Assume $f>0$; replacing it by $|f|$ gives the thickness for either hemisphere. With $W=(u-U)+iv$, the steady anomaly equations reduce to
$$
\nu W''=ifW,
\qquad W(0)=-U,
\qquad W(\infty)=0