7.2.2: (Semi-) empirical derivations

Bruun

Bruun (1954) was one of the first coastal engineers to introduce the existence of a dynamic equilibrium profile. He proposed an empirical equation for the equilibrium beach profile based on an analysis of beach profiles along the Danish North Sea coast and the California coast. The equation consists of a simple power law relating the water depth \(h\) to the offshore distance \(x'\) (where \(x' = 0\) is around the mean waterline and is positive in the offshore direction), according to:

\[h = A(x')^m\label{eq7.2.2.1}\]

where:

\(m\)	exponent equal to \(m = 2/3\)	-
\(h\)	water depth	[m]

Interestingly, the value of \(m = 2/3\) corresponds to the theoretical derivation of Bowen (1980), see Eq. 7.5.3.9. The dimensional constant \(A\) (the dimension of which depends on the magnitude of the exponent \(m\); with \(m = 2/3\) the dimension of \(A\) is \(m^{1/3}\)) is a so-called shape factor that depends on the “stability characteristics of the bed material” (what this actually means is somewhat unclear). Bruun found that \(A = 0.135m^{1/3}\) provided the best correlation for North Sea beaches in the Thyborøn area in Denmark. Hughes and Chiu (1978) show that \(A = 0.10m^{1/3}\) provided the best correlation for beaches along the coast of Florida. The larger \(A\) is, the steeper the profile. So, apparently North Sea beaches are somewhat steeper than Florida beaches. Whether this can be explained physically will be discussed next.

Dean

The Bruun equation – a simple power law – was supported by Dean (1977) on semi-empirical grounds by reasoning that for a certain grain size, nature strives towards a uniform energy dissipation \(\varepsilon (D_{50}) = D/h\) (loss in wave power) per unit volume of water across the surf zone (in \(W/m^3\)). The reasoning behind it is that the energy dissipation per unit volume is a measure for the “destructive forces” (causing offshore sediment transport) acting on a sediment particle. With Dean’s equilibrium assumption and for waves normally incident to an alongshore uniform coast \(\theta = \varphi = 0\), the energy balance Eq. 5.2.1.2 can be written as:

\[\dfrac{d(Ec_g)}{dx} = -D = -h \varepsilon (D_{50})\label{eq7.2.2.2}\]

where:

\(E\)	wave energy per unit sea area	\(J/m^2\)
\(c_g\)	group velocity	\(m/s\)

We again use the simple dissipation model for the surf zone \(H = \gamma h\) (see Ch. 5). If we further assume shallow water (\(c_g = c = \sqrt{gh}\)) Eq. \(\ref{eq7.2.2.2}\) leads to:

\[\begin{array} {rcl} {\dfrac{d}{dx} (1/8 \rho g \gamma^2 h^2 \sqrt{gh})} & = & {-h \varepsilon (D_{50}) \Rightarrow } \\ {\varepsilon (D_{50})} & = & {-\dfrac{1}{h} \dfrac{d}{dx} (1/8 \rho g \gamma^2 h^{5/2} \sqrt{g}) =} \\ {} & = & {\dfrac{5}{16} \rho g^{3/2} \gamma^2 \sqrt{h} \dfrac{dh}{dx'}} \end{array}\label{eq7.2.2.3}\]

Notice that in the last step in this equation we have reverted to the cross-shore coordinate \(x'\) which is positive offshore. The depth is the only parameter that varies with \(x'\). If we integrate Eq. \(\ref{eq7.2.2.3}\) we get:

\[h(x') = \left (\dfrac{24 \varepsilon (D_{50})}{5 \rho g^{3/2} \gamma^2} \right )^{2/3} (x')^{2/3}\]

Apparently, based on monochromatic waves and a constant breaker index across the surf zone \(\gamma = H/h\), the magnitude of the exponent \(m\) in Eq. \(\ref{eq7.2.2.1}\) can be derived and is found to be 2⁄3 in agreement with Bruun’s suggestion. Furthermore, the dimensional shape factor \(A\) (in \(m^{1/3}\)) is given by:

\[A = \left (\dfrac{24 \varepsilon (D_{50})}{5 \rho g^{3/2} \gamma^2} \right )^{2/3}\]

in which the equilibrium energy dissipation rate \(\varepsilon (D_{50})\), in \(W/m^3\), depends on the particle diameter. Hence, in Dean’s approach the shape factor \(A\) is some function of the particle diameter. Assuming that the breaker index does not vary with the wave conditions, \(A\) is independent of the deep water wave conditions. The magnitude of \(A\) is seen to vary from \(0.079\ m^{1/3}\) to \(0.398\ m^{1/3}\) (Dean, 1977).

The shape parameter \(A\) was empirically related to the median grain size by Moore (1982), showing that a coarser grain size implies a larger value of and thus a steeper cross-shore profile. Dean (1987) showed that this relation could be transformed to a relation using the fall velocity \(w_s\) (in cm/s) as a parameter, viz.:

\[A = 0.067 w_s^{0.44}\]

Or for \(w_s\) in m/s:

\[A = 0.5 w_s^{0.44}\]

If this result is accurate and universally valid we have a powerful tool. Once we know the fall velocity of the sediment (which depends largely on the grain size, Sect. 6.2.3) we can predict the equilibrium profile after for instance nourishment with borrow material of a different grain size than the native material (see also Sect. 10.7.2). Qualitatively, this result is in line with the finding that coarser beaches are generally steeper than finer beaches.