An exploratory numerical model of sediment deposition over tidal salt marshes

Estuarine, Coastal and Shelf Science (1995) 41, 515-543

A n Exploratory N u m e r i c a l M o d e l of S e d i m e n t D e p o s i t i o n Over Tidal Salt Marshes

S. J. W o o l n o u g h % J. R. L. A l l e n b a n d W. L. W o o d ~ aDepartment of Mathematics, The University, P.O. Box 220, Whiteknights, Reading RG6 2AX, U.K. and bPostgraduate Research Institute for Sedimentology, The University, P.O. Box 227, Whiteknights, Reading RG6 2AB, U.K. Received 6 June 1994 and in revised form 9 September 1994

Keywords: tides; salt marshes; sediment movement; sediment accretion; mathematical model; numerical analysis This paper describes an exploratory numerical model of sediment transport and deposition on an idealized tidal salt marsh. The mass transport equation is simplified to make it hyperbolic in one space dimension and time. The box scheme is used for the numerical solution and this is satisfactorily checked against an exact solution for a particular case. Difficulties encountered in the numerical solution for the flood and ebb tides are discussed. The results from using the programme on sediment made up of a population of different sized particles are in qualitative agreement with observations on the Severn Estuary and the Lincolnshire and Norfolk coasts. ,~ 1995 Academic Press Limited

Introduction I n m i d a n d high latitudes, tidal salt m a r s h e s ( C h a p m a n , 1960; A d a m , 1990) are w i d e s p r e a d on the m a r g i n s o f estuaries a n d tidal e m b a y m e n t s , on b a r r i e r - i s l a n d coasts and, in places, even on the o p e n shore (e.g. D i j k e m a , 1987; B u r d , 1989). Salt m a r s h e s serve as m a j o r stores for fine s e d i m e n t a n d for c o n t a m i n a n t s released into the coastal zone, a n d are a significant ecological resource; they play a vital role in the n a t u r a l system o f coastal defence against flooding b y the sea, w h i c h is n o w rising p o s s i b l y b e c a u s e o f global w a r m i n g . T h e p u r p o s e o f this p a p e r is to e n h a n c e u n d e r s t a n d i n g o f the m o r p h o d y n a m i c s o f tidal salt m a r s h e s b y d e s c r i b i n g a simple n u m e r i c a l m o d e l for the d e p o s i t i o n o f s e d i m e n t over an idealized salt marsh. T h e p a r t i c u l a r aim o f this p a p e r is to identify the spatial variation o f s e d i m e n t d e p o s i t i o n rate a n d grain size in relation to the effective s e d i m e n t source. As this m o d e l is simple a n d e x p l o r a t o r y , a n d n o t site-specific, it clearly reveals the physical p a r a m e t e r s involved in the d e p o s i t i o n a l process a n d their roles. T i d a l salt m a r s h e s are v e g e t a t e d p l a t f o r m s t h a t exist high in the tidal frame w h i c h are m a i n l y d i s s e c t e d b y d e a d - e n d networks o f tidal channels, creeks a n d gullies (Ashley & Zeff, 1988; Zeff, 1988; Allen & Pye, 1992). T h e channel n e t w o r k s , b r a n c h i n g like a tree, 'Author to whom correspondence should be addressed. 0272-7714/95/050515+29 $12.00/0

© 1995 Academic Press Limited

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permit the flooding tide, carrying sediment with it, to enter and then drown the marsh, and the ebbing tide to withdraw so that the marsh platform once again emerges. Given appropriate forcing factors, the platforms are observed to build up very rapidly at first and then more gradually (Pethick, 1980a, 1981), while the drainage lines lengthen, narrow and deepen and the platform tends to overall horizontality (Steers, 1964; Beeftink & Rozema, 1988; French et al., 1990). The strength of flow in the marsh creek depends on whether the level of water in it is below, at or above the position of the adjoining platform (Boon, 1975; Bayliss-Smith et al., 1979). Field observations in a variety of marshes show that during the flood, the gentle flow of undermarsh tides gives way to much faster currents as the stream spreads on to and drowns the platform (Myrick & Leopold, 1963; Pestrong, 1965; Boon, 1975; Bayliss-Smith et al., 1979; Healey et al., 1981; Reed et al., 1985; Green et al., 1986; Reed, 1987, 1988; Stoddart et al., 1987; French & Stoddart, 1992). This pattern is reversed at the ebb tide. Modelling confirms that these velocity 'pulses' reflect the role of the marsh platform as a topographic threshold, at which the volume to be either filled or emptied by the tidal discharge changes by many times (Boon, 1975; Pethick, 1980b; Allen, 1994). A lag, linked to the development of a friction-related water-surface slope, is observed in the field between a velocity peak and the time when the water surface in a channel is at the height of the platform (Healey et al., 1981; Stoddart et al., 1987; French & Stoddart, 1992). Goodwin et al. (1992) successfully modelled these slopes together with the evolution of the associated wetting and drying fronts, Modelling also points to very low velocities on the drowned platform while vigorous currents sweep through the adjoining creeks (Neves, 1988). However, very few observations have so far been made to confirm this result (Green et al., 1986), and what patterns of streamlines might typify flow over marsh platforms are at present unknown, although bounds can be suggested (Allen, 1994). Recent field work has emphasized the turbulence characteristics of creek flows (French & Clifford, 1992a, b). Modelling confirms Pethick's (1980a, b) observation of the steeply non-linear build-up of a marsh platform through the tidal deposition and plant growth (Randerson, 1979; Krone, 1987; Allen, 1990; French, 1991, 1993). As developed so far, these models treat a marsh as though reduced to a point, but give helpful insights into the general response of marshes to changing sea level and to sediment autoconsolidation and vertical crustal movement. Rates of deposition of mineral plus biogenic material, typically of a few millimetres annually but ranging up to about 20 m m , have been measured by a wide variety of methods from scattered points on many marshes (for review see Allen, 1990; also Berry, 1967; Van Eerdt, 1985; Bricker-Urso et al., 1989; DeLaune et al., 1989; Knaus & Van Gent, 1989; Oertel et al., 1989; Reed, 1989; Wood et al., 1989; Dijkema et al., 1990; Orson et al., 1990; Patrick & DeLaune, 1990; Allen, 1991; Gu~n~gou et al., 1991; Kearney & Stevenson, 1991; Plater & Poolton, 1992; Anderson et al., 1993; Craft et al., 1993; Fletcher et al., 1993; Hutchinson, 1993; Kearney et al., 1993; Shi, 1993; Hutchinson & Prandle, 1994). There is much less data on spatial patterns of deposition rate (tidal silt) on marshes, measured either on transects or areally. Observations from transects and their equivalents point to a trend of declining rate with increasing distance away from the coast and away from channels (Pestrong, 1972; Carling, 1982; Stumpf, 1983; Collins et al., 1987; Reed, 1988; Oenema & DeLaune, 1988). Areal studies in small contemporary marshes show that marsh-wide as well as more local trends of decline occur (Hartnall, 1984; French & Spencer, 1993). These patterns point to the importance (French & Spencer, 1993), but not the exclusive role, of advection in the

Exploratory numerical model of sediment deposition

Water surface

',

L

Landward end .. b,\\\'%

!

