The influence of diffusive convection on sedimentation from buoyant plumes

Ž .Marine Geology 159 1999 205–220www.elsevier.nlrlocatermargeo

The influence of diffusive convection on sedimentation frombuoyant plumes

David C.J.D. Hoyal a,), Marcus I. Bursik b, Joseph F. Atkinson c

a Exxon Production Research, P.O. Box 2189, Houston, TX 77252-2189, USAb Department of Geology, State UniÕersity of New York at Buffalo, Buffalo, NY 14260, USA

c Department of CiÕil Engineering, State UniÕersity of New York at Buffalo, Buffalo, NY 14260, USA

Received 16 July 1997; accepted 16 December 1998

Abstract

Diffusive convection driven by the differential diffusion of density altering fluid properties may enhance the scavengingŽ .of particles from natural buoyant plumes. For single-phase fluid–solute–heat systems this phenomenon has been

extensively studied because salt fingering generated at the oceanic thermocline is a major mechanism of salt transport in theoceans. However, the influence of this process on particle laden plumes, for example, fluvial plumes in lakes and estuariesvolcanic clouds and seafloor hydrothermal plumes is largely unknown. In this paper, we present direct experimentalmeasurements of the interfacial particle flux at the plume base which can be applied to predict particle scavenging fromnatural buoyant plumes. Particle flux is measured using a light attenuation technique employing a chain of photodiodeswhich average concentration over a large number of fingers. The results are in good general agreement with earlier studiesbased on finger velocity. Flux measurements cover a wide range of conditions from those where diffusive convectiondominates to those where settling and diffusive convection are of a similar magnitude. For very small particles double

Ž . wdiffusive salt finger theory is applicable to two component particulate systems as suggested by earlier studies Green, T.,x1987. The importance of double diffusion to the settling of suspended material. Sedimentology 34, 319–331 . Two

component diffusive convection theory is extended to three components in order to predict particle scavenging from marinefluvial plumes which involve the diffusion of sediment, salt and heat. For larger particles which settle significantly the fluxcan be approximated by adding the double-diffusive and settling fluxes. A theory to predict particle transport through thelower layer and sedimentation at the bed is developed based on the observation of strong convection below the plume.Application of our theory and experimental results indicates that while double diffusion may significantly influence thelongitudinal distribution and vertical sorting of deposits from lacustrine plumes, the diffusive convection process is generallyinsignificant in marine plumes. Observational evidence for lacustrine double diffusion based on water column measurementsis presented. q 1999 Elsevier Science B.V. All rights reserved.

Keywords: sedimentation; convection; river; plumes; estuaries; lakes

) Corresponding author. Fax: q1-713-431-6336; E-mail: [email protected]

0025-3227r99r$ - see front matter q 1999 Elsevier Science B.V. All rights reserved.Ž .PII: S0025-3227 99 00005-5

( )D.C.J.D. Hoyal et al.rMarine Geology 159 1999 205–220206

1. Introduction

The development of dynamic models for particleŽ .dispersal from buoyant hypopycnal plumes is cur-

rently hindered by a poor understanding of the pro-cesses of particle scavenging from the base of theplume. Examples of particle laden hypopycnalplumes include river mouth effluent issuing into

Žlakes and estuaries Wright and Coleman, 1974;Sturm and Matter, 1977; Syvitski et al., 1988; Bur-

. Ž .sik, 1995 , coastal ocean currents Drake, 1971 , lakeŽepilimnion algal blooms Casamitjana and Schladow,

. Ž1993 , black smokers at ocean ridges Feely et al.,

. Ž1994 , and volcanic clouds Bursik et al., 1992;.Woods et al., 1995 . The dynamics considered in this

paper are directly applicable to all these cases, butwe focus particularly on examples drawn from rivermouth environments. River plume models usuallyassume that the only particle flux is by settlingŽ .Syvitski et al., 1988; Bursik, 1995 although it isgenerally recognized that diffusive convection mayalso be an important particle scavenging mechanismŽBell, 1942; Green, 1987; Chen, 1997; Parsons and

.Garcia, 1998 . Modelers need to know if and wherethis diffusive–convective mechanism may be signifi-cant, for example how important is the process inmarine salt wedge plumes compared to heat stratified

Ž .lacustine plumes or marine MOR hydrothermalplumes. Each of these environments has differentgradients of density altering fluid properties whichcontrol the rate of diffusive convection. In addition,the interaction of the settling and diffusive convec-tion fluxes is not well known, nor is the influence ofdiffusive convection on particle transport below theplume.

Diffusive convection involves the development ofunstable density gradients due to the differentialmolecular diffusion of density altering fluid proper-ties. The best known example is salt fingering in theocean where heat and salt are the diffusing propertiesŽ .hence, the name double diffusion . As the surface ofthe ocean is warmed and evaporates, a saltier warmerlayer develops above a cooler and less salty layer.

Ž .Near the interface thermocline fast heat diffusiondue to the proximity of the upper layer and lowerlayers causes a fingering instability which developsinto convection away from the interface. Simpletheoretical expressions of the interfacial flux have

been developed and tested against accurate, highfrequency measurements of temperature and conduc-tivity in small-scale experimental tanks using sugar

Žand salt solutions e.g., Turner, 1967; Stern and.Turner, 1969; Schmitt, 1979 . It seems reasonable to

expect that particle suspensions should follow doubleŽ .diffusion salt finger theory, at least at the limit of

very fine particles. Particle double diffusion may beeven more common than solute double diffusion

Ž .because the Brownian diffusion coefficient of parti-cles is lower. However, there are at present noaccurate high-resolution measurements of sediment

Ž .flux particle scavenging to compare to the saltfinger measurements. Measurements of sediment fin-gers are semi-quantitative based on measurements of

Ž .finger velocity growth rate . Using this method,Ž . Ž .Green 1987 and Parsons and Garcia 1998 com-

Ž .pared the velocities growth rates of salt and sedi-Ž .ment fingers with good agreement. Chen 1997

presents direct measurements of the particle flux butfor a restrictive number of cases.

