The Bruun Rule is a formula for estimating the magnitude of the retreat of the shoreline of a sandy shore in response to changes in sea level. Originally published in 1962 by Per Bruun, the Bruun Rule was the first to give a relationship between sea level rise and shoreline recession. The rule is a simple, two dimensional mass conversion, and remains in common use to estimate shoreline recession in response to sea level rise, despite criticism and modification, and the availability of more complex alternate models.
The rule
The Bruun Rule gives a linear relationship between sea level rise and shoreline recession based on equilibrium profile theory, which asserts that shore face profile maintains an equilibrium shape, and as sea level rises the increasing accommodation space forces this equilibrium profile landward and upward to preserve its shape relative to the new sea level. As such, the Bruun Rule analysis assumes that the upper beach is eroded as the shore profile moves landward, and that the volume of eroded material is deposited offshore, resulting in a rise of the nearshore bottom which maintains a constant water depth. The Bruun Rule predicts coastal recession to be as much as 10 to 50 times sea level rise, depending on the slope of the beach.
The mathematical notation of the Bruun Rule is:
Where,
R is shoreline recession, in metres
S is sea level rise, in metres
L is the horizontal length of the bottom affected by the sea level rise (from the dune peak to depth of closure), in metres h is the depth of closure (the water depth beyond which significant sediment transport does not occur), in metres B is the dune height above sea level, in metres β is the average slope of the active profile
History
In 1954, Per Moller Bruun published a paper describing beach profiles and cross shore equilibrium. Developing on these ideas, in 1962 he published his paper "Sea-Level Rise as a Cause of Shore Erosion", which first gave a relationship between sea level rise and the shoreline recession of equilibrium beach profiles. His rule wasn't given a name until 1967, when Schwartz published "The Bruun Theory of Sea-Level Rise as a Cause of Shore Erosion" which detailed laboratory and field tests of the theory and concluded that "the concept henceforth be known as "Bruun's Rule.""
Per Bruun has since published works to clarify the limitations and practicality of his model, including "Review of Conditions for Uses of the Bruun Rule of Erosion" in 1983, and "The Bruun Rule of Erosion by Sea-Level Rise: A Discussion on Large-Scale Two- and Three- Dimensional Usages" in 1988. These publications outline the rule's limitations and practicality and stress that, though this model is intentionally two dimensional, it is applied to three dimensional environments in practice. Bruun states that the rule "must be subjected to realistic adjustments" for practical use.
As Bruun acknowledges, the Bruun Rule is a characteristically simple rule and has therefore been modified many times over, to account for more factors and in order to be accurately applicable in more cases. The Bruun Rule has been most often modified to account for longshore transport, overwash, and Aeolian sediment transport. For example, Dean and Maurmeier in 1983 modified the rule to be applicable to barrier shores and islands, some of which experience accretion instead of erosion. Rosati et al. in 2013 and Dean and Houston in 2016 have modified the rule to account for onshore transport, and cross shore movement over the supposed depth of closure. Further, Ashton et al. in 2011 modified the Bruun Rule for use on cliffed coasts, and Hinkel et al. in 2013 used the rule as part of a wider methodology to describe the effects on sea level rise on and around tidal inlets. The biggest challenge that remains seems to be isolating the effect of sea level rise on beach morphology from the coupled effects of wave energy, tidal currents, wind action, sediment supplies, sediment types and grain size, among others.
In various forms and coupled with other models, the Bruun Rule has now been used to estimate shoreline responses to sea level rise worldwide. The Bruun Rule has been applied to coasts including of the Caspian Sea, Korean Peninsula, Shuidong Bay, Norfolk, Rhode Island, Florida, Accra, and Hawaii.