9.5 Biharmonic friction

The previous derivations were all concerned with second order, or Laplacian, friction. It is often useful to consider another method of dissipating momentum through use of a fourth order, or biharmonic, friction. Such friction acts more strongly on the small scales than the Laplacian friction, and less strongly on the large scales (see Section 33.4 for more discussion). Each property is desirable, especially when aiming to realize some form of a quasi-geostrophic turbulent cascade in which enstrophy cascades to the small scales and energy to the large scales. Biharmonic friction is therefore commonly used in ocean modeling. It should be noted, however, that biharmonic friction is motivated mostly from numerical reasons and has no first principle derivation.

The goal of this section is to derive the appropriate form of the biharmonic operator which both dissipates kinetic energy yet does not introduce spurious sources of angular momentum. As with the previous derivations, some work is necessary in order to realize these properties on a sphere with a generally non-constant viscosity.

Recall that the quasi-hydrostatic approximation allowed for the separation of the transverse or horizontal subspace from the vertical subspace. Consequently, the vertical term $(\kappa {\bf u}_{,z})_{,z}$ was isolated from the other terms in the friction vector. As a result, the following will focus solely on deriving the biharmonic operator for use in the two-dimensional transverse subspace.