35.1.8.7 Choosing the biharmonic coefficient

The choice of the biharmonic diffusivity is based on the need to maintain linear stability in steep sloped regions, and the interest to damp grid scale features within a reasonable time. The previous section mentioned the slope constraints. The smallest ``resolvable'' wave on a grid has size $2 \Delta$ and wavenumber $k=\pi/\Delta$. For such a wave, as mentioned above, the biharmonic scheme will act to damp it within a time scale $\tau = (\Delta/2)^{4}/B$. Again, this relation is derived in Section 33.4 for the analysis of the traditional biharmonic scheme. Its application to the present scheme is approximate.