In this study, we use a light attenuation methodinvolving a chain of calibrated photodiodes and com-puter data acquisition system to obtain high fre-quency spatially integrated measurements of particleconcentration and flux. The structure of the presenta-tion is as follows. First we review the two compo-

Ž .nent salt finger theory adapted by Green 1987 fromsalt fingers and extend it to three components byincluding the retarding effect of salt density but

Žignoring the enhancing effect of salt diffusion Sec-. Ž .tion 2 . This pseudo two component approximation

appears reasonable due to the vast diffusion speeddifferences between salt and heat. In Section 3, theseflux expressions are used to develop theoretical mod-els for the sediment concentration in the fluid layerbelow the plume. The models are a 1D approxima-

Žtion based on the assumption of a convecting well-.mixed lower fluid layer with a particle flux in at the

top from the plume and settling flux at the bed.Experiments designed to test these expressions arepresented in Section 4. In the first series of experi-ments, very small particles are used to assess the

Ž .double diffusion salt finger theory for particles.Measurements of the interfacial flux from these ex-periments were in good agreement with the adaptedsalt finger theory. Larger particles with a similarsettling and diffusive convective flux are used in the

( )D.C.J.D. Hoyal et al.rMarine Geology 159 1999 205–220 207

second series of experiments to test the combinedinterfacial flux by settling and convective diffusion.It was found that the flux from these experimentscould be adequately estimated by simply adding thediffusive–convective and settling fluxes. In Section5, the theory is applied to fluvial plumes in lakes andestuaries. This analysis indicates that while convec-tive diffusion may significantly influence the longi-tudinal distribution and vertical sorting of depositsfrom lacustrine plumes the effect may be insignifi-cant in marine plumes. Heat diffusion has the poten-tial to scavenge particles faster than typical plumeparticle fall velocities while salt diffusion which isapproximately two orders of magnitude slower, doesnot. As a consequence salt effectively hinders doublediffusive scavenging by heat and has a stabilizingeffect that needs to be taken into account. Observa-tional evidence for lacustrine double diffusion basedon water column measurements is presented. Thediscussion and conclusions are presented in Section6.

2. Theory of diffusive convection

A typical example of particle double diffusiveconvection develops when a warm suspension ofparticles is emplaced above a clearer, cooler anddenser lower fluid layer as in a fluvial plume issuing

Ž .into a lake Kindle, 1927; Bell, 1942 . This system isoriginally gravitationally stable but because heat dif-fuses much faster than the sediment particles, sedi-ment particles are retained in the upper layer whileheat diffuses out, causing an instability and drivingthe downward motion of sediment laden water. Simi-larly, the heat gained by the upper regions of thelower layer causes this fluid to rise in the vicinity ofthe interface because it becomes less dense than thefluid above. These upward and downward motionsform thin vertical fingers at the density interface. Asthese fingers propagate up and down away from theinterface they gain and lose heat very effectively dueto their increased surface area, thus reinforcing thefingering mechanism. Most double diffusive experi-ments have used a salt–heat or sugar–salt system.The fastest interfacial flux occurs when the density

Ž .interface is sharp i.e., a step and lowers as thesystem runs down and the interfacial gradient spreads.

A theory of particle double diffusion was devel-Ž .oped by Green 1987 by modifying salt finger the-

ory for the analogous particle–heat stratification. Foran originally sharp interface, double diffusive fingerswill grow when:

3r2kT

RF 1Ž .ž /kp

Ž . ŽWhere the stability ratio R is defined as Huppert.and Mannins, 1973; Green, 1987 :

aDTRs 2Ž .

bDCm

Here, C is the particle concentration of the suspen-m

sion in units of massrmass, a is the volumetricexpansion coefficient for heat, DT is the temperaturedifference between the upper and lower layers, k isT

the diffusion coefficient for heat and b is the volu-metric expansion coefficient for a particle suspensiondefined as

1 Er r yrs fbs ; 3Ž .ž /r E C rm sC f0m

Here, r is the density of the suspension, r is thes

density of the solid particles and r is the interstitialf

fluid density. The Brownian diffusion coefficient ofŽ .the particles k is defined as:p

k T0 ak s 4Ž .p 3pr n df

where k is the Boltzmann constant, T is the abso-0 aŽ .lute temperature K , d is the particle diameter and n

is the kinematic viscosity. Because the diffusionŽ y10 2coefficient for particles k s10 cm rs for 20p

. Ž y3mm particles is much smaller than heat k s10t

2 . Ž y5 2 . Ž .cm rs or salt k s10 cm rs Eq. 1 suggestss

that particle double diffusion should be very com-mon for either heat or salt diffusion into the overly-ing plume.

Ž .The diffusive convective flux F modified todd

accommodate particles rather than a dissolved sub-Ž .stance in the upper layer Schmitt, 1979 is:

F sKDC4r3 5Ž .dd m

where K is:1r3Ks f R r gkb 6Ž . Ž . Ž .


here k is the diffusion coefficient of the fastestdiffusing substance. This 4r3 power law was origi-

Ž .nally developed by Turner 1967 based on an anal-ogy with the parallel plate convection problem withheating from below. This theory has been success-fully applied to many salt–heat and sugar–salt exper-

Žiments Turner, 1967; Stern and Turner, 1969; Green.and Kirk, 1971; Schmitt, 1979 and semi-quantita-Žtively to particle suspensions Green and Schettle,

.1986; Green, 1987; Green and Mogahed, 1994 .These experiments imposed an initial step densitystructure that spread due to diffusion and convectionŽ . Ž .rundown experiments . The function f R is anexperimentally determined coefficient ranging from

Ž .0.051 for R ) 3.5 and f R ™ 0.1 as R ™ 1Ž .Schmitt, 1979 for saltrheat double diffusion whichmay relate the smearing of the horizontal interface to

Ž .the overall stratification. Parsons and Garcia 1998present an alternate derivation by considering eachfinger as an entraining gravity current. For a dilute

Ž . Ž .particle suspension e.g., -50 grl the density r

can be approximated by:

rsr 1qbC 7Ž . Ž .f m

and sediment concentration defined as massrmassŽ . Ž .C can be converted to concentration C inm

massrvolume by:

y1C r yrŽ .s f

r C sC 1q (C 8Ž .f m ž /r rs f

For a particle–heat system, these equations indicatethat the particle concentration gradient DC and thenature of the fastest diffusing substance k can have alarge effect on diffusive convective scavenging. For

Ž .example, Eq. 5 predicts that the flux will be 40=

greater for heat than for salt because of the differ-Ž y3 y5 2 .ence in diffusion coefficients 10 vs. 10 cm rs .

