Zonal Boundary Layers

Neglect small contributions from other boundary layer terms
Choose $\epsilon'^2 = \epsilon_S$ to balance terms

This reduces to the dominant balance $$ \frac{\partial^2 \phi_N}{\partial \alpha^2} + \frac{\partial \phi_N}{\partial \hat{x}} = 0 $$

The boundary conditions necessary to complete the solution are

The total streamfunction (interior plus boundary correction) must vanish at the northern boundary $\phi_N(\hat{x}, \alpha = 0) = -\psi_I(\hat{x}, \hat{y} = 1) = -(1-\hat{x})$
The boundary solution should match to the interior streamfunction far from the boundary $\phi_N(\hat{x}, \alpha \to -\infty) = 0$
At the eastern boundary $\phi_N$ should also vanish, for otherwise it would provide a velocity into the eastern wall $\phi_N(1, \alpha) = 0$

Obtaining the solution at the southern boundary analytically is quite algebraically tedious and is easier done numerically. The figure shows solutions to the Stommel problem with a wind stress that increases linearly from $y = 0$ to $y = 1$. The interior solution is $\psi_I = (1-x)$, or $v_I = -1$, necessitating zonal boundary layers at $y = 0$ and $y = 1$, as well as a western boundary layer at $x = 0$.
The nondimensional thickness of the zonal boundary layers is $\epsilon_S^{1/2}$. This is thicker than the western boundary layer because it scales as the square root of a small parameter. The thickness comes from the leading balance in the dimensional vorticity equation $$ r \frac{\partial^2 \psi}{\partial y^2} + \beta \frac{\partial \psi}{\partial x} = 0 $$

¹Vallis, G.K. (2017) Atmospheric and Oceanic Fluid Dynamics: Fundamentals and Large-Scale Circulation. 2nd edn. Cambridge: Cambridge University Press.

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