosystem condition over a

larger scope [R Pawlowicz, A Sastri, S E Allen, D Cassis, O Riche, M Halverson and

J F Dower; R Pawlowicz, Susan E Allen and M Halverson; unpublished works].

209

Bibliography

Aiken, J., Hardman-Mountford, N. J., Barlow, R., Fishwick, J. and Hi-

rata, T. (2008). Functional links between bioenergetics and bio-optical traits of

phytoplankton taxonomics groups: an overarching hypothesis with applications for

ocean color remote sensing. Journal of Phytoplankton Research 30, 165–181.

Anderson, L. A. (1995). On the hydrogen and oxygen content of marine phyto-

plankton. Deep Sea Research Part I: Oceanographic Research Papers 42, 1675–1680.

Arrigo, K. R., DiTullio, G. R., Dunbar, R. B., Robinson, D. H., VanWo-

ert, M., Worthen, D. L. and Lizotte, M. P. (2000). Phytoplankton taxonomic

variability in nutrient utilization and primary production in the Ross Sea. Journal

of Geophysical Research 105, 8827–8846.

Arrigo, K. R., Robinson, D. H., Worthen, D. L., Dunbar, R. B., DiTullio,

G. R., VanWoert, M. and Lizotte, M. P. (1999). Phytoplankton community

structure and the drawdown of nutrients and CO2 in the southern ocean. Science

283, 365–367.

Austin, J. A. (2002). Estimating the mean ocean-bay exchange rate of the Chesa-

peake Bay. Journal of Geophysical Research 107, 131–138.

Baker, P. and Pond, S. (1995). The low-frequency residual circulation in Knight

Inlet, British Columbia. Journal of Physical Oceanography 25, 747–763.

210

Bibliography

Barwell-Clarke, J. and Whitney, F. (1996). Institute of Ocean Sciences nutri-

ent methods and analysis. Canadian Technical Report of Hydrography and Ocean

Sciences 182, 1–43.

Boyer, J. N., Christian, R. R. and Stanley, D. W. (1993). Patterns of phy-

toplankton primary production in the Neuse river estuary, North Carolina. Marine

Ecology Progress Series 97, 287–297.

Brzezinski, M. (1985). The Si:C:N ratio of marine diatoms: Interspecific vari-

ability and the effect of some environmental variables. Journal of Phycology 21,

347–357.

Brzezinski, M. A., Dickson, M. L., Nelson, D. M. and Sambrotto, R.

(2003a). Ratios of Si, C and N uptake by microplankton in the Southern Ocean.

Deep-Sea Research II 50, 619—-633.

Brzezinski, M. A., Jones, J. L., Bidle, K. D. and Azam, F. (2003b). The

balance between silica production and silica dissolution in the sea: Insights from

Monterey Bay, California, applied to the global data set. Limnology and Oceanog-

raphy 48, 1846—-1854.

Brzezinski, M. A., Jones, J. L. and Demarest, M. S. (2005). Control of silica

production by iron and silicic acid during the Southern Ocean Iron Experiment

(SOFeX). Limnology and Oceanography 50, 810–824.

Bugden, G. L. (1981). Salt and heat budgets for the Gulf of St. Lawrence.

Canadian Journal of Fisheries and Aquatic Sciences 38.

Caffrey, J. (2003). Production, respiration and net ecosystem metabolism in U.S.

estuaries. Environmental Monitoring and Assessment 81, 207–219.

211

Bibliography

Cameron, W. M. and Pritchard, D. W. (1963). Estuaries in M. N. Hill

(editor): The Sea, volume 2. John Wiley and Sons, New York.

Campbell, R. W., Boutillier, P. and Dower, J. F. (2004). Ecophysiology of

overwintering in the copepod Neocalanus plumchrus : changes in lipid and protein

contents over a seasonal cycle. Marine Ecology Progress Series 280, 211–226.

Chatwin, P. C. (1976). Salinity distribution in estuaries. Estuarine, Coastal and

Shelf Science 4, 555–566.

Clavero, V., Garcia, M., Fernandez, J. A. and Niell, F. X. (1993).