In contrast the temperature gradient DT has only aminor effect on the flux through its influence on RŽ Ž .i.e., F R only varies by a factor of two in the

Ž ..empirical fit of Schmitt 1979 . The fastest fluxoccurs for the smallest R when the density contrastbetween the layers is smallest and the flux decreases

Ž .with increasing R. Chen 1997 found that there is aminimum concentration below which diffusive con-vection was not observed and transport was domi-nated by settling. This critical concentration may be

due to viscosity hindering convection and might becalculated through a Rayleigh number.

Because most environmental scale buoyant plumesare turbulent due to the shear developed at theirboundaries, a consideration of the effect of turbu-lence on diffusive convective scavenging of particlesis necessary. Evidence from the experiments of Lin-

Ž .den 1971 and corroborated by our experimentssuggests that fingers continue to form even if themixing is quite strong in the two layers due todamping of turbulence near the density interface.

Ž .Linden 1971 used an apparatus with oscillatinggrids on each side of the interface and continued tomix during the experiment. He showed that as turbu-lence increased the diffusive salt flux is lowered byup to 1r2 due to the disruption of the fingers butthen begins to increase again due to transport across

Ž .the density interface by mixing i.e., entrainmentŽ .Linden, 1971, fig. 4 . In contrast the heat fluxincreased continuously with the mixing intensity astransport shifted from fingering to turbulent mixingŽ .Linden, 1971, fig. 5 . Therefore, although turbu-lence may retard double diffusive convection to somedegree, in the long run, the interface sharpeningeffects of turbulence generated by free convection ormechanical mixing may allow faster interfacial dou-ble diffusive fluxes than for quiescent layers.

The foregoing analysis developed from salt fingertheory is only applicable to sediment fingers so longas the suspension behaves as a fluid continuum andparticles follow the flow in which they are embed-

Ž .ded. Green 1987 suggests that, in order that theŽ .suspension act like a continuum like a solute , the

characteristic vertical time scale of the fingers mustbe much greater than a settling timescale. A typicaltimescale for fingering may be Drw, where D is thecharacteristic diameter of the fingers and w is acharacteristic vertical finger velocity while a charac-teristic settling timescale is the particle relaxation

2Ž .y1time r d 18r n . An estimate of w analogous tos fŽ .that of Stern and Turner 1969 is:

D2 g bCmw; 9Ž .

n

Ž .and the criterion, following Green 1987 becomes:

18r n 2f2Cd < 10Ž .

gD r yrŽ .s f


If continuum conditions apply the slow settling ofparticles should not interfere with the double diffu-sive flux. The two fluxes can be assumed to actindependently and can be added linearly in this

Ž .region. Green 1987 presents a criterion for whethersettling or double diffusive convection will dominateparticle transport across the interface based on the

Ž .ratio F of the flux by double diffusive convectionŽ . Ž .F to the flux by settling F :dd s

FddFs 11Ž .

Fs

Ž .The flux by Stokes settling F across the densitys

interface is:

F sw C 12Ž .s s

where, assuming laminar flow around the settlingparticles:

gd2 r yrŽ .s fw s 13Ž .s 18r nf

which may require some modification in the marineenvironment due to the flocculation of particles.

Ž Ž ..Based on the value of F Eq. 11 , three arbitraryregimes of interfacial transport can be defined. ForF410, double diffusion dominates the interfacialparticle flux, for 0.1-F-10, both double diffusiveconvection and settling may be important while forF<0.1 settling dominates. In this study, we willfocus on the first two regimes, since the case wheresettling dominates the interfacial flux has been con-

Ž .sidered in another paper Hoyal et al., 1999 .Previous work on the same heat–particle system

Žwas semi-quantitative based on finger velocity e.g.,.Green, 1987 . Green et al. showed that the velocity

of sediment fingers is comparable to the velocity ofŽ .salt fingers measured by Green and Kirk 1971 and

Ž . Ž .Houk and Green 1973 . Green and Schettle 1986observed fingering developed when a warm suspen-sion of fine particles was released over a clear coldwater layer to form a density current. They notedthat the shape of the fingers was a vortex ringpattern, unlike that commonly observed for salt fin-

Ž .gers. Green and Mogahed 1994 considered the caseof larger particles with a significant settling fluxacross the density interface and suggested that set-

tling seems to make the originally sharp interfaceless stable. The high frequency spatially integratedmeasurements made in our experiments were de-signed to compare directly with salt flux measure-ments made using conductivity probes and thermis-

Ž .tors e.g., Schmitt, 1979 . The flux for larger parti-cles is considered in more detail in our experiments.

In many particle laden plumes of geological inter-est, there are more than two significant density alter-ing fluid properties, and the theory outlined above is

Žnot directly applicable e.g., marine fluvial plumes.may have salt, heat and particles . Ideally,

diffusive–convective particle scavenging is possiblewhen the gradients drive the salt into, or heat out ofthe buoyant particle-laden plume. However, becauseof the significantly lower diffusion rate of salt com-

Ž y5 y3.pared to heat salt;10 , heat;10 , and itsŽlower double diffusive convective flux i.e., 40=

Ž . Ž ..lower from Eq. 4 or Eq. 5 , salt diffusion into thebase of the plume may have very little influence onthe loss of particles in most natural situations. Thiscan be demonstrated by decomposing the doublediffusive convective flux into the product of a con-

Žcentration and a velocity i.e., assuming Fs1 in Eq.Ž . Ž . Ž . .11 and solving Eqs. 11 and 12 for w . Thiss

velocity is representative of the settling velocity ofthe largest particles that are settling as fast as theyare scavenged by diffusive convection. All particles

Ž .below this size the ‘equilibrium’ size will be trans-ported predominantly by diffusive convection andlarger particles will tend to settle. Assuming a typicalplume particle concentration of 5 mgrl and particlespecific gravity of 2.5 the equivalent ‘equilibrium’diameter for salt flux is 2 mm while the diameter forheat flux is 260 mm. Thus only very fine mudparticles will be scavenged by the diffusion of saltwhile much larger particles will be scavenged byheat diffusion. However, because fine natural parti-cles flocculate readily in the presence of salt, parti-cles will remain in this size range for a brief timeand salt diffusion will play an even smaller role inparticle scavenging than our above calculation wouldsuggest. As a consequence, natural particle ladenplumes in which enhanced particle loss rates will bemost readily observed are those where the excessdensity of the particles is supported by a temperaturegradient. In fact, the areas of the plume in whichenhanced loss rates will be found may be spatially


restricted, since temperature, salinity and sedimentconcentration vary along the plume axis.

