Transactions on the Built Environment vol 27, © 1997 WIT Press, www.witpress.com, ISSN 1743-3509
Morphodynamic modelling in the
nearshore area
J.P. Sierra, I. Azuz, F. Rivero, A. Sanchez-Arcilla
& A. Rodriguez
Laboratory d'Enginyeria Maritima, Universitat Politenica de
(bmW)/67, C C/mM C6zp/Y(7 ^/M,
E-mail: sierra@etseccpb. upc. es
Abstract
A numerical model is presented to determine the bottom topography
changes in the nearshore area. The model is divided in three modules
which compute the wave propagation, the wind and waves induced
currents and the sediment transport and associated morphodynamical
changes. Some examples extracted from the literature are employed
to calibrate the model. The comparison shows that the model gives
accurate estimates of values and trends of bottom evolution under different
hydrodynamic conditions and therefore it can be a usefull tool for
engineering purpouses.
1 Introduction
The nearshore zone is a very active area, where a series of dynamic
processes occurs. These processes involve the action of waves, currents,
wave-currents interaction, sediment transport and bathymetry changes.
Inside the nearshore zone, the surf-zone is the most dynamic one, with
great energy disssipation, complex fluid motions, a great movement of
sand and significative changes in the bottom topography.
In this paper, a computer model developed for the simulation of
these processes is described. The model is structured as a compound
morphodynamic model and its intrinsic behaviour can be divided in
two clearly differentiated blocks. In the first one, a fixed bottom is
considered and with this hypothesis the waves, currents and sediment
Transactions on the Built Environment vol 27, © 1997 WIT Press, www.witpress.com, ISSN 1743-3509
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Computer Modelling of Seas HI
transport fields are computed. In the second step, the hydrodynamic
conditions are assumed stationary and, taking into account the divergence
of the sediment transport, the bottom changes are evaluated. The new
bathymetry is introduced in the model as a fixed bottom and the cycle is
reinitiated again.
The model here described can be considered as a medium term
model (e.g. De Vriend [6]). This kind of models reproduce different
physical phenomena including the most relevant processes in the nearshore
zone and usually are divided in equilibrium and non-equilibrium models
according to the way of computing the sediment transport. The first type
assumes an instantaneous response to the forcing actions while the nonequilibrium models consider the existence of a lag between the mobilizing
force and the sediment motion. This last type of models are generally
employed when the suspended sediment transport is predominant and they
solve an advective-diffusive equation to obtain the vertical distribution of
the sediment concentration.
The equilbrium models like the one described here are simpler, because
they introduce a sediment transport formula in the solution of the sediment
continuity equation. Nevertheless their development and application is
recent and much work remains to be done in this line.
2 Background
One of the first models showing the characteristic conceptual structure
of the compound morphodynamic models was presented in 1982 by
Coeffe and Pechon [3]. In this work, the authors implicitly recognized
the difference between the hydrodynamic and morphodynamic variation
scales.
In the same decade, the research in the field of bottom evolution
modelling was dynamized by researchers like Watanabe [16] and Horikawa
[8]. In these years, there was a significant increase in the knowledge
about the numerical aspects -e.g. discretization of the equations- and
the analytical ones -e.g. stability theoretical criteria- (e.g. Van Rijn,
[15], De Vriend, [4]). On the other hand, the conceptual structure of the
morphodynamic models and their spatial and temporal scales were also
established (e.g. De Vriend [5], O'Connor [9]).
An interesting exercise was carried out in the frame of the European
Union Mast Programme, by comparing the performances of several
hydrodynamic and morphodynamic models working on the same test
cases (e.g. De Vriend et al. [7]). In this way, the capabilities and
limitations of the different models were determined and several aspects
of the morphodynamic modelling process were clarified.
In spite of this, the bottom evolution modelling in the nearshore zone
is still an open and active research area, with many subjects to be studied
and with the necessity to get measured field data in order to validate and
calibrate the models.