Adsorption-desorption of phosphate and its availability in the sediment of a saline

lake (Fuente de Piedra, southern Spain). International Journal of Salt Lake Re-

search 2, 153–163.

Clifford, P. J., Yin, K., Harrison, P. J., St. John, M. A., Goldblatt,

R. H. and Varela, D. E. (1992). Plankton production and nutrient dynamics

in the Fraser River Plume, 1991. Volume 1. Manuscript Report. Department of

Oceanography, The University of British Columbia 59, 157.

Coale, K. H., Johnson, K. S., Chavez, F. P., Buesseler, K. O., Barber,

R. T., Brzezinski, M. A., Cochlan, W. P., Millero, F. J., Falkowski,

P. G., Bauer, J. E., Wanninkhof, R. H., Kudela, R. M., Altabet, M. A.,

Hales, B. E., Takahashi, T., Landry, M. R., Bidigare, R. R., Wang, X.,

Chase, Z., Strutton, P. G., Friederich, G. E., Gorbunov, M. Y., Lance,

V. P., Hilting, A. K., Hiscock, M. R., Demarest, M., Hiscock, W. T.,

Sullivan, K. F., Tanner, S. J., Gordon, R. M., Hunter, C. N., Elrod,

V. A., Fitzwater, S. E., Jones, J. L., Tozzi, S., Koblizek, M., Roberts,

A. E., Herndon, J., Brewster, J., Ladizinsky, N., Smith, G., Cooper,

212

Bibliography

D., Timothy, D., Brown, S. L., Selph, K. E., Sheridan, C. C., Twining,

B. S. and Johnson, Z. I. (2004). Southern Ocean Iron Enrichment Experiment:

Carbon cycling in high- and low-Si waters. Science 304, 408–414.

Codiga, D. L. and Aurin, D. A. (2007). Residual circulation in eastern Long

Island Sound: Observed transverse-vertical structure and exchange transport. Con-

tinental Shelf Research 27, 103–116.

Collins, K., Allen, S. E. and Pawlowicz, R. (2009). The role of wind in

determining the timing of the spring bloom in the Strait of Georgia. Canadian

Journal of Fisheries and Aquatic Sciences 66, 1597–1616.

Crawford, D. W. and Tore, L. (1997). Some observations on vertical distribution

and migration of the phototrophic ciliateMesodinium rubrum (=Myrionecta rubra)

in a stratified brackish inlet. Aquatic Microbial Ecology 13, 267–274.

Culberson, C. H. (1991). Dissolved oxygen. In WHPO 91-1: WOCE Operations

Manual WoodsHole USA. .

Dale, A. W., Prego, R., Millward, G. E. and Gomez-Gesteira, M. (2004).

Transient oceanic and tidal contributions to water exchange and residence times in

a coastal upwelling system in the NE Atlantic: the Pontevedra Ria, Galicia. Marine

Pollution Bulletin 49, 235–248.

D′Asaro, E. A. and Dairiki, G. T. (1997). Turbulence intensity measurements

in a wind-driven mixed layer. Journal of Physical Oceanography 27, 2009–2022.

Davenne, E. and Masson, D. (2001). Water properties in the Straits of Georgia

and Juan de Fuca. Technical Report, Institute of Ocean Sciences, Sidney, BC,

Canada , 40.

213

Bibliography

Dewey, R., Tunnicliffe, V., Round, A., Macoun, P. and Vervynck, J.

(2010). VENUS: The Victoria Experimental Network Under the Sea. CMOS Bul-

letin 37, 76–82.

Dyer, K. R. (1973). Estuaries: a physical introduction. John Wiley and Sons,

London.

Efron, B. and Tibshirani, R. J. (1993). Introduction to the bootstrap. Chapman

& Hall.

El-Sabaawi, R. W., Dower, J. F., Kainz, M. and Mazumder, A. (2009).

Characterizing dietary variability and trophic positions of coastal calanoid cope-

pods: insight from stable isotopes and fatty acids. Marine Biology 15, 225–237.

El-Sabaawi, R. W., Sastri, A. R., Dower, J. F. and Mazumder, A. (2010).

