 Solution

Represent a horizontal vector $(u,v)$ by the complex number $W=u+iv$, let $s=\mathrm{sgn}f$, and define the two Ekman layer depths and drag coefficients byThe atmospheric and oceanic departures from their respective geostrophic flow satisfyThe solutions that decay away from the ice areandHere $W_a^{\infty }$ and $W_o^{\infty }$ denote the atmospheric and oceanic geostrophic velocities.

The viscous stress exerted on the ice by the atmosphere and ocean is, respectively,Because the ice is an infinitesimally thin, freely moving sheet, its horizontal force balance is ${\tau }_a+{\tau }_o=0$. The common Ekman turning factor cancels, leavingThus the ice moves along the weighted mean of the two geostrophic currents. In particular, as ${\nu }_a\to 0$ one has $r_a\to 0$ and $W_i\to W_o^{\infty }$: an atmosphere with vanishing viscosity transmits no finite stress to the ice.

The atmospheric Ekman transport relative to its geostrophic current isThe atmospheric stress isIf the two geostrophic currents are parallel but unequal, both directions are obtained by rotating their velocity difference: in the Northern Hemisphere the stress lies $45^{\circ }$ anticlockwise from $W_a^{\infty }-W_o^{\infty }$, while the atmospheric transport lies $45^{\circ }$ clockwise from its negative. Both rotations reverse in the Southern Hemisphere. If the two currents are identical, the shear, stress, and relative Ekman transport all vanish.