Prediction of Long-Term Coastal Evolution

Stochastic prediction of long-term coastal evolution

D.E. Reeve1 & M. Spivack2 1Department of Civil Engineering, University of Nottingham, UK 2Department of Applied Mathematics and Theoretical Physics, University of Cambridge, UK

Abstract

Abstract sent already

1 Introduction

In recent years the general public have become increasingly aware of the threat posed by coastal erosion and flooding. In response, national and local authorities are preparing strategic management plans in many parts of the world. A good understanding of the coastal processes is a crucial ingredient on which the planning process depends. This paper describes results from a study to initiate the development of statistical methods for predicting changes in coastal morphology over the long term.

While there are many methods available for predicting the short-term (weeks to a year) coastal response to physical processes, the same is not true for predicting long-term (decades to a century) changes in coastal morphology. Researchers have employed a variety of statistical techniques to analyse historical beach records for trends and quasi-periodic behaviour, eg Winant et al.[1], Li et al.[2]. The results of such analyses have also been used in combination with empirical formulae as a basis on which to predict future beach response to wave action, Hsu et al. [3]. An alternative approach has been to develop predictive equations based on a description of the dominant physical processes that drive changes in beach shape. Pelnard-Considere [4] developed an equation, based on observations in laboratory experiments; to forecast changes in coastline position. This was termed a ‘one-line’ equation as it predicted the position of a single contour line on the beach. Subsequent researchers have extended this approach to simulate multiple depth contours, eg. Bakker [5], Perlin & Dean [6]. Such models, however, do not forecast the mean or variance of the beach position likely to be experienced in real situations.

To overcome this deficiency it is necessary to employ stochastic forecasting methods. For example, Vrijling & Meijer [7] used a one-line equation to perform Monte Carlo simulation of beach positions; while Reeve & Fleming [8] used a one-line model and historical shoreline positions to infer the distribution of time-averaged sediment sources and thence to estimate likely future shoreline movement.

A drawback of this approach is that to obtain meaningful results very many realisations are needed.

In this paper we describe and solve an equation for the first moment (or mean) of shoreline position. The equation describes the averaged or long-term solution and its dependence on wave-climate; and eliminates the need for computationally intensive Monte-Carlo simulations. 2 Summary of theory

We summarise here the derivation of the first moment equation, which describes the mean shoreline.

The starting point is the governing equation first derived by Pelnard-Considere [4]. Under the assumption that the beach has parallel depth contours, and that there exists a depth D beyond which no changes in the cross-shore profile of the beach occurs a consideration of the conservation of beach material yields

Q y  D (1) x t where Q is the longshore sediment transport rate, D is known as the ‘depth of closure’, t is time, x is the distance along a reference line and y is the distance of the shoreline measured perpendicularly from the reference line.

An empirical formula relating the sediment transport to wave angle and wave power has been developed by the US Army Corps [9]:

Q  Q0 sin(2 b ) (2)

where Q0 is the amplitude of the longshore transport rate. b is the angle between the wave front and the shoreline, and may be written as

1  y   b   0  tan   (3)  x 

where 0 is the angle between the wave front and the x-axis. Substituting equations (2) and (3) into (1), and assuming both 0 and y/x are small yields

y  2 y  K (4) t x 2 where K = 2Q0/D. Equation (4) has the form of a linear diffusion equation, where K is a parameter that depends on the wave climate and beach material and has the role of a diffusion coefficient. In practice, K will be a function of both time and position, in which case a more complicated governing equation results, Larson et al [10]. Here, we let K(t) vary randomly as a function of time. In physical terms this corresponds to accounting for variations in wave height with time. The position of the shoreline, y(x,t) is therefore considered to be a stochastic variable.

