 Solution

To close a depth-integrated bottom-drag model, approximate the bottom horizontal velocity by the depth mean, ${\mathbf{u}}_b\simeq \mathbf{T}/H$. This is the barotropic closure needed to express the stated stress solely in terms of the transport. DefineThe coefficient $\gamma$ in a physical stress law is not itself a drag rate; $r$ has units of inverse time. The ocean basin equation with bottom drag and lateral viscosity follows by taking curl of the integrated momentum equation and yields the combined Stommel boundary layer and Munk boundary layer equation,The bottom contribution is negative because the vertically integrated stress difference is surface stress minus bottom stress. Set the constant wall streamfunction to zero. No normal flow and no tangential slip impose $\bar{\psi }=0$ and ${\partial }_n\bar{\psi }=0$ on the walls.

The inviscid interior solution has ${\bar{\psi }}_I=S(y)x/\beta +C_0(y)$. In a thin east/west layer, $x$ derivatives dominate the smaller $y$ derivatives. Holding $y$ as a parameter givesEquivalently the boundary correction $\varphi =\bar{\psi }-{\bar{\psi }}_I$ obeys $\nu {\varphi }^{\prime \prime \prime \prime }-r{\varphi }^{\prime \prime }-\beta {\varphi }^{\prime }=0$. In a stretched inward coordinate $X=x/\delta$ at the west wall and $X=(1-x)/\delta$ at the east wall, this becomes respectivelyThe first-derivative sign reversal distinguishes east from west. Vanishing $W$ and its derivatives at the north/south boundaries makes leading forcing-dependent solutions compatible with those boundaries, avoiding extra leading meridional layers. These are leading boundary-layer equations, not an exact deletion of every meridional derivative in the full basin PDE.