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The effect of geometry and bottom friction on local bed forms in a tidal embayment

01 Jul 2002-Continental Shelf Research (CONTINENTAL SHELF RESEARCH)-Vol. 22, Iss: 11, pp 1821-1833

AbstractUsing a 2DH idealized local morphodynamic model for a tidal channel, it is demonstrated that estuarine bars with typical length scales on the order of the tidal excursion length can develop as the result of a positive feedback between water motion, sediment transport and the sandy bottom. The water motion is modelled by the depth-averaged shallow water equations and driven by an externally prescribed M2 tide. Sediment is mainly transported as suspended load due to advective processes. Convergences and divergences of the tidally averaged sediment fluxes result in the evolution of the bed. It is shown that the combined effect of bottom friction and advective processes can trigger instabilities that lead to the formation of bottom patterns. Bed slope effects are required in order to prevent infinite braiding of these features. With bed slope effects, bars with longitudinal length scales of the order of the tidal excursion length are most likely to become unstable. This result is found to be independent of the ratio of the width to the tidal excursion length as well as the adopted formulation of the bed shear stress. In the case that the width is much smaller than the tidal excursion length and non-linear bottom friction is used, there is good qualitative agreement with results from 3D models reported in literature which were applied to the same parameter regime. Qualitatively, the results are recovered when bottom friction is linearized. Quantitatively, only small modifications occur: the critical friction parameter is decreased and the longitudinal length scale of the most unstable bed form increases. r 2002 Elsevier Science Ltd. All rights reserved.

Topics: Length scale (56%), Suspended load (54%), Sediment transport (53%)

Summary (3 min read)

1. Introduction

  • The geomorphology of semi-enclosed tidal embayments with a sandy bed often consists of a complex network of channels and shoals.
  • On the other hand, deeper embayments, e.g. those located in the Dutch and German Wadden Sea, are characterized by a fractal pattern of channels (cf. Cleveringa and Oost, 1999; Ehlers, 1988) which appear to scale with the length of the embayment.
  • Their model results apply to narrow, frictionally dominated tidal channels.
  • The results from the two approaches should qualitatively agree in the appropriate limits.
  • Here, the intermediate model will be compared with the results by Seminara and Tubino (1998).

2. Model description

  • The features studied in this paper have length scales which are small compared with the tidal wavelength, the embayment length and the length scale on which variations of the channel width occur.
  • Within the local model, tidal velocities are of the order of 1 m s 1: Since the amplitude of the sea surface elevations is assumed to be much smaller than the undisturbed water depth, the socalled rigid lid approximation can be adopted.
  • Using a procedure first proposed by Lorentz (1922), see also Zimmerman (1992), the non-linear bottom friction can be linearized in such a way that averaged over one tidal cycle the same amount of energy is dissipated in both formulations.
  • The first term on the right-hand side of Eq. (4) models the sediment pick-up function, and the second term the tendency of sediment to settle due to gravity effects.
  • This seems to be consistent with literature where it is suggested that the transport due to the bed slope terms cannot be neglected (see Parker, 1978; Talmon et al., 1995), even if suspended-load transport dominates.

3. Linear stability analysis

  • For realistic values of the parameters the 2D system of equations, as described in Section 2, allows for a morphodynamic equilibrium solution Weq ¼ ðu; v;rz;C; hÞeq; which is spatially uniform, i.e. they are independent of both the x- and the ycoordinate.
  • Here, ueq ¼ ðueq; 0Þ and u0 ¼ ðu0; v0Þ are the equilibrium and perturbed velocity vector.
  • The ratio of the tidal period and the morphologic timescale is typically of the order of 10 2–10 4: Since Eqs. (8a)–(8d) evolve on the tidal timescale, the bed perturbation h0 in these equations can be considered fixed.
  • The first two terms on the right-hand side of Eq. (13) give the contribution of the divergence of the advective sediment flux Fadv; while the last two terms model the divergences of fluxes due to diffusive processes (Fdiff ) and bedslope effects (Fbed), respectively.
  • The real part of the eigenvalue RðoÞ denotes the growth rate of the perturbation and IðoÞ=k its migration speed.

4. Results

  • In this section results from the local 2D channel model will be described.
  • Default values which are characteristic for the Western Scheldt will be used, see Table 1.
  • In the remainder of this paper, the authors will only consider advective modes, i.e. horizontal dispersion terms in the momentum and concentration equations are neglected ðm ¼.
  • This is justified since the ratio of dispersive to advective fluxes is of the order 10 1–10 3 for the bed form length scales that are considered in this paper.
  • This is also done in the model adopted by Seminara and Tubino (1998).