Using equation (4) above we formulate an evolution equation, for the averaged coastal plan shape. The first moment gives the mean value of y as it evolves with time. In order to form the evolution equation we must first specify the statistics of K. Assuming that K has stationary statistics and we may write

K(t)  K  (t) (5) where the mean < K > is constant and the perturbation  has mean zero and stationary Gaussian statistics, which are known. Angled brackets denote an ensemble average. Following the derivation given by Reeve & Spivack [11], we take Fourier transforms of each side of equation (4) to obtain

~y   2 K~y (6) y where the tilde denote the Fourier transform with respect to x. This has the following solution over any time step [t, t+t]:

tt ~y( ,t  t)  exp{ 2 (t)dt}~y( ,t) (7) t

Taking the average of this equation and substituting (4) into the result we obtain, after some manipulation (see Reeve & Spivack [11]) this can be written

 ~y( ,t  t)  exp{ 2  K  t}.   4 2 tt tt      ~ exp   (t  t )dt dt y( ,t) (8)  2 t t 

Where  and  are the given autocorrelation function and standard deviation of  respectively. 3 Solution in specific cases

We consider the case when the correlation function for (t) is Gaussian, ie

2  tt    (t  t)  e  T  (9)

The extent to which values of the coefficient are similar is governed by the `correlation time', T, while the statistics of (t) obey a Gaussian distribution.

In practice the initial coastline configuration will depend on alongshore position. In this case the statistics of y(x,t) will not be stationary, even though (t) is stationary. Equation (8) is valid for an arbitrary initial coastline configuration.

Direct comparison with previous results that have been obtained using equation (4) in a deterministic manner, (ie with K not random), are not straightforward. However, equation (4) is often used for predictive purposes with a ‘best estimate’ of K and the resulting shoreline position interpreted as being representative of the typical shoreline position. If we equate ‘best estimate’ and ensemble average then the results of the two approaches may be compared. In layman’s terms we will investigate whether the average shoreline position under varying wave conditions is the same as the shoreline position obtained by assuming perpetual mean wave conditions.

For purposes of comparison we consider the case of a tapered rectangular beach recharge scheme on a straight beach, see Figure 1. This comprises a rectangular block extending a distance a either side of the origin on the x-axis and protruding a distance V from the rest of the shoreline. At the end of each rectangular block is a triangular ‘taper’ extending to a distance b along the shore. The initial condition is therefore given by  0 x  b (x  b)V  b  x  a  b  a  V  a  x  0 y(x,0)   (10)  V 0  x  a (b  x)V  a  x  b  b  a  0 x  b V

-b -a 0 a b

Figure 1: Initial condition of tapered beach nourishment.

The solution of equation (4) subject to given initial condition may be written in the form of an integral, Walton [12], and for the initial condition above a closed form solution may be obtained as:

2y(x, t)  [erf (AX  A)  erf (AX  A)]  V  B  AX   [erf (AX  A)  erf (AX  B)]   B  A   B  AX   [erf (AX  B)  erf (AX  A)]   B  A  1 [exp{(AX  B)2 } exp{(AX  A)2 }]  (B  A) 1 [exp{(AX  B)2 } exp{(AX  A)2 }] (11) (B  A) where A=a/(2(Kt)), B = b//(2(Kt)) and X = x/a.

Given values of a, b and K equation (11) yields the solution for the shoreline position at any time t after the initial condition. The corresponding solution of the stochastic equation may be obtained by inserting the Fourier transform of (10) and equation (9) into equation (8). This yields an expression for the Fourier transform of the ensemble average shoreline position after time t. The result is generally not accessible by analytical methods and the inverse Fourier transform must be evaluated numerically. Here, we have used a discrete Fourier transform for this purpose.

Computations have been performed for the correlation function for a range of temporal correlation scales. In order to provide some tangible measure of the impact of the presence of temporal variations in diffusion coefficient, comparisons are made against the analytical solution for the instantaneous shoreline position for the same initial condition with the diffusion coefficient set equal to its ensemble average value.