Because of the great difference in time scales forsalt and heat diffusion these processes are essentiallyde-coupled. Consequently, although theoreticallyboth heat and salt diffusion can promote the down-ward flux of suspended sediment by inducing con-vection, the salinity gradient in fact tends to inhibitenhanced transport due to its stabilizing effect on thedensity gradient in the presence of a more rapidlydiffusing temperature gradient. To determine thisstabilization effect we develop a simplified model byincluding the effect of salinity on the bulk densityfield, but ignoring its diffusion. Taking into accountthe stabilizing effect of salinity we would expect thediffusive convective flux of sediment out of theplume to become important in an estuary in anyregion where the density differences caused by tem-perature are greater than the density differencescaused by particles plus density differences causedby salt, i.e.:

r gS r gSs 1 s 2aDT) y 14Ž .ž / ž /C gS yr C gS yr1 1 s 2 2 s

Ž .where both sides of Eq. 14 are positive and:

1 Er 1 Eras and gs 15Ž .ž / ž /r E T r ES

In any plume satisfying this criterion, the plumeremains buoyant due to temperature difference andwill sink after the temperature difference relaxes out.In this regime, we might expect that the doublediffusive flux of sediment can be described by a

Ž Ž .modified version of the two component flux Eq. 5Ž ..or Eq. 6 of the form:

r gSs 11r3bF s f R r gkŽ . Ž .dd ž /C gS yr1 1 s

4r3r gSs 2

y 16Ž .ž /C gS yr2 2 s

where:

aDTRs 17Ž .

r gS r gSs 1 s 2yž / ž /C gS yr C gS yr1 1 s 2 2 s

and ksk .T

Ž . Ž .A preliminary analysis of Eqs. 14 – 17 indicatesthat the maximum diffusive convective scavengingshould occur proximal to the river mouth. Our rea-soning is as follows. Firstly, there is a tendency tolower the diffusive convective particle flux downinlet as the bulk density gradient of the slowest

Ž Ž ..diffusing substances right hand side of Eq. 14decreases due to salt entrainment into the upper layerand particle transport into the lower layer. This term

Ž .should overcome the increase in f R as R de-creases due to faster entrainment and conduction of

Ž Ž ..heat top of Eq. 17 compared to the bulk densityŽof the slow substances bottom right hand side of Eq.

Ž ..17 . In the proximal region, we can simplify theanalysis by assuming a pure freshwater sedimentladen plume overlying saltwater, assuming minimalsalt has been entrained into the upper layer. In this

Ž .case, Eq. 14 suggests that thermal diffusive convec-tion can occur if the density due to sediment in theplume is greater than the density due to salt in the

Ž .layer below the plume i.e., C )gS rb . This con-1 2

dition is quite rare although it may occur at certainŽriver mouths during flood events Mulder and Syvit-

.ski, 1995 . For a plume to remain buoyant underthese conditions the temperature gradient must be atleast as large as the particle density minus the salt

Ž .density i.e., aDTsbCygS . However, a fullanalysis of this problem in the context of fluvialplumes would require detailed consideration of ob-served or modeled down-inlet gradients of tempera-ture, concentration and salinity. Although it appearsthat convective diffusive particle scavenging may berare in marine estuaries, it may be much more com-

Ž .mon in hydrothermal plumes e.g., black smokersdue to the high loading of dissolved salts and precip-

Žitates and strong initial temperature gradients Feely.et al., 1994 . Sedimentation models for these plumes

have been developed but have not included a diffu-sive convective flux.

3. Sedimentation model development

ŽApart from the flux through the interface particle.scavenging which has been covered in Section 2, a

model of sedimentation from fluvial plumes needs toconsider the particle transport mechanism in the


region below the plume. Compared to a systemdominated by particle settling, convective transportin the water beneath the plume may alter the arrival

Ž .rate of particles at the bed bed sorting and influ-ences the driving gradient between the plume and thefluid below. In this section, we develop simple phe-nomenological models based on the assumption of awell-mixed fluid layer below the plume as com-

Ž .monly observed in solute e.g., Turner, 1967 andŽ .particle Chen, 1997; Hoyal et al., 1999 convective

diffusion experiments. To simplify the differentialequations we make the approximation that rsr inf

Ž . Ž .Eq. 7 i.e., the first term in the Maclaurin seriessince the sediment concentrations are small. The

Ž .initial conditions we apply are C sC 0 at ts01 1Ž .and C s0 at ts0. Three cases are considered: 12

Žboth the double diffusive and settling fluxes 0.1-F. Ž . Ž .-10 ; 2 double diffusive flux only F)10 ; and

Ž . Ž .3 settling flux only F-0.1 .If double diffusive fingering and settling are both

significant the fluxes at the density interface can beadded which leads to:

dC k w1 s4r3sy C yC q CŽ .1 2 1ž /d t h h1 1

0.1-F-10 18aŽ .dC k w w2 s s4r3s C yC q C y CŽ .1 2 1 2d t h h h2 2 2

0.1-F-10 18bŽ .where C , h and C , h are the concentrations and1 1 1 2

depths of the upper and lower layers, respectively.For a constant concentration in the upper layer, Eqs.Ž . Ž .18a and 18b reduces to:

dC k w2 s4r3s C 0 yC q C 0 yC ,Ž . Ž .Ž . Ž .1 2 1 2d t h h2 2

0.1-F-10 19Ž .Or, if we neglect the concentration build-up in the

Ž .lower layer C ;0 the solution is:2

y3w ws st t3h y1r3 3h13 1C sw ke qw C 0 e ykŽ .1 s s 1ž /

0.1-F-10 20Ž .