4.1. Advective instabilities for linear bottom friction

  • 0Þ: Fig. 3 shows the dimensionless growth rate as a function of the dimensionless longitudinal wavenumber k for various values of the lateral number n:.
  • This result is reminiscent from river morphodynamics where this mode is also found to be the most unstable one if bed slope effects are neglected (Callander, 1969).
  • For long waves, sediment transport is mainly driven by the residual velocity perturbation /u0S:.
  • These features will now be explained in more detail.
  • Expression (17) shows that the growth of long-wave perturbations is primarily governed by the residual perturbed velocity /u0S: Fig. 4 shows a typical example of the behaviour of this quantity as the longitudinal wavenumber k varies.

4.2. Bedslope effects

  • The most unstable mode now occurs for finite n: Eigenfunctions with high modenumber n (i.e. fast spatial oscillations in the lateral direction) are damped.
  • For friction values above the neutral curve, bedforms have positive growth rates.
  • The minimum of the neutral curve is referred to as the critical mode for the specified lateral modenumber n and is characterized by the critical wavenumber kcr and friction parameter values rcr:.
  • Since this destabilizing effect has its maximum value for a finite value of k (see Fig. 3), it is to be expected that both rcr and kcr have finite (non-zero) values.

4.3. Non-linear friction and the influence of channel width

  • The authors will extend their model by including non-linear bottom friction, which means that sb in Eq. (2) reads sb ¼ r#rjjujju; ð20Þ where #r ¼ 3pr=ð8UÞ follows from the Lorentz linearization procedure that was mentioned below Eq. (2).
  • The effect of non-linearity on the growth of bedforms can be inferred from Fig. 7 which shows the neutral curves for both linear and nonlinear friction.
  • Also, the most unstable wavenumber shifts towards a higher value, i.e. the critical mode occurs on a shorter longitudinal length scale.
  • The explicit dependence of the non-linear friction parameter #rjjujj on velocity thus yields a decrease of bottom friction above shallow (deep) parts of the channel.
  • So far, the authors have considered a socalled wide channel for which width and tidal excursion length are of the same order of magnitude.

5. Discussion and conclusions

  • The formation of bottom patterns that scale with the tidal excursion length has been studied within a 2D idealized model.
  • In the case that sediment diffusion can be neglected, this instability is mediated by advective processes, in particular through residual flows that arise from tide-topography interactions.
  • Bed slope effects act as a means to prevent the emergence of both longitudinal and lateral smallscale features.
  • All these discrepancies may in principle yield qualitatively different outcome.

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Continental Shelf Research 22 (2002) 18211833
The effect of geometry and bottom friction on local bed forms
in a tidal embayment
G.P. Schramkowski*, H.M. Schuttelaars
1
, H.E. de Swart
Institute for Marine and Atmospheric Research Utrecht, Utrecht University, Princetonplein 5, 3508 TA Utrecht, The Netherlands
Abstract
Using a 2DH idealized local morphodynamic model for a tidal channel, it is demonstrated that estuarine bars with
typical length scales on the order of the tidal excursion length can develop as the result of a positive feedback between
water motion, sediment transport and the sandy bottom. The water motion is modelled by the depth-averaged shallow
water equations and driven by an externally prescribed M
2
tide. Sediment is mainly transported as suspended load due
to advective processes. Convergences and divergences of the tidally averaged sediment fluxes result in the evolution of
the bed. It is shown that the combined effect of bottom friction and advective processes can trigger instabilities that lead
to the formation of bottom patterns. Bed slope effects are required in order to prevent infinite braiding of these features.
With bed slope effects, bars with longitudinal length scales of the order of the tidal excursion length are most likely to
become unstable. This result is found to be independent of the ratio of the width to the tidal excursion length as well as
the adopted formulation of the bed shear stress. In the case that the width is much smaller than the tidal excursion
length and non-linear bottom friction is used, there is good qualitative agreement with results from 3D models reported
in literature which were applied to the same parameter regime. Qualitatively, the results are recovered when bottom
friction is linearized. Quantitatively, only small modifications occur: the critical friction parameter is decreased and the
longitudinal length scale of the most unstable bed form increases. r 2002 Elsevier Science Ltd. All rights reserved.
Keywords: Bed forms; Tides; Estuaries; Estuarine dynamics; Mathematical models; Suspended load
1. Introduction
The geomorphology of semi-enclosed tidal
embayments with a sandy bed often consists of a
complex network of channels and shoals. In the
channels, strong tidal currents are observed (of
order 1 m s
1
). Field data indicate that different
types of bed forms can exist. On the one hand,
near the entrance of shallow embayments (e.g.
those located along the east coast of the United
States) the so-called estuarine bars are often
observed (Dalrymple and Rhodes, 1995). These
rhythmic bars have wavelengths which are related
to the embayment width and do not migrate. On
the other hand, deeper embayments, e.g. those
located in the Dutch and German Wadden Sea,
are characterized by a fractal pattern of channels
(cf. Cleveringa and Oost, 1999; Ehlers, 1988)
which appear to scale with the length of the
embayment. Often both types of bottom patterns
can be observed simultaneously. An example is the
Western Scheldt, a tidal estuary located at the
*Corresponding author. Fax: +31-30-254-3163.
E-mail address: g.p.schramkowski@phys.uu.nl
(G.P. Schramkowski).
1
Also at Delft University of Technology, Stevinweg 1, 2628
CN Delft, The Netherlands.
0278-4343/02/$ - see front matter r 2002 Elsevier Science Ltd. All rights reserved.
PII: S0278-4343(02)00040-7