The initial condition is defined with a = 5000m, b = 1000m, V = 20m. The ‘best estimate’ and mean diffusion coefficient, , are set to 105 m2/year. The temporal correlation function is defined by equation (9) with a correlation time, T, of 0.1 years.

Results are shown for times t = T/2, T, 2T and 4T. Here, the full line is the shoreline position assuming no fluctuations, and the dashed line is the solution when fluctuations are taken into account. This demonstrates, in this case, that neglecting the fluctuations results in an underestimate of the rate at which 'beach nourishment' is spread along the shoreline.

Picture!!!

4 Conclusions

Equations (4) and (9) are very useful as a means of simulating changes in beach plan shape. The first moment allows us to look at long-term evolution of the mean beach shape and quantify the dependence on the statistics of . However, it does not provide any information about the typical variation about this mean. Reeve & Spivack [11] illustrate how the variance of shoreline position may be obtained as a function of its first and second moments.

We can define the second moment of the shoreline position:

 y(x1 ,t)y(x2 , y)  (12)

which is the autocorrelation of the shoreline, where x1 and x2 are any two positions along the datum line. From the equations above we can form an exact evolution equation for the second moment. This will be discussed in a separate paper.

Moment equations provide an efficient and rigorous basis on which to examine long term shoreline evolution from a stochastic perspective. Numerical solution of the equations to cover a wider range of situations is the subject of ongoing work.

Acknowledgements

The authors acknowledge the support of the Institution of Civil Engineers, UK and the Engineering and Physical Sciences Research Council of the UK for support of this work via a Connectivity Research Fellowship, Grant No. GR/N20973.

References

[1] Winant, C.D., Inman, D.L. & Nordstrom, C.E., 1975. Description of seasonal beach changes using empirical eigenfunctions, J. Geophys. Res., 80(15), p1979-1986. [2] B Li & D E Reeve, 'Analysis of long-term changes in nearshore morphology', Proceedings of IAHR Symposium on River, Coastal and Estuarine Morphodynamics, Genoa, 6-10 Sept., 1999, p477-486. [3] Hsu, T-W., Ou, S-H & Wang, S-K., 1994. On the prediction of beach changes by a new 2-D empirical eigenfunction model, Coastal Engineering, 23, p255-270. [4] Pelnard-Considere, R., 1956. Essai de theorie de l'evolution des formes de rivage en plages de sables et de galets, Societe Hydrotechnique de France, IV'eme Journees de L'Hydraulique Question III, rapport 1, 74-1-10. [5] Bakker, W.T., 1968. The dynamics of a coast with a groyne system, In. Proc. 11th ICCE, London, p492-517. [6] Perlin, M. and Dean, R.G., 1979. A numerical model to simulate sediment transport in the vicinity of coastal structures, CERC, Vicksburg, MS, Misc. Rept. No. 83-10, 119pp. [7] Vrijling, J.K. & Meyer, G.J., 1992. Probabilistic coastline position computations, Coastal Engineering, 17, p1-23 [8] Reeve, D.E. & Fleming, C.A., 1997. A statistical-dynamical method for predicting long term coastal evolution, Coastal Engineering, Vol 30 (3-4), p259-280. [9] US Army Corps of Engineers, 1984. Shore Protection Manual, Coastal Engineering Research Centre, Washington. [10] Larson, M., Hanson, H. & Kraus, N.C., 1997. Analytical solutions of one- line model for shoreline change near coastal structures, J. Wtrwy., Port, Coast., and Oc. Engrg., ASCE, July/Aug, p180-191. [11] Reeve, D.E. & Spivack, M., 2001. Prediction of long-term coastal evolution using moment equations, in Proceedings of Coastal Dynamics 2001, Lund, Sweden, 9-15 June, 2001. [12] Walton, T. L., 1993. Shoreline solution for tapered beach fill, J Wtrwy, Port, Coastal & Oc. Engrg, ASCE 120 (6), p651-655.