Deciphering the seasonal cycle of copepod trophic dynamics in the Strait of Georgia,

Canada, using stable isotopes and fatty acids. Estuaries and Coasts 33, 738–752.

Elskens, M., Brion, N., Buesseler, K., Van Mooy, B. A. S., Boyd, P.,

Dehairs, F., Savoye, N. and Baeyens, W. (2008). Primary, new and export

production in the NW Pacific subarctic gyre during the vertigo K2 experiments.

Deep Sea Research Part II: Topical Studies in Oceanography 55, 1594–1604.

England, L. A., Thomson, R. E. and Foreman, M. G. G. (1996). Estimates

of seasonal flushing times for the southern Georgia Basin. Canadian Technical

Report of Hydrography and Ocean Sciences .

Eppley, R. W. (1972). Temperature and phytoplankton growth in the sea. Fishery

Bulletin 70, 1063–1085.

214

Bibliography

Eppley, R. W. and Peterson, B. J. (1979). Particulate organic matter flux and

planktonic new production in the deep ocean. Nature 282, 677–680.

Fisher, T. R., Harding, L. W., Stanley, D. W. and Ward, L. G. (1988).

Phytoplankton, nutrients, and turbidity in the Chesapeake, Delaware, and Hudson

estuaries. Estuarine, Coastal and Shelf Science 27, 61–93.

Forbes, J. R., Denman, K. L. and Mackas, D. L. (1986). Determination of

photosynthetic capacity in coastal marine phytoplankton: effects of assay irradiance

and variability of photosynthetic parameters. Marine Ecology Progress Series 32,

181–191.

Fraga, F., Rıos, A. F., Perez, F. F. and Figueiras, F. G. (1998). Theoret-

ical limits of oxygen:carbon and oxygen:nitrogen ratios during photosynthesis and

mineralization of organic matter in the sea. Scientia Marina 62, 161–168.

Fu, F.-X., Zhang, Y., Leblan, K., Sanudo-Wilhelmy, S. A. and Hutchins,

D. A. (2005). The biological and biogeochemical consequences of phosphate scav-

enging onto phytoplankton cell surfaces. Limnology and Oceanography 50, 1459—

-1472.

Garay, M. J. H. and Soberanis, L. H. (2008). Dominance shift of zooplankton

species composition in the central Strait of Georgia, British Columbia during 1997.

Hydrobiologica 18, 53–60.

Gay, P. and O’Donnell, J. (2009). Comparison of the salinity structure of the

Chesapeake Bay, the Delaware Bay and Long Island Sound using a linearly tapered

advection-dispersion model. Estuaries and Coasts 32, 68–87.

215

Bibliography

Geider, R. J., MacIntyre, H. L. and Kana, T. M. (1998). A dynamical

regulatory model of phytoplantonic acclimation to light, nutrients, and temperature.

Limnology and Oceanography 43, 679–694.

Geyer, W. R. and Farmer, D. M. (1989). Tide-induced variation of the dy-

namics of a salt wedge estuary. Journal of Physical Oceanography 19, 1060–1072.

Godin, G., Candela, J. and de La Paz-Vela, R. (1981). On the feasibility of

detecting net transports in and out of the Strait of Georgia with an array of current

meters. Atmosphere-Ocean 19, 148–157.

Hagy, J. D., Sanford, L. P. and Boynton, W. R. (2000). Estimation of net

physical transport and hydraulic residence times for a coastal estuary using box

models. Estuaries 23, 328–340.

Haigh, R., Taylor, F. and Sutherland, T. (1992). Phytoplankton ecology of

Jervis Inlet, a fjord system on the British Columbia coast. I. general features of the

nano- and microplankton. Marine Ecology Progress Series 89, 117–134.

Halverson, M. and Pawlowicz, P. (2008). Estuarine forcing of a river plume by

river flow and tides. Journal of Geophysical Research 113, 1–15.

Harding, L. W., Mallonee, M. E. and Perry, E. S. (2002). Toward a predictive

understanding of primary productivity in a temperate, partially stratified estuary.

Estuarine, Coastal and Shelf Science 55, 437–463.

Harris, R. J. (2001). A Primer of Multivariate Statistics 3rd Edition. Lawrence

Erlbaum Associates.

