Estuarine, Coastal and Shelf Science (1985) 21,337-356

Modelling of Tidal Motion in Shoaling Waters: The Estuary of Milford Haven

D. J. Gunn and 0. Yenigiin Department of Chemical Engineering, University College Swansea, Singleton Park, Swansea SA28PP, U.K.

Received 8June 1984 and in revisedform 5 November 1984

Keywords: estuarine circulation; current measurements; horizontal currents; mathematical models; numerical analysis; Milford Haven Estuary

An experimental survey of the estuary of Milford Haven has been carried out in which velocity, temperature and salinity have been measured over the full extent of the estuary during a complete spring and a complete neap- tidal cycle. The measurements were taken from three survey vessels, each of which was used to service six or seven survey points so that the measurements comprised a total of about 40 vertical profiles distributed over a grid of 20 survey points for each tide. Tide level measurements at two points were made simultaneously. The experimental measurements of tidal level have been used to calibrate a two- dimensional mathematical model based upon a new numerical representation of the shallow water equations. The predictions of velocity for the model have been compared with vertically integrated velocities calculated from experiments, and good agreement has been found over the full extent of the estuary.

The effect of a strong south-westerly wind in the estuary has been examined in numerical simulation.

Introduction

The motion of tides in waters of the continental shelf is dominated by coastal hydro- graphy with the courses of flooding and ebbing tides channelled by estuaries, rivers and creeks, sometimes imposing a sharply varying pattern upon the water motion, yet with relatively small variations in depth, so that vertical components of velocity are not large. The emergence and submergence of sand banks and mud flats at different stages of the tide are important in major estuaries, so important in some cases that channel widths at high tide are several times greater than at low tide.

In constructing a mathematical model to simulate tidal motion, tidal variation of the shore line is not important if the scale of variation is smaller than the grid size of the model. However, in the estuary of Milford Haven, for example, extensive mud flats and sand banks are exposed at low tide so that the hydrography of the region of flow changes continuously throughout the tidal cycle.

One of the earliest methods proposed for the calculation of changing coastal bound- aries was that of Reid and Bodine (1968) who omitted advection in the two-dimensional equations of motion and calculated water depths and velocities by an explicit finite difference scheme. When the elevation of water in a grid was greater than land level in a neighbouring dry block, the normal component of flow on to the dry block was determined from an empirical formula representing flow over submerged barriers, and

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338 D. J. Gum & 0. Yenigiin

water was allowed to flow on to the dry region. The model was used to simulate water motion under hurricane conditions in Galveston Bay, Texas. In the first 12 h of the storm period there were discrepancies between computed and observed surge levels, after which a satisfactory agreement was achieved, until the subsiding period of the storm when there was again divergence between observed and computed levels. The model was adjusted to fit the observed data by choosing friction coefficients, but while fair agreement was obtained over the middle of the storm period, there were significant differences in the early and late stages.

An implicit method for the solution of the two-dimension shallow water equations was proposed by Leendertse (1970) and applied to the stimulation of water motion in Jamaica Bay. The method could also be applied to the simulation of flooding of tidal flats but the calculation procedures were complicated by the implicit nature of the solution.

Sielecki and Wurtele (1970) applied three numerical schemes to the solution of the shallow water equations, a two-step Lax-Wendroff method, Arakawa’s nine-point for- mulation and a two-step quasi-implicit method. The moving shoreline was located by a search from deep to shallow water, but for two-dimensional systems in particular the location of the shoreline was a cumbersome part of the computational procedure.

More recently Flather and Heaps (1975) have given a detailed description of a moving boundary tidal model and its application to Morecambe Bay. In the first scheme the non-linear convective terms were omitted, and three displaced numerical grids were employed for the two velocity components and water-levels; in a second scheme the convective terms were evaluated separately using the ‘ angled derivative ’ representation, while in a third scheme the angled derivative was used to represent convective terms within the body of the equations of motion. The three schemes were used to simulate tidal motion in Morecambe Bay with a grid spacing of one nautical mile and a time step of 46.6 s. The inclusion of the convective terms gave difficulties and irregularities in the solution, so that in order to obtain a stable solution convective terms were omitted from the equations within three grid lengths of the open sea boundary. Qualitative agreement with the low water shoreline was reported but a reasonable agreement with two single station measurements could only be obtained when tidal inputs were provided from another set of computations for the Irish Sea.

A similar unsatisfactory experience with the representation of convective terms in the equations of motion by the angled derivative has been reported by Prandle (1978) in numerical modelling of the North Sea. A three-level input scheme was employed by Grubert (1976) and several studies of two-dimensional models of larger-scale areas have been reported where shoreline changes could be neglected in comparison with a grid spacing (Miles &Worthington, 1981; Pingree & Griffiths, 1980).