Dutch–Belgium border. Its marine part has a length
of about 90 km and can be divided into six separate
sections. Within these sections, bars are observed
which are related to the embayment width, see
Jeuken (2000); Van den Berg et al. (1997).
Knowledge about the behaviour of these bar–
shoal systems is relevant, both for estuarine
management and scientific purposes. For example,
the morphology influences ship routing, but it also
plays an important role in determining the flushing
characteristics of the water motion and thereby the
ecological diversity of the area (Verbeek et al.,
1999). In semi-empirical models, see e.g. Di Silvio
(1989); Van Dongeren and de Vriend (1994), the
shoal surface area is often parametrically ac-
counted for. From a process-oriented perspective
the evolution of bars has been successfully
simulated by Wang et al. (1995) and Ranasinghe
et al. (1999). These models are rather complex and
have not been designed to gain fundamental
understanding about the physical mechanisms
controlling the channel–shoal dynamics. For the
latter objective idealized models, which focus on
isolated processes, are useful tools.
It has been demonstrated by Seminara and
Tubino (1998) and Schuttelaars and De Swart
(1999) that bars in tidal channels can form as
inherent morphologic instabilities. Seminara and
Tubino (1998) analyse a local model, which in
general is designed to deal with phenomena that
scale on a length scale which is small compared
with both the tidal wavelength and channel length.
In this approach the water motion must be
prescribed by specifying external pressure gradi-
ents which result from the dynamics on the global
scale. In their study Seminara and Tubino (1998)
use a three-dimensional model, based on the
shallow water equations. They also a priori assume
that the embayment width is the controlling length
scale of the bed forms. Their model results apply
to narrow, frictionally dominated tidal channels.
Schuttelaars and De Swart (1999) on the other
hand study a global model of a semi-enclosed tidal
basin where the water motion is driven by a
prescribed vertical tide at the seaward boundary.
This choice implies that they put emphasis on
bottom patterns that occur on the length scale of
the entire domain. Their model is based on the
depth-averaged shallow water equations and
assumes that the ratio of the tidal excursion length
and the embayment length is small. Results are
presented for the case of a short embayment (with
a length being much smaller than the tidal
wavelength), in which sediment transport is
dominated by diffusive processes and advective
terms in the flow and sediment equations can be
neglected. However, the model can be generalized
in a straightforward sense, as is demonstrated in
Schuttelaars and De Swart (2000) (1D equilibrium
dynamics in a long embayment) and in Van
Leeuwen and De Swart (2001) (2D non-linear
bed forms in a short embayment).
Thus, global and local models may be viewed as
describing morphodynamics for opposite limits
with regard to the length scales of bottom patterns.
The results from the two approaches should
qualitatively agree in the appropriate limits. This
is not straightforward, because the local model of
Seminara and Tubino (1998) and the non-linear
global model of Schuttelaars and De Swart (2000)
use different formulations and assumptions. This
motivates the set up of an intermediate model, in
which both the embayment width and the tidal
excursion length are retained as relevant length
scales. Hence, this model can deal with phenomena
that scale on a length scale which is small
compared with both the tidal wavelength and
embayment length, but not necessarily small
compared with the tidal excursion length. More-
over, in the intermediate model discussed in this
paper the ratio of the frictional timescale and tidal
period can be arbitrarily varied, which is an
important generalisation with respect to the model
of Seminara and Tubino (1998). This intermediate
model serves as a link between the two models
which are already available. Here, the intermediate
model will be compared with the results by
Seminara and Tubino (1998). It will be shown
that in the limit of a narrow and frictionally
dominated channel, the bed forms found by the
latter authors will be recovered. In a forthcoming
paper, the connection with the bed forms found in
a global model (see Schuttelaars et al., 2001) will
be made.
The paper is organized as follows. In Section 2
the local model is described. In Section 3 the
G.P. Schramkowski et al. / Continental Shelf Research 22 (2002) 182118331822

