Bruun rule for shoreface adaptation to sea-level rise
Contents
The equilibrium profile concept
The basic idea of Per Bruun when he proposed in 1954 the rule for shoreface adaptation to sea-level rise refers to the concept of equilibrium profile [1][2]. The morphologically active 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]. With this assumption, the active shoreface of an unconsolidated sedimentary coast is shaped by adjustment to the prevailing waves, currents and sediment supply, see the article Shoreface profile. The seaward limit of the morphologically active shoreface is time-scale dependent. However, even if the deeper lower shoreface retains inherited morphology, it participates so slowly in cross-shore sediment exchange that its influence on shoreline adaptation over management time scales can be neglected.
The Bruun rule assumes that sea-level rise is slow compared with the adjustment of this average profile. If climate change affects only the sea level, while leaving the wave climate unchanged, Bruun reasoned that the equilibrium shoreface profile [math]Z_{eq} (x) [/math] then should also remain the same, while migrating landward and upward.
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. A dynamic equilibrium is reached when onshore directed sediment transport and offshore directed transport are equal on average over a period exceeding the morphodynamic adaptation time.
Sand can also be supplied or taken away as a result of gradients in the longshore sand transport. However, because sea level rise is a slow large-scale process, we may consider large temporal and spatial scales. A temporal scale is considered which is much larger than time scales associated with shoreline adaptation to fluctuations in the wave climate, including major storms. A spatial scale is considered which is much larger than the spatial scales associated with variations in longshore sediment transport. As the average gradient of the longshore sand transport decreases with increasing longshore length scale, it can become almost insignificant. Taking 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 quasi-equilibrium profile then should be preserved while moving landward and upward with sea level rise according to the Bruun rule. In reality, coastal stretches without significant net long-term erosion or accretion by longshore transport gradients are more the exception than the rule.
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.
Mathematical expression of the Bruun rule
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 initial equilibrium profile [math]Z_{eq} (x)[/math] with an upward shift equal to the sea-level rise [math]\Delta h[/math], and a landward shift [math]\Delta x[/math] :
[math]Z^{new}_{eq} (x) = Z_{eq} (x+\Delta x) + \Delta h . \qquad(1)[/math]
Conventions are (see Fig. 1): [math]x[/math] increases seaward; [math]Z_{eq}[/math] is an elevation measured upward from the closure-depth level, [math]Z_{eq}(x_{cl})=0[/math]. The shoreline retreat [math]\Delta x[/math] can be computed by the requirement that the total sand volume in the initial active zone ([math][x_0,x_{cl}][/math]) and final active zone ([math][x_0 - \Delta x ,x_{cl}-\Delta x][/math]) remains the same. This implies
[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]
We further assume that the shoreline retreat [math]\Delta x[/math] is small enough to justify the approximation
[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]
Conditions and limitations for application of the Bruun rule
Evidence for the Bruun hypothesis
Laboratory experiments of shoreface profile adaptation to sea-level rise show reasonable agreement with the Bruun rule if time scales are considered which are sufficiently long for the development of an equilibrium profile[5].
In addition to long time scales, the application of the Bruun rule also requires large spatial scales. At local scales, natural processes can cause trends in coastal change that deviate significantly from the average trend over long coastal stretches - examples are given in the article Erosion hotspots.
No convincing field validation or falsification of the original Bruun rule is presently available. Long-term observations generally do not permit the small sea-level-rise contribution to be separated reliably from storm variability, longshore transport, aeolian transport and sediment exchange with the lower shoreface[6]. Several field studies show shoreline behaviour very different from a stand-alone Bruun prediction, but these coasts do not demonstrably satisfy the assumption of a closed, alongshore-uniform active profile.
External sand sources and sinks
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 [7]. 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[8][9][10]. Let [math]\Delta V[/math] [[math]m^3/m[/math]] be the net sediment loss per unit alongshore length from the active profile during the period over which the relative sea-level change is [math]\Delta h[/math]. Sediment loss is defined as positive and sediment gain as negative. This volume change 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]
The volume [math]\Delta V[/math] may include:
- Gradients in longshore transport;
- Artificial coastal nourishment;
- Sand loss and sand supply across the boundaries of the active coastal zone, for example, sand loss to the dunes by aeolian transport, incidental overwash of low coastal barriers by overtopping waves, transport across the closure depth contour.
However, where a persistent sediment-budget imbalance is large enough to alter the average profile shape, the assumption of simple profile translation is no longer valid and equation (5) provides only an approximate volume-balance estimate. 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.
Knowledge of the closure depth
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.
Human interventions
It is clear that the Bruun rule cannot be applied without modification 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.
Coastal squeeze
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. The Bruun translation cannot occur in its original form where a seawall, revetment or other fixed landward boundary prevents erosion and landward migration of the upper profile, a situation usually referred to as 'coastal squeeze'. The beach may then narrow or lower as sea level rises, a common manifestation of coastal squeeze. If there is no sand supply from landward or seaward sources (e.g. nourishment), sea level rise will lead to a gradual ongoing submergence of the subaerial beach, starting with beach lowering adjacent to the sea wall[11]. A resistant natural cliff can impose a similar geometrical constraint, although this is more generally described as geological limitation of profile migration.
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[12]. Another condition implies that cliff erosion should not release coarse material that changes the beach sediment characteristics and thus modifies the equilibrium profile.
Wave climate
The original Bruun rule assumes that the long-term wave climate, and therefore the equilibrium profile shape, remains unchanged. If wave heights, periods or directions change substantially, shoreline response includes both sea-level-rise-driven translation and adjustment to the new wave climate. These effects must be modelled separately or with a more complete shoreline model[13].
Related articles
- Sea level rise
- Shoreface profile
- Active coastal zone
- Closure depth
- Natural causes of coastal erosion
- 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: 1-79
- ↑ Bruun, P. 1962. Sea-level rise as a cause of shore erosion. Proc.Am.Soc.Civ.Eng., J.Water Harbors Div. 88: 117-130.
- ↑ Cornaglia, P. 1889. Delle Spiaggie. Accademia Nazionale dei Lincei, Atti.Cl.Sci.Fis., Mat.e Nat.Mem. 5: 284-304
- ↑ Fenneman, N. M. 1902. Development of the profile of equilibrium of the subaqueous shore terrace. J. Geol., 10(1), 1–32.
- ↑ 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
- ↑ McLean, R., Thom, B., Shen, J. and Oliver, T. 2023. 50 years of beach–foredune change on the southeastern coast of Australia: Bengello Beach, Moruya, NSW, 1972–2022. Geomorphology 439, 108850
- ↑ Cooper, J.A. and Pilkey, O.H. 2004. Sea-level rise and shoreline retreat: time to abandon the Bruun Rule. Global Planetary Change 43: 157-171
- ↑ 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: 71-81
- ↑ Dean, R.G. and Houston, J.R. 2016. Determining shoreline response to sea level rise. Coastal Engineering 114: 1–8
- ↑ 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 wave-dominated 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 sea-level rise – Examples from the exposed and macrotidal north coast of SW England. Marine Geology 450, 106853
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