Although two-dimensional numerical modelling of estuaries and seas has thus been attempted by several workers, validations of numerical models by comparison with experimental measurements of velocity and level are fewer in number. When the geographical area of the numerical model is relatively large, so that tidal changes in the shoreline may be neglected, friction coefficients may be adjusted in the numerical scheme to obtain agreement with water levels over the region. In this way satisfactory models for tidal level prediction in the North Sea, for example, have been developed and used to predict the propagation of tidal surges under stormy conditions (Prandle, 1978). Agreement of model predictions for velocity with experiment is less certain, partly because of the greater experimental difficulty of measuring accurate vertically integrated velocities, because of the equipment required to measure velocities at several stations,

Modelling of tidal motion in shoaling waters 339

0 I 2 km

MILFORD HAVEN Cresswell Rwe

Head River

7 PEMBROKE

Figure 1. The estuary of Milford Haven. (- - ) Low water mark with respect to the Admiralty Chart Datum; (x) location of tidal level gauges; (- ) boundaries of the numerical model.

particularly when fixed station measurements are employed, and because of the diffi- culty of adjusting models for velocity when the degrees of freedom for adjustment have been taken up in matching water levels.

When the area of the numerical model is relatively small, so that tidal changes in the shoreline are important, the problems of model development are different in emphasis, because the tidal shoreline and therefore the geometry of the region of integration changes continuously throughout the tidal periods, and because the pattern of water level variation is simpler, often not including amphidromic points for example. Although feasible numerical techniques have been devised and employed (Flather & Heaps, 1975) comparison with experiments have shown disagreements, and numerical predictions of velocities have been affected by apparent difficulty in the inclusion of con- vective terms over the entire region of integration, while the small number of stations examined for velocity has not been suflicient to give a comparison between experiment and prediction over the whole region.

The purpose of this paper is to present the application of a new method of modelling the shallow water equations to the prediction of level and two-dimensional velocity in the estuary of Milford Haven, and to compare the predictions with measurements of velocity and level. The development and validation of the method is shown to give a model for the estuary in which the prediction of level and velocity are in close agreement with experimental over the full extent of the estuary.

The estuary of Milford Haven

The estuary, in West Wales, is located at the intersection of the Severn estuary and the Celtic Sea on the South Pembrokeshire coast; the local geography of the estuary is shown in Fig. 1. The principal rivers are the Eastern and Western Cleddau that flow west and east to Picton Point, from whence the combined rivers, now named the Daugleddau, flow south to Mill Bay, and westwards, increasing in width to form the main reach of the

340 D. 7. Gum & 0. Yeni~iin

Figure 2. Sediments of the outer Milford Haven estuary.

estuary. The distance from Picton Point to the mouth of the estuary at St Ann’s Head along the reach is about 27 km.

The estuary is subjected to a semi-diurnal co-oscillating tide with an average range of 6.4 m for spring tides and 2.7 m for neap tides at Dale Fort and Milford Haven. From St Ann’s Head to Newton Noyes the width of the estuary at low water is similar to the width at high water, but mud flats and sand banks emerge at Angle Bay, Sandy Haven and Dale Roads in the lower reaches. Between Newton Noyes and Hobbs Point there are mud flats at low water along the southern shore of the estuary, while at Pennar Gut, the inlet of the Pembroke River to the estuary, there are extensive mud banks with the river width reduced to a small fraction at low water. There are variations in depth in the main channels with gullies and shallows, and some dredged trenches. The deepest regions of about 2 m at low water lie off Chapel Bay.

Prevailing winds are southerly or south-westerly throughout the year. Williams and Jolly (1975) report that water velocities in a spring tide are of the order of 1 m s- ’ between Newton Noyes and Carr Jetty, increasing beyond this value as the channel nar- rows into the Daugleddau. The freshwater discharge into the estuary is about 2 m3 s- ‘, and as the tidal flow in the main estuary at Hobbs Point is of the order of 10 000 m3 s- ’ at mid-tide both vertical salinity variations and variations along the reach of the main body of the estuary are very small. A survey carried out by the Welsh Water Authority in 1981 showed that vertical salinity differences were less than 1%0 in spring tides and 5%0 in neaps, with similar longitudinal variations between the mouth of the estuary and Hobbs Point. Salinity variations in the Daugleddau and both Cleddau rivers were large with the limits of saline intrusion about 35 km from the mouth of the estuary.

