A STOCHASTIC MODEL FOR SHORELINE EVOLUTION
Mark Spivack1 and Dominic Reeve2
The problem of coastline evolution in the presence of varying waveclimate is studied. Statistical moments are introduced as mean of investigating
shoreline evolution and variability. Equations for these quantities are developed
and solved, for given wave-climate statistics. These equations describe
analytically the time-dependent averaged solution, and its dependence on waveclimate; and circumvents computationally intensive Monte-Carlo simulations.
Abstract:
INTRODUCTION
Strategic planning for the management of our coasts relies implicitly upon an
understanding of the physical processes responsible for shaping coastal morphology.
Morphodynamic models can play a key role in assessing the likely impact of coastal
developments. Predictive models used in current practice are usually employed in a
deterministic fashion and do not forecast the variability likely to be experienced in real
situations. In this paper we will describe results from a research project that has developed a
theoretical framework for stochastic shoreline prediction.
A review of the main difficulties in modelling long term morphological change and of a
number of alternative modelling techniques has been presented in De Vriend et al (1993).
Morphological forecasting methods are in use, based on solving detailed dynamic equations.
This deterministic approach to morphological prediction was pioneered by Pelnard-Considere
(1956) who presented an equation to forecast changes in coastline position. The approach is a
one-line model (as it predicts the position of a single depth contour) and has also become
known as influence function theory. However, a key drawback is that it requires as input
1
Department of Applied Mathematics and Theoretical Physics, Silver Street, Cambridge, CB3 9EW, UK
Professor of Environmental Fluid Mechanics, Division of Environmental Fluid Mechanics, School of Civil
Engineering, Coates Building, University of Nottingham, Nottingham, NG7 2RD, UK
2
1
Spivack and Reeve
exact values of environmental parameters unlikely to be available in detail; and gives no
measure of the variability or estimate of likelihood of "extreme cases".
One alternative to the deterministic modelling approach is to employ stochastic
forecasting methods, such as Monte Carlo methods. That is, using independent realisations of
wave conditions, a corresponding set of realisations of shoreline evolution can be computed.
This set of results can then be used to calculate the `ensemble average' statistics of shoreline
position. If a 1-line model is used the computational overhead of the Monte Carlo method
may be restrained to a modest level, allowing predictions of the likely variability of the
shoreline to be obtained, eg Vrijling & Meijer (1992). A key input to this type of model is the
statistical description of the driving forces. In the case of a 1-line model this is a description
of the wave climate. To obtain physically meaningful results this approach requires a large
amount of wave data to define accurately the joint statistical characteristics of the wave
climate and water level over the prediction period.
SUMMARY OF EQUATIONS
In order to overcome the above difficulties a different approach has been adopted. By an
appropriate formulation of the governing equations, the ensemble average can be obtained
analytically, using well-established techniques (Spivack & Reeve 2002). This procedure
provides a closed-form expression for the average position of the shoreline that may be
evaluated directly. A corresponding expression may also be derived for the variance of the
shoreline position. The theory describes the evolution of an arbitrary initial coastline subject
to a varying wave attack. Some examples for the evolution of beaches with localised beach
recharge have been given by Reeve & Spivack (2001).
In this paper we summarise theory and present some illustrative results. The quantities
derived give an estimate of mean, and allow an examination of "worst-case" solution.
Our starting point is the following equation for beach plan shape y:
or in operator notation
where K is the stochastically varying wave climate, which is a known function of physical
parameters, and which can be written as a mean (deterministic) part plus a random component
with mean zero:
2
Spivack and Reeve
Here the angled brackets denote the ensemble average. The function y(x,t) can therefore be
treated and solved as a stochastic variable, in order to obtain its statistical behaviour over time.
The statistics of the random part δ(t) are known or can be approximated. In particular we require
the autocorrelation function, denoted ρ(t). A typical record of wave heights and angles over a
short period is shown in the following graph:
6
4
2
0
0
400
800
1200
SOLUTION AND RESULTS
The formal solution of time evolution of
respectively as
y(x,t) and its Fourier transform can be written
The statistics of the equation above can be found analytically. For example the mean
profile can be written
3
Spivack and Reeve
in which the unknown middle term can be evaluated to obtain
In this expression f(t) is the integral of the varying part δ(t’) up to time t and σt is its
variance:
This approach, as described above, allows us to quantify mean beach shape, and its
variability, as functions of time and wave-climate statistics. For example the following graph
shows the autocorrelation of y(x,t) as a function of spatial separation and time, and from this
the variance can be obtained directly, thus providing an estimate of likelihood of extreme
events.
 ex
2

4
Spivack and Reeve
ACKNOWLEDGEMENTS
This work was supported under grants from the EPSRC, the London Mathematical
Society grant 4615, and the Institution of Civil Engineers under Research & Development
Enabling Grant 0101.
REFERENCES
De Vriend, H.J., Capobiance, M., Chesher, T., de Swart, H.E., Latteux, B. & Stive, M., 1993.
Approaches to long-term modelling of coastal morphology: a review, Coastal Engineering,
21, p225-269.
Pelnard-Considere, R., 1956. Essai de theorie de l'evolution des formes de rivage en plages de
sables et de galets, Soc. Hydrotech. de France, IV'eme Journees de L'Hydraulique Question
III, rapport 1, 74-1-10.
Perlin, M. & Dean, R.G., 1983. A numerical model to simulate sediment transport in the vicinity
of coastal structures, CERC Report MR 83-10.
Reeve, D. E. & Spivack, M., 2001. Prediction of long-term coastal evolution using moment
equations, presented at Coastal Dynamics 2001, Lund, Sweden.
Spivack, M. & Reeve, D.E., 2002. Moment equations for coastal evolution, submitted for
publication.
Vrijling, J.K. & Meijer, G.J., 1992. Probabilistic coastline position computations, Coastal
Engineering, 17, p1-23.
5
Spivack and Reeve