When the settling flux is small only the diffusiveconvective fluxes need to be considered and:

dC k1 4r3sy C yC F)10 21aŽ . Ž .1 2d t h1

dC k w2 s4r3s C yC y C F)10Ž .1 2 2d t h h2 2

21bŽ .

or assuming negligible density effect of the sedimentŽ .in the lower layer C ;0 :2

y3kt y1r3C s qC 0 F)10 22Ž . Ž .1 1ž /3h1

Ž Ž .The validity of the analytical solutions Eqs. 20 andŽ ..22 depends on the h rh ratio. These solutions1 2Ž .Fig. 1 , approximate the full numerical solutionŽ Ž . Ž ..Eqs. 18a and 18b very well for low h rh1 2

Ž Ž ..Fig. 1. Comparison of approximate analytical solution Eq. 20w x Ž Ž . Ž ..dashed line and full numerical solution Eqs. 18a and 18bŽ . Ž .solid line for the concentration in the upper layer. a h rh s1 2

Ž .0.8, b h rh s0.08.1 2


Ž .h <h , e.g., a deep glacial lake but very poorly1 2Ž .for high h rh h ;h , e.g., the experiments .1 2 1 2

If settling dominates the interfacial transport anddouble diffusive convective transport is negligiblethe solution for the upper layer is:

yw st

h1C t sC 0 e F-0.1 23Ž . Ž . Ž .1 1

Ž .e.g., Martin and Nokes, 1988 and the solution forŽ .the lower layer is Hoyal et al., 1999 :

yw yws st tC 0Ž .1 h h1 2C s e ye F-0.1 24Ž .2 ž /h2

1yh1

As the concentration in the plume and the waterbelow approach the same value, at some point con-vection must cease. This critical concentration can beestimated by setting C sC and solving for time.1 2

An analytical solution to this problem is available forthe simplest cases, for example a settling only fluxŽ Ž . Ž ..e.g., Eqs. 23 and 24 but when both fluxes are

Ž Ž . Ž ..included e.g., Eqs. 18a and 18b a closed solu-tion is not available and numerical methods must beapplied.

Apart from the concentration of particles in thewater column as a function of time, another problemof great importance to geologists and engineers is therate of accumulation of particles on the bed. This canbe obtained by numerically integrating the flux to thebed w AC from 0 to t, i.e.:s 2

M 0 yM t w AŽ . Ž . t1 ss C d tH 2M 0 M 0Ž . Ž . 01 1

w tss C d t 25Ž .H 2h C 0Ž . 01 1

4. The experiments

We now describe the experiments designed tomeasure the interfacial particle flux for very fineparticles which can be compared directly to the

Žsolute flux theory i.e., settling much less than dou-.ble diffusive convective flux and for larger particles

where both fluxes need to be included.

4.1. Fine particle flux experiments

The first series of experiments were designed tomeasure the vertical interfacial flux of very fineparticles which have a negligible interfacial settlingflux compared to their double diffusive flux. Theexperiments were carried out in a 12.5=12.5=15cm3 transparent Perspex box with an upper warmerparticle suspension overlying a cooler clearer lower

Ž .layer Fig. 2 . The two layers were initially separatedby a sliding plastic sheet, which was removed at thestart of each experiment, ensuring a relatively sharpinterface as the initial condition. This sharp interface

Ž .degraded during the experiments rundown and nomechanical mixing was introduced. The particlesused in these experiments were microcrystalline sil-

Ž .ica Sigma because of their ready availability andsmall size. Suspensions of the particles were allowedto stand for 4 h which, assuming particles are settlingby Stokes law gives a mixture of particles -2 mmin diameter. It was assumed that all particles re-mained in suspension in the fluid during the experi-ment. Water in both the upper and lower layerscontained 5 grl hexa-metaphosphate to reduce thecoagulation efficiency of the particles. Once theexperiment was started a lid was placed on the boxto reduce heat loss through the water surface. Theparticle concentration in the upper and lower layerswas determined by measuring the attenuation of lightpassing through the box at different levels. Thistechnique involved photodiodes set behind brasstubes so that each photodiode received a cone of

Ž .light from the source Fig. 2 . The concentrationŽ .light attenuation readings were made at good verti-

Ž .cal resolution 1 cm and readings of the entire depthprofile were made every 15 s by a computer dataacquisition system. Thus, highly accurate, integratedŽ .over many fingers flux measurements were made.

The flux of particles into the lower layer wasmeasured by depth integrating the concentrationmeasured by the lower layer photodiodes and calcu-lating the mass increase in the lower layer as afunction of time. The data were processed in thefollowing way:

Ž .1 The photodiode readings were converted toparticle concentration. Each photodiode is calibratedseparately and the correlation coefficients of thecalibration equations, fitted by least squares were all


Fig. 2. Schematic of the 12.5=12.5=15 cm3 Perspex box used to measure the interfacial flux of particles.

greater than 0.99. A plot of typical concentration–depth profiles illustrating the transfer of mass intothe lower layer as a function of time is presented inFig. 3.

Ž .2 The integrated mass in the lower layerŽ . ŽM rM 0 i.e., normalized by the initial mass in the2 1

.upper layer was calculated by depth integratingŽconcentration from photodiodes in lower layer i.e.,

.six photodiodes from the tank bottom . The inte-

Fig. 3. Typical vertical concentration profiles for the fine particlestill water experiments, run JP15, initial condition: 2 grl micro-

Ž . Ž .crystalline silica, T 0 s22.08C, T 0 s41.48C. Upper layer1 2Ž .convection damped by sedimentation density interface as ob-

Ž .served by Chen 1997 .

grated mass in the lower layer as a function of timeis presented in Fig. 4.

Ž .3 The numerical derivative was obtained bywŽ Ž ..differencing the time series M rM 0 y2 1 tyD t

Ž Ž .. xM rM 0 rD t. A typical plot of the normalized2 1 t

flux across the interface is presented as a function oftime in Fig. 5.

These graphs show a low initial flux, whichreaches a peak and then declines more slowly. Theincreasing limb of the curve can be explained, as thetime needed to set up the double diffusive convec-tion and the transport of the fingers into the lowerfluid layer. The declining limb is due to the spread of

w Ž .xFig. 4. Normalized particle mass in the lower layer M rM 02 1

as a function of time.


w Ž .xFig. 5. Mass flux into the lower layer d M rM 0 rd t as a2 1

function of time.

the gradients around the interface. The maximumflux was then read off these graphs, since this is theflux that would occur at a continuously sharp densityinterface.