The distribution of sediments on the bed of the lower estuary is shown in Fig. 2 drawn up on the basis of grab samples from a variety of sources. Sediment size analyses obtained from eight transects have been documented by West (1978). The analyses con- firm that size distributions in the mouth of the estuary lie in the range greater than 1 mm, while silts dominate in the eastern regions. This pattern of deposition is significantly different from the classical pattern in which silt is deposited from rivers as the river velocity is reduced in the widening reaches of the estuary.

ModellinE qf tidal motion in shoaling waters 341

Figure 3. Drogue trajectories near the outer boundary. E, ebb release; F, flood release.

The measurement of velocity and salinity in the estuary

A series of experiments in the outer estuary was carried out during June 1981. The survey area from St Ann’s Head to Hobbs Point is about 15 km long and varies in width from 1 to 5 km. The experiments included measurement of velocity salinity and temperature.

The main purpose of the experiments was to provide experimental measurements to be compared with the numerical predictions of a two-dimensional mathematical model of tidal level and velocity. Three vessels were provided by the Welsh Water Authority. The Survey Vessel Procyon, 10m in length, was equipped with Decca Navigation Equipment and an electrically operated winch for the lowering and raising of the Braystoke current meter and attached salinometer that provided readings of both salinity and temperature. Two dories, 4 m and 5 m in length were equipped with hand-operated winches and similar instrumentation. As measurements from three fixed measuring stations, even when taken over full spring and neap tidal cycles, would be insufficient to provide a comprehensive comparison with the numerical model over the full extent of the estuary, 21 measurement stations in the estuary were surveyed and identified by landmarks and bearings; the survey stations were chosen to be as close to nodes on the mathematical model as possible. During the tidal survey measurements were taken from each vessel at each of the seven stations in turn by reading velocities and salinities at five points 05 m from bed and surface and three points equally spaced in the water column, while anchored or moored at a buoy. A 500-ml surface water sample was also taken. As three to four measurements could be taken at each grid point during the tidal cycle, a total of about 40 velicity profiles were available at 19 different locations over a full neap and a full spring tidal cycle. No measurements were taken at two locations originally chosen because of local difficulties in obtaining satisfactory profiles. All measurements were taken when wind velocities corresponded to Force 4 (6-8 m s-l) or less. Tidal levels during the survey were measured by means of two tidal gauges installed at Dale, about 2 km north of the outer boundary of the numerical model, and at Carr Jetty, close to the inner boundary of the model. Both gauges were levelled to O.D. Newlyn during the period of the survey.

342 D. J. Gum & 0. Yenigt’in

Water velocities near the outer boundary were investigated in a separate drogue sur- vey, in which drogues were released, and charted with time using the Decca Navigation system. Drogues were released just inside the estuary during the ebb tide, and just out- side the estuary on the flood tide. As the drogue trajectories in Fig. 3 show, releases in the main flow of the Bristol Channel were strongly influenced by the dominant east-west tidal flows in the channel. Thus drogues released on the flood tide, particularly near the western part of the boundary, were transported eastwards, and not into the estuary of Milford Haven, while drogues released on the ebb, particularly near the west boundary, were transported westwards. Drogue tracks, however, near the boundary of the numeri- cal model, were approximately co-linear across the boundary showing that close to the boundary of the numerical model, the influence of the east-west tidal flows was subordinate to the configuration of the estuary.

The experimental measurements of velocity and level provided by the survey documented the principal distributions of flow and shoaling areas during various stages of the tide, and therefore provided a comprehensive basis for comparison with the predictions of the numerical model.

Numerical modelling of the estuary

The vertically integrated equations of motion and continuity may be expressed as:

au au au ah a,p, W2 cost3

i% +u---&+v-+g--52v-

gu(u2 + v2)‘i2 +

C2H =o

ay ax PH

av av av ah a,p, W2 sin0

z +z4;ix+v-+g-++u-

gv (2.42 + v2y +

C2H =o

a~ ay PH

ah a(Hu) + (Hv) Tt+- -=

0 ax ay

(1)

(2)

(3)

In the shallow water eqns shown as (l), (2) and (3), the estuary is taken to be well mixed, and the water body is subjected to Coriolis forces characterized by the Coriolis coefficient 0, surface drag due to the wind velocity vector W and bottom friction characterized by the Chezy coefficient C. The barometric pressure gradient was neglected.