Harrison, P. J., Clifford, P. J., Cochlan, W. P., Yin, K., St. John, M. A.,

Thompson, P. A., Sibbald, M. J. and Albright, L. J. (1991). Nutrient and

216

Bibliography

plankton dynamics in the Fraser River plume, Strait of Georgia, British Columbia.

Marine Ecology Progress Series 70, 291–304.

Harrison, P. J., Fulton, J. D., Taylor, F. J. R. and Parsons, T. R. (1983).

Review of the biological oceanography of the Strait of Georgia - pelagic environment.

Canadian Journal of Fisheries and Aquatic Sciences 40, 1064–1094.

Hasle, G. R. (1978). Using the inverted microscope. In A Sournia (Ed.), Phy-

toplankton Manual, Number 6 in Monographs on Oceanographic Methodology. UN-

ESCO (News York) , 161–178.

Heip, C. H. R., Goosen, N. K., Peter, M. J., Kromkamp, H. J., Middle-

burg, J. J. and Soetaert, K. (1995). Production and consumption of biological

particles in temperate tidal estuaries. Oceanography and Marine Biology: an Annual

Review 33, 1–149.

Henriksen, P., Knipschilt, F., Moestrup, O. and Thomsen, H. A. (1993).

Autecology, life history and toxicology of the silicoflagellate Dictyocha speculum

(silicoflagellata, dictyochophyceae). Phycologia 32, 29–39.

Hetland, R. D. and Geyer, W. R. (2004). An idealized study of the structure of

long, partially mixed estuaries. Journal of Physical Oceanography 34, 2677–2691.

Ho, T.-Y., Quigg, A., Finkel, Z. V., Milligan, A. J., Wyman, K.,

Falkowski, P. G. and Morel, F. M. (2003). The elemental composition of

some marine phytoplankton. Journal of Phycology 39, 1145–1159.

Howarth, R. W., Marino, R., Swaney, D. P. and Boyer, E. W. (2006).

Wastewater and watershed influences on primary productivity and oxygen dynamics

217

Bibliography

in the lower Hudson River estuary. In: J S Levinton and J R waldman (eds.). The

Hudson River Estuary, Cambridge University Press , 121–139.

Huang, Y., Schmitt, F. G., Lu, Z. and Liu, Y. (2009). Analysis of daily river

flow fluctuations using empirical mode decomposition and arbitrary order hilbert

spectral analysis. Journal of Hydrology 373, 103–111.

Hughes, F. W. and Rattray, M. (1980). Salt flux end mixing in the Columbia

River Estuary. Estuarine and Coastal Marine Science 10, 479–493.

IOC, SCOR and IAPSO (2010). The international thermodynamic equation of

seawater – 2010: Calculation and use of thermodynamic properties. Intergovern-

mental Oceanographic Commission, Manuals and Guides, UNESCO 56, 196.

Johannessen, S. C., Macdonald, R. W. and Paton, D. W. (2003). A sed-

iment and organic carbon budget for the greater Strait of Georgia. Estuarine,

Coastal and Shelf Science 56, 845–860.

Johannessen, S. C., Mary, T., OBriena, C., Denman, K. L. and Macdon-

ald, R. W. (2005). Seasonal and spatial variations in the source and transport

of sinking particles in the Strait of Georgia, British Columbia, Canada. Marine

Geology 46.

Kara, B. A., Wallcraft, A. J. and Hulburt, H. E. (2005). A new solar

radiation penetration scheme for use in ocean mixed layer studies: An application

to the Black Sea using a fine-resolution hybrid coordinate ocean model (HYCOM).

Journal of Physical Oceanography 35, 13–32.

Kristiansen, S., Farbrot, T. and Naustvoll, L. J. (2001). Spring bloom

nutrient dynamics in the Oslofjord. Marine Ecology Series 219, 41–49.

218

Bibliography

Lagus, A., Suomela, J., Weithoff, G., Heikkila, K., Helminen, H. and

Sipura, J. (2004). Species-specific differences in phytoplankton responses to N

and P enrichments and the N:P ratio in the Archipelago Sea, northern Baltic Sea.

