 Ocean basin equation with bottom drag and lateral viscosity

With depth transport $(-{\psi }_y,{\psi }_x)$ and a depth-mean bottom-drag closure, the wind-driven equation is $\beta {\psi }_x=S-r{\nabla }^2\psi +\nu {\nabla }^4\psi$, where $S$ is wind-stress curl divided by density and $r=\gamma /({\rho }_{0}H)$ is the drag rate. Its zonal boundary modes solve $\nu {\lambda }^3-r\lambda -\beta =0$. The Munk boundary layer and Stommel boundary layer are different limits of this combined equation.