Geostrophic adjustment of a finite-width current ($\lambda =c/f$)

An initial current $U{1}_{|y|<a}$ with flat surface has two oppositely signed vortex sheets. On an f-plane, its local final geostrophic balance hasThe Rossby deformation radius controls the edge layers. The height is continuous and odd; the along-strip velocity is $-g{\eta }_f^{\prime }/f$ and has the sheet jumps. Outgoing inertia-gravity waves carry the energy not retained in this balanced component.

For an initially uniform strip current of half-width $a$, the final balanced energy fraction isThe fraction tends to $1-r+O(r^2)$ for a narrow strip and to $1/(2r)$ for a wide strip. The difference is outgoing inertia-gravity wave energy, consistent with conservation of energy; the infinite-strip energies are interpreted per unit along-strip length.