Transactions on the Built Environment vol 27, © 1997 WIT Press, www.witpress.com, ISSN 1743-3509
Computer Modelling of Seas III
435
3 Model description
The developed model includes three different modules which compute
several physical phenomena. The first module carries out the wave
propagation solving in a coupled way the irrotationality of the wavenumber
vector and the wave action density conservation equation.
Once the wave propagation is solved in all the domain, the second
module computes the 2DH currentfieldthrough the Mass and Momentum
Conservation Equations, obtained after applying depth-integration and
time- averaging operators.
The last module computes the sediment transport and the beach
topography changes solving the sediment transport continuity equation
by means of a second order Lax-Wendroff numerical scheme.
3.1 Wave propagation
The wave propagation module is based on a numerical model (Rivero
and Sanchez- Arcilla [12], Rivero et al. [13]) which simulates the wave
propagation over an arbitraryfluiddomain with irregular bathymetry and
in the presence of currents. The module can reproduce the following
physical phenomena associated to the wave propagation: shoaling,
refraction, diffracction, waves-current interaction, non-linear frequency
dispersion, energy disipation produced by bottom friction and wave
breaking. The model does not consider wave reflection and other nonlinear phenomena like amplitude dispersion or wave- wave interactions.
The wave propagation direction a. is obtained from the wavenumber
irrotationality condition:
VxK
=Q
(1)
with the following relationships (corresponding to linear wave theory):
u^ff + K-U
(2)
0-2 - gk tanh(fc/i)
(3)
where K is the wavenumber vector, w the absolute frequency, <j the
intrinsic frequency, U the current velocity, fc the separation factor, h the
depth, C the wave celerity, Cg the group celerity and H the wave height.
The wave height is computed from the wave action conservation
equation, including the energy disipation terms due to wave breaking (-D&)
and bottom friction (£)/):
+ 2t.,
(5,
Transactions on the Built Environment vol 27, © 1997 WIT Press, www.witpress.com, ISSN 1743-3509
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Computer Modelling of Seas HI
where E/a is the wave action and E is the energy density:
E=\pgH'
(6)
The differential equations (1) and (5) are solved by finite differences.
As a consequence of the parabolic character of these equations a marchingin-space numerical scheme is applied with the advantage that a boundary
condition in the last row is not needed as it would be the case in elliptic
or hyperbolic equations.
3.2 Nearshore circulation
The nearshore circulation module is based on a numerical model which
computes the circulation induced by waves and wind (Rivero and SanchezArcilla [11]), Rivero [10]), taking into account the results of the wave
propagation code. This module is of average flux type, namely it
has a 2DH resolution and does not give a current vertical description.
It solves the average flux non-linear equations, which are obtained by
vertical integration and temporary average of the mass and momentum
conservation equations:
„
(7)
f
(8)
dx
dy
(9)
where 77 is the mean water level variation (set-up/set-down), 17, V
the average flux velocity components, d the depth relative to the still
water level, g the gravity aceleration, p the fluid density, /. the Coriolis
parameter, JF., Fy the wave induced force components, W*, Wy the wind
stress components, B=, By the bottom stress components and J?^ the
Reynolds stresses components.
These equations are solved by means of an ADI implicit numerical
scheme infinitedefferences over a staggered grid. From a numerical point
of view the only existing limitation is due to Af, which has to be selected
so that the C our ant number be less than 1.
3.3 Morpho dynamic module
Once the hydrodynamic field is known, the following step is the
computation of the sediment transport. There are numerous models in the
Transactions on the Built Environment vol 27, © 1997 WIT Press, www.witpress.com, ISSN 1743-3509
Computer Modelling of Seas III
437
literature to estimate the sediment transport, most of them empirically
derived (e.g. Van Rijn [14]). In the module here described, several
sediment transport expressions have been introduced, in order to compare
their capabilities and to have the possibility to select the most suitable one
in each case (e.g. currents or currents and waves; only bed load transport,
only suspended transport or total sediment transport).