'

~,~\~

Seaward 0 ~

Deep water

517

~

u

~H ~ ~,~

Salt marsh

~

~

Figure 1. The region of investigation. deposition of tidal sediment over salt marshes, and confirm that each segment of the complex channel network is effectively a ' source ' of sediment (Allen, 1994)'. Even less is understood of the variation of grain size over marsh platforms, although there are indications (Stumpf, 1983; Collins et al., 1987; Allen, 1992), as may be expected theoretically (Allen, 1994), that grain size also tends to decline (and the organic matter tends to increase) with increasing distance from the coast and from the channels. A real salt marsh will be idealized in this paper in order to try to model this complexity of morphology and dynamics and to establish the role of some of the fundamental controls on deposition rate and texture of sediments deposited. A n idealized salt marsh and its water-sediment supply A real salt marsh (Allen & Pye, 1992; Pethick, 1992) may be taken to consist of a marsh platform bounded on one side by open water or a major channel and on the other side by higher ground, e.g. outcropping bedrock or the rear of a sandy barrier island (Figure 1). The platform may be cut by a number of branching channel networks, each of which services the part of the platform immediately surrounding it. The boundaries on each network may be taken as the inner (seaward) and outer (landward) edges of the platform and the water confluence/parting between it and the adjoining networks. Just as there is a water confluence/parting between two neighbouring networks, so can a hierarchy of confluences/partings between neighbouring channel segments or groups of segments within each network be sketched. The positions of these confluences/partings may vary from tide to tide, together with the associated patterns of streamlines, and there may be flow from one channel to another. However, the existence of confluences/partings in a statistical sense is undeniable from the very occurrence of separate channel networks on marshes. Hence the essential geometry of interest is a two-dimensional, horizontal platform bounded on one side by ' d e e p ' water, from which tidal discharge issues and to which it returns, and on the other side by an actual or virtual barrier defining the limits of the tidal flow (Figure 1). This idealized marsh platform has a width of the order of I km if

518

S. .7. Woolnough et al.

the whole marsh is under consideration, falling to a scale of 10-100 m when a part of the platform lying between channel segments of a low order is of interest. The deep water lying in the open area or channel at the seaward edge of the platform is the source of the aqueous discharge over the platform, and the suspended fine sediment is taken with the aqueous discharge to be advected to the interior by the rising tide. In real estuaries and tidal embayments, as exemplified by the Severn Estuary which has perhaps been studied more thoroughly than any other estuary in Britain (Hydraulics Research Station, 1981; Kirby, 1986; Severn Tidal Power Group, 1993), the distribution of suspended fine sediment is complex and difficult to predict. The concentration varies vertically, with strong stratification evident at times, and also spatially, with a suspended-sediment front and a turbidity maximum. There are also significant temporal variations on a number of time and seasonal scales. Therefore, in this model, the concentration (Co) of suspended sediment at the seaward edge of the platform is assumed to be steady and vertically uniform. The absolute sediment fractional volume concentration in the Severn Estuary ranges in the general order 10 5-10 3 (Hydraulics Research Station, 1981; Kirby, 1986). The suspended fine sediment itself is a complex mixture of individual mineral grains, organic debris and loose aggregates (flocs) of mineral particles and organic matter (e.g. Bryant & Williams, 1983). From observations and flow modelling carried out in real salt marshes, as summarized above, it can be concluded that the high-momentum water available at the seaward edge of the platform will become ' over-charged ' with suspended fine sediment upon entering the low-momentum environment of the platform itself. The sediment particles, being unsupported by flow-derived forces, will therefore proceed to settle. This model assumes with Bagnold (1966) that these particles are travelling horizontally with the speed of the ambient fluid, and that they are settling towards the surface of the platform at their terminal fall velocity in still water. Although settling velocity varies with particle concentration (Maude & Whitmore, 1958), the effect is negligibly small at the sediment concentrations of interest here. A more substantial effect is likely to arise from the seasonal variations in water temperature and therefore viscosity, but this study will be limited to a single condition. The possible effects of residual (i.e. surviving from the high-momentum state) and new turbulence in the overmarsh tide on the settling of the particles will also be neglected. Both turbulence intensity and particle properties appear to control this effect, which in terms of present knowledge cannot be satisfactorily predicted (Nielsen, 1993). The mathematical model The generalized mass balance equation for suspended sediment is:

(1)

where C is the fractional volume of sediment in the fluid, t the time, ui the components of the fluid velocities in the coordinate directions xi, with x 3 =z vertically upwards, vp the particle settling velocity, defined as positive downwards, and ei the sediment diffusion coefficients. All notation used in this paper is summarized in Appendix A.

Exploratory numerical model of sediment deposition

519

The settling velocity of a single, smooth, spherical particle in a stagnant unbounded fluid for small particle Reynolds numbers, R e = p D v o # l , is given by Stokes' Law:

vo =

1 (~-p)gD 2

18

(2)

~/

where v o is the settling velocity of the single particle, a the particle density, p the fluid density, g the acceleration due to gravity, D the particle diameter and ~/ the dynamic viscosity of the fluid. In this paper, vp is taken equal to v o. To solve equation (1) the fluid velocities are required. These can be obtained from analytical or numerical solutions of the equations of motion of the fluid or approximations to them, and will be specific to the problem being considered. The sediment diffusion coefficients are related to the diffusivities for the m o m e n t u m of the fluid. There are many ways of modelling turbulence and calculating diffusion coefficients, based on results from both theory, e.g. Prandtl's mixing length theory (Graf, 1971) or experiments, e.g. Rajaratnam and Ahmadi (1981). These results will also be dependent on the problem being considered. The region's boundary conditions must also be specified. N o sediment may be transferred across the water surface. A known concentration profile may be specified at a boundary of the region. At a solid boundary, e.g. the bed of the region, the rate of transfer is defined by the probabilities of a particle being deposited on reaching the boundary and of a particle on the boundary being eroded. At a vertical solid boundary, it can usually be assumed that there is no transfer of sediment. At the bed of the region, the probabilities can be estimated in many ways; James (1987) ignores the possibility of erosion and defines the probability of deposition by the complement of the erosion probability defined by Einstein (1950). This paper investigates a simplified model of sediment deposition over a tidal salt marsh. For simplicity, the salt marsh is assumed to be horizontal and alongside a body of tidal water, i.e. the sea, an estuary or a major channel, straight enough and long enough that only variations in the transverse direction need to be considered. Figure 1 shows the two dimensional approximation to the region of investigation. The marsh is bounded at one end, hereafter known as the ' seaward end ' by the body of water which is the source of the sediment, and at the ' landward ' end a distance L away, by a vertical barrier. It is assumed that the water surface remains horizontal across the marsh; this implies that the tide, when it reaches the level of the marsh, instantaneously covers it and the depth of the water, H, over the marsh is a fi.mction of time alone. The simplifications made here allow the removal of one of the dimensions from equation (1). The mass balance equation for suspended sediment in two dimensions is:

~t+~x(UC3 az(~C3--~ ~XTx ~ ~=~

~vpC3

(3)

where x is the transverse direction and u and co are velocities in x and z directions respectively with the origin on the marsh at the seaward end. In order to simplify the equations to be solved, further assumptions are made about the flow over the marsh. Firstly, it is assumed that there is no vertical motion of the fluid,

S.J. Woolnough et al.

