Ocean basin equation with bottom drag and lateral viscosity ($\nu {\lambda }^3-r\lambda -\beta =0$)

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.

If $r^3\gg {\beta }^2\nu$, the broad western Stommel boundary layer has thickness $r/\beta$, but small viscosity supplies no-slip skins of thickness $(\nu /r{)}^{1/2}$ at both east and west walls. The western solution therefore contains two decaying exponentials, while the east has the rapidly decaying skin. Setting viscosity identically to zero removes the modes needed to satisfy all four lateral no-slip conditions.