Once the computation of the sediment transport is performed, the
change in bed level is determined according to the sediment continuity
equation, which states that the change in bed level in an elementary
volume is related to the divergence of the sediment flux:
"-»£-£-£<«>
where z& is the bed level, p the porosity of the bed and <fc, g% the sediment
transport rates.
The effect of bed slope needs to be taken into account in order to
avoid instabilities because it modifies the morphodynamic evolution rates.
This effect can be included directly in the sediment transport formulas
(through motion initiation criterions) without any structural change in
the continuity equation. Another way to include the bed slope effect is to
consider it as a correction factor for the sediment transport computed for
flat bottom. Mathematically, this latter alternative can be expressed as:
r\
3x = 9o=- f|#»|i
(lla)
where <fc is the sediment transport for a flat bed and e is a parameter to
be calibrated.
Introducing equations (lla) and (lib) in equation (10) and using
vectorial notation, the following hyperbolic-parabolic (ad vective- diffusive)
equation is obtained (Azuz [1]):
(l-?)|jr-eV.(|fib|Vz,) = -V.«b
(12)
The selection of a suitable numerical scheme for the solution of the
sediment continuity equation will be subordinate to the type of equation.
But, in both cases, i.e. the linear equation (10) or the ad vective- diffusive
equation (12), the numerical scheme usually employed is a second order
Lax-Wendroff scheme be it a single step (e.g. Chesher et al. [2]) or a
two steps (predictive-corrective) scheme. A complete description of the
different numerical schemes can be found in Azuz [1].
Transactions on the Built Environment vol 27, © 1997 WIT Press, www.witpress.com, ISSN 1743-3509
Computer Modelling of Seas III
4 Model application
The different modules of the model have been calibrated for different cases
from the literature (e.g. Rivero and Sanchez-Arcilia [12], Rivero et al. [13],
Rivero [10], Azuz [1]). At this time a process of validation employing field
measurements is being carried out. Some of the tests performed in Azuz
[1] will be shown here.
The first test case is the simulation of a sumerged dune (longitudinally
uniform) in a channel. The initial bed comprises a sinusoidally varying
sand dune with peak height 1m and base width 140m. The channel has
a length of 1000m and the water depth is 10m. The sediment diameter is
0.2mm. The hydrodynamic conditions are given by a flow with a velocity of
Im/s. The results are presented in figure 1. The plots show the evolution
of the bathymetry over a 150 hours period at 10 hours intervals, without
and with bed slope effects included in sediment transport. The results
are better when considering the bed slope effects. The main feature of
this simulation is the elongation of the upstream side of the dune and the
decrease of the initial bedform height. The results are quite similar to the
ones presented by Chesher et al. [2].
DISTANCE (m)
u
S
a5
S
cd
DISTANCE (m)
Figure 1. Sumerged dune evolution (above: without slope effect).
The second test is the simulation of the evolution of a conical dune in
a channel. The same parameter seetings are specified, namely an initially
sinusoidally shaped (in profile) sand dune, but now circular in plan, with
peak height 1m and base width 60m, on a sandy bed of grain size 0.2mm.
Unidirectional flow conditions are specified with input water depth of 10m
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Computer Modelling of Seas III
and velocity (depth-averaged) of Im/s. The simulation time is 100-hours.
The evolution of the model bathymetry after 25, 50 and 100 hours is
presented in figure 2. The results from the model indicate a number of
complex features associated with the flow in the vicinity of the hill and
the bedform also appears to propagate laterally.
Figure 2. Sumerged dune evolution
1.0 m /$
Figure 3. Groin. Circulation pattern
In the third test case, the marine bottom behaviour around a costal
structure oriented in the onshore-offshore direction (like a groin for
example) is examined. The structure has a length of 210m, a width of
20m and it is placed on a beach with an initial slope of 1:50. The wave
Transactions on the Built Environment vol 27, © 1997 WIT Press, www.witpress.com, ISSN 1743-3509
Computer Modelling of Seas HI
440
characteristics, at a point with a depth of 10m are Hrms = 1m, Tp — 8s
and incidence angle of 30°.
