9.3.7 Quasi-hydrostatic assumption

As MOM is designed for large-scale ocean modeling, it is a good approximation to assume motions maintain the hydrostatic balance. So far as the stress tensor is concerned, this assumption boils down to setting the viscosity coefficient $\nu$ to zero (Smagorinsky 1993),

$$\nu = 0.$$ (9.51)

It also amounts to approximating the following off-diagonal strains as

$$2 \, e_{13} \approx u_{1 \, , 3}$$ (9.52)

$$2 \, e_{23} \approx u_{2 \, , 3}.$$ (9.53)

The resulting stress tensor is given by

  

$$\tau^{mn} \rho \, \left( \begin{array}{ccc} A \, (e_{11} - e_{22}) & 2 \, A... ..._{23} \\ 2 \, \kappa \, e_{13} & 2 \, \kappa \, e_{23} & 0 \end{array}\right),$$ = (9.54)

$$\rho \, \left( \begin{array}{ccc} A \, (u_{1,1} - u_{2,2}) & A \,... ...pa \, u_{2,3} \\ \kappa \, u_{1,3} & \kappa \, u_{2,3} & 0 \end{array}\right),$$ = (9.55)

which exposes the familiar two viscous degrees of freedom. The scales

$$A >> \kappa \ge 0$$ (9.56)

are relevant for large-scale stratified GFD flows. Generalizations for non-hydrostatic applications are given in Williams (1972).