4.2. Larger-particle fluxrsedimentation experiments

Ž .To test the sedimentation models Section 3 ,based on the addition of the two dominant fluxesacross the density interface, a series of experimentswas performed using larger particles in a deep 30=

30=150 cm3 Perspex tank. Silicon carbide particlesŽ .specific gravity 3.2 , with a size of 600 meshŽ .median diameter d s9.3, w s0.55 cmrmin , 800m s

Ž .mesh d s6.5, w s0.27 cmrmin , and 1000 meshm sŽ .d s4.5, w s0.13 cmrmin were used and all ofm s

the water contained 5 grl hexa-metaphosphate toreduce coagulation effects. The tank was separatedinto a 34 cm upper layer and a 102 cm lower layerand a flexible plastic barrier, similar to that depictedin Fig. 2, was used to separate the layers prior to thestart of the experiment. Once the lower layer of cool

Ž .water ;208C was poured into the tank and theŽ .barrier put in place a hot particle suspension ;408C

was added to form the top layer. The walls surround-ing the upper layer were insulated to prevent heatloss. Following a quick stirring of the upper layer toensure an initially uniform distribution or particles inthe upper layer, the barrier was removed, and animpeller was started to mix the upper layer. Watersamples were collected through taps in the tank walls

and the concentration determined by a light attenua-tion technique. All efforts were made to minimize

Ž .fluid entrainment interfacial mixing in these experi-ments.

4.3. Comparison of experimental and natural condi-tions

Because the experiments were designed to under-stand particle scavenging from natural plumes itseems important to link the experimental and naturalplume conditions. The experimental particles werewell sorted and chemically disaggregated in order tosimplify the dynamics. They range in size from 2–10

Žmm with settling velocities in the range 0.0003–.0.005 cmrs which are typical of disaggregated clay

particles in natural fluvial plumes. For example parti-cles in the plume of the Aare river 2 km into Lake

Ž .Brienz Switzerland average 6.8 mm in diameterŽ .Sturm and Matter, 1977, fig. 12 . In marine fluvialplumes particles grow by coagulation to attain a sizeand settling velocity much greater than the primary

Žparticles ds125–1000 mm w s0.01–1 cmrs;sŽ .Sternberg et al., 1996 . However, because thetimescale of flocculation is significantly larger than

Ž .that for convective diffusion, Green 1987 showedthat flocculation can usually be ignored for the pur-poses of calculating the diffusive convective flux.Although the experimental particles were bettersorted than natural plume particle distributions thediffusive convective flux is not related to particlesize and the flux for each particle size class shouldbe dependent on its concentration in the flow. Inaddition particle-laden plumes particles are segre-gated down-inlet so that at any distance from thesource the flux is dominated by particular sizes likein our experiments.

Another important scaling parameter is the parti-cle concentration which has a major control on the

Ž Ž . Ž ..interfacial flux Eqs. 5 and 6 . Although theŽ .concentrations used in the experiments 0.1–5 grl

are high they are possible in natural plumes. ForŽ .example Syvitski et al. 1985 measured 1 grl for

the proximal regions of the Homathko River plumeŽ .and Hoskin and Burrell 1972 give a similar value

for a hypopycnal plume in Glacier Bay, Alaska.During flood events the particle concentrations at


some river mouths can be much greater, overcomingŽ .the salt wedge density 42.6 grl and developing a

Ž .hyperpycnal flow Mulder and Syvitski, 1995 . Be-cause temperature gradient has only a minor effect

Ž Ž . Ž ..on the flux Eqs. 5 and 6 the high temperatureŽ .gradient in the experiments ;208C need not de-

tract from the applicability of the experiments. Typi-Žcal DT values for lakes are 5–188C Pharo and

.Carmack, 1979, p. 537 and fjords are 6–128CŽ .Picard, 1967, p. 1487 . Values of R for the experi-

Ž .ments high DC to high DT are actually quiteŽ .comparable to natural flow low DC to low DT . In

the second series of experiments we imposed a ve-locity gradient ;5 cmrs which may be typical ofsome natural flows where shear develops at thedensity interface due to the relative motion of theplume and underlying water. The diffusive convec-tive flux predictions worked adequately for this case

Ž .and in the work of Linden 1971 for mixed layers.Another factor that must be considered in comparingthe experiments to nature is the depth and cross-sec-tion of the experimental tank. Because of the small

Ž .size of the fingers diameter -1 cm the smallPerspex boxes used the experiments seem adequate

Žto capture the average behavior spatially integrated.flux of many fingers. However the slight concentra-

tion increase at larger times at the bottom of thesmall box indicates that there may be some dampingof mixing by the bottom of the tank. This effect wasnot observed in the large tank experiments.

5. Experimental results

5.1. Fine particle flux experimental results

The measured vertical particle fluxes can be com-pared directly to the vertical fluxes of salt into thelower layer from heat–salt experiments and to the

Ž .flux theory e.g., Schmitt, 1979 . The value of RŽ Ž ..Eq. 2 was calculated for the experiments based onthe initial uniform temperatures of the upper andlower layers and the initial concentration of particlesin the upper layer. Values of R ranged from 2.96 to41.39. Because the time to reach the maximum fluxwas generally short -5 min for initial concentra-

tions -0.75 grl and -10 min for all concentra-tions the temperature of the layers could be approxi-mately characterized by the initial condition.

In Fig. 6 the flux data from all 39 experiments areplotted against the density of the upper layer bC.

Ž Ž ..The line is the theoretical 4r3 power law Eq. 5Ž . Ž .presented by Schmitt 1979 with a coefficient f R

sconstants0.05. In general the agreement betweenthe theory and our data is as good as for salt finger

Ž .data see Schmitt, 1979, Fig. 6 . The data tend tohave a higher slope than the theory for the lower R

Ž .values because we have assumed that f R is aconstant when in fact it increases as R lowers. Thisis illustrated in Fig. 7 where our sediment finger

Žfluxes are plotted divided by the 4r3 power law Eq.Ž .. Ž .5 but assuming F R s1. These data, plotted as a

Ž .function of R show the coefficient F R obtainedfrom the experiments. The observed trend and scatter

Žis very similar to that observed for salt fingers see. Ž .Schmitt, 1979, Fig. 7 where Schmitt 1979 showed

Ž . Ž .that f R is constant around 0.05 for R)3.5 andincreases for R-3.5 to a maximum value of 0.1.This asymptotic value is a little lower than that

Ž .obtained by Schmitt 0.05 and closer to that ob-Ž . Ž .tained by Linden 1973 i.e., 0.04 .