The equations are determined by an initial condition, in which the water level within the estuary was set at a uniform value. The initial distribution of velocity was taken to be zero. Boundary conditions within the estuary were set at a harmonic condition for the M, tide at the open sea boundary,

H = HOsin (cot), (4)

Modelling of tidal motion in shoaling waters 343

and at a harmonic condition for the upstream boundary, based upon measurements from West (1978):

u = f, sin (cot + .s,) (5)

v = f,sin (it + cl), (6)

with f, and f, as experimentally determined amplitude and ei as the experimentally determined phase difference between the open sea and upstream boundaries. The landward boundary conditions were taken to be the slip-boundary conditions, i.e. the velocity normal to the boundary was set at zero. Advection terms were retained at all nodes within the estuary domain.

The key element in the numerical formulation of eqns (l), (2) and (3) was the numeri- cal representation of the derivatives au/&, I%/& and c?H/c%. In the time-split method adopted, the partial derivative with respect to time was expressed as:

where n s is n x in the first time-split application, and Ly in the second. The integral in eqn (7) was expressed numerically as:

?f at= f “+l - cf?-, + Tf:+f:+,)/(T+2) /At

>

The principal difficulty in numerical differentiation is the growth of error due to the loss of significant figures on forming the numerical derivative, a process that is reversed on numerical integration, as this is accompanied by a gain in significant figures. The value of T in eqn (8) determines the particular formula for integration; T= 0 gives trapezoidal integration over a double interval, T= 2 gives trapezoidal integration over single inter- vals, while T= 4 determines Simpson-rule integration over the interval. Greater values of T may be associated with higher order quadratures, and it will be noticed that the usual definition of df/at is approached as T becomes large.

The partial differentials in eqns (l), (2) and (3) were expressed by central difference equivalents so that when time-splitting is applied, as illustrated by Roache (1976), for example, the finite difference representations of eqns (l), (2) and (3) become:

n+: - Vij - & (Vl-l,j + TV”, + vl+l,j) - Uyjat(v:+lj - V:-,,j) 2Ax

344 D. J. Gum & 0. Yenigiin

while for the second part of the time split:

(uyJ+‘)2 + (v;l+‘)2

+ ntQv”,+~ -g atuy,+: (12) cz Hl.+i *I

(uyj+:)2 + (4’92

gat - -(h:,;\, - ry.$,) -g ntv:,+~ (13) 2AY C2 H:,‘:

Equations (9) to (14) were calculated in sequence over the entire domain of integration for one or a number of tidal cycles. The formulation above does not include wind stress which, when required, was added as further terms in the same manner.

Values of the Chezy coefficient were chosen as characteristic of the bed of the estuary. Chow (1959) has compiled values of the Manning roughness parameter, n for different materials in various channels, and as n is related to the Chezy constant by C= H”6/n, the appropriate values of C for the estuary may be calculated from Chow’s tabulation. From the sediment distribution shown in Fig. 2, it is clear that the dominant distribution is silt in high friction regions of the estuary. Chow’s table gives IZ to be 0.017 for earthen channels, corresponding to a Chezy coefficient of 62 rnt s- ‘. Hence for depths of channel exceeding 2 m C was given by the following equation:

C= 18.0 log,,H+59.6, (1%

while for depths of 2 m or less, a constant value of C was set at 65 rnt s- ‘. According to Dronkers (1964), measurements in silted rivers gave values of C between 60 and 65 mfs-‘, consistent with values chosen for the estuary of Milford Haven.

The horizontal components of the Coriolis acceleration were calculated from expressions given by Dronkers (1964).

Modelling of tidal motion in shoaling waters 345

60

50

h 40

30

20

0 I 2 3 4 5 6 7 8 9 IO II I2 I3 Time interval (s)

Figure 4. Stability diagram for the Local Integral method.

Comparison of the numerical solution with experiment

The Courant et al. (1967) stability criterion for the shallow water equations gives the upper limit to the time interval of integration, thus

(14 + (2gfLl)“*) At It, (16)

Ax where ZY,,, is the maximum water depth. Since the maximum value of u was about Im s-’ and H,,, was about 20m, for a chosen grid size of 250m, the maximum time interval of integration was about 12 s. However, the stability behaviour of the solution was also strongly affected by the magnitude of the parameter T.

The development of instability in the solution took the form of oscillations in veloci- ties and water levels that at some point in the solution increased sharply without bounds. A set of numerical experiments was performed in which both T and LI t, the time inter- val, were changed. The stability behaviour of the solution is illustrated in Fig. 4, in which the stable and unstable regions are shown; the relationship between the maximum value of the parameter T and the maximum value of the time interval of integration is apparently approximately hyperbolic.

Since the numerical integration gave converged values of velocity and level over several tidal cycles for a wide range of values of T, the best value of T was chosen as that providing the best agreement between experimental and theoretical tidal levels over the full extent of the estuary. The calculations showed that the hydraulic gradient was sensitive to the value of T.