method of analysis that is used to describe the
formation of bed forms is described. In Section 4
the results from the local analysis will be presented.
Finally, the results will be discussed in Section 5.
2. Model description
The features studied in this paper have length
scales which are small compared with the tidal
wavelength, the embayment length and the length
scale on which variations of the channel width
occur. On the other hand, their length scales are
large with respect to the water depth. Hence, the
geometry of the model is represented as an
infinitely long channel having a constant width
B: The banks are straight and non-erodible, with
only the bottom being subject to erosion (see
Fig. 1). The total water depth is denoted by D ¼
H h where H is a reference depth, while uðxÞ and
vðyÞ are the along-channel and cross-channel
velocities (coordinates), respectively.
The model is forced by a prescribed external
tide. This explicitly reflects the local nature of the
model which by definition cannot solve the global
tidal motion self-consistently. It will be assumed that
the external tide contains only an M
2
component.
The water motion is described by the depth-
averaged shallow water equations (see Vreugden-
hil, 1994). Within the local model, tidal velocities
are of the order of 1 m s
1
: Since the amplitude of
the sea surface elevations is assumed to be much
smaller than the undisturbed water depth, the so-
called rigid lid approximation can be adopted.
This implies that the water level z and its
derivatives may be neglected everywhere except
for terms that describe tidal forcing, i.e. gz
x
and
gz
y
: These latter terms describe, for instance, the
forcing due to the prescribed external tide.
Furthermore, since the bed changes on a timescale
that is long compared to the tidal period, the
bottom can be considered time independent on the
short tidal timescale. The resulting equations read
ðDuÞ
x
þðDvÞ
y
¼ 0; ð1aÞ
u
t
þ uu
x
þ vu
y
þ F
b
1
¼gz
x
; ð1bÞ
v
t
þ uv
x
þ vv
y
þ F
b
2
¼gz
y
; ð1cÞ
where subscripts denote differentiation, g is the
gravitational acceleration and F
b
¼ðF
b
1
; F
b
2
Þ the
bottom friction. The friction vector is defined as
F
b
¼
s
b
rðH hÞ
; ð2Þ
with s
b
the bed shear stress vector, which is in
general a non-linear function of velocity. Using a
procedure first proposed by Lorentz (1922), see
also Zimmerman (1992), the non-linear bottom
friction can be linearized in such a way that
averaged over one tidal cycle the same amount of
energy is dissipated in both formulations. Since
linearized bottom friction gives the same qualitative
results as the non-linear formulation, while it is
easier to analyse, we will hereafter use the linearized
shear stress unless stated otherwise. Hence,
s
b
¼ r ru;
where r is a friction parameter with dimensions
ms
1
:
The impermeability of side walls to water
motion implies that the cross-channel velocity v
should vanish at the banks, i.e.
v ¼ 0aty ¼ 0; B: ð3Þ
The bed of the estuaries considered consists of fine
non-cohesive sediment (typical grain size of
H
ζ
x
z
u
v
u
B
x
y
(a)
(b)
h
Fig. 1. The side view (a) and top view (b) of the embayment. For an explanation, see the text.
G.P. Schramkowski et al. / Continental Shelf Research 22 (2002) 18211833 1823

2 10
4
m) which is mainly transported as
suspended load. The dynamics of the suspended
sediment is described by an advection–diffusion
equation for the so-called volumetric depth-inte-
grated concentration C (with dimension m), which
is defined as the depth-integrated sediment con-
centration divided by the sediment density. The
evolution equation for C reads (Van Rijn, 1993)
C
t
þðuC mC
x
Þ
x
þðvC mC
y
Þ
y
¼ S a ðu
2
þ v
2
ÞgC; ð4Þ
where m denotes the horizontal coefficient for
sediment diffusion and S the difference between
erosion and sedimentation at the top of the active
layer. Typically, m ¼ Oð102 100Þ m
2
s
1
for tidal
flows in estuaries. The first term on the right-hand
side of Eq. (4) models the sediment pick-up func-
tion, and the second term the tendency of sediment
to settle due to gravity effects. The adopted values
of the coefficients a ðOð10
5
210
7
Þ sm
1
Þ and g
ðOð10
3
210
2
Þ s
1
Þ are representative for fine sand
(see e.g. also Dyer, 1986).
The non-erodibility of the banks implies that
there is no sediment flux through the side walls, i.e.
vC mC
y
¼ 0aty ¼ 0; B: ð5Þ
The evolution of the bottom follows from the
conservation of sediment. Since the bed in general
evolves on a timescale (typically weeks to months)
which is long compared to a tidal period, the
dynamics of bottom patterns is not sensitive to
instantaneous rates of erosion and deposition, but
rather to their tidal mean values. The resulting
equation for the bottom evolution thus reads
h
t
þ /rS
b
S ¼/SS; ð6Þ
where /S denotes the time average over an M
2
tidal period. The sedimentation function S is
defined in Eq. (4). The volumetric bed-load sedi-
ment flux in the active layer is denoted by S
b
and
parameterized as Engelund (1975)
S
b
¼
#
sjjujj
b
u
jjujj
k
%
rh