Ž . ŽFig. 6. Maximum flux F i.e., the maximum from plots like Fig.s.5 as a function of initial particle concentration in the upper layerŽ .bC , for microcrystaline silica particles.


w Ž .x ŽFig. 7. Coefficient f R of double diffusive flux expression Eq.Ž .. Ž .6 calculated by dividing measured flux by Eq. 5 but with anŽ . Ž .f R in Eq. 6 of unity.

5.2. Larger particle sedimentation experimental re-sults

ŽConcentration depth profiles see for example Fig..8 show that the assumption of a lower layer uni-

formly mixed by convective motions away from theinterface is reasonable in some cases. We observedthis strong mixing in all of our deep tank experi-ments. The concentration in the upper layer is some-what noisy for large concentrations due to the non-linearity of the light–concentration calibration curvewhich asymptotes at high concentrations. A compari-son of the concentration measured in the upper andlower layers as a function of time and a numerical

Ž Ž . Ž ..solution to the model Eqs. 18a and 18b , solved

Fig. 8. Concentration–depth profiles in the deep tank using siliconŽ . Ž . Ž .carbide particles. T 0 s20.98C, T 0 s41.18C, C 0 s4.7 grl,1 2 1

ds6.5"1.0 mm, w s0.27 cmrmin, Fs18.73. Initial concen-s

tration ‘noise’ in the upper layer is due in large part to thenon-linear light–concentration calibration.

Fig. 9. A comparison of models including a double diffusiveŽ . Ž .convective flux thin solid line , a settling flux dashed line andŽ . Ž Ž . Ž . Ž . Ž . Ž .both these fluxes solid line Eqs. 18a , 18b , 21a , 21b , 23

Ž ..and 24 for the upper and lower layer with experimental concen-tration. The settling flux is small compared to double diffusive

Ž . Ž . Ž .flux, i.e., Fs39.08, T 0 s26.78C, T 0 s40.88C, C 0 s4.71 2 1Ž .grl, ds4.5"0.8 mm, w s0.13 cmrmin, Rs3.0, F R esti-s

mateds0.1. The numerical solution breaks down when C ap-1

proaches C . Initial deviation of the upper layer concentration2

from the model due to non-linear calibration.

by commercially available software using a 4th orderRunge–Kutta technique for two different F values

Ž . Ž .are shown in Fig. 9 Run T1 and 10 Run T3 . InFig. 9 the settling flux is small compared to double

Ž .diffusive flux Fs39.08 while in Fig. 10 the set-

Fig. 10. A comparison of models including a double diffusiveŽ . Ž .convective flux thin solid line , a settling flux dashed line andŽ . Ž Ž . Ž . Ž . Ž .both these fluxes thick solid line Eqs. 18a , 18b , 21a , 21b ,

Ž . Ž ..23 and 24 for the upper and lower layer with experimentalconcentration. The settling flux and double diffusive flux of

Ž . Ž . Ž .similar order, i.e., Fs9.15, T 0 s22.98C, T 0 s42.18C, C 01 2 1Ž .s4.7 grl, ds9.3"1.0 mm, w s0.55 cmrmin, Rs3.6, F Rs

estimateds0.1. The numerical solution breaks down when C1

approaches C . Upper layer data are all at a higher concentration2

than lower layer data at the same time.


tling flux and double diffusive flux are closer inŽ .magnitude Fs9.15 . Since these experiments have

Ž .low R values we selected a value of F R s0.1 forboth of the model fits in accordance with Fig. 7. Inboth cases the fit between experiment and data isreasonable, there is no evidence that interfacial trans-port is inhibited by the effect of turbulence on thefingers. The fit of the model and experiment suggeststhat the double diffusive flux across the interfaceholds even when the settling of particles across theinterface is significant. Thus, it appears that theassumption that the two fluxes can be combinedlinearly is reasonable so long as the continuum con-

Ž Ž ..dition Eq. 10 is met and conditions are favorableŽ Ž ..for finger formation e.g., Eq. 1 .

6. Application of the sedimentation models tobuoyant fluvial plumes

6.1. Particle scaÕenging mechanisms

Ž Ž ..Typical values of F Eq. 11 for isothermalmarine fluvial plumes based on salt diffusion are ofthe order of 0.14–0.014 assuming Cs1–1000 mgrl,ds10 mm, SGs2.5 and Ss30 psu, indicatingthat diffusive convection may be an insignificantparticle scavenging mechanism in these environ-ments. Consequently settling dominates unless theparticle concentration is supported by a temperaturegradient, and settling models such as those of Hoyal

Ž .et al. 1999 are applicable. In contrast it appears thatdiffusive convection may be very common in ther-mally stratified lacustrine fluvial plumes. Values ofF for natural lakes indicate that both diffusive con-vection and settling scavenging mechanisms are im-

Žportant, e.g., Kamloops lake, Fs1.43 Pharo and. ŽCarmack, 1979 ; Lake Brienz, Fs2.33, Sturm and

.Matter, 1977 .

6.2. Particle transport below the plume

Water column measurements from lake plumesindicate that convective processes may dominatetransport below the plume as we observed in ourexperiments and modeled in Section 3. Natural lake

Žplumes e.g., Kamloops Lake—Pharo and Carmack,.1979; Lake Brienz—Sturm and Matter, 1977 are

characterized by sharp near horizontal bases at thethermocline depth and step-like vertical concentra-tion profiles with much lower particle concentrationbelow than inside the plume. The form of theseprofiles is not in agreement with a settling model,which predicts a descending front of particle-ladenfluid below the plume at a concentration similar tothe plume. The problem of reconciling a settlingmodel with these lake observations has been recog-nized by Lake Researchers. For example, in Kam-loops Lake, British Columbia, the Thompson Rivercreates a classic interflow, with step like verticalturbidity profiles. The interflow has been going forover 7 days, but in the words of Pharo and CarmackŽ .1979 : ‘‘There is no indication from the hydro-graphic measurements of temperature and turbidityof material transfer from the interflowing river waterto the bottom of the lake. Instead, the lake waterbelow the thermocline maintains relatively low val-ues of turbidity throughout the stratified period. Isthis not in direct conflict with our conjecture thatmaterial settling from the interflowing plume deter-mines the open lake sediment distributions.’’

