Bragg scattering

As waves propagate from deep towards shallow water they interact with the seabed. Because of the large scale variations of the bottom topography (at the scale of many tens or hundreds of wave lengths), waves tend to shoal and refract towards the coast. On the other hand, irregular bed variations at shorter scales (i.e. within a few wave lengths) result in forward and backward scattering of waves, known as the Bragg scattering. In general, forward scattering counteracts the directional narrowing caused by refraction and leads to a broadening of the directional spreading, whereas backward scattering attenuates the incident wave field (Ardhuin et al., 2003).


Ardhuin and Herbers (2002) developed a theory for the Bragg scattering of surface waves and proposed a source term that can be implemented in spectral wave models. This source term describes the lowest order resonant interaction between a triad of two wave components with the same frequency but different wave number vectors $\vec{k}$ and $\vec{k}'$ (and thus the associated directions $\theta$ and $\theta'$), and a bottom component that has the difference wave number $\vec{l} = \vec{k} - \vec{k}'$. This source term is given by

  $$S_{\rm bragg} (\sigma,\theta) = \chi\, \int_0^{2\pi}\, \cos^2(\th... ...iny B}(\vec{l}) \left [ E(\sigma,\theta') - E(\sigma,\theta) \right ] d\theta'$$ (2.156)

where $F^{\tiny B}$ is the bed elevation spectrum representing the random (small-scale) variability of the seabed, and $\chi$ is the coupling coefficient and is expressed as follows

  $$\chi = \frac{2\pi\,\sigma^2\,k^3}{c_g\,\sinh^2(2kd)}$$ (2.157)

with $k = \vert\vec{k}\vert$. (Note that in contrast to the study of Ardhuin and Herbers (2002), the present source term is formulated in $(\sigma,\theta)-$space.)


The source term $S_{\rm bragg}$ is estimated by means of a bottom spectrum $F^{\tiny B}$ at the difference wave number $\vec{l}$. In SWAN, two options are available to input this spectrum.


The first option is to input a detailed bottom topography that captures the irregular variations on top of a gently sloping bed. Using a bilinear fit, the large-scale bottom is separated from the high-resolution bathymetric data. Here, the large-scale bathymetry is represented by the mean bed elevation $d(\vec{x})$ given at the computational grid points, so that refraction is resolved properly. The remainder, that is, the small-scale bed modulation, is used to compute the bottom spectrum $F^{\tiny B}$ based on a Fourier transform from $\vec{x}$ to $\vec{k}$.


With the second option, the bottom spectrum $F^{\tiny B}(\vec{k})$ is assumed to be obtained elsewhere and is the same at all computational grid points. Furthermore, the inputted bathymetry $d$ may vary on a scale at which refraction is dominant.


For evaluating the Bragg scattering, an upper limit to the bathymetric variability is imposed. The cutoff $(k/l)_{\rm max}$, displaying the ratio between surface and bed elevation wave numbers with $l = \vert\vec{l}\vert$, is set to 5 (Ardhuin and Herbers, 2002).