:
Typical values for b; k
%
and
#
s are bB 3; k
%
B2
and
#
sB3 10
4
s
2
m
1
while it is assumed that
the critical shear stress is effectively zero in view of
the strong tidal currents. In the situations under
study, the fluxes associated with the bed-load
transport S
b
are much smaller than those related
to the suspended-load transport S; typically by a
factor of 0.1–0.01. This implies that bed-load
transport is small compared to suspended-load
transport and hence can be neglected. However,
the bed slope correction term ðpk
%
rhÞ cannot be
ignored in case of bed forms with short spatial
oscillations and has to be incorporated in the
model. This seems to be consistent with literature
where it is suggested that the transport due to the
bed slope terms cannot be neglected (see Parker,
1978; Talmon et al., 1995), even if suspended-load
transport dominates.
3. Linear stability analysis
For realistic values of the parameters the 2D
system of equations, as described in Section 2,
allows for a morphodynamic equilibrium solution
W
eq
¼ðu; v; rz; C; hÞ
eq
; which is spatially uniform,
i.e. they are independent of both the x- and the y-
coordinate. In case that the forcing of the water
motion (due to the externally prescribed pressure
gradient) consists of one single tidal constituent (an
M
2
component with frequency s ¼ 1:4 10
4
s
1
),
it reads
W
eq
¼ðU cosðstÞ; 0; rz
eq
; C
eq
; 0Þ
with
ðz
eq
Þ
x
¼
sU
g
sinðstÞ
r
sH
cosðstÞ
hi
;
ðz
eq
Þ
y
¼ 0; ð7aÞ
C
eq
¼
aU
2
2g
1 þ
g
2
cosð2stÞþ2sg sinð2stÞ
g
2
þ 4s
2

: ð7bÞ
This describes a tidal flow in a channel with an
undisturbed water depth H and a horizontal
bottom. The concentration consists of a residual
component as well as of a component which
oscillates with twice the basic tidal frequency. This
equilibrium solution is in general not stable with
respect to perturbations having a structure in the
cross-channel direction. This means that such
perturbations can grow due to a positive feedback
between the water motion and the erodible bed.
G.P. Schramkowski et al. / Continental Shelf Research 22 (2002) 182118331824

The dynamics of the perturbations is analysed by
substitution of
Wðx; y; tÞ¼W
eq
ðtÞþW
0
ðx; y; tÞ
in the full equations of motion. Linearizing these
equations with respect to the small perturbations
results in
Hu
0
x
þ Hv
0
y
¼ u
eq
h
0
x
; ð8aÞ
u
0
t
þ u
eq
u
0
x
þ F
b
0
1
¼gz
0
x
; ð8bÞ
v
0
t
þ u
eq
v
0
x
þ F
b
0
2
¼gz
0
y
; ð8cÞ
C
0
t
þ u
eq
C
0
x
þðu
0
x
þ v
0
y
ÞC
eq
þ mr
2
C
0
¼ 2au
eq
u
0
gC
0
; ð8dÞ
h
0
t
lr
2
h
0
¼/2au
eq
u
0
gC
0
S ð8eÞ
with l ¼
#
s/ju
eq
j
b
Sk
%
B10
4
m
2
s
1
and F
b
0
the
perturbed bottom friction vector. In the linearized
context this vector reads
F
b
0
¼
r
H
u
0
þ
ru
eq
H
2
h
0
;
r
H
v
0

