Bruun rule for shoreface adaptation to sealevel rise
The basic idea of Per Bruun when he proposed in 1954 the rule for shoreface adaptation to sealevel rise refers to the concept of equilibrium profile ^{[1]}^{[2]}. The shoreface of a sedimentary coast should not be considered a geologically inherited feature, but the result of the natural coastal response to modern hydrodynamic conditions, as noted already more than a century ago by Cornaglia (1889) ^{[3]} and Fenneman (1902) ^{[4]}. For a more detailed explanation, see the article Shoreface profile.
If climate change affects only the sea level, while leaving the wave climate unchanged, Bruun reasoned that the slope of the equilibrium shoreface profile [math]Z_{eq} (x) [/math] then should also remain the same. The profile should move landward and upward, however, to keep the same position with respect to sea level as before.
An equilibrium profile is a theoretical concept that is never realized in practice at a given time and location. Sand is transported all the time up or down the active zone of the shoreface in response to fluctuations in the wave climate. Sand can also be supplied or taken away as a result of gradients in the longshore sand transport. However, we may consider larger temporal and spatial scales because sea level rise is a slow process on a large spatial scale. Fluctuations in the wave climate generally occur on much shorter time scales. Considering long coastal stretches with similar characteristics (far from headlands or coastal inlets), the average gradient of the longshore sand transport decreases with increasing longshore length scale and can become almost insignificant. Hence, if we take the average of the coastal profiles over temporal and spatial scales which are sufficiently long, the resulting profile may be close to equilibrium. This averaged quasiequilibrium profile then should be preserved while moving landward and upward with sea level rise if the averaging scales are long enough but much shorter than the temporal and spatial scales of sea level rise.
Lifting the profile requires supply of sand. Bruun assumed that sand supply from offshore is negligible if the shoreface profile extends seaward up to the closure depth. The sand should then be supplied from the coastal profile that lies inland from the active coastal profile, for example a high berm, a (fore)dune or a sand barrier. This can only be the case if the sea level does not exceed the elevation of the inland coastal profile. It also must be assumed that the sediment supplied by the inland part of the coastal profile has characteristics similar to the sand of the active coastal zone.
Sand supply from the inland coastal profile implies shoreline retreat (designated [math]\Delta x[/math]) and a corresponding landward shift of the equilibrium shoreface profile. The new translated equilibrium profile [math]Z^{new}_{eq} (x)[/math] is then equal to the old equilibrium profile [math]Z_{eq} (x)[/math] with an upward shift equal to the sealevel rise [math]\Delta h[/math], and a landward shift [math]\Delta x[/math] (see Fig. 1):
[math]Z^{new}_{eq} (x) = Z_{eq} (x+\Delta x) + \Delta h . \qquad(1)[/math]
The shoreline retreat [math]\Delta x[/math] can be computed by the requirement that the total sand volume in the active zone (between [math]x_0[/math] and [math]x_{cl}[/math], see Fig. 1) remains the same:
[math]\int_{x_0}^{x_{cl}} [Z_{eq} (x+\Delta x) + \Delta h] dx = \int_{x_0}^{x_{cl}} Z_{eq} (x) dx . \qquad(2)[/math]
Note that the equilibrium shoreface profile [math]Z_{eq} (x)[/math] is defined here relative to the closure depth. It is assumed that [math]\Delta x[/math] is small ([math]\Delta x \lt \lt x_{cl} – x_0[/math]). We therefore have ignored the small shift of the limits of the active zone in Eq. (2) and we further approximate
[math]Z_{eq} (x+\Delta x) \approx Z_{eq} (x) + \Delta x \; \Large \frac{d Z_{eq} (x)}{dx} \normalsize . \qquad(3)[/math]
Substitution in equation (2) then yields for the horizontal landward shoreline shift [math]\Delta x[/math] the Bruun rule
[math]\Delta x \approx \Delta h \; \Large \frac{x_{cl} – x_0}{Z_{eq} (x_0)} \normalsize . \qquad(4)[/math]
The Bruun rule has been widely applied, but the results did often not compare very well to observations. Critical comments have been written on the Bruun rule ^{[5]}. A major reason for discrepancies results from ignoring the assumptions underlying the derivation of the Bruun rule. This holds in particular for the assumption that the sand volume in the active coastal profile remains unchanged: sand supply from other sources and sand losses are not considered. However, the Bruun rule can be extended to include to some degree other sand sources and sand losses that occur during profile translation^{[6]}^{[7]}^{[8]}. Therefore the volume change [math]\Delta V[/math] [[math]m^3/m[/math]] of the active coastal profile due to sand loss (positive) or sand supply (negative) during sea level rise should be added in the left hand side of Eq. (2). We then find for the horizontal landward shoreline shift [math]\Delta x[/math]:
[math]\Delta x \approx \Delta h \; \Large \frac{x_{cl} – x_0}{Z_{eq} (x_0)} + \frac{\Delta V}{Z_{eq} (x_0)} \normalsize . \qquad(5)[/math]
Causes of sand loss and sand supply are:
 Sand loss to the dunes, by aeolian transport;
 Incidental overwash of low coastal barriers by overtopping waves;
 Transport across the closure depth contour;
 Gradients in longshore transport;
 Artificial coastal nourishment.
