Linear-drag boundary-current consistency (${\delta }_S=r/\beta$)

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.