Journal of Plankton Research 26, 779–798.

Laws, E. A. and Bannister, T. T. (1980). Nutrient- and light-limited growth of

Thalassiosira fluviatilis in continuous culture, with implications for phytoplankton

growth in the ocean. Limnology and Oceanography 25, 457–473.

LeBlond, P. H. (1983). The Strait of Georgia: functional anatomy of coastal sea.

Canadian Journal of Fisheries and Aquatic Sciences 40, 1033–1063.

LeBlond, P. H., Ma, H., Doherty, F. and Pond, S. (1991). Deep and In-

termediate Water Replacement in the Strait of Georgia. Atmosphere-Ocean 29,

288–312.

Lebo, M. E. (1991). Particle-bound phosphorus along an urbanized coastal plain

estuary. Marine Chemistry 34, 225–246.

Legendre, L., Rassoulzadegan, F. and Michaud, J. (1999). Identifying the

dominant processes (physical versus biological) in pelagic marine ecosystems form

field estimates of chlorophyll a and phytoplankton production. Journal of Plankton

Research 21, 1643–1658.

Leterme, S. C., Ellis, A. V., Mitchell, J. G., Buscot, M.-J., Pollet, T.,

Schapira, M. and Seuront, L. (2010). Morphological flexibility of Cocconeis

placentula (bacillariophycea) nanostructure to changing salinity levels. Journal of

Phycology 46, 715–719.

219

Bibliography

Li, M., Gargett, A. and Denman, K. L. (1999). Seasonal and interannual

variability of estuarine circulation in a box model of the Strait of Georgia and Juan

de Fuca Strait. Atmosphere-Ocean 37, 1–19.

Li, M., Gargett, A. and Denman, K. L. (2000). What determines seasonal

and interannual variability of phytoplankton and zooplankton in strongly estuarine

system? Estuarine, Coastal and Shelf Science 50, 467–488.

Lucotte, M. and D’Anglejan, B. (1983). Forms of phosphorus and phosphorus-

iron relationships in the suspended matter of the St. Lawrence estuary. Canadian

Journal of Earth Science 20, 1880–1890.

MacCready, P. and Geyer, W. R. (2010). Advances in estuarine physics.

Annual Review of Marine Science 2, 35–58.

Mackas, D. L. and Harrison, P. J. (1997). Nitrogenous nutrient sources and

sinks in the Juan de Fuca Strait/Strait of Georgia/Pugent Sound estuarine system:

Assessing the potential for eutrophication. Esturiane, Coastal and Shelf Science

44, 1–21.

Marchetti, A., Varela, D. E., Lance, V. P., Johnson, Z., Palmucci, M.,

Giordano, M. and Armbrust, E. V. (2010). Iron and silicic acid effects on phy-

toplankton productivity, diversity, and chemical composition in the central equato-

rial Pacific Ocean. Limnology and Oceanography 55, 11–29.

Marinone, S. G. and Pond, S. (1996). A three-dimensional model of deep water

renewal and its influence on residual currents in the Central Strait of Georgia,

Canada. Estuarine, Coastal and Shelf Science 43, 183–204.

220

Bibliography

Marra, J. and Barber, R. T. (2004). Phytoplankton and heterotrophic respira-

tion in the surface layer of the ocean. Geophysical Research Letters 31, 1–4.

Masson, D. (2002). Deep water renewal in the Strait of Georgia. Estuarine,


Masson, D. (2006). Seasonal water mass analysis for the Straits of Juan de Fuca

and Georgia. Atmosphere-Ocean 44, 1–15.

Masson, D. and Cummins, P. (2004). Observations and modeling of seasonal

variability in the Straits of Georgia and Juan de Fuca. Journal of Marine Research

62, 491–516.

Masson, D. and Pena, A. (2009). Chlorophyll distribution in a temperate estu-

ary: The Strait of Georgia and Juan de Fuca Strait. Estuarine, Coastal and Shelf

Science 82, 19—-28.

McDowell, R. W. and Sharpley, A. N. (2003). Uptake and release of phospho-

rus from overland flow in a stream environment. Journal of Environmental Quality

32, 937–948.

