A historical overview of limitation on $c_\theta$

The problem with refraction showing some inaccurate results has been known for a long time. This issue had received some attention by Nico Booij for the first time in November 1998. His basic idea to fix this problem is as follows. We consider a case with parallel depth contours within one sector, see Figure 3.6.

Figure 3.6: Geographic grid with parallel depth contours.

We assume that grid point $(i,j)$ is in shallow water. The other two grid points $(i-1,j)$ and $(i,j-1)$ are in deeper water. Let $n$ be the coordinate along the wave rays. Then according to Snel's law (Holthuijsen, 2007, NOTE 7A, pg. 207), we have

  $$\frac{d\theta}{dn} = \frac{1}{c}\frac{dc}{dn}\tan \theta\, .$$ (3.74)

Here $\theta$ will be of the order of 45$^o$. So, we get

  $$\frac{d\theta}{dn} = \frac{1}{c}\frac{dc}{dn} \, .$$ (3.75)

The slope at grid point $(i,j)$ determines the value of $d\theta/dn$. In shallow water, $c = \sqrt{gh}$, so

  $$\frac{d\theta}{dn} = \frac{1}{2h}\frac{dh}{dn} \, .$$ (3.76)

This may be approximated as follows

  $$\frac{d\theta}{dn} \approx \frac{1}{2h_{i,j}}\frac{h_*-h_{i,j}}{\Delta n}$$ (3.77)

with $h_*$ the water depth in one of the neighbouring grid points $(i-1,j)$ and $(i,j-1)$. In the numerical procedure $d\theta/dn$ is constant over a spatial step, so the change in direction over a step is

  $$\frac{d\theta}{dn} \Delta n = \frac{h_* - h_{i,j}}{2h_{i,j}} \, .$$ (3.78)

In order to maintain stability the change of direction must remain below 90$^o$. Consequently, we obtain

  $$h_* - h_{i,j} \leq \pi h_{i,j} \, .$$ (3.79)

In the program the factor $\pi$ is replaced by a user-determined factor $\beta$. Hence, the depths in surrounding grid points are reduced to $\beta h_{i,j}$, if they are larger than this value. It should be noted that this approach was outlined in an unpublished note. Our experience with this approach is that it seems not effective enough.