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.

Solution

Past exam of the mathematics course of the University of Cambridge / 2021 / iii / Paper 333 / 2 / Solution

{c}
{title2=$\delta_E$}
{wiki}

An Ekman layer is a viscous boundary layer in a rotating fluid. For constant <kinematic viscosity> $\nu$ and <Coriolis parameter> $f$, its characteristic thickness is $\delta_E=\sqrt{2\nu/|f|}$, and its velocity turns with distance from the boundary.

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