McQuoid, M. R. and Hobson, L. A. (1996). Review: Diatom resting stages.

Journal of Phycology 32, 889–902.

Miller, C. B. (2004). Biological Oceanography . Blackwell Publishing.

Montagnes, D. and Franklin, D. (2001). Effect of temperature on diatom vol-

ume, growth rate, and carbon and nitrogen content: reconsidering some paradigms.

Limnology and Oceanography 46, 2008–2018.

221

Bibliography

Moore, T. S., Nuzzio, D. B., Toro, D. M. D. and III, G. W. L. (2009).

Oxygen dynamics in a well mixed estuary, the lower Delaware Bay, USA. Marine

Chemistry 117, 11–20.

Newton, J., Albertson, S. L., Voorhis, K. V., Maloy, C. and Siegel, E.

(2002). Washington state marine water quality, 1998 through 2000. Environmental

Assessment Program Olympia, Washington 98504-7710 , 111.

Ono, T., Midorikawa, T., Watanabe, Y. W., Tadokoro, K. and Saino,

T. (2001). Temporal increases of phosphate and apparent oxygen utilization in

the subsurface waters of western subarctic pacific from 1968 to 1998. Geophysical

Research Letters 28, 3285–3288.

Parsons, T. R., Maita, Y. and Lalli, C. M. (1984). A manual of chemical and

biological methods for seawater analysis (1st ed.). Pergamon Press Canada Ltd. .

Pawlowicz, R. (2001). A tracer method for determining transport in two-layer

systems, applied to the Strait of Georgia/Haro Strait/Juan de Fuca Strait estuarine

system. Estuarine, Coastal and Shelf Science 52, 491–503.

Pawlowicz, R., Beardsley, R., Lentz, S., Dever, E. and Anis, A. (2001).

Software simplifies air-sea data estimates. EOS Transactions of American Geophys-

ical Union 82.

Pawlowicz, R. and Farmer, D. M. (1998). Diagnosing vertical mixing in a

two-layer exchange flow. Journal of Geophysical Research 103, 30,695–30,711.

Pawlowicz, R., Riche, O. and Halverson, M. (2007). The circulation and

residence time of the Strait of Georgia using a simple mixing-box approach.

Atmosphere-Ocean 45, 173–193.

222

Bibliography

Pennock, J. R. and Sharp, J. H. (1986). Phytoplankton production in the

Delaware estuary: Temporal and spatial variability. Marine Ecology Progress Series

34, 143–155.

Peterman, R. M. and Steer, G. J. (1981). Relation between sport-fishing catch-

ability coefficients and salmon abundance. Transactions of the American Fisheries

Society 110, 585–593.

Petrie, B. and Yeats, P. (1990). Simple models of the circulation, dissolved met-

als, suspended solids and nutrients in Halifax Harbour. Water Pollution Research

Journal Canada 25, 325—-349.

Pond, S. and Pickard, G. L. (1978). Introductory Dynamic Oceanography .

Pergamon International Library.

Program, B. I. E. A. (1996). Modelling the fate of contaminant discharges in

Burrard Inlet , 1–89.

Quigg, A., Finkel, Z. V., Irwin, A. J., Rosenthal, Y., Ho, T.-Y., Rein-

felder, J. R., Schofield, O., Morel, F. M. and Falkowski, P. G. (2003).

The evolutionary inheritance of elemental stoichiometry in marine phytoplankton.

Nature 425, 291–294.

Redfield, A. C., Ketchum, B. H. and Richards, F. A. (1963). The influence

of organisms on the composition of seawater. In: The Sea (MN Hill ed.). Wiley

Interscience: NY 2, 26–79.

Roson, G., Alvarez-Salgado, X. A. and Perez, F. F. (1997). A non-stationary

box model to determine residual fluxes in a partially mixed estuary, based on both

223

Bibliography

thermohaline properties: Application to the Ria de Arousa (NW Spain). Estuarine,


Roubeix, V., Becquevort, S. and Lancelot, C. (2008). Influence of bacteria

and salinity on diatom biogenic silica dissolution in estuarine systems. Biogeochem-

istry 88, 47–62.