In figure 3, the circulation pattern generated by this wave field is
shown. The results of the bottom evolution after a simulation period of
84 hours are presented in figure 4. In this figure three main features can
be observed. First of all, in the exposed side of the structure, an acretive
process is detected, as expected. Moreover, in the head of the structure,
the bottom contours curvature indicates that scouring is happening. This
scouring of the groin head has been described by some researchers. Finally,
in the lee side of the structure, the depth contours curvature towards the
shoreline suggests a light erosion of this area, as it is observed in nature.
-06-0'9-OY-09-OG-
-9.0-8.0-OY-0'9-O'G-
-9.0-8.0-7.0-09-O'G-
Figure 4. Groin. Bottom evolution
Figure 5. Detached breakwater. Circulation pattern
The last model application deals with the bottom evolution around
a coastal structure which is parallel to the shoreline (e.g. a detached
breakwater). The structure is 300m long and 20m wide. It is placed on a
beach with an initial bottom slope of 1:50 at a distance of 210m from the
shoreline. The wave parameters at a depth of 10m are Hm» — 2m, T? = 8s
and normal incidence with respect to the structure. The circulation
pattern originated by these wave conditions is presented in figure 5. The
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Computer Modelling of Seas III
results of the bathymetric evolution after a simulation period of 5 days are
<
shown infigure6. In thisfigure,a small advance of the depth contoursinm
the lee side of the structure is observed, suggesting the incipient formation
of a tombolo, as expected.
These examples show the model ability to predict the impact of coastal
structures on nearshore area.
§
§
§
OD
8
U3
8
I
Figure 6. Detached breakwater. Bottom evolution
5 Summary and conclusions
In this paper, an integrated coastal area model has been presented.
The model includes three modules. The first one (wave propagation)
and the second one (nearshore circulation) correspond to two previously
developed numerical models. The third one is a sediment transport and
morphodynamic module recently developed.
Several tests have been carried out. The model has shown to behave
physically and a suitable ability to reproduce the short term coastal
processes. On the other hand, the numerical scheme has shown stability
and improved behaviour when the effect of the bed slope was considered.
Nevertheless, the numerical modelling of nearshore areas still requires
a great deal of work, including the calibration of the models with field
measurements and the variation of simulation intervals.
References
[1] Azuz, I. Modelo morfodindmico de evolucion del fondo en zona
costeraj Ph. D. Thesis, Universitat Politecnica de Catalunya, Barcelona,
Spain (in press).
Transactions on the Built Environment vol 27, © 1997 WIT Press, www.witpress.com, ISSN 1743-3509
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Computer Modelling of Seas HI
[2] Chesher, T.J., Wallace, H.M., Meadowcroft, I.C. & Southgate, H.N.
Pisces: A Morphodynamic Coastal Area Model, HR Wallingford, Report
SR 337, Waffingford, UK, 1993.
[3] Coeffe, Y. & Pechon, P. Modelling of sea-bed evolution under waves
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Coastal Engineering Conference, Capetown, South Africa, 1982, ASCE,
New York, 1983.
[4] De Vriend, H.J. 2DH Mathematical modelling of morphological
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[5] De Vriend, H.J. Mathematical modelling and large-scale coastal
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[8] Horikawa, K. Nearshore Dynamics and Costal Processes, University
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[9] O'Connor, B. Suspended sediment transport in the coastal zone,
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[11] Rivero, F.J. & Sanchez-Arcilla, A. Modelo quasi-3D delflujoen
la zona de rompientes, Programa de Clima Maritime, MOPU, publication
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[12] Rivero, F.J. & Sanchez-Arcilla, A. Propagation of linear gravity
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[13] Rivero, F.J., Rodriguez, M. & Sanchez-Arcilla, A. Propagation
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77 Jornadas Espanolas de Ingenieria de Puertos y Costas, Gijon, Spain,
1993.
[14] Van Rijn, L.C. Sediment transport, part III: bed forms and
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