:
Finally, we note that Eq. (8d) can be used to rewrite
Eq. (8e) in the following form:
h
0
t
¼r/FS; ð9Þ
where
F ¼ F
adv
þ F
diff
þ F
bed
with
F
adv
¼ u
eq
C
0
þ u
0
C
eq
;
F
diff
¼mrC
0
; F
bed
¼lrh
0
denoting the advective, diffusive and bedslope
contributions to the sediment flux, respectively.
Here, u
eq
¼ðu
eq
; 0Þ and u
0
¼ðu
0
; v
0
Þ are the equili-
brium and perturbed velocity vector. From Eq. (9)
we see that the bed level rises (descends) when the
total net sediment flux converges (diverges).
The structure of the linearized Eqs. (8) and
the boundary conditions allow for solutions
written as
ðu
0
; z
0
; C
0
; h
0
Þ
¼ R½ð
#
uðtÞ;
#
zðtÞ;
#
CðtÞ;
#
hðtÞÞ cosðl
n
yÞe
ikx
; ð10aÞ
v
0
¼ R½
#
vðtÞ sinðl
n
yÞe
ikx
: ð10bÞ
Here l
n
¼ np=B where n ¼ 0; 1; 2; y is the cross-
channel mode number and k is an arbitrary
longitudinal wavenumber. R denotes the real part
of the solution. Hereafter,
#
h is assumed to be real:
this may be done without loss of generality
because the equilibrium state is spatially uniform.
The timescale associated with the evolution of
the bed is much longer than the tidal timescale.
The ratio of the tidal period and the morphologic
timescale is typically of the order of 10
2
–10
4
:
Since Eqs. (8a)–(8d) evolve on the tidal timescale,
the bed perturbation h
0
in these equations can be
considered fixed. Now the fast variables can be
calculated for a given bed perturbation as a
Fourier series:
#
uðtÞ;
#
vðtÞ;
#
zðtÞ;
#
CðtÞÞ ¼
#
h
X
p
ð *u; *v;
*
z;
*
CÞe
ipst
: ð11Þ
Substituting Eq. (10) in the bed evolution equation
(8e), which is determined by the tidally averaged
transport, results in
q
t
#
h ¼ o
#
h; ð12Þ
where the eigenvalue o reads
o ¼½ik/u
eq
#
CS /ðik
#
u þ l
n
#
vÞC
eq
S
mðk
2
þ l
2
n
Þ/
#
CS lðk
2
þ l
2
n
Þ
#
h=
#
h ð13Þ
and ð
#
uðtÞ;
#
vðtÞ;
#
zðtÞ;
#
CðtÞÞ is defined in Eq. (11). The
first two terms on the right-hand side of Eq. (13)
give the contribution of the divergence of the
advective sediment flux F
adv
; while the last two
terms model the divergences of fluxes due to
diffusive processes (F
diff
) and bedslope effects
(F
bed
), respectively. For a more detailed descrip-
tion of the method used, see e.g. Hulscher et al.
(1993).
For given model parameters, the eigenvalue o
of the associated perturbation can be calculated
for every mode number k and l
n
: The real part
of the eigenvalue RðoÞ denotes the growth rate
of the perturbation and IðoÞ=k its migration
speed. If the growth rate is positive ðRðoÞ > 0Þ;
the perturbation grows and the basic state is
unstable. It turns out that if the system is only
forced with an M
2
tide, the migration speed is
always zero.
We will now give a more geometric interpreta-
tion of the advective contributions to Eq. (13). To
G.P. Schramkowski et al. / Continental Shelf Research 22 (2002) 18211833 1825

Figures (11)
Citations
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Journal ArticleDOI
Abstract: In this review we discuss the morphodynamics of tidal inlet systems that are typical of barrier coasts formed during a period of continuous sea-level rise during the Holocene. The morphodynamics concerns feedbacks between three major components: the hydrodynamics of tidal currents and wind waves; the erosion, deposition, and transport of sediment under the action of the former hydrodynamic agencies; and the morphology proper, which results from the divergence of the sediment transport. We discuss the morphodynamics of the different units that characterize a tidal inlet system: the overall system, the ebb-tidal delta, the tidal channels, channel networks, tidal bars and meanders, and finally the intertidal zone of tidal flats and salt marshes. In most of these units, stability analysis is a major guide to the establishment of equilibrium structures.

249 citations


Journal ArticleDOI
Abstract: [1] The research objective is to investigate long-term evolution of estuarine morphodynamics with special emphasis on the impact of pattern formation. Use is made of a two-dimensional (2-D), numerical, process-based model. The standard model configuration is a rectangular 80 km long and 2.5 km wide basin. Equilibrium conditions of the longitudinal profile are analyzed using the model in 1-D mode after 8000 years. Two-dimensional model results show two distinct timescales. The first timescale is related to pattern formation taking place within the first decades and followed by minor adaptation according to the second timescale of continuous deepening of the longitudinal profile during 1600 years. The resulting longitudinal profiles of the 1-D and 2-D runs are similar apart from small deviations near the mouth. The 2-D results correspond well to empirically derived relationships between the tidal prism and the channel cross section and between the tidal prism and the channel volume. Also, comparison between the current model results and data from the Western Scheldt estuary (in terms of bar length, hypsometry, percentage of intertidal area and values for the ratio of shoal volume and channel volume against the ratio of tidal amplitude and water depth) shows satisfying agreement. On the basis of the model results a relationship for a characteristic morphological wavelength was derived on the basis of the tidal excursion and the basin width and an exponentially varying function was suggested for describing a dimensionless hypsometric curve for the basin. Furthermore, special attention is given to an analysis of the numerical morphodynamic update scheme applied.

217 citations


Cites background or result from "The effect of geometry and bottom f..."

  • ...Seminara and Tubino [2001], Schuttelaars and De Swart [1999], Schramkowski et al. [2002] and Dronkers [2005] extensively describe the prevailing processes in bar development for the linear domain where the bars themselves do not yet significantly influence the velocity field....

    [...]

  • ...The reason why a rectangular configuration was chosen is twofold.First,it makes comparison possible with earlier research carried out by Schuttelaars and De Swart [1996, 1999, 2000], Seminara and Tubino [2001], Schramkowski et al. [2002, 2004] and Hibma et al. [2003b] who all used rectangular basins or assumedthatnosignificantchangesofthebasinwidthoccurred over a typical length scale of the developing bars....

    [...]