Sarmiento, J. L. and Gruber, N. (2006). Ocean Biogeochemical Dynamics .

Princeton University Press. Princeton, Woodstock.

Sarthou, G., Timmermans, K. R., Blain, S. and Teguer, P. (2005). Growth

physiology and fate of diatoms in the ocean: a review. Journal of Sea Research 53,

25–42.

Sastri, A. R. and Dower, J. F. (2009). Interannual variability in chitobiase-

based production rates of the crustacean zooplankton community in the Strait of

Georgia. Marine Ecology-Progress Series 388, 147–157.

Savenkoff, C., Vezina, A. F., Roy, S., Lovejoy, C., Therriault, J. C.,

Legendre, L., Rivkin, R., Berube, C., Tremblay, J.-E. and and, N. S.

(2000). Export of biogenic carbon and stucture and dynamics of the pelagic food

web in the Gulf of St. Lawrence: Part 1. seasonal variations. Deep-Sea Research II

47, 585–607.

Savenkoff, C., Vezina, A. F., Smith, P. C. and Han, G. (2001). Summer

transports of nutrients in the Gulf of St. Lawrence estimated by inverse modelling.

Estuarine, Coastal and Shelf Science 52, 565–587.

Shaha, D. C., Cho, Y. K., Seo, G. H., Kim, C. S. and Jung, K. T. (2010).

Using flushing rate to investigate spring-neap and spatial variations of gravitational

224

Bibliography

circulation and tidal exchanges in an estuary. Hydrology and Earth System Sciences

14, 1465–1476.

Smith, S. V. and Hollibaugh, J. T. (1997). Annual cycle and interannual

variability of ecosystem metabolism in a temperate climate embayment. Ecological

Monographs 67, 509–533.

Sorokin, Y. I. and Sorokin, P. Y. (1996). Plankton and primary production in

the lena river estuary and in the south-eastern Laptev Sea. Esturiane, Coastal and


St. John, M. A., Marinone, S. C., Stronach, J., Harrison, P. J., Fyfe, J.

and Bearnish, R. J. (1993). A horizontal resolving physical-biological model of

nitrate concentration and primary productivity in the Strait of Georgia. Canadian

Journal of Fisheries and Aquatic Sciences 50, 1456–1466.

Stacey, M. W., Pond, S. and LeBlond, P. H. (1991). Flow dynamics in the

Strait of Georgia, British Columbia. Atmosphere-Ocean 29, 1–13.

Stoecker, D. K., Johnson, M. D., de Vargas, C. and Not, F. (2009). Ac-

quired phototrophy in aquatic protists. Aquatic Microbial Ecology 57, 279–310.

Strathman, R. R. (1967). Estimating the organic carbon content of phytoplank-

ton from cell volume or plasma volume. Limnology and Oceanography 12, 411–418.

Stronach, J. A. (1981). The Fraser River plume, Strait of Georgia. Ocean Man-

agement 6, 201–221.

Sutherland, D. A., MacCready, P., Banas, N. S. and Smedstad, L. F.

(2011). A model study of the Salish Sea estuarine circulation. Journal of Physical

Oceanography , 29.

225

Bibliography

Sweeney, C., Smith, W. O., Hales, B., Bidigare, R. R., Carlson, C. A.,

Codispoti, L. A., Gordon, L. I., Hansell, D. A., Millero, F. J., Park, M.

and Takahashi, T. (2000). Nutrient and carbon removal ratios and fluxes in the

Ross Sea, Antarctica. Deep-Sea Research II 47, 3395–3421.

Takahashi, K. (1987). Seasonal fluxes of silicoflagellates and actiniscus in the

subarctic Pacific during 1982-1984. Journal of Marine Research 45, 397–425.

Testa, J. M. and Kemp, W. M. (2008). Variability of biogeochemical processes

and physical transport in a partially stratified estuary: a box-modeling analysis.

Marine Ecology Progress Series 356, 63–79.

Therriault, J. C. and Levasseur, M. (1985). Control of phytoplankton produc-

tion in the Lower St. Lawrence estuary: light and freshwater runoff. Le Naturaliste

Canadien 112, 77–96.

