9.3.8 Cartesian form of the friction vector

The friction vector in Cartesian coordinates is given by the divergence of the frictional stress tensor

$$\rho \, F^{m} = \tau^{mn}_{\, ,n}.$$ (9.57)

Performing the divergence leads to the components

$$\rho \,F^{1} \nabla_{h} \cdot (\rho \, A \, \nabla_{h} \, u_{1}) + \hat{z} \cd... ...} \, u_{2} \wedge \nabla_{h} \, \rho \, A + [\rho \, \kappa \, (u_{1,3}) ]_{,z}$$ = (9.58)

$$\rho \,F^{2} \nabla_{h} \cdot (\rho \, A \, \nabla_{h} \, u_{2}) - \hat{z} \cd... ...} \, u_{1} \wedge \nabla_{h} \, \rho \, A + [\rho \, \kappa \, (u_{2,3}) ]_{,z}$$ = (9.59)

$$\rho \, F^{3}$$ = 0. (9.60)

In these expressions, the horizontal divergence operator $\nabla_{h} = (\partial_{1}, \partial_{2}, 0)$ was introduced, and $z=\xi^{3}$ is the vertical coordinate. The factors of density cancel out trivially upon making the Boussinesq approximation. Note that when making the quasi-hydrostatic approximation, the vertical friction F³ is set to zero so that the vertical momentum equation reduces to the inviscid hydrostatic equation. The extra cross-product terms appearing in the transverse friction vanish when using a constant viscosity. Their importance when using a spatially nonconstant viscosity is briefly highlighted in the next section.