  • ...The relation of a typical morphological length to the tidal excursion is also found by Hibma et al. [2003b, 2003c] and Schramkowski et al. [2002] ....

    [...]

  • ...Swart [1999], Seminara and Tubino [2001], Schramkowski et al. [2002] and Van Leeuwen and De Swart [2004] describe initial channel/shoal formation in a highly schematized tidal environment....

    [...]

  • ...…is that they can be related to the (short) basin length, the embayment width and the relative importance of diffusive and advective transports [Van Leeuwen and De Swart, 2004], or the tidal excursion length implicitly taking into account the impact of friction and depth [Schramkowski et al., 2002]....

    [...]


Journal ArticleDOI
Abstract: Over decades and centuries, the mean depth of estuaries changes due to sea-level rise, land subsi- dence, infilling, and dredging projects. These processes produce changes in relative roughness (friction) and mixing, resulting in fundamental changes in the char- acteristics of the horizontal (velocity) and vertical tides (sea surface elevation) and the dynamics of sediment trapping. To investigate such changes, a 2DV model is developed. The model equations consist of the width- averaged shallow water equations and a sediment balance equation. Together with the condition of mor- phodynamic equilibrium, these equations are solved analytically by making a regular expansion of the vari- ous physical variables in a small parameter. Using these analytic solutions, we are able to gain insight into the fundamental physical processes resulting in sediment trapping in an estuary by studying various forcings separately. As a case study, we consider the Ems es- tuary. Between 1980 and 2005, successive deepening of the Ems estuary has significantly altered the tidal and sediment dynamics. The tidal range and the surface sediment concentration has increased and the position of the turbidity zone has shifted into the freshwater zone. The model is used to determine the causes of these historical changes. It is found that the increase of the tidal amplitude toward the end of the embayment is the combined effect of the deepening of the estuary and a 37% and 50% reduction in the vertical eddy viscosity and stress parameter, respectively. The phys- ical mechanism resulting in the trapping of sediment, the number of trapping regions, and their sensitivity to grain size are explained by careful analysis of the var- ious contributions of the residual sediment transport. It is found that sediment is trapped in the estuary by a delicate balance between the M2 transport and the residual transport for fine sediment (ws = 0.2 mm s −1 ) and the residual, M2 and M4 transports for coarser sediment (ws = 2 mm s −1 ). The upstream movement of the estuarine turbidity maximum into the freshwa- ter zone in 2005 is mainly the result of changes in tidal asymmetry. Moreover, the difference between the sediment distribution for different grain sizes in the same year can be attributed to changes in the temporal settling lag.

144 citations


Cites background from "The effect of geometry and bottom f..."

  • ...Hence, the partial slip condition can be rewritten as Avuz = su (Schramkowski et al. 2002)....

    [...]

  • ...Following Friedrichs and Hamrick (1996) and Schramkowski et al. (2002), this dependency is taken to be linear in the local water depth, i.e., s = s0 H(x)H0 ....

    [...]


Journal ArticleDOI
Abstract: The formation of channel and shoal patterns in a schematic estuary is investigated using a 2-D depth-averaged numerical model based on a description of elementary flow and sediment transport processes. The schematisations apply to elongated inland estuaries, sandy, well-mixed and tide-dominated. The model results show how, due to non-linear interactions, a simple and regular pattern of initially grown perturbations merges to complex larger-scale channel/shoal patterns. The emerging patterns are validated with field observations. The overall pattern agrees qualitatively with patterns observed in the Westerschelde, The Netherlands, and in the Patuxent River estuary, Virginia. Quantitative comparison of the number of channels and meander length scales with observations and with an analytical model gives reasonable accordance. Complementary to other research approaches, this model provides a tool to study the morphodynamic behaviour of channels and shoals in estuaries.

113 citations


Journal ArticleDOI
Abstract: The morphodynamic system in alluvial, coastal plain estuaries is complex and characterized by various timescales and spatial scales. The current research aims to investigate the interaction between these different scales as well as the estuarine morphodynamic evolution. Use is made of a process-based, numerical model describing 2-D shallow water equations and a straightforward formulation of the sediment transport and the bed level update. This was done for an embayment with a length of 80 km on a timescale of 3200 years, with and without bank erosion effects. Special emphasis is put on analyzing the results in terms of energy dissipation. Model results show that the basins under consideration evolve toward a state of less morphodynamic activity, which is reflected by (among others) relatively stable morphologic patterns and decreasing deepening and widening of the basins. Closer analysis of the tidal wave shows standing wave behavior with resonant characteristics. Under these conditions, results suggest that the basins aim for a balance between the effect of storage and the effect of fluctuating water level on wave celerity with a negligible effect of friction. Evaluating the model results in terms of energy dissipation reflects the major processes and their timescales (pattern formation, widening, and deepening). On the longer term the basin-wide energy dissipation decreases at a decreasingly lower rate and becomes more uniformly distributed along the basin. Analysis by an entropy-based approach suggests that the forced geometry of the configurations prevents the basins from evolving toward a most probable state.

