18.5 Topographic instability

When using option stream_function, it sometimes happens that steep topographic gradients as described in Killworth (1987) are a source of problems. Essentially, a restriction on the external mode time step dtsf is given by the condition

$$dtsf < \Delta \tau$$ (18.13)

where

$$\Delta \tau \frac{(\dxti\;\cstj)^2}{A_m}\cdot\frac{(8\;h_1\;h_2)/(h_1+h_2)^2 + (\dxti\;\cstj)^2/(dyt_{jrow})^2}{4 + (\dxti\;\cstj)^2/(dyt_{jrow})^2}$$ = (18.14)

h₁ = zw_k1 (18.15)

h₂ = zw_k2 (18.16)

k₁ = kmu_i+1,jrow-1 (18.17)

k₂ = kmu_i+1,jrow (18.18)

The above condition on time step $\Delta \tau$ must be met at each point with index (i,jrow) within the domain. Since analysis requires knowledge of time step length and mixing coefficients, the above time step restriction is not detected in module topog. Rather, detection is left to MOM where all the information is known. Notably, the above condition is most restrictive in polar regions where convergence of meridians makes the effective zonal grid length $\dxti\;\cstj$ go toward zero. The time step restriction can be ameliorated by making the mixing coefficient a function of latitude to compensate for the convergence of meridians. For using variable mixing coefficients, refer to Sections 33.6 and 33.6.2.

Locally smoothing the topography can also help by lessening topographic gradients in the meridional direction and thereby increasing the factor $(8\;h_1\;h_2)/(h_1+h_2)^2$. This can be accomplished with option smooth_topo. When enabled, the smoothing is applied northward of a specified latitude by successively applying a 2D^18.5 finite impulse response symmetric filter where 1D weights are given by (1/4, 1/2, 1/4). Additional conditions are applied to insure that the original coastline is not changed^18.6 and topographic depths can only by decreased^18.7 by applying this filter. When option smooth_topo is enabled, latitudes northward of 85N are filtered by 10 passes of this filter. To change the smoothing latitude or the number of passes (both are arbitrary), look at the USER INPUT section of topog.F under the options scripps_kmt and etopo_kmt.

In the test case resolution with Scripps topography, options varhmix and am_cosine along with the above mentioned topographic filter will eliminate the Killworth time step restriction. Another alternative is to change the external mode solution technique from option stream_function to option implicit_free_surface.