Step-like water column sediment distributions mayindicate either one of two things: either the particlesare transported undetected from the base of theplume to the near bed region, or the sediment isbeing mixed into and diluted by the deeper lowerlayer. Based on our laboratory observations we sug-gest the latter is occurring and attempted to explainthe lower layer concentration with our 1D-transportmodels. We used a model with a constant flux from

Ž Ž ..the upper layer Eq. 19 because the interflows weconsider last a few days so that an instantaneousemplacement model may be unrealistic. We assumea constant concentration in the section of the plumewe are modeling and by using the particle fall veloc-ity as a fitting parameter, we adjusted w to obtains

the observed lower layer concentration at the correcttime. The characteristics of the interflow incorpo-rated in the model were: Lake Brienz: h s20 m,1

Ž .h s80 m, C 0 s40 mgrl, k s0.001, ts4 days;2 1 tŽ .Kamloops Lake: h s20 m, h s110 m, C 0 s101 2 1

mgrl, k s0.001, ts7 days. These times weret

selected based on observations of how long theplume had been overlying a particular part of the

Ž .lake Kamloops Lake or how long a time hadelapsed since the plume had reached that part of the


w Ž .xFig. 11. Deposition in a hypothetical lake with 7f particles and heat stratification. Deposition at the bed i.e., depositedsM rM 0deposited 1Ž Ž . Ž .. Ž .as a function of time for the convective model Eqs. 18a and 18b solid thick line compared to the established settling-hypolimnion

Ž Ž .. Ž .model Eq. 24 dashed line for a well-mixed upper layer. Interfacial flux includes both double diffusive convection and settling terms.Ž .The thin solid line is deposition once convection stops estimated to be when C sC at which point the particles will be transported1 2

through the lower layer by settling.

Ž .lake where the samples were taken Lake Brienz .The results of this analysis indicate that a convectionmodel accounts well for the observed lake concentra-tions. The modeled and actual particle sizes arewithin 73% and 82% of one another.

ŽThe substantial difference in particle loss sedi-.mentation from the fluid column for a convecting

model compared to the traditional settling model canŽbe calculated by integrating the flux to the bed i.e.,

Ž Ž .substituting one of the models for C Eqs. 18a ,2Ž . Ž . Ž . Ž . Ž ..18b , 19 , 21a and 21b or Eq. 24 into Eq.Ž .25 . This case is illustrated in Fig. 11 where wesimulate the sedimentation from an instantaneouslyemplaced plume in an idealized lake w s0.00445s

cmrs, h s10 m, h s100 m. The dotted line1 2w Ž .xrepresents deposition i.e., M rM 0 fordeposited 1

mixed settling in the upper layer and settling in theŽ Ž ..lower layer Eq. 24 . The thick solid line represents

deposition while the lower layer is convecting due tofingering at the interface. As indicated in Section 3,convection must cease as the upper and lower layers

Žapproach the same value. Sedimentation after C s1.C is represented by the thin solid line. Convection2

in the bottom layer may be inhibited by shear due to

horizontal advection in the lower layer which is lowin most lakes but may be strong in tidal estuaries.

7. Discussion and conclusions

We have presented detailed experimental mea-surements of the flux of particles across a densityinterface by diffusive convection integrated over alarge number of fingers. For small particle–heat

Ž .diffusion double diffusive our flux measurementsagree reasonably with salt finger theory and labora-tory experiments as suggested by the work of GreenŽ .1987 . An analysis of the magnitude of the diffusiveconvective and settling flux suggests that doublediffusive particle scavenging should be common inwarm lacustrine fluvial plumes. We extended twocomponent salt finger theory to include the effect ofsalinity gradients on diffusive convective particle

Ž .scavenging three component system and consideredthe implications for marine–river plume sedimenta-tion. Apparently diffusive convective scavenging willbe negligible in most marine–fluvial plumes exceptin some very extreme cases. Although either heat or


salt diffusion can drive diffusive–convective particlescavenging, our analysis suggests that marine parti-cles tend to settle faster than any scavenging effectof salt diffusive convection. In addition the stronglystabilizing salinity gradients in estuaries tend to op-pose and prevent any heat diffusive convection whichmight occur when warmer particle laden effluentoverlies a cooler salt wedge. Consequently heatdriven diffusive convective particle scavenging mayonly be possible in marine fluvial plumes in rarecases when sediment loads are extremely high andthe temperature gradients are favorable.

Experiments with larger particles with a signifi-cant settling flux across the density interface indicatethat the total flux can be determined by simplyadding the flux by double diffusive convection andthe flux by settling, provided that continuum condi-tions for the particle suspension are met. Both diffu-sive convection and settling can drive fingering andconvection below the plume. The experimental re-sults are in good agreement with sedimentation mod-els based on double diffusive convective and settlingflux at the interface, a settling flux to the bed, and awell-mixed upper and lower layer. Double diffusiveconvection not only causes sedimentation from

Žplumes to be more rapid than previously thought as.suggested by Green, 1987 but, due to the effect of

convection in the lower layer, leads to differentpatterns of sediment distribution in the water columnand deposition at the bed. This effect is stronglysupported by water column measurements from lakes,which are not consistent with a model based onsettling below the plume base.

8. List of symbols

F Flux ratioF Double diffusive convective flux throughdd

the density interfaceF Flux through the density interface by set-s

tlingr Density of the suspensiong Acceleration due to gravityk Diffusion coefficient of the fastest diffus-

ing substancek Diffusion coefficient for heatT

k Brownian diffusion coefficient of the par-p

ticlesa Volumetric expansion coefficient for heatb Volumetric expansion coefficient for sedi-

ment suspensionC Sediment concentration in massrmassm

C Sediment concentration massrvolumer Fluid densityf

r Solid particle densitysŽw Terminal fall velocity of a particle Stokes’s

.lawn Kinematic viscosityd Particle diameterw Velocity of the fingers

Ž .D Typical finger diameter e.g., 0.2 cmt TimeM Mass of particles in the upper layer1

M Mass of particles in the lower layer2

C Concentration of particles in the upper1

layerC Concentration of particles in the lower2

layerh Depth of the upper layer1

h Depth of the lower layer2

A Horizontal cross-sectional area

Acknowledgements

This Research was supported by NSF grantEAR9316656 and by funding from Science Applica-tions International. We would like to thank AlexEisen for help in designing and building the turbid-ity-measuring device.

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The influence of diffusive convection on sedimentation from buoyant plumes

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