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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.

The nonconstant exponential modes of the leading basin equation satisfyThus a leading uniformly valid form is $\bar{\psi }\simeq S(y)x/\beta +C_0(y)+{\sum }_{j=1}^3{C}_j(y)e^{{\lambda }_{j}x}$, with exponentials referenced to the wall where they decay. The coefficients enforce the four east/west no-slip conditions; their values are not needed to identify the scales.

If $r^3\ll {\beta }^2\nu$, horizontal viscosity dominates drag in the boundary layers. The Munk boundary layer scale is ${\delta }_M=(\nu /\beta {)}^{1/3}$, assumed much smaller than basin width. The roots are ${\delta }_M^{-1}$ and $(-1\pm i\sqrt{3})/(2{\delta }_M)$ to leading order, soThe east layer has exponential thickness ${\delta }_M$ and the west oscillatory layer has envelope thickness $2{\delta }_M$, both of order ${\delta }_M$. The west correction supplies the order-one return transport, while the east no-slip correction is typically smaller in amplitude. Setting $r=0$ still leaves enough viscous modes to impose no-slip.

If $r^3\gg {\beta }^2\nu$, linear drag dominates the broad vorticity layer. Definewith ${\delta }_S$ small compared with basin width. The roots are approximately $-{\delta }_S^{-1}$ and $\pm {\delta }_v^{-1}$; the fast roots have smaller correction $\beta /(2r)$. The drag-dominated basin solution with no-slip layers isThere is a broad western Stommel boundary layer of thickness ${\delta }_S$, a thinner western no-slip layer of thickness ${\delta }_v$, and an eastern no-slip layer of thickness ${\delta }_v$. The narrow layers can have small streamfunction amplitude while supplying an order-one change in wall velocity.

If viscosity is set identically to zero, only the particular solution, a constant and the western Stommel exponential remain. That second-order model cannot generally satisfy both streamfunction and derivative conditions at both walls. Small nonzero viscosity must be retained in the wall skins when no-slip is required. In the question's stress notation, the limits compare $[\gamma /({\rho }_{0}H){]}^3$ with ${\beta }^2\nu$. Neither limit applies at the crossover, where all cubic terms contribute. These composite forms are leading asymptotic solutions of the basin problem; meridional derivatives and interior diffusion supply higher-order corrections.

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.

Use $u=-{\psi }_y$, $v={\psi }_x$. Since $v=W(y)/\beta$ and the eastern boundary may be chosen as the zero streamline,The lower half-basin has southward interior flow and a northward western return current, giving a clockwise ocean gyre. The upper half has northward interior flow and a southward western return current, giving a counterclockwise gyre. The interior solution cannot satisfy no slip at every wall, so viscous layers complete the circulation.

With constant lateral viscosity $\nu$, the vorticity equation contains $\nu {\nabla }^4\psi$. At the western and eastern walls, balancing $\beta {\psi }_x$ against four $x$ derivatives gives the Munk boundary layer scaleAt the southern and northern walls, $x$ variation remains on the basin scale $L_X$ while four $y$ derivatives occur across the layer, so