The outer tide recording gauge at Dale Fort was fairly close to the outer boundary, and therefore the agreement between computed and experimental levels was close because the amplitude at the outer boundary had been chosen to produce agreement between experimental and computed levels at Dale Fort. Thus comparison between cal- culated and measured levels at the inner boundary at Carr Jetty is the principal measure of agreement between experiment and theory for hydraulic gradient over the entire range of the estuary since the dominant hydraulic gradient is that between Dale Fort and Carr Jetty.

Figure 5 shows the calculated tidal levels at Dale Fort over a full spring tide, calculated tidal levels at Carr Jetty for various values of T, and experimental tidal levels measured

346 D. J. Gum & 0. Yenigiin

5r

- Computed levels at Carr Jetty for ~(lr~ous T --- Computed levels at Dale Fort for T=9 u

0 Measured levels at Carr Jetty 0 Measured levels at Dale Fort

-4 I I I 1 I I I I I I I I I 0 I 2 3 4 5 6 7 8 9 IO II 12 I3

Time (h)

Figure 5. The effect of T upon tidal levels at Carr Jetty and a comparison between measured and computed tidal levels at Dale Fort and Carr Jetty for a spring tide.

3 i I

2

I 7

t 0

h w

-I

0 Dole Fort, measured X Can Jetty, measured

-41 1 I I I 1 I I I I I I I I 0 I 2 3 4 5 6 7 8 9 IO II 12 13

Time ( h )

Figure 6. A comparison between measured and computed tidal levels at Dale Fort and Carr Jetty for a neap tide.

at Carr Jetty. It may be seen that the longest values of hydraulic gradient are found for low values of T, and indeed at mid-tide for T= 0 the computed difference between levels at Carr Jetty and Dale Fort was about 4 m, while the experimental value was about 0.8 m. AS the value of T is increased, however, the disparity between theory and experiment is reduced and agreement of experiment and theory is very close for T = 9.

A comparison between experimental and computed levels for T = 9 at both Dale Fort and Carr Jetty for the neap tide is shown in Fig. 6. The agreement is generally satisfac- tory although for the second part of the tidal cycle there is a difference for both between

Modelling of tidal motion in shoaling waters 347

km Pembroke River f

I ” s-1

Figure 7. Computed velocity field at high tide (spring tide).

experimental and computed levels of about 0.20m. Given that the difference between experimental and computed levels at Dale Fort is due to imperfect matching between the experimental and computed levels as the sea boundary and not to the model, it is apparent that the agreement between experiment and theory at Carr Jetty is quantitat- ively similar to the agreement at Dale Fort. Therefore, for T=9, hydraulic gradients are satisfactorily represented by the model. It should be emphasized that the values of the Chezy coefficient were chosen on the basis of the roughness of the estuarine bed according to eqn (15) and not as a tuning parameter in the model.

Depth-mean velocities were found from the current meter measurements by calcu- lating the vector components of velocity from each instrument reading and determining the depth mean velocity by integration over the water column using the trapezoidal rule. Each mean velocity was based upon five experimental estimates for most of the survey points, except at a small number of locations where the water depth was too shallow and three readings only were taken.

During the course of the computation the extent of the wetted area was assessed at intervals during the rising and the falling tide. During the rising tide dry grid points were examined by comparing the dry height with water levels at neighbouring wet nodes. If the dry height was 0.10 m less than a neighbouring water level, that node was taken as wetted from the next time interval. The time interval between the examination of neighbouring nodes for wetting was chosen from a consideration of velocities of flooding during the rising tide. The time required for the tide to flood from one node to the next, given necessary conditions of water level and land height, is about equal to the grid dimension/flooding velocity. Hence, for a grid spacing of 250 m and an estimated flooding velocity of 1 m s- ‘, a period of 250 s is required to wet the adjacent grid, and this condition was met by examining levels at intervals of 250 s on the rising tide. The same principle was applied on the falling tide.

Vector plots of water velocity over a spring tidal cycle are shown in Figs 7,8,9 and 10, representing the distribution at high tide, mid-tide, low-tide and mid-tide. On each dia- gram the high water line is shown as the continuous perimeter, while vectors represent- ing the velocity appear at each node. Nodes within the high water boundary that are marked by a dot show that water is present but the velocity is very low. Nodes within the high water contour that are not marked have dried up in the course of the falling tide and have not been rewetted at that time. Thus areas within the outer boundary that are otherwise marked represent the emergence of mud flats. On Fig. 9, at low tide, the shaded areas of mud flats are compared with the low water contour as represented on

348 D. J. Gunn & 0. Yenigiin

Hobbs Potnt

St Ann’s Head

Angle Boy

km

ILL Pembroke River

f I m s-1

Figure 8. Computed velocity field 3.7 h after high water (spring tide).