However, these loss and supply volumes should not be very important, as they would otherwise alter the average shape of the coastal profile. With this restriction, Eq. (5) suggests that sand nourishment of sufficient volume (depending on the rate of sea level rise and the width of the active coastal zone), can neutralize shoreline retreat if applied to any location of the active coastal profile over a sufficient longshore length.
Laboratory experiments of shoreface profile adaptation to sealevel rise show reasonable agreement with the Bruun rule if time scales are considered which are sufficiently long for the development of an equilibrium profile^{[9]}.
Application of the Bruun rule requires knowledge of the closure depth under natural conditions with respect to the considered time scale. Determining the closing depth is not straightforward as explained in the article Closure depth. Several formulas have been proposed, but there is no universal prescription that applies to every type of coast. The problem of choosing the correct closure depth introduces an important uncertainty in the application of the Bruun rule.
It is clear that the Bruun rule cannot be applied if the equilibrium conditions for the coastal profile are significantly changed due to human interventions, for example:
 change of the local wave climate due to construction of artificial islands, offshore windfarms, dredging of navigation channels or offshore deposition of dredged material;
 change in sand supply due to dam building in nearby rivers, construction of jetties or groynes.
Application of the Bruun rule assumes that the sand needed for an upward shift of the beach profile can be provided by a landward source. Therefore, the Bruun rule for shoreline retreat cannot be applied in cases where the landward boundary of the active zone is bounded by a hard structure, for example a rocky cliff or an artificial sea wall. If there is no sand supply from landward sources, sea level rise will lead to a gradual ongoing submergence of the subaerial beach, starting with beach lowering adjacent to the sea wall.^{[10]}.
The Bruun rule can be applied under certain conditions if the active zone is bounded by a soft cliff. Cliff erosion during sea level rise should yield an amount of sand that is sufficient for lifting the active coastal profile; cliff retreat then depends on the cliff slope and the sand content^{[11]}. Another condition implies that cliff erosion should not release coarse material that changes the beach sediment characteristics and thus modifies the equilibrium profile.
If climate change causes not only a rise in sea level, but also a change in the wave climate, then the Bruun rule cannot be applied^{[12]}.
Related articles
 Sea level rise
 Shoreface profile
 Active coastal zone
 Closure depth
 Natural causes of coastal erosion
 Effect of climate change on coastline evolution
 Dealing with coastal erosion
 Dune erosion
References
 ↑ Bruun, P. 1954. Coast erosion and the development of beach profiles. Beach Erosion Board, US Army Corps of Eng., Tech.Mem. 44: 179
 ↑ Bruun, P. 1962. Sealevel rise as a cause of shore erosion. Proc.Am.Soc.Civ.Eng., J.Water Harbors Div. 88: 117130.
 ↑ Cornaglia, P. 1889. Delle Spiaggie. Accademia Nazionale dei Lincei, Atti.Cl.Sci.Fis., Mat.e Nat.Mem. 5: 284304
 ↑ Fenneman, N. M. 1902. Development of the profile of equilibrium of the subaqueous shore terrace. J. Geol., 10(1), 1–32.
 ↑ Cooper, J.A. and Pilkey, O.H. 2004. Sealevel rise and shoreline retreat: time to abandon the Bruun Rule. Global Planetary Change 43: 157171
 ↑ Stive, M.J.F. 2004. How Important is Global Warming for Coastal Erosion? Climatic Change 64: 27–39
 ↑ Rosati, J.D., Dean, R.G. and Walton, T.L. 2013. The modified Bruun Rule extended for landward transport. Mar. Geol. 340: 7181
 ↑ Dean, R.G. and Houston, J.R. 2016. Determining shoreline response to sea level rise. Coastal Engineering 114: 1–8
 ↑ Atkinson, A.L., Baldock T. E., Birrien, F., Callaghan, D. P., Nielsen, P., Beuzen, T., Turner, I.L., Blenkinsopp, C. E. and Ranasinghe, R. 2018. Laboratory investigation of the Bruun Rule and beach response to sea level rise. Coastal Engineering 136: 183–202
 ↑ Beuzen, T., Turner, I.L., Blenkinsopp, C.E., Atkinson, A., Flocard, F. and Baldock, T.E. 2018. Physical model study of beach profile evolution by sea level rise in the presence of seawalls. Coast. Eng. 136: 172–182
 ↑ Wolinsky, M.A. and Murray, A.B. 2009. A unifying framework for shoreline migration: 2. Application to wavedominated coasts. J. Geophys. Res. Earth Surf. 114: F01009
 ↑ Masselink, G., Brooks, S., Poate, T., Stokes, C. and Scott, T. 2022. Coastal dune dynamics in embayed settings with sealevel rise – Examples from the exposed and macrotidal north coast of SW England. Marine Geology 450, 106853
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