# Parametric equilibrium models

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Parametric equilibrium models represent the shape of the coastline or its response to forcing through simple equations that have been derived through a mixture of curve-fitting and theoretical considerations. They are necessarily simplistic, but quick to apply.

## Contents

### Equilibrium Beach Profile

Bruun (1954)[1] examined beach profiles in Denmark and California and concluded that the cross-shore profile in the vertical could be expressed in the form: [math]h \, = \, A \, y^{2/3} \, ,[/math] where [math]h[/math] = water depth, [math]A[/math] is a sediment scale parameter and [math]y[/math] is the cross-shore distance from the shoreline. In 1977 Dean examined the forms of equilibrium beach profiles that would results from different dominant forcing mechanisms and concluded that equilibrium beach profiles would take the form shown above if the dominant destructive force was wave energy dissipation per unit volume (Dean et al., 2002 [2]). The sediment scale parameter can be related to sediment size or fall speed [2] so the equation above can be used to make predictions about beach profiles (see also Shoreface profile).

Alternative forms of the equilibrium beach profile have been developed by other authors, but these have more free parameters and so are less suited to making predictions as calibrations tend to be site-specific (Dean et al., 2002[2]). The main problems with the equilibrium beach profile are that the slope is infinite at the water line and the profile does not allow for bars.

### Bruun rule for coastal retreat

Bruun (1962) proposed the following equation for the equilibrium shoreline retreat, [math]R[/math], of sandy coasts that will occur as a result of sea level rise, [math]S[/math]:

[math]R = S L /(h_{cl}+B)[/math].

Here [math]L[/math] is the cross-shore width of the active profile (i.e. cross-shore distance from closure depth to furthest landward point of sediment transport), [math]h_{cl}[/math] is the closure depth (maximum depth of sediment transport) and [math]B[/math] is the elevation of the beach or dune crest (maximum height of sediment transport). The equation balances sediment yield [math]R(h_{cl}+B)[/math] from the horizontal retreat of the profile with sediment demand, [math]SL[/math], from a vertical rise in the profile (Dean et al., 2002). The magnitudes of [math]h_{cl}[/math] and [math]B[/math] are difficult to determine, however and the actual seabed will need time to respond to a change in sea level.

The Bruun rule does not depend on a particular coastal profile, but does assume that no sediment is lost from the coastal system (which is likely to happen if there are fines in the area eroded). It assumes a coast of unconsolidated sediment, mainly sand, with (originally) a coastal dune and makes no allowances for gradients in the longshore or cross-shore transport of sand. However, the Bruun rule has been extensively modified, developed and used. See also Bruun rule.

In the coastal regions where the Bruun rule can be said to apply, the rate of shoreline retreat ([math]dR/dt[/math]) is directly proportional to the rate of sea level rise ([math]dS/dt[/math]). It follows that the ratio of future shoreline retreat rate to present day shoreline retreat rate (the shoreline retreat rate multiplier) will be the same as the ratio of future sea level rise rate to present day sea level rise rate.

Such predictions should be treated with some caution, however, as the Bruun rule is a very simplistic analysis tool and difficult to validate. Bray and Hooke (1997)[3] modified it to look at the erosion of soft cliffs by adding sediment exchange and considered it particularly suitable for assessing the sensitivity of eroding soft cliffs to future climate change. On the other hand both Cooper and Pilkey (2004b)[4] and Stive (2004)[5] cautioned against its use due to its simplicity and restrictions.

Dickson, Walkden and Hall (2007)[6] compared the predictions of recession from the modified Bruun rule and the systems model SCAPE (Walkden and Hall, 2005[7]) for 50km of the soft rock shoreline of northeast Norfolk on the east coast of England. They found that the systems model SCAPE predicted a rather more complex response, with lower overall vulnerability to sea level rise, than the Bruun rule. Where beaches overlie shore platforms both SCAPE and the Bruun rule gave accelerated recession rates in response to sea level rise. However, in some areas with large beaches and gradients in the longshore transport rates the Bruun rule predicted recession where SCAPE predicted accretion due to sediment transport from eroding up-drift stretches of the coastline. This indicated the inadequacy of the Bruun rule in regions where there is a significant variability in the longshore transport rates.

Therefore Bruun rule predictions of shoreline retreat rates due to accelerated sea level rise should be treated with some caution as they may well be too small or high (see also Bruun rule). It seems probable that the shoreline recession rate will increase in many places if the rate of sea level rise increases.

### Pocket beaches

Sand beaches are often limited by headlands or other fixed points. These beaches (often called pocket beaches) form bays with a crescentic shape, particularly when there is a dominant wave direction. In time, such a bay may reach a form of static equilibrium and cease to erode further (providing conditions do not change and no sediment is lost by processes other than longshore drift). The plan shape of a static equilibrium bay is predictable, for a given deep-water wave obliquity. Wave diffraction at the headlands and depth-induced refraction of waves when entering shallow water produce a log-spiral shoreline shape (Silvester and Hsu, 1997[8]). This can be used to provide equilibrium bays shapes based on the structure control points and predominant wave direction. For further details, see Embayed beaches.

Shoreface profile
Bruun rule
Active coastal zone
Closure depth
Embayed beaches

### References

1. Bruun, P., 1954, ‘Coast erosion and the development of beach profiles’, Beach erosion board technical memorandum. No. 44. U.S. Army Engineer Waterways Experiment Station. Vicksburg, MS.
2. Dean, R.G., Kriebel, D.L. and Walton, T.L., 2002. Cross-shore secdiment transport processes. Chapter 3 of Part III of the Coastal Engineering Manual, EM 1110-2-1100.
3. Bray, M.J. and Hooke, J.M., 1997, ‘Prediction of soft-cliff retreat with accelerating sea-level rise’, Journal of Coastal Research 13 (2), 453– 467.
4. Cooper, J.A.G. and Pilkey, O.H., 2004b, ‘Sea level rise and shoreline retreat: time to abandon the Bruun rule’, Global Planet Change, 43: 157 – 171.
5. Stive, M.J.F., 2004. How important is global warming for coastal erosion. Climatic Change. 64: 27 – 39.
6. Dickson, M.E., Walkden, M.J.A. and Hall, J.W., 2007. Systematic impacts of climate change on an eroding coastal region over the twenty-first century. Climatic Change, in press.
7. Walkden, M.J.A. and Hall, J.W., 2005. A predictive mesoscale model of the erosion and profile development of soft rock shores. Coastal Engineering 52, 535-563.
8. Silvester, R. and Hsu, J.R.C., 1997. Coastal Stabilisation. Advanced Series on Ocean Engineering – Volume 14, World Scientific. .

 The main author of this article is James, SutherlandPlease note that others may also have edited the contents of this article. Citation: James, Sutherland (2021): Parametric equilibrium models. Available from http://www.coastalwiki.org/wiki/Parametric_equilibrium_models [accessed on 14-06-2024] For other articles by this author see Category:Articles by James, Sutherland For an overview of contributions by this author see Special:Contributions/J.Sutherland