21 St Ann’s

Head

Milford HoVeIl

-.I Hobbs point

km

c lm 5-l

Pembroke Rlver

Figure 9. Computed velocity field at low tide showing mud flats, (shaded as shown by the Ordnance Survey map), spring tide.

Dole

F-L

St. Ann’s Head

f Pembroke River

Im s-1

Figure 10. Computed velocity field 3.7 h after low water (spring tide).

Modelling of tidal motion in shoaling waters 349

Station

3 18

4 19

5 10

L--z3 Angle Bay

Time after HW Station

0 h 56 min 9 1 h 3min 11 1 h 18 min 8 1 h 44 min 14 2h Omtn 2 2 h 50 min

Pwllcrachan

(5

Time after HW Pembroke River

3 h 18 min 3 h 30 mun 3h48min 4h Omn 4h 8m1n

Figure 11. Comparison between computed and measured velocity vectors in the period high water to high water+4.15 h. Spring tide: (-) computed, (- - - -) measured.

Ordnance Survey maps of the region. The agreement is clearly as good as the grid size will allow.

The computed vector velocities may be compared with experimental depth-averaged vectors obtained from the vertical velocity profiles for both spring and neap cycles. Since the experimental velocity profiles had been obtained at varying times during the survey, it was not possible to produce vector plots similar to Figs 7-10. Therefore, comparison of experiment and theory for the spring tide have been shown on three vector plots that each cover a period of about one third of a tidal cycle. On each plot the computed veloci- ties in magnitude and direction are shown as full arrows, and the experimental measure- ments corresponding in position and time are shown as dotted arrows. The number shown at each point of measurement identifies the survey grid point, and the times of each experimental measurement and the corresponding interpolated computer value are shown on each figure.

Figure 11 shows the comparison between experiment and theory in the first 4 h of the tida1 cycle from high water. The agreement is close, and the vectors often almost identi- cal with the exception of station 10. This station was close to Thorn Island, a small island that was of the order of 50 x 50 m in extent, although in the model the island occupied a full grid, and was therefore taken as 250 x 250 m in extent. Therefore, the effect of the island was much larger in the model, causing the computed velocity at station 10 to be almost 30” north of the measured velocity.

The comparison for the second third of the spring tidal cycle is shown in Fig. 12. Agreement over the estuary between experiment and theory is very close for all points in this period of the tide. There was another experimental measurement at station 10, but this was taken after the turn of the tide, and the agreement between experiment and theory is close.

The comparisons for the final third of the tidal cycle are shown in Fig. 13. With the exception of station 10 agreement between experiment and theory is close over the full extent of the estuary. Indeed Figs 11, 12 and 13 all show good uniform agreement over

350 D. J. Gunn & 0. Yenigiin

Head Angle Bay

Pembroke River

Station Time after HW Station Time after HW

1 4 h 24 min 9 7 h 4 min 7 4 h 25 min 10 7 h 26 min

15 4 h 50 ml” 8 7 h 29 min 17 5h Omin 2 7 h 57 min 13 5 h 30 min 11 8h 4min

6 5 h 31 min

Figure 12. Comparison between computed and measured velocity vectors in the period HW+4.15htoHW+%lOh.Springtide:(-) computed, (- - - -) measured.

St. Ann’s Heod

Angle Boy Pembroke Rover

Station Time after HW

1 8 h 13 min 7 8 h 40 min

14 9h Omin 6 9 h 20 min

15 9 h 25 min 18 9 h 25 min

3 9 h 50 min 13 10h 8mm

4 10 h 22 ml”

Statton Time after HW

19 10 h 22 min 5 10 h 43 min 9 10 h 52 min 8 11 h20 min

10 11 h 29 min 2 11 h40min

21 11 h58min 12 12h 1 ml”

Figure 13. Comparison between computed and measured velocity vectors in the period HW f8.10 h to HW + 125 h. Spring tide: (---+) computed, (- - - +) measured.

the estuary at all states of the tide, except at station 10 where it is apparent that differ- ences at this isolated point are due to an overestimate of the effect of Thorn Island. It is a property of numerical models that resolution of differences on a scale smaller than one grid size is indeterminate. Thorn Island is therefore associated with an inherent diffi- culty in characterizing the effect of features smaller than one grid in extent. The difficulty may be resolved in principle by adopting a finer grid scale locally if there is a need for improved resolution. The effect of omitting Thorn Island altogether was

Modelling of tidal motion in shoaling waters 351

Head Angle Boy

Pembroke River

Station Time after HW Station Time after HW

21 0 h 50 min 18 2 h 7mcn 10 1 h 7 min 11 2 h 25 min 17 1 h 33 min 19 2 h 35 ml” 15 1 h 35 min 13 2 h 53 ml” 14 2 h Omin 12 3 h 14 n-1”

Figure 14. Comparison between computed and measured velocity vectors in the period HW to HW + 3.5 h. Neap tide: (-) computed, (- - - 4) measured.