Therriault, T. W., HAY, D. E. and Schweigert, J. F. (2009). Biological

overview and trends in pelagic forage fish abundance in the Salish Sea (Strait of

Georgia, British Columbia). Marine Ornithology 37, 3–8.

Thomson, R. E. (1981). The Oceanography of British Columbia coast . Canadian

Special Publication of Fisheries and Aquatic Sciences 56.

Thomson, R. E. (1994). Physical oceanography of the Strait of Georgia-Puget

Sound-Juan de Fuca system. Canadian Technical Report of Fisheries Aquatics

Sciences. Review of the Marine Environment and Biota of Strait of Georgia, Puget

Sound and Juan de Fuca Strait, Eds. R Wilson, R Beamish, F Aitkens and J Bell.

Symposium on the Marine Environment 1948, 36–100.

226

Bibliography

Thomson, R. E., Mihaly, S. F. and Kulikov, E. A. (2007). Estuarine versus

transient flow regimes in Juan de Fuca Strait. Journal of Geophysical Research

112, 1–25.

Tian, R. C., Vezina, A. F., Legendre, L., Ingram, R. G., Klein, B.,

Packard, T., Roy, S., Savenkoff, C., Silverberg, N., Therriault, J.-C. and

Tremblay, J.-E. (2000). Effects of pelagic food-web interactions and nutrient

remineralization on the biogeochemical cycling of carbon: a modeling approach.

Deep-Sea Research II 47, 637–662.

UNESCO (1981 a). The Practical Salinity Scale 1978 and the International Equa-

tion of State of Seawater 1980. Technical Papers in Marine Science 36, 1–25.

UNESCO (1981 b). Background papers and supporting data on the Practical

Salinity Scale, 1978. Technical Papers in Marine Science 37, 1–144.

UNESCO (1983). Algorithms for computation of fundamental properties of sea-

water. Technical Papers in Marine Science 44, 1–53.

Waldichuk, M. (1957). Physical oceanography of the Strait of Georgia, British

Columbia. Journal of Fisheries Research Board Of Canada 14, 321–486.

Wanninkhof, R. and McGillis, W. R. (1999). A cubic relationship between

air-sea CO2 exchange and windspeed. Geophysical Research Letters 26, 1889–1892.

Williams, P. J. L. B. and Lefevre, D. (2008). An assessment of the measure-

ment of phytoplankton respiration rates from dark 14C incubations. Limnology and

Oceanography: Methods 6, 1–11.

Wilson, J. G. (2002). Productivity, fisheries and aquaculture in temperate estu-

aries. Estuarine, Coastal and Shelf Science 55, 953–967.

227

Bibliography

Wong, K. C. and Valle-Levinson, A. (2002). On the relative importance of

the remote and local wind effects on the subtidal exchange at the entrance to the

Chesapeake Bay. Journal of Marine Research 60, 477–498.

Woolf, D. K. (2005). Parametrization of gas transfer velocities and sea-state-

dependent wave breaking. Tellus 57B, 87–94.

Wua, J. T. and Chou, T. L. (2003). Silicate as the limiting nutrient for phy-

toplankton in a subtropical eutrophic estuary of Taiwan. Estuarine, Coastal and


Wunsch, C. (1996). The Ocean Circulation Inverse Problem. Cambridge Univer-

sity Press.

Wunsch, C. (2006). Discrete Inverse and State Estimation Problems: With Geo-

physical Fluid Applications . Cambridge University Press.

WWF (2006). Free-flowing rivers - economic luxury or ecological necessity? WWF

Global Freshwater Programme, Project 2085, Netherlands , 1–41.

Yin, K., Goldblatt, R. H., Harrison, P. J., St. John, M. A., Clifford, P. J.

and Beamish, R. J. (1997). Importance of wind and river discharge in influencing

nutrient dynamics and phytoplankton production in summer in the central Strait

of Georgia estuary. Marine Ecology Progress Series 161, 173–183.

Yin, K., Harrison, P. J., Goldblatt, R. H. and Beamish, R. J. (1996).

Spring bloom in the central Strait of Georgia: interactions of river discharge, winds

and grazing. Marine Ecology Progress Series 138, 255–263.

228