Stommel boundary layer

The Stommel boundary layer closes a Sverdrup interior using linear drag. Balancing beta-effect advection against drag gives widthfor the normalization in which quasi-geostrophic drag is $-\gamma {\nabla }^2\psi$.

For streamfunction $\psi =\Psi (x)\cos (my)$ with impermeable walls at $x=0,L$, let $A=-F_0/(r{m}^2)$ and positive exponents satisfy $\lambda -\gamma =\beta /r$, $\lambda \gamma =m^2$. The exact profile is $\Psi =A[1-C_W{e}^{-\lambda x}-C_E{e}^{\gamma (x-L)}]$, where $C_W=(1-e^{-\gamma L})/(1-e^{-(\lambda +\gamma )L})$ and $C_E=(1-e^{-\lambda L})/(1-e^{-(\lambda +\gamma )L})$. Substitution proves the ordinary differential equation and both boundary conditions. The usual narrow western boundary current limit requires $\lambda L\gg 1$ and $\gamma L\ll 1$, with $\lambda \simeq \beta /r$ and $\gamma \simeq r{m}^2/\beta$; the opposite first inequality does not describe weak drag.

If the dimensional linear drag acceleration is $-r\mathbf{u}$, then $r$ has units of inverse time and Stommel boundary layer balance $\beta v=-r{v}_x$ gives ${\delta }_S=r/\beta$. For returned volume flux $Q$ in depth $H$, the boundary speed is $Q/(H{\delta }_S)$ and relative vorticity is of order $Q/(H{\delta }_S^2)$. A gyre return varying along a distance $B$ has a generic nonlinear-to-planetary vorticity advection ratio $V_b/(\beta B{\delta }_S)$. Large relative vorticity alone does not force nonlinear advection: a perfectly parallel $v=v(x)$, $u=0$ has zero material relative-vorticity advection.