Dole

Head Angle Boy

Pembroke River Pembroke River

Station Station Time after HW Time after HW Station Time after HW Station Time after HW

10 10 3 h 3 h 38 min 38 min 11 11 7 7 h h 7 7 min min 17 17 3 h 3 h 43 min 43 min 6 6 7 7 h h 11 11 min min 14 14 4 h 4 h 37 min 37 min 15 15 7 7 h h 36 36 min min 21 21 4 h 4 h 53 min 53 min 3 3 7 7 h h 39 39 min min

9 9 5h 5h 9min 9min 19 19 7 7 h h 46 46 mi” mi”

Figure 15. Comparison between computed and measured velocity vectors in the period HW + 3.5 h to HW + 8-O h. Neap tide: (-) computed, (- - - +) measured. Figure 15. Comparison between computed and measured velocity vectors in the period HW + 3.5 h to HW + 8-O h. Neap tide: (---+) computed, (- - - +) measured.

examined, but while this gave slightly better agreement between experiment and model close to Thorn Island, agreement was worse in the intermediate vicinity, and therefore Thorn Island was retained at one grid in extent.

The comparison between theory and experiment for the first 3.5 h of the neap tide is shown in Fig. 14. Agreement is good except for station 10 as for the spring tide, but there is some discrepancy at points 18 and 19. These points lie between Milford Haven and the confluence with the Pembroke River, and differences between experiment and theory in this section of the estuary were experienced throughout the tidal cycle. The period 3.5-8 h is shown in Fig. 15 where agreement is close except at stations 19 and 21. The final period of the tide is shown in Fig. 16 where again agreement is quite good except at stations 17, and 18. Stations 17, 18, 19 and 21 were serviced by the 4-m dory.

352 D.J. Gunn @ 0. Yenigiin

Dole

r

t St. Ann’s

HP”Ci

Angle Bay I Pembroke River

Station Time after HW Station Time after HW

4 8 h 15 min 10 10 h 18 m,n 13 8 h 23 min 21 10 h 25 min

9 8 h 50 min 15 11 h 2min 17 8 h 55 ml” 14 11 h30min 18 9 h 23 m,n

Figure 16. Comparison between computed and measured velocity vectors in the period HW + 8.0 h to HW + 12.5 h. Neap tide: (----+) computed, (- -- -) measured.

The operating log of the 4-m dory showed that the crew started off the day with a faulty cable connecting the meter indicators with the Braystroke current meter, and the cable was replaced. But it is clear from the general nature of the computed velocities and the hydrography of the estuary in the region of Milford Haven to the Pembroke River and that the measured velocities are not consistent with the hydrography. The possi- bility of local hydrographic effects may be discounted, because the same survey points were examined in the spring tide, where local conditions might be expected to have a more severe effect, particularly as low water is approached. There remains the possibility of wind effects although wind was never stronger than force 4 (6-8 m s-i), or the possi- bility of some malfunction in the direction sensor while measuring vertical profiles. There is an evident inconsistency that we feel is due to malfunction of equipment on the 4-m dory, most probably related to the instrument difficulties experienced at the begin- ning of the survey for the neap tide. If these experimental estimates and measurements near Thorn Island are omitted, the ratio velocity expt./velocity model was calculated as 0.95 + 0.14 standard error with a mean of directional error of 3” k 6” standard error.

The effect of wind stress

The influence of wind stress may be incorporated in the equations of motion as shown in eqns (1) and (2). In nature wind is variable, affected both by topography and variations due to turbulence, but in the treatment of this paper the effect of a steady wind has been considered, south-westerly in direction for a given tidal variation at the sea boundary. In practice there will be a gathering effect of a south-westerly wind over the open sea that may have a considerable effect upon tidal levels at the open sea boundary. But the magnitude of this effect on the tidal range at the open sea boundary has not been made explicit in the model calculations.

