# Talk:Parametric equilibrium models

## Review by Dominic Reeve (January 2013)

This article provides an introduction to Bruun’s rule and equilibrium model concepts. Searches for ‘Bruun’s rule’, ‘Equilibrium profile’ and Equilibrium beach profile’ are all redirected here which is not very helpful.

I suggest the following rearrangements:
Retain the existing section on Bruun’s rule for coastal retreat under a heading ‘Bruun’s rule for coastal retreat’. Remove references to specific software packages – the appropriate URL’s and sources can be included in the references section.

The section on ‘Equilibrium beach profile’ should be removed and included in a separate article titled ‘Equilibrium beach profile’. The existing text can be retained but expanded to include:

(1) an additional deficiency is that the equilibrium profile tend to infinite depth without tending towards an asymptote equivalent to a depth of closure. The proposed alterations by Bodge (1992) and Komar & McDougal (1994) address this;

(2) the treatment of barred profiles by Pruszak & Rozynski (1998), Inman e al (1993) and Wang & Davis (1998);

(3) extension of the concept beyond the surf zone (viz. Bernabeu et al 2003); treatment of the case where sediment size varies along the profile (as per Dean & Dalrymple 2002); an example illustrating the level of agreement that can be expected between the equilibrium profile and one (or a set) of profiles.

Some of these comments are included in the article Shoreface profile.

The section on log spiral beaches should be removed and included in a separate article titled ‘Equilibrium bay shape’. The existing text can be retained but needs to be extended to include:

(1) some discussion of the historical development of the concept (include the logarithmic spiral model (Silvester, 1960, 1970, 1976; Yasso, 1965); the parabolic model (Hsu and Evans, 1989); and more recently the hyperbolic-tangent model (Moreno and Kraus, 1999);

(2) Modifications such as those proposed by Silvester and Hsu (1997) and Tan and Chiew (1994) to reduce the number of parameters from three to one;

(3) discussion of reducing uncertainties eg. Moreno and Kraus (1999) and González and Medina (2001);

(4) application of stochastic ideas to quantify uncertainties Reeve & Li (2009), Li & Reeve (2009).

**References**

Bodge, K R 1992. Representing equilibrium beach profiles with an exponential expression, J Coastal Res., 8(1), pp47-5 Dean, R.G. & Dalrymple, R.A. 2002. Coastal processes with engineering applications. Cambridge University Press, Cambridge, UK.

González, M. and Medina, R. (2001). “On the application of static equilibrium bay formulations to natural and man-made beaches.” Coastal Eng., 43, 209-225.

Hsu, J.R.C. and Evans, C. (1989). “Parabolic crenulate shaped bays and applications.” Proc. Instn. Civil Engrs. 87, 557-570.

Komar, P D & McDougal, W G. 1994. The analysis of exponential beach profiles, J Coastal Res., 10(1), pp58-69. Inman, D.L., Elwany, M.H.S. and Jenkins, S.A., 1993. Shorerise and Bar–Berm Profiles on Ocean Beaches, J. Geophys. Res., 98, C10, p18,181–18,199.

Martino, E., Moreno, L. and Kraus, N.C. (2005). “Uncertainties in design guidance for headland-bay beaches”. Proc. Coastal Dynamics’05, CD-ROM, ASCE.

Moreno, L.J. and Kraus, N.C. (1999). “Equilibrium shape of headland-bay beaches for engineering design.” Proc., Coastal Sediments’99, 860-875.

Li Y & Reeve D E (2009). A stochastic method for predicting average beach shape, Proceedings of the ICE, Maritime Engineering, 162, p97-103.

Pruszak, Z. & Rozynski, G., 1998. Variability of multi-bar profiles in terms of random sine functions, Journal Waterway,Port, Coastal and Ocean Engineering, 124(2), p48-56.

Reeve D E & Li Y (2009), Stochastic description of quasi-static beach behaviour, ASCE J. Waterway, Port, Coastal & Ocean Engineering, 135(4), p144-153, 2009.

Silvester, R. (1960). “Stabilization of sedimentary coastlines”. Nature, 188, 467-469.

Silvester, R. (1970). “Growth of crenulate shaped bays to equilibrium”. J. of Waterways and Harbours Div., 96(2), 275-287.

Silvester, R. (1976). “Headland defense of coasts”. Proc. 15th Coastal Eng. Conf., ASCE, 1394-1406.

Silvester, R. and Hsu, J.R.C. (1997). Coastal stabilization, Advanced Series on Ocean Engineering, Vol. 14, World Scientific, Singapore.

Tan, S.K., and Chiew, Y.M. (1994). “Analysis of bayed beaches in static equilibrium.” J. of Waterw., Coastal, Port and Ocean Eng., 120(2), 145-153.

Yasso, W.E. (1965). “Plan geometry of headland bay beaches.” J. of Geology, 73, 702-714.