520

i.e. oJ=O. Secondly, it is assumed that the flow is non-turbulent and hence, that the sediment diffusion coefficients are zero. Finally, it is assumed that there is no variation in the vertical direction of the horizontal velocity, u. These assumptions result in a one-dimensional equation for sediment transport:

?C ? ? Ot + cn~ (uC) - ~-cz(vpC) = 0

(4)

It is further assumed that the settling velocity of each species of sediment is constant and that the water entering the region at the seaward end of the marsh has a uniform, steady concentration of sediment, Co, distributed throughout its depth. T h e b o u n d a r y condition at the bed of the region is approximated by assuming that there is no erosion from the bed and that all the suspended sediment reaching the bed is deposited on it. Equation (4) can be integrated over the depth of the flow, H(t), giving:

fo n'''pC ~ (uC)dz- fore" ~? (vpC)dz=O [~t dz + fo n''~ ~'x

(5)

Using Leibniz's rule for differentiation under an integral sign and omitting zero terms

gives: --?t

Cdz - --dt C(H(t)) + ~x

(uC)dz -

--?z (v.C)dz = 0

(6)

Assuming that at (x,t) the sediment is distributed through a column of height y with a uniform concentration, i.e.:

c = { C o if O _ < z _ y if y < z < H

(7)

?C ?z - - Co6(y)

(8)

implies

where d(y) is the Dirac delta distribution (see e.g. W o o d , 1993). T h e form of the concentration profile given by equation (7) also implies that at every point except x = 0 , the concentration at the water surface is zero, so that for x>0, dH/dt C(H(t))=O, which means that equation (6) becomes:

?'t + ~ (uy) + vp = 0

(9)

Equation (9) is the continuity equation for the suspended sediment that will be used in this paper to provide a numerical simulation of the deposition of sediment on a salt marsh. Only values o f y > 0 have any physical significance in the solution of this problem. In order to solve equation (9), the velocity over the salt marsh is required. Since it is assumed that the sea surface is always horizontal, if u(x,t) is the horizontal velocity of the

Exploratory numerical model of sediment deposition

521

water at a distance x along the marsh when the depth is H(t), from Figure 1 the rate at which water enters the right hand section of the region is given by: d

uH=--:-at [ ( L - x ) H ] = ( L - x )

dH d--T

(10)

T h e depth of the water above the marsh is given by the height of the salt marsh above mean sea level subtracted from the height above mean sea level of the water in the main channel, which is governed by the tidal regime. Assuming a sinusoidal tide, the depth of water above the marsh is given by:

H(t) = Asin(oo(t+to) ) - b

(11)

where b=Asin(o2to) is the height of the marsh above mean sea level, A the tidal amplitude, one-half the tidal range, and T=2z~/co the period of the tide. T h e time, t, is defined such that t= 0 when the water instantaneously covers the marsh. The marsh will be covered by the tide whilst: O
(12)

U)

An analytic solution The model equations can be solved analytically for the case where b=0, i.e. H=Asin(o2r), by the method of characteristics described by Wood (1993). This analytical solution is used to check the results obtained from the numerical solution. From equations (10) and (11):

u = (L - x)co cot(co0

(13)

and from equation (9):

u

(14)

The characteristics of equation (14) are given by:

dx

L-xdH

--=u dt

H

dt

(15)

Integrating equation (15) gives:

foXLl~_x

dx =

f~

~lId H

(16)

o

where Ho=Asina , i.e. at x = 0 , a=cot. Hence the equation of the characteristic lines (Figure 2) is:

x=L (1

sin-

sin (tot)}

(17)

522

S.J. Woolnoughet al.

0.5

0.4

0.3

c5

c5

~

~

w

~

~

0 o 0

c5

0.2

0.1

i

0

0.25

0.5

x/L

i

I

0.75

i

i

i

1.0

Figure 2. Characteristics of the model equation with b= 0. b, height of salt marsh above mean sea level; t, time; T, tidal period; x, horizontal coordinate, L, length of salt marsh. On the characteristic given by equation (17):

dy_ ( y g ' U ) dt

(18)

?~+vp

which implies from equation (13) that:

@ d t - (~o cot

oJt)y=

- vp

(19)

Multiplying equation (19) by a factor cosec (¢vt) gives: d(

y

)=

dt s~n-(o)t)

vp sin (wt)

(20)

Given that, for 0 < t_< ~z/2 at x= 0, a = o~t and y =Asina, integration of equation (20) gives: y = A sin 0ot) where a is given by equation (17).

[

% (tan ~ 1-~o.~ In \ ~ / / ]

(21)

Exploratory numerical model of sediment deposition

-(a)

523

- (b)

5.0 0.5 o~ 2.5

co 0.25

~

I

25 x (m)

50

_

0

25 x(m)

50

Figure 3. Equivalent thickness of sediment deposited on the marsh (S) by one tide with b=0, A = S m , L = 5 0 m , T=12h, Co=3× 10 3 with (a) % = 3 x 10 4ms 1, (b) vp= 3 x 10 - 5 ms - i. b, height of salt marsh above mean sea level; A, amplitude of tide; L, length of salt marsh; T, tidal period; Co, fractional volume of sediment in tidal water; vp, particle settling velocity; x, horizontal coordinate.

T h e rate o f d e p o s i t i o n ,

R(x,t),

o f s e d i m e n t o n the salt m a r s h is g i v e n by:

R={ C°vpify>Ootherwise

(22)

so, in o r d e r to b e able to d e t e r m i n e t h e d e p t h o f s e d i m e n t d e p o s i t e d o n t h e m a r s h , t h e t i m e , rE(x), at w h i c h y = 0 is n e e d e d . T h e p o s i t i o n , xE(t), at w h i c h y = 0 c a n b e f o u n d f r o m e q u a t i o n (21) b y s u b s t i t u t i n g an e x p r e s s i o n for a f r o m e q u a t i o n (17), giving:

xE(x) =

L

{

1

sin [2 a r c t a n { e - Y , tan (~)] sin (~ot)

}

(23)

R e a r r a n g i n g a n d u s i n g t r i g o n o m e t r i c i d e n t i t i e s gives: 2

rE(x) = -

[e,-V~ (1 - ~ ) - 1] A~ [ 1 - e - ~-w ( 1 - ~ ) 1

arctan

oJ

(24)

It m u s t b e n o t e d t h a t for x > L [ l - e A,o/vp],the e x p r e s s i o n d o e s n o t h o l d as t h e r e is n o t i m e for w h i c h y = 0 at t h e s e x values. F r o m e q u a t i o n (22), t h e d e p t h o f s e d i m e n t , S, d e p o s i t e d b y o n e tidal s e q u e n c e is g i v e n by:

S(x) = lr

~, - 2t 0

j0

R(x,t)dt

(25)

w h i c h gives: S(x)=

CovptE(x)

(26)

524

S.J. Pgoolnough et al.