For purposes of illustration a wind speed of 50 knots (25 m s-i) was selected for the same tidal levels as considered in the spring tide calculations shown in Figs 7-10. The

Modelling of tidal motion in shoaling waters 353

Figure 17. The effect of a 25 m s-l, south-westerly wind (50 knots) and computed velocity fields 3.7 h after high water.

St. Ann’s Head

Pembroke River

FigurelKTheeffectofa25ms-‘, south-westerly wind and computed velocity field at low water.

wind stress was calculated from a relationship given by Reid and Bodine (1968). It was found that the magnitudes and directions of depth mean currents in deeper water were not greatly affected. The velocity vectors are illustrated in Fig. 17 at mid-tide from high water; this figure may be compared with Fig. 8, where the effect of wind has not been included. The currents in the shallow regions show considerable changes in both magnitude and direction under the influence of wind. In the shoals of Angle Bay and the Pembroke River there are pronounced velocities in a north-easterly direction with some circulation also in the shallow water of Dale Roads at the north-west entrance of the estuary. Figure 18, which may be compared with Fig. 9, shows that large-scale circu- lations are generated particularly in shoaling regions of Dale Roads, Angle Bay and Pennar Gut. After the turn of the tide, flood currents in the mouth of the estuary are increased, even though the depths of water are greatest in this region. Circulations that were evident at low water have been reduced in this period by the flooding tide. The period between low water and high water is illustrated in Fig. 19.

The tidal levels computed at Carr Jetty show significant differences because of wind. Water levels were generally higher with a minimum level of 1.65 m below O.D. Newlyn, while the level without wind was 3.0 m below O.D. Newlyn. At high water there was a difference of 0.20 m, although when the influence of wind on reaches outside the estuary

354 D. J. Gunn t3 0. Yenigiin

St. Ann’s Head

Angle Bay

Figure 19. The effect of a 25 m s-l, south-westerly wind and computed velocity field 3.7 h after low water.

Roods

St Ann’s Head

I I Angle Bay

Figure 20. Enlargement of vector diagram at the mouth of the estuary for slack water on the spring tide (computed vectors).

is taken into account, this effect may be considerably increased. At low water it may be seen that there is some increase in the extent of mud flats in Angle Bay. At lower wind velocities the effects are reduced even in shoaling regions with wind effects becoming noticeable at velocities in excess of 10 m s-i, For this region the most critical conditions are experienced in prevailing south-westerly winds, and it appears that strong winds can have considerable effects in water level and erosion on the north shoreline of the estuary.

Silt distribution and water velocity

As may be seen from Fig. 2. the distribution of silt in the estuary lies mostly east of the western entrance of Angle Bay although, on balance, it appears that the primary origin of the fine sediment is fresh water run off into the estuary, and bed erosion. In the mouth of the estuary, grab samples have shown very little trace of fine sediment, an unusual pattern since it might be expected that there might be some deposition of water-borne solids as water velocity is reduced in the outer reaches of the estuary.

However, in the estuary, the reduction in velocity from upper to lower reaches is small, and therefore the conditions in the outer regions are not particularly conducive to the deposition of water-borne silt in the upper reaches. Indeed some features of the flow

ModellinE of tidal motion in shoaling waters 355

suggest that silt that might be expected to deposit in the outer regions at the turn of the tide may be subjected to a scouring action inhibiting deposition even at slack water.

Figure 20 is an enlargement of the high water vector diagram that shows water veloci- ties in the mouth of the estuary. The particular feature of interest is the circulation that follows from west to east in the mouth of the estuary. As the velocities even at high water are of the order of 0.2 m s- ’ in this region conditions for deposition of tine silt in the mouth would not be favourable and the effect of this current would be to carry fme silts into the main flows of the Bristol Channel. The regions dominated by this outer circulation correlate fairly closely with silt free regions in the mouth.

Acknowledgements

We are particularly indebted to the Welsh Water Authority for the surveys on which the validation of the model was based.

Notation

a, At

Ax

AY

El 6

P P, Q 0

Chezy’s coefficient variable representing u, v or h amplitude of velocity wave in equation 5 amplitude of velocity wave in equation 6 acceleration due to gravity total water depth tidal amplitude tidal elevation x-coordinate of computational node y-coordinate of computational node time level, Manning’s roughness parameter local integral coefficient time depth mean velocity component in x-direction depth mean velocity component in y-direction wind velocity longitudinal axis lateral axis

wind stress coefficient time increment grid spacing along x-axis grid spacing along y-axis phase difference angle of wind direction with respect to x-axis density of water density of air Coriolis coefficient tidal frequency

356 D. J. Gunn &?J 0. Yenigiin

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