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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.