The ' depth ' of sediment deposited is shown in Figure 3 for two different values of %. Assuming a voids to solids ratio of 0-5, the equivalent sediment thickness is obtained by multiplying by 1"5. The n u m e r i c a l m o d e l The tidal period over which equation (9) is solved can be split into two distinct parts. The first part is whilst the tide is rising and water, full of sediment from the channel, is coming into the region over the salt marsh. The second part is after the tide has turned and the water, possibly with some of the mainly finer sediment remaining, is flowing out of the region. Equation (9) could be solved approximately by the numerical method of characteristics which would involve the repeated numerical solution of the characteristic equation (15). This could be done by using either an explicit method, which would create difficulties because of the need to satisfy the stability condition of the method, or by using an unconditionally stable method, which would involve the repeated solution of a non-linear equation which could be time-consuming. For these reasons, the numerical method of characteristics was not used to solve the equations in general. However, the method is helpful in the solution of the equations during the ebb tide, where it is used once at each time level to provide one height, y, of the column of sediment at the new time level. Apart from this, the box scheme (Preissmann, 1961) is used to solve the equations numerically. The salt marsh is divided into M equal sections of length A x = L / M with xi=jAx. The time discretization is done by using 2 N equal time-steps, At= Qr/2~o - to)/N , with t,, = nAt. The solution y(xj, t,) is approximated by y~'and [uy]y =u(x~,t,)y~'. The depth of sediment deposited, S(xi, t,), on the marsh is approximated by S~" When the box scheme is applied to equation (9), the time derivatives are approximated by a weighted average of the finite difference form at two spatial points: ~y (1 -~o) ~0 . --~ (y~j + ' - y T ) + (y~/++~ - y 1 + ), ?t At ~t '

(27)

and the space derivatives are replaced by a weighted average of the finite difference forms at two time levels:

~-cx(uy)-~

0 ([uy]~+ I -- [uy]~) + ~xx ([uy]"i:~--[uy]~ +l)

(28)

where 0 and ~0 are user-defined parameters between 0 - I . The box scheme is second-order accurate with ~o=0=0"5. With ~0:~0'5 it is only first-order accurate in time, and with (p:~ 0'5 it is only first-order accurate in space. The box scheme is unconditionally stable for ~o,0> 0"5 (Wood, 1993).

The flood tide During the period of time in which the tide is flooding the marsh, the height of the column of sediment at the seaward boundary is known from equation (11). This means that if the box scheme is used, stepping along the region from the seaward end, then there is only one unknown in each equation. Since the differential equation is linear in

Exploratory numerical model of sediment deposition

525

y, i.e. the velocity, u, is independent of y, the equation p r o d u c e d is linear and can be solved easily. Equation (9) is discretized using the box scheme with the parameters set at 0=q~=0.5. Rearranging the finite difference form, to solve for Yn+ i+ 1 explicitly gives:

,,,,+1 .Y j+

[YJ"-yj'+I+yJ'+,IAx+[[uyl'-[uy]'~+~+[uy]'+1]At-2vpAtAx""

1 --

"~

"

""

"

""

"

(29)

[Ax + u(xi+ l,t,,+ 1)At]

This equation is then solved for 0 < j < M 1 to step along the region from the seaward end for each time step O < n < N 1, until the turn of the tide. A numerical approximation x'/ to xE(t,,) is obtained by using linear interpolation between the two points across which the sign o f y y changes. At each time level, there will be a JE such that y'j~->O and y~.~..+~ <0 and x~ is defined by:

x~ =jEAx +

3~jEAx

(30)

Y~E--YTE+, T h e approximation to the thickness of sediment deposited, S~', is u p d a t e d at each time level using the following algorithm:

Sj + CvpAt

if xj < x~+ 1

) ,, ( x ~ - x j ) C v v A t S~ + 1 = ~Sj + x ~ - x~ + ~ |"S~

- - -

.~ ,,+1< - , , 11 x~ _ xj ~ xE

(31)

otherwise

The initial time step Initial values o f y are given at t = 0 . Initial values of uy are not so obvious; initially from equation (10), u is infinite and the product, uy, is not so simply evaluated. T w o m e t h o d s for dealing with the p r o d u c t uy were examined. Firstly [uy]°was set to zero for all j. T h e second m e t h o d was based on the assumption that for very small t, the horizontal velocities are large c o m p a r e d to the settling velocities of the sediment. This means that for very small t, the c o l u m n of sediment in the water is approximately the same depth, H, as the water. T h e n , using the expression for the velocity from equation (10), the p r o d u c t uy, for small t, is given by: •. . , - x d H uy . . . . H dt

H=(L-x

dH dt

(32)

Both these expressions for uy were tried for the initial nme-step in the numerical scheme applied to the problem with b = 0 and the results c o m p a r e d with the analytic solution for this case. It can be seen from Figure 4(a,b) that neither of these initial conditions gave results which agreed well with the analytic solution and so both m e t h o d s were discarded. However, the p r o d u c t uy only appears in the space-derivative in equation (9), so if the box scheme is used with 0= 1-0 instead of 0=0.5, then an expression for uy(x,O) is not

S.J. Woolnough et al.

526

(a)

(b)

(e)

0.5

0.5

0.5

0.25

0.25

0.25

t

o

i

,,i,~ 0 25 50 0 25 50 25 50 x (m) x (m) x (m) Figure 4. The position of xE for A=Sm, vp=3x 10-4ms - I and T=12h with different assumptions for the initial time step as explained in the text. xe, position at which no sediment is suspended above the marsh; A, amplitude of tide; vp, particle settling velocity; T, tidal period; t, time; x, horizontal coordinate.

needed. In o r d e r to preserve the s e c o n d - o r d e r accuracy o f the m e t h o d for the other time steps, 0= 1.0 is only used for the first time step a n d for s u b s e q u e n t time steps the original 0 = 0 . 5 is used. T h i s m e t h o d can be seen to provide sufficiently accurate results b y c o m p a r i n g the solution o b t a i n e d with the analytic solution [Figure 4(c)]. Using this m e t h o d , the b o x scheme for the first t i m e - s t e p b e c o m e s : Y"+' j+l

--

[y,~_y~+ l + y,~+ ,]Ax + 2[uy]~+ i A t _ 2vvAtAx Ax + 2u(xi+ 1,t,, + 1)At

(33)

The landward boundary W h e n the m o d e l is run with the smaller particle sizes in the range for which the s c h e m e is required here, the a p p r o x i m a t i o n Y~t to the height of the c o l u m n of s e d i m e n t at the l a n d w a r d end of the region is positive. T h e analytic solution gives y - ~ - c6 as x ~ L a n d this implies that at the very e n d of the region, there can be no s e d i m e n t s u s p e n d e d above the marsh. F o r this reason, if the a p p r o x i m a t i o n y ] t > 0 then the n u m e r i c a l m o d e l sets its value to zero. If the value were set to a large negative n u m b e r to m a t c h the analytic solution m o r e closely, the linear interpolation used to find the a p p r o x i m a t i o n x~. to xLwould be biased towards the p o i n t x M ,. In the cases where this difficulty at the l a n d w a r d end occurs, the solution to the e q u a t i o n has a very steep g r a d i e n t near the e n d of the region and so the linear interpolation b e t w e e n the two points is n o t a valid m e t h o d of obtaining the position of x~.. Since the d e p o s i t i o n on the m a r s h is defined by the position of x~., it is m o r e i m p o r t a n t to have this value accurately evaluated t h a n to have the height of the c o l u m n at the end of the region accurately defined, b e c a u s e only values of y > 0 have any physical significance in this p r o b l e m a n d the value o f y ~ affects the solution at s u b s e q u e n t time levels only at x~vI and, therefore, does n o t affect the solution in the interior of the region.

The ebb tide D u r i n g the ebb tide, the b o x scheme c a n n o t be used quite as simply as d u r i n g the flood tide. Since there are no b o u n d a r y data, no values are k n o w n at the n e w time'level a n d it is therefore necessary to evaluate at least one of the heights, y j~ t + l at the new t i m e level by some other m e t h o d . If a position towards the l a n d w a r d e n d o f the region is chosen, then this new value can be used like a b o u n d a r y c o n d i t i o n a n d the b o x s c h e m e can be used to step towards the seaward e n d from this p o i n t in the same way as it is u s e d d u r i n g the flood tide.

