= Solution
Represent a horizontal vector $(u,v)$ by the <complex number> $W=u+iv$, let $s=\operatorname{sgn}f$, and define the two <Ekman layer> depths and drag coefficients by
$$
\delta_j=\sqrt{\frac{2\nu_j}{|f|}},
\qquad
r_j=\frac{\rho_j\nu_j}{\delta_j}
=\rho_j\sqrt{\frac{\nu_j|f|}{2}},
\qquad j\in\{a,o\}.
$$
The atmospheric and oceanic departures from their respective <geostrophic flow> satisfy
$$
\nu_jW_{j,zz}=ifW_j.
$$
The solutions that decay away from the ice are
$$
W_a-W_a^\infty=(W_i-W_a^\infty)
\exp\left[-\frac{(1+is)z}{\delta_a}\right],
\qquad z>0,
$$
and
$$
W_o-W_o^\infty=(W_i-W_o^\infty)
\exp\left[\frac{(1+is)z}{\delta_o}\right],
\qquad z<0.
$$
Here $W_a^\infty$ 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,
$$
\tau_a=(1+is)r_a(W_a^\infty-W_i),
\qquad
\tau_o=(1+is)r_o(W_o^\infty-W_i).
$$
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, leaving
$$
\boxed{
W_i=\frac{r_aW_a^\infty+r_oW_o^\infty}{r_a+r_o}
}.
$$
Thus the ice moves along the weighted mean of the two geostrophic currents. In particular, as $\nu_a\to0$ one has $r_a\to0$ 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 is
$$
\mathcal M_a
=\int_0^\infty(W_a-W_a^\infty)\,dz
=\frac{\delta_a}{1+is}(W_i-W_a^\infty)
=-\frac{\delta_a r_o}{2(r_a+r_o)}
(1-is)(W_a^\infty-W_o^\infty).
$$
The atmospheric stress is
$$
\tau_a=\frac{(1+is)r_ar_o}{r_a+r_o}
(W_a^\infty-W_o^\infty).
$$
If 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.
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