Diffraction

To accommodate wave diffraction in SWAN simulations, a phase-decoupled refraction-diffraction approximation is suggested (Holthuijsen et al., 2003). It is expressed in terms of the directional turning rate of the individual wave components in the 2D wave spectrum. The approximation is based on the mild-slope equation for refraction and diffraction, omitting phase information. This approach is thus consistent with the assumption of a quasi-homogeneous wave field. However, see Section 2.7 for the discussion of the inclusion of wave statistics of inhomogeneous wave fields due to diffraction.


In a simplest case, we assume there are no currents. This means that $c_\sigma = 0$. Let denotes the propagation velocities in geographic and spectral spaces for the situation without diffraction as $c_{x,0}$, $c_{y,0}$ and $c_{\theta,0}$. These are given by

  $$c_{x,0} = \frac{\partial \omega}{\partial k} \cos\theta\,, c_{y,... ...= -\frac{1}{k}\frac{\partial \omega}{\partial h} \frac{\partial h}{\partial n}$$ (2.178)

where $k$ is the wave number and $n$ is perpendicular to the wave ray. We consider the following eikonal equation

  $$K^2 = k^2 (1+\delta)$$ (2.179)

with $\delta$ denoting the diffraction parameter as given by

  $$\delta = \frac{\nabla (c c_g \nabla \sqrt{E})}{c c_g \sqrt{E}}$$ (2.180)

where $E(x,y)$ is the total energy of the wave field ($\sim H^2_s$). Due to diffraction, the propagation velocities are given by

  $$c_x = c_{x,0} \overline{\delta}\,, c_y = c_{y,0} \overline{\delt... ...}}{\partial x} c_{y,0} + \frac{\partial \overline{\delta}}{\partial y}c_{x,0}$$ (2.181)

where

  $$\overline{\delta} = \sqrt{1+\delta}$$ (2.182)

In early computations, the wave fields often showed slight oscillations in geographic space with a wavelength of about 2$\Delta x$ in $x-$direction. These unduly affected the estimations of the gradients that were needed to compute the diffraction parameter $\delta$. The wave field was therefore smoothed with the following convolution filter:

  $$E_{i,j}^n = E_{i,j}^{n-1} - 0.2 [E_{i-1,j}+E_{i,j-1}-4E_{i,j}+E_{i+1,j}+E_{i,j+1}]^{n-1}$$ (2.183)

where $i,j$ is a grid point and the superscript $n$ indicates iteration number of the convolution cycle. The width of this filter (standard deviation) in $x-$direction $\varepsilon_x$, when applied $n$ times is

  $$\varepsilon_x \approx \frac{1}{2} \sqrt{3n} \Delta x$$ (2.184)

By means of computations, $n=6$ is found to be an optimum value (corresponding to spatial resolution of 1/5 to 1/10 of the wavelength), so that $\varepsilon_x \approx 2\Delta x$. For the $y-$direction, the expressions are identical, with $y$ replacing $x$. Note that this smoothing is only applied to compute the diffraction parameter $\delta$. For all other computations the wave field is not smoothed.


Diffraction in SWAN should not be used if,

• an obstacle or coastline covers a significant part of the down-wave view, and
• the distance to that obstacle or coastline is small (less than a few wavelengths), and
• the reflection off that obstacle or coastline is coherent, and
• the reflection coefficient is significant.

This implies that the SWAN diffraction approximation can be used in most situations near absorbing or reflecting coastlines of oceans, seas, bays, lagoons and fjords with an occasional obstacle such as (barrier) islands, breakwaters, or headlands but not in harbours or in front of reflecting breakwaters or near wall-defined cliff walls. Behind breakwaters (which may be reflecting), the SWAN results seem reasonable if the above conditions are met.