34.2.4 Summary: viscosity on the sphere

On a sphere using spherical coordinates, the grid spacing in the zonal direction changes according to $\cos\phi$. From many numerical and physical perspectives, it is useful to employ isotropic grids in which the latitudinal resolution is kept abreast with the converging meridions

$$\Delta \phi = \cos\phi \, \Delta \lambda.$$ (34.37)

In this way, a grid cell is roughly square, or isotropic, with squares becoming smaller as one moves poleward. As discussed in Section 16.1.3, MOM provides an option which constructs grids satisfying this equation. Consistent with the desire to employ isotropic grids, it might be useful to prescribe a momentum friction which damps a particular grid scale anomaly with the same time scale regardless of the position on the sphere. As shown in Section 33.4, the damping times for a constant viscosity used for Laplacian and biharmonic friction in one Cartesian dimension is given by

$$\tau_{Lap} (\Delta/2)^{2}/ A$$ = (34.38)

$$\tau_{Bih} (\Delta/2)^{4}/ B,$$ = (34.39)

where $\Delta$ is the grid spacing and B is the biharmonic viscosity. Preserving the damping time as $\Delta$ changes on the sphere suggests letting A have a $\cos^{2}\phi$ dependence and B have a $\cos^{4}\phi$ dependence. Besides providing for a constant damping time, a latitudinally dependent friction can be prescribed that relieves the time step constraint given in Section 33.2.2 which ensues when employing a constant viscosity over the extent of the sphere. That constraint becomes more restrictive on the size of $A \, \Delta t$ when moving towards the poles. Again, letting A have a $\cos^{2}\phi$ dependence relieves this constraint. Similar stability considerations with biharmonic friction leads to a biharmonic coefficient with $\cos^{4}\phi$ dependence. The above considerations neglect the lower bound considerations given in Sections 33.2.1 and 33.2.3. Notably, if the viscosity gets too small, the flow will become numerically unstable. Therefore, as a compromise, instead of a $\cos^{2}\phi$ dependence, the Laplacian viscosity is typically given a $\cos\phi$ dependence

$$A = A_{o} \, \cos\phi,$$ (34.40)

where A_o has no latitudinal dependence. This form for the viscosity is furthermore not carried all the way to the pole. In practice, the $\cos\phi$ dependence appears sufficient to alleviate the time step restrictions arising from friction. This form for the viscosity is enabled through option varhmix and the suboption am_cosine as discussed in Section 33.6.2. For biharmonic friction, the analogous viscosity is given by

$$B = B_{o} \, \cos^{3}\phi,$$ (34.41)

where B_o has no latitudinal dependence. In summary, there are three main constraints which are placed on the viscosity. The Reynolds number and Munk boundary layer constraint provide a lower bound on viscosity, whereas the linear diffusion equation constraint provides an upper bound. In general, modelers hope to reduce the viscosity to its lowest value consistent with these constraints. The reason is that it will allow the flow to become more advectively dominant, which is more realistic. Unfortunately, that effort is often difficult to achieve, largely due to the Reynolds number and Munk constraints. As proposed in Section 33.7, the Smagorinsky scheme provides the most general means of satisfying each of the above three constraints with only one adjustable constant. This scheme, originally used in the GFDL model by Rosati and Miyakoda (1988), is currently being employed more frequently in experiments at GFDL.