9.3.9 The case of nonconstant viscosity

It is not uncommon for ocean modelers to employ a nonconstant viscosity for various numerical reasons. As emphasized by Wajsowicz (1993), some implementations of the corresponding friction vector often ignore the importance of formulating friction as the divergence of a symmetric stress tensor. Namely, what is sometimes done is to simply take the friction appropriate for a constant viscosity for a Boussinesq fluid

$$\nabla_{h} \cdot (A \, \nabla_{h} \, u_{1}) + [\kappa \, (u_{1,3}) ]_{,z}$$ F¹_const = (9.61)

$$\nabla_{h} \cdot (A \, \nabla_{h} \, u_{2}), + [\kappa \, (u_{2,3}) ]_{,z}$$ F²_const = (9.62)

and then letting A be nonconstant. That is, the cross-product terms derived above are dropped. Focusing on the two-dimensional transverse sub-space, doing so amounts to employing the non-symmetric stress tensor

$$\tau^{mn}_{NS} = \rho_{o} \, A \, \left( \begin{array}{cc} u_{1,1} & u_{1,2} \\ u_{2,1} & u_{2,2} \end{array}\right).$$ (9.63)

It is easy to show that the chosen friction dissipates kinetic energy since it is written as a Laplacian. However, for a fluid in uniform rotation

$${\bf u} = {\bf\Omega} \wedge {\bf x},$$ (9.64)

where ${\bf u} = (u_{1},u_{2},0)$, ${\bf x} = (x_{1},x_{2},0)$, and ${\bf\Omega}$ is spatially constant, the horizontal friction vector takes the form

$${\bf F}_{h} - \nabla \wedge (A \, {\bf\Omega}),$$ = (9.65)

and it vanishes only when A is a constant. As such, by using friction derived from a non-symmetric stress tensor and with a non-constant viscosity, a uniformly rotating fluid will feel a nonzero stress. Conversely, such a stress tensor can introduce uniform rotation; i.e., it can act as an internal source or sink of angular momentum. Unless one has a physical reason for doing so, such viscosity dependent sources of angular momentum should be avoided.