Munk's wind-driven circulation model

Remarks

Example: in a boundary layer at a western wall $x = 0$, the equation reduces to $\beta \psi_x = A_H \psi_{xxxx}$. Its exponential solutions $e^{\lambda x}$ satisfy $\lambda^3 = \delta_M^{-3}$, so $\lambda \in \set{1, e^{2\pi i/3}, e^{-2\pi i/3}} \cdot \delta_M^{-1}$. The two roots with negative real part give decaying solutions $e^{-x/(2\delta_M)} \cos\big(\sqrt{3} x / (2 \delta_M)\big)$ and $e^{-x/(2\delta_M)} \sin\big(\sqrt{3} x / (2 \delta_M)\big)$, enough to meet both boundary conditions at the western wall. At an eastern wall only one decaying root is available, so the eastern layer is weaker and cannot carry the return flow.

Non-example: Stommel's model has bottom friction $-\varepsilon \nabla^2 \psi$ instead of lateral friction $A_H \nabla^4 \psi$. It is a second-order problem, it takes only the condition $\psi = 0$, and its boundary layer has width $\varepsilon / \beta$.

Away from the boundaries the friction term is small, and $\psi$ follows the Sverdrup balance. With free-slip walls, the condition $\partial \psi / \partial n = 0$ is replaced by $\nabla^2 \psi = 0$.