110 citations


References
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Book
01 Jan 1993

1,902 citations


Additional excerpts

  • ...The evolution equation for C reads (Van Rijn, 1993) Ct þ ðuC mCxÞx þ ðvC mCyÞy ¼ S aðu2 þ v2Þ gC; ð4Þ where m denotes the horizontal coefficient for sediment diffusion and S the difference between erosion and sedimentation at the top of the active layer....

    [...]


Book
06 May 1986
Abstract: The movement of mud, sand and gravel on the continental shelf, in the nearshore zone, on beaches and in estuaries can be significant in economic and environmental terms. From ease of navigation, to liability to flooding, from sewage and waste disposal to fish populations changes in the deposition and erosion of sedimentary material can have an effect on man's activities in the shoreline zone. Coastal and Estuarine Sediment Dynamics discusses such movements using the different viewpoints of the marine geologists, the oceanographer and the engineer, and integrates them into an essentially multidisciplinary treatment. Quantified descriptions of the physical processes causing sedimentary movement and response are emphasised in the context of natural systems. Among the features included in this book are: Establishment of physical concepts governing sediment movement in the sea with the minimum of mathematics; Essential background material and up-to- date research results; Information on the measurement and prediction of sea transportation of sediment; Detailed physical processes within regional sediment circulation patterns.

966 citations


Additional excerpts

  • ...The adopted values of the coefficients a ðOð10 5210 7Þ s m 1Þ and g ðOð10 3210 2Þ s 1Þ are representative for fine sand (see e.g. also Dyer, 1986)....

    [...]


Book
01 Jan 1994
Abstract: Preface. 1. Shallow-water flows. 2. Equations. 3. Some properties. 4. Behaviour of solutions. 5. Boundary conditions. 6. Discretization in space. 7. Effect of space discretization on wave propagation. 8. Time integration methods. 9. Effects of time discretization on wave propagation. 10. Numerical treatment of boundary conditions. 11. Three-dimensional shallow-water flow. List of notations. References. Index.

490 citations


"The effect of geometry and bottom f..." refers background in this paper

  • ...The water motion is described by the depthaveraged shallow water equations (see Vreugdenhil, 1994)....

    [...]


Journal ArticleDOI
Abstract: Rivers and canals with perimeters composed of non-cohesive sand and silt have self-formed active beds and banks. They thus provide a most interesting fluid flow problem, for which one must determine the container as well as the flow. If bed load alone occurs across the perimeter of a wide channel, gravity will pull particles down the lateral slope of the banks; bank erosion is accomplished and the channel widens. In order to maintain equilibrium, this export of material from the banks must be countered by an import of sediment from the channel centre.The mechanism postulated for this import is lateral diffusion of suspended sediment, which overloads the flow near the banks and causes deposition. The model is formulated analytically with the aid of a series of approximate but reasonable assumptions. Singular perturbation techniques are used to define the channel geometry and obtain rational regime relations for straight channels. A comparison with data lends credence to the model.It is hoped that a first step has been made towards a more general treatment, which would include various complicating factors that are important features of natural rivers but are not essential to the maintenance of channel width. Among these factors are meandering, sediment sorting and seepage.

272 citations


"The effect of geometry and bottom f..." refers result in this paper

  • ...This seems to be consistent with literature where it is suggested that the transport due to the bed slope terms cannot be neglected (see Parker, 1978; Talmon et al., 1995), even if suspended-load transport dominates....

    [...]


Journal ArticleDOI
Abstract: Laboratory experiments have been conducted to provide data for modelling the direction of sediment transport on a transverse sloping alluvial bed. Conditions with prevailing bed-load transport, and conditions in which a significant part of the bed material is transported as suspended-load are studied. The effect of a sloping bed on the direction of sediment transport is determined by conducting bed-levelling experiments. Comparison of the results with data of curved flume experiments and experience gained with numerical computation of the bed topography in natural rivers yields the conclusion that, at least for bed-load transport, a distinction should be made between laboratory conditions and natural rivers. For conditions with suspended sediment transport the transverse slope effect can not be modelled identical as for bed-load transport.

240 citations


"The effect of geometry and bottom f..." refers result in this paper

  • ...This seems to be consistent with literature where it is suggested that the transport due to the bed slope terms cannot be neglected (see Parker, 1978; Talmon et al., 1995), even if suspended-load transport dominates....

    [...]


Frequently Asked Questions (1)
Q1. What are the contributions mentioned in the paper "The effect of geometry and bottom friction on local bed forms in a tidal embayment" ?

In the case that the width is much smaller than the tidal excursion length and non-linear bottom friction is used, there is good qualitative agreement with results from 3D models reported in literature which were applied to the same parameter regime.