Inertial boundary-current width (${\delta }_I\sim \sqrt{V_I/\beta }$)

An inertial closure balances advection of relative vorticity with planetary vorticity advection. If a basin of width $L$ and meridional size $B$ has interior speed $V_I$, transport continuity gives $V_b\delta \sim V_{I}L$. Equating $V_b^2/(B\delta )$ with $\beta V_b$ yields ${\delta }_I^2\sim V_{I}L/(\beta B)$, reducing to $V_I/\beta$ when $B$ and $L$ are comparable. This is a scale estimate rather than a full boundary-current profile.