Exploratory numer@al model of sediment deposition

527

Several points need to be considered when calculating this ' b o u n d a r y value ', the most important of which is that in order to get a complete solution for the part of the marsh which still has sediment suspended above it, the first value at the new time level must be in the region where there is no sediment suspended above the marsh. One possible method of obtaining the first value at the new time level is by using the implicit three-point scheme obtained from the box scheme by setting the parameters to O=0, ~0= 1. Remembering that we are now stepping in the negative x direction and that u is negative, this gives: .+, _y~'_ ,Ax + ([uy] ~_ ,-- [uy]y)At-vpAxAt Y~- ~ Ax

(34)

There are two main difficulties with this method; firstly the explicit scheme given in equation (34) is only conditionally stable, which will mean that dx, dt will have to be varied to provide stability. Secondly, once a value has been obtained at the new time level, a check will have to be made to ensure that the value is negative and if not a new Ax, dt chosen to secure a negative value. An alternative method is to trace along the characteristic from a point at the old time level, n, where the solution is non-positive to the new time level and evaluate a numerical solution at this new point. This method guarantees that the value at the new time level will be negative and does not create the same problems over choice of Ax anti At. It does, however, require the characteristic equation (15) to be solved numerically, which may itself create difficulties with stability conditions. The method used here was to trace the characteristic from the point where the numerical solution is zero, x~., to the next time level, n+ 1, by numerically solving the characteristic equation:

dx dt =

L-xdH H

de

(35)

and simultaneously to update the values o f y on this characteristic from the equation:

y(yOU) de

-~x + %

(36)

by use of a numerical method. T o avoid problems with stability, the trapezium rule was chosen because it is unconditionally stable. Equations (35) and (36) are integrated from t~ to r,+l using P time steps dtc=At/P. The position of the characteristic through x~ is given by x~(t) and is approximated by at a time tp=t,,+pAt~. The trapezium rule then gives: X~¢+ 1 ----- X,~C+ -1 / I t c [ u ( X ~ c +1 ' ~ p + 1) + U (xpc,~p) l

(37)

This equation is solved iteratively by making an initial guess: ~ + , xo) = # + , a t c u ( G , tp)

(38)

528

S.J. Woolnough et al.

and then using the iteration:

~+,,~+ ,~=~+ ½atAU(~c "(*~,t~+,) + u(~,tA]

(39)

until two consecutive approximations differ by less than a specified tolerance. If the value of y on the characteristic is approximated by y~, then application of the trapezium rule to equation (36) gives: p yp + 1 =YP -- [vv + 12Y~p ux(x~,to)] At~ [1 + ½At~u.~(xV~ + l,t~+ 1)]

(40)

With ~ : and yl, it is possible to use the box scheme to integrate along the region towards the seaward end. T h e first step must be done with a steplength defined by Ax'=x~, - x l c where j, is such that xjc
n n P P y,j,+ 1 _- [Ys~. +Y~'-Y,"p ]Ax i + [ [uy] j~.u(x~P ,t,,)y,.,- u(x,.,t,,+ A t - 2vpAtAx' 1)Y,.] [ A x ' - u(xj,t, + l)At]

(41)

For subsequent horizontal time steps the box scheme is given by:

y~j+~ _ [y~j-, +y~-y~j + 1lAx+ [[uy];- l -- [uy]"j- [uy]; + ' ] A t - - 2 v p A t A x ( A x - u ( x j _ ~,t, + ~)At

(42)

Ify(O,t,,+ 1)>0 then the position of x~.+~ and the thickness of deposited sediment are evaluated in the same way as for the flood tide. The seaward boundary If the characteristic being traced goes out of the region, i.e. x c < 0, then the integration of the characteristic equation stops; the time at which xc=0 and the value of y, are approximated using linear interpolation between tp and tp+ 1. In this case or ify~;+a<0, the m e t h o d evaluates the time, ts, at which y(O,t)--0 by using linear interpolation along the time axis at x = 0 , updates the deposited sediment thickness for the shortened time step, ts - t,, and stops.

The final time step If the numerical solution continues until t=Tr/co-t o without giving y ~ + J < 0 then the method forces Yo2 N _ -_ 0 and interpolates between x~ N - l and zero to evaluate the deposition for the final time step.

Exploratory numerical model of sediment deposition

529

3

2

1

0

10

20

30

40

50

x(m) Figure 5. The numerical solution for y at t=4.5 h when A = 5 m, L= 50 m, T= 12 h and vp=3 × 10 ~ms ~withM=N=4OO.y, heightofcolumnofsedimentinwater;t, time; A, amplitude of tide; L, length of salt marsh; T, tidal period; vp, particle settling velocity; M, number of space steps; N, half the number of time steps; x, horizontal coordinate. Numerical results

T h e n u m e r i c a l m e t h o d was tested b y using it to solve the p r o b l e m with the analytic solution given in e q u a t i o n s ( 2 1 ) - ( 2 6 ) . T h e testing was d o n e with the p a r a m e t e r s o f the p r o b l e m set as; A = 5 m, L = 5 0 m, T=12h and vp=3x 10-4ms i and vp=3x 10-Sms 1 c o r r e s p o n d i n g to q u a r t z density particles at o r d i n a r y t e m p e r a t u r e s o f a p p r o x i m a t e l y 20 ~tm a n d 5 ~tm d i a m e t e r . T h e s e two values were c h o s e n b e c a u s e they give solutions with different forms [Figure 2(b)], b o t h o f w h i c h the m e t h o d m u s t be able to r e p r o d u c e . T h e n u m e r i c a l solution is visually a l m o s t i n d i s t i n g u i s h a b l e from the analytic solution. It does s m o o t h the c o m e r , at x=L, for the case with v p = 3 x 10 5 m s - 1 w h i c h c o r r e s p o n d s to the n u m e r i c a l s o l u t i o n u n d e r - e s t i m a t i n g x E. T h i s is b e c a u s e the linear i n t e r p o l a t i o n u s e d to find x E u n d e r estimates the steepness o f the g r a d i e n t in the s o l u t i o n for y at the very e n d o f the region. D u e to the high curvature o f y on the integration for the e b b tide, small oscillations which are n o t in the analytic solution are g e n e r a t e d b y the s c h e m e ( F i g u r e 5). T h i s p h e n o m e n o n is n o t u n c o m m o n w h e n s e c o n d - o r d e r s c h e m e s are u s e d to solve h y p e r b o l i c p r o b l e m s in which there is a steep gradient. Since the oscillations r e m a i n small a n d the d e p t h o f s e d i m e n t d e p o s i t e d is d e p e n d e n t on the p o s i t i o n o f x E, w h i c h is n o t affected b y these oscillations, no action n e e d s to be taken to c o u n t e r this. M e a s u r e m e n t s o f the error b e t w e e n the n u m e r i c a l s o l u t i o n a n d the analytic solution o f the p r o b l e m were m a d e . A n a v e r a g e d absolute error in the equivalent thickness o f d e p o s i t e d s e d i m e n t , Es, is defined by:

1

ES=M+ 1

/M (S2N ~ J -S(xs))2 i=0

(43)

T h e error in the solution was e v a l u a t e d for varying space a n d t i m e discretizations, i.e.

M , N , for the p a r a m e t e r s given above, a n d the results are s h o w n in F i g u r e 6. T h i s brings o u t s o m e i m p o r t a n t points a b o u t the error in the equivalent thickness o f s e d i m e n t

S..7. Woolnoughet al.

530

(a)

M=50

.M=50

-7.5~ -

M= 100

-7.5 ~

-

--...._

M = 200

~

M = 400

~

-10.0

100 M = 200 M = 400 M = 800

M

=

- 1 0 " 0 ~ 1 , ,

M = 800

,I,,,, 4

5

0

6 ln(N)

5

6 ln(~

Figure 6. The variation of error, E, with N for fixed values of M, with (a) vp=3x 10 4ms i and (b) %=3x 10 5 m s - 1 N, half the number of time steps; M, number of space steps; %, particle settling velocity. deposited; firstly, it can be seen that for fixed M the error can not be reduced greatly, if at all, by increasing the value of N b e y o n d that of M. Secondly, for a fixed value of N t h e error improves if M is increased, even beyond the value of N, although this effect is reduced for M>>N. These two properties show that the error can only be improved by reducing the time step if the space step is already small, but that the error can be improved by reducing the space step, even for large time steps. F r o m this it can be concluded that the error in the sediment thickness is dominated by the error due to the space discretization. It is possible to obtain an estimate for the order of accuracy for the scheme by calculating the gradient of the graphs of ln(E,) against In(N) and in(M). T h e results show that the scheme is at least first order in time and space, i.e.:

(1)

(44)

= O(•t) + o(Llx)

(45)

Es~.O ~

+0

T h e scheme used is first order because although the second-order box scheme is used to solve for y, linear interpolation which is only first order is used to calculate the approximations to x E and hence the scheme becomes first order overall.

Experiments T h e model was used to examine how the equivalent thickness of sediment deposited on the marsh varies with the parameters A,b,%,L for a fixed tidal period T= 12 h. T h e problem is non-dimensionalized by making the following changes of variables in the equations: Horizontal coordinate x ' = -

X

L

Height of the column of sediment y' = A

T i m e f = tot

(46)

(47)

(48)

Exploratory numerical model of sediment deposition

531

H Height of water above marsh H' = ~

(49)

b Height of the marsh above sea level b' = - A

(50)

Phase of tide at which water crosses the marsh go= O~to S Equivalent thickness of sediment S' = A , - - Vp

Particle settling velocity Vp Aco

Horizontal fluid velocity u ' =

U

Lw

(51)

(52)

(53)

(54)

Substituting these into the model equations leads to a non-dimensional problem defined by the equation:

~Y'+ ~ (u'y') +v'.~0 ?~

(55)

?x'

with the boundary condition: /[

y'(O,t')=H'(f)

for all O<_t'<~-to

(56)

where:

u'(x',t') =

1 --X'

dH'

/-/'

df

H ' ( t ' ) = s i n ( t ' + t'o) - b'

(57)

(58)

From this non-dimensional form of the equation, it can be seen that the solution is dependent on only two non-dimensional parameters, b/A and vp/Aco. Holding A,b,L and co constant and varying vp shows how the thickness of sediment deposited on the marsh depends on the settling velocity. Figure 7 shows that the sediment with small settling velocities is uniformly distributed over the salt marsh and that as the settling velocity increases a gradient in the sediment thickness develops across the marsh and the deposited sediment does not extend across the whole width of the marsh. Figure 8 shows that increasing the ratio b/A reduces the sediment thickness for a given concentration and also caused sediment of a given settling velocity to be less evenly distributed over the marsh.

S. ft. IVoolnough et al.

532

(a)

~0.25

-(b)

'

0.5

25 x (m)

25 x(m)

50

:(d)

(e)

1

r~

50

o~ 1 I

0

25 x(m)

I

i

50

0

25 x (m)

50

Figure 7. The equivalent thickness of sediment deposited (S) for A = 5 m, b=3 m, L=50m, T=12h and Co=l x 10 3 with (a) v p = 3 x 10 Sins- l, (b) v p = 7 x 10 5 m s - l , (c) v p = 2 x 10 4ms 1 and (d) v p = 3 x 1 0 - 4 m s '. A, amplitude of tide; b, height of salt marsh above mean sea level; L, length of salt marsh; 7", tidal period; Co, fractional volume of sediment in tidal volume; vp, particle settling velocity, x, horizontal coordinate. (a)

(b)

- (c)

1.0 1.0

E

E 0.5

I

I

t

i

I

i

25 x (m)

i

t

1 50

0.5 0.25 L

0

I

t

~

I

I

25 x(m)

t

50

0

25 x (m)

50

Figure 8. The equivalent thickness of sediment deposited (S) for A = 5 m, L=50 m, t = 1 2 h , Co=l x 10 -3 and vp=8× 10 5ms -1 with (a) b = 2 m , (b) b = 3 m and (c) b=4 m. A, amplitude of tide; L, length of salt marsh; t, time; Co, fractional volume of sediment in tidal water; vp, particle settling velocity; b, height of salt marsh above mean sea level; x, horizontal coordinate.

S e d i m e n t s u s p e n d e d i n t i d a l w a t e r is n o t m a d e u p o f p a r t i c l e s o f a u n i q u e size a n d s e t t l i n g v e l o c i t y b u t o f p a r t i c l e s o f m a n y d i f f e r e n t sizes w h i c h f o r m a h y d r a u l i c a l l y d e t e r m i n e d p o p u l a t i o n . T h i s m o d e l c a n b e u s e d t o i n v e s t i g a t e t h i s c a s e as it h a s b e e n assumed that each species of particle behaves independently and, therefore, the solution for e a c h p a r t i c l e size is i n d e p e n d e n t o f t h e o t h e r s . T a b l e 1 s h o w s t w o s y n t h e t i c populations of suspended sediment used in the numerical model. The choice of p o p u l a t i o n is d i s c u s s e d later.

Exploratory numerical model of sediment deposition

533

TABLE 1. Properties of experimental sediment populations Settling velocity ( m s - ') Fraction of total Class mid-point sediment concentration (a) Coarse population (b) Fine population SD +3.0 +2-5 +2.0 +1"5 +1.0 +0-5 0 -0"5 - 1"0 - 1"5 -2"0 -2"5 -3"0

0'0022 0-0088 0-0270 0.0648 0.1211 0.1762 0.1997 0.1762 0-1211 0"0648 0"0270 0"0088 0"0022

2.511 x 2"310 x 2.109 x 1-908 x 1.707 x 1"506 x 1.205 x 1-105 x 9"038 x 7"030 x 5"021 x 3"013x 1"004 X

10 - 3 10 - 3 10 - 3 10 - 3 10 3 10 - 3 10 - 3 10 - 3 10 4 10 - 4 10 - 4 10 4

6.276 x 5-774 x 5.272 x 4.770 x 4-268x 3.766 x 3.264 x 2.761 X 2"259 X 1"757 x 1"255 x 7-530x 2.510x

10 -4

10 - 4 10 - 4 10 - 4 10 - 4 10 - 4 10 - 4 10 - 4 10 -4 10 -4

10 - 4 10 4 10 - ~ 10 5

,10 -3 0.01 5.0 r~

0.005

10

,,,, 30 40

20

50

0

10

20

X

,10

50

-4

2

0.0025 L

1

I ,

0

40

30 X

10

20

',

30

,

I

,

,

40

50

0

10

X

I

,

20

I

,

30

I

,

40

50

X

Figure 9. T h e equivalent thickness of sediment deposited (S) for the coarse population and (a) b = 3 m, (b) b = 4 m, (c) b=4.5 m and (d) b=4.75 m. b, height of salt m a r s h above m e a n sea level; x, horizontal coordinate.

F i g u r e s 9 a n d 10 s h o w t h e t h i c k n e s s mean

deposited

for values of b (height of marsh

t i d e l e v e l ) e q u a l t o 3, 4 , 4 . 5 , 4 . 7 5 m f o r p o p u l a t i o n s

As before, assuming multiplying

these

a voids to solids ratio of 0.5, the actual thicknesses results

by

settling velocities, defined by

1.5. Figures

11 a n d

above

(a) a n d Co) g i v e n i n T a b l e

12 s h o w

the

are obtained

corresponding

1. by

mean

S. ft. Woolnough et al.

534

1 1

0

I0

20

30

40

0

50

10

2O

3O

4O

50

X

X

*10-3 ~ e ) ~

l I l l l l ~ j l 0.5

5 " 1 0 - 4 f~- , ~I , I ~

,

I

,

I, 0

10

20

30

40

50

10

0

20

30

40

50

x

X

Figure 10. The equivalent thickness of sediment deposited (S) for the fine population and (a) b=3 m, (b) b=4 m, (c) b=4.5 m and (d) b=4.75 m. b, height of salt marsh above mean sea level; x, horizontal coordinate.

(a)

(b)

:~ 0.001

=

~ 0

I , I , I , I,x-r-10 20 30 40 50

0.001

I 0

10

20

x

40

,) 50

X

(c)

(d)

z~ 0.001

0.001

_

0

30

,

10

20

30 X

i

40

i

, I i,

i

50

0

10

20

30

40

50

X

Figure 11. Mean settling velocities (V) for the deposits shown in Figure 9. x, horizontal coordinate.

Exploratory numericalmodel of sediment deposition

535

(a)

(b)

2 . 5 . 1 0 -4

2 . 5 . 1 0 -4

, 0

I , I 10 20

30

40

10

50

20

(c)

40

50

30

40

50

(d)

= 2 . 5 , 1 0 -4

= 2.5,10-4

,

0

30 x

x

I

10

,

I

,

20

I

30

,

I

40

~

50

0

10

20

x Figure 12. Mean settling velocities x, horizontal coordinate.

(v)

for the deposits shown in Figure 10.

= ~i" diviS

(5 9)

where d i is the depth of species i deposited in the marsh and v i is the settling velocity of species i. S e d i m e n t d e p o s i t i o n o v e r real salt m a r s h e s

This model, developed for a single sediment species and then generalized to an arbitrary population of mixed settling velocity values, makes no claim to detailed physical realism, but is sufficiently comprehensive and simple to clearly reveal the role of the main variables that govern sediment deposition over natural salt marshes. T h e essential process is the advection of sediment with the tidal waters that first flood over and then drain from a marsh. T h e introduction of appropriate non-dimensional variables [equations (46)-(54)] shows that, whereas the velocity field over an idealized marsh is governed by the non-dimensional parameters x / L (4-4) and ogt (4-6), the sedimentation field (deposition rate, grain size) depends on the height ratio b/A and the velocity ratio vp/Am. T h e values of u and S (single and mixed species) and of ~p (mixed species) all diminish landward over a marsh with increasing relative distance x / L from its seaward margin (Figures 7-12). Similarly, these quantities will decline landward with increasing distance from some channel within a marsh, the effective source of the sediment being carried to the adjoining sector of the marsh. T h e other effect of the principal boundary condition is to ensure that the spatial gradients of these quantities all steepen as marshes

536

S . J . Woolnough et al.

diminish in width and as the spacing of the creeks within the marsh becomes less. These general conclusions are supported by observations from real salt marshes. Most spatial studies of sediment deposition rate and grain size over real salt marshes have taken place over transects or what amounts to them. Pestrong (1972) measured a silt content of 15-25% from marsh sediments over a profile 135 m long in San Francisco Bay. The mean grain size of the deposits ranged from 8.9tp (~0-- - log2D, where D is the grain diameter in millimetres) at the seaward edge of the marsh to 10-5~0 at the landward margin. Elsewhere in the bay, Collins et al. (1987) observed a three-fold increase in the organic content of the marsh sediments, pointing to a ' landward ' decline in both grain size and deposition rate of the imported, mainly non-organic debris. A declining trend of deposition rate landward was measured by Carling (1982) from a salt marsh in the Loughor Estuary, South Wales, both over 500 m from the marsh front and over 150 m from the bank of a major creek. Similar trends with respect to creeks, steepening as these channels became more closely spaced, typify the marshes around Bridge Creek on the Dengie Peninsula in eastern England, where there is a general decline away from the seaward edge of the system related to height and flood frequency (Reed, 1988). The levees of creeks in the Rattekaai Marsh of the eastern Scheldt Estuary (The Netherlands) were building up about twice as fast as the ' backmarshes ' some tens of metres distant (Oenema & DeLaune, 1988). Stumpf (1983) measured a decrease in grain size from the levee adjoining a major creek (>16 ~tm, 32%; 4-16 ~tm, 26%; <4~tm, 42%) to a backmarsh some 70 m away (>16 ~tm, 25%; 4-16 ~tm; 28%; <4 ~m, 47%) in the Delaware River estuary. Two-dimensional spatial studies, attempted by few workers, also lend support to the model. Harmall (1984), in a pioneering study, measured deposition at 81 points distributed over a small (c. 25 ha), largely enclosed marsh drained by a fourth- or fifth-order creek system at the entrance to The Wash in eastern England. Little deposition, possibly attributable to wave effects, was observed in a broad zone along the seaward margin of the marsh, but landward from this occurred an irregular decline in the rate. The largest additions were found in zones bordering the higher-order creeks. Deposition over a somewhat larger (c. 54 ha), mainly enclosed marsh in eastern England, but drained by two creek systems, was measured by French and Spencer (1993). The accretion rate declines exponentially with increasing distance from the larger creeks. Two overall trends can be discerned. On a large area of marsh, drained only by a number of small, independent creeks, the rate decreases landward over a distance of about 300 m. In the main part of the marsh, drained by the two channel systems, the maximum deposition rate occurs some distance from the seaward margin, as in Hartnall's (1984) New Marsh. Two-dimensional grain-size studies have been undertaken on large reclaimed marshes in the Severn Estuary alone (Allen, 1992). Grain size declines steeply over a marsh about 1 km wide in the inner estuary, but the chief process of sediment transport here may have been diffusion rather than advection. A marsh 2'5 km wide in the middle estuary reveals only weak reductions related to the seaward boundary and to a major channel. This exploratory model was designed to reveal the role in salt-marsh sedimentation played by a number of fundamental controls and deliberately ignored many significant natural circumstances and real physical effects. Real salt marshes vary in cross-section between convex-up, through horizontal-planar (as in our model), to concave-up, partly as an expression of increasing maturity but, in some cases, as a reflection of the shape of the surface on which the continuously

Exploratory numerical model of sediment deposition

537

consolidating marsh was built. Hence in the real marshes to which the results of this model are compared, local effects have arisen due to differences m marsh height and frequency of inundation (e.g. French & Spencer, 1993). Furthermore, the important effects likely to arise from seasonal variations of water temperature and viscosity (a seasonal variation between c. 5 °C and c. 20 °C is typical in Britain) have not been explicitly observed, despite claims of seasonal lamination in salt marsh deposits (e.g. Allen, 1990). Implicitly, however, these experiments with particle populations described in settling velocity terms amount to such an examination. A given distribution of settling velocities will be specified by a population of larger grains at a low temperature, when the viscosity is high, than at a higher temperature, when the viscosity is significantly less [equation (2)]. Hence, in Britain, ' w i n t e r ' layers should be coarser grained than those of summer. T h e next stage in the modelling of these important coastal environments should see the incorporation of greater physical realism. It will be necessary, for example, to model the friction-related evolution of the tidal wetting and drying fronts and associated water-surface slopes on the marsh, and the complex evolution of the boundary layer and (largely decaying) turbulence, with which the advective transport of sediment is strongly coupled, as the overmarsh flow, transferring debris and m o m e n t u m from the highenergy channel environment, flows first through, then over and finally through (again) the tall and dense vegetation that typically covers marshes. T h e as yet uncertain, time-dependent properties of the boundary layer will be crucial to these endeavours.

Specification of the sediment populations In order to exploit this model as realistically as possible, numerical experiments were performed with two populations of mixed sediment species, such as occur in the tidal waters affecting real salt marshes. T h e fine suspended sediment in tidal waters has a complex composition (e.g. Eisma & Kalf, 1979; Bryant & Williams, 1983), consisting typically of clay mineral flakes and needles, quartz silt and organic detritus loosely aggregated in floccules, individual quartz sand and silt grains, whole to partly disintegrated faecal pellets and larger organic particles. M a n y factors influence the settling of these unstable mixtures (For reviews, see McCave, 1984; Eisma, 1986; Van Leussen, 1988; Nielsen, 1993), so the measurements are of limited reliability (Pulset al., 1988; Kineke et al., 1989) and predictions uncertain. Modal particle diameters, however, range in the field from a few microns up to about the silt-sand boundary (63 ~tm) and only toward the coarser end of this spectrum does the modal diameter of the deflocculated (artificially dispersed) sediment compare with that of the ' natural ' flocculated material (e.g. Kranck, 1973). In terms of concentration and settling velocity, a broadly Gaussian size-frequency distribution appears to typify the flocculated sediment, to judge from a range of field and laboratory studies (Migniot, 1968; Puls et al., 1988; Kineke et al., 1989; Lau & Krishnappan, 1992). This seems compatible with the nearly Gaussian distribution of turbulent fluctuating velocities in shear flows (Bradshaw, 1971) and with the dependence of suspended sediment transport on the small anistropy of the vertical turbulence component (Bagnold, 1966; Wei & Willmarth, 1991). Accordingly, two experimental sediment populations were defined (Table 1) in which the particle settling velocity is distributed normally between 13 one-half standard deviation classes in the range 4-3.25 SD from the mean. In the fine population, the settling velocity equivalent to +3-25 SD corresponds to a smooth quartz sphere of diameter 30 ~m settling in still water at 12 °C. T h e corresponding upper size limit for the

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coarse p o p u l a t i o n is 60 pm. Figures 9 a n d 10 show the c o r r e s p o n d i n g results for the thickness of sediment deposited. Figures 11 and 12 show the c o r r e s p o n d i n g m e a n settling velocities.

Conclusions A simple o n e - d i m e n s i o n a l e q u a t i o n to represent s e d i m e n t deposit o n a tidal salt m a r s h has been formulated and its numerical solution tested against the analytic solution for a special case. Once the model had been tested, it was used to examine how the equivalent thickness of sediment deposited on the marsh depends o n the characteristics of the m a r s h a n d sediment. With b = 0 it was f o u n d that the form of the solution was d e p e n d e n t only o n the n o n - d i m e n s i o n a l parameter vp/Ao9 a n d a n o n - d i m e n s i o n a l form of the m o d e l equation was f o u n d which verified this. T h e solution of the n o n - d i m e n s i o n a l form of the e q u a t i o n is d e p e n d e n t on two parameters, the height ratio, b/A, and the velocity ratio, vr,/Aco. Increasing b/A is f o u n d to reduce the thickness of s e d i m e n t deposited on the m a r s h a n d also to cause the sediment to be less evenly spread across the marsh. Increasing vp/Ao9 causes a higher proportion of the sediment to be deposited on the marsh and also causes the s e d i m e n t to be less evenly distributed on the marsh. T h e results show that s e d i m e n t m a d e up of different sized particles will be deposited on the marsh to a greater thickness at the seaward end, and that there will be a higher p r o p o r t i o n of the larger particles at the seaward end than at the landward end. This pattern of s e d i m e n t deposition is in agreement with observations o n salt marshes in the Severn Estuary (Allen, 1992) a n d the Lincolnshire a n d Norfolk Coasts (Hartnall, 1984; F r e n c h & Spencer, 1993).

Acknowledgements SJW was supported by a S E R C A d v a n c e d Course Studentship. F o r JRLA, this paper is Reading University PRIS c o n t r i b u t i o n No. 358.

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Appendix A Notation A b

b'

C

Co D E

/-/

/4'

J

Amplitude of tide (m) H e i g h t o f salt m a r s h a b o v e m e a n s e a level ( m ) Non-dimensionalized b Fractional volume of sediment in fluid Fractional volume of sediment in tidal water Diameter of sediment particle (m) Absolute error in the depth of sediment deposited (m) Height of water above salt marsh (m) Non-dimensionalized H Label of space grid points

542

L L M N n

P P R S S' T t tE t' l"n

tp to t to U U' Ui

[uyl;' vp t Vp 73o W X Xr Xc

< X~ ,(k)

x~ XE Xi

y yi Yc Yc"

y; Z (1

q 0 P ~7

S..7. Woolnough et al.

Node before x~ Node before x s Length of salt marsh (m) N u m b e r of space steps Half the number of time steps Exponent or time level N u m b e r of time steps in characteristic solving routine Time level in characteristic solving routine Rate of deposition on salt marsh (ms - ~) Equivalent thickness of sediment deposited on the salt marsh (m) Approximation to equivalent thickness of sediment deposited at (xi, t,, ) (m) Non-dimensionalized S Tidal period (s) Time (s) Time at which no sediment is suspended above a point on the marsh (s) Non-dimensionalized t Time at nth time level (s) T i m e at pth level of characteristic solving routine (s) Phase of tide at which water covers the salt marsh (s) Non-dimensionalized to Horizontal fluid velocity (ms - ~) Non-dimensionalized u Fluid velocity components (ms - 1) Approximation to uy(xs, t,,) (m 2 s - ~) Particle settling velocity (ms 1) Average settling velocity of population (m e s 1) Non-dimensionalized v p Settling velocity of single smooth sphere (ms ~) Vertical fluid velocity (ms 1) Horizontal coordinate (m) Non-dimensionalized x Position of characteristic through x E (m) Approximation to x~.(tp) (m) kth iteration of ~ (m) Position o f j t h node (m) Position at which no sediment is suspended above the marsh (m) Components of position vector (m) Height of column of sediment in water Non-dimensionalized y Value o f y on characteristic x,. (m) Approximation to y(x~,t,,) (m) Approximation to y(xi, t,,) (m) Vertical coordinate (m) Label of characteristics Sediment mixing coefficients (m e s - ~) Dynamic viscosity of fluid (l'q sm 2) Parameter in box scheme Density of fluid ( k g m - 3) Density of sediment (kgm - 3) Parameter in box scheme

Exploratory numerical model of sediment deposition

O9

At Ate Ax ~x'

Angular frequency of tide (s - 1) Time step (s) Time step in characteristic solving routine (s) Space step (m) Space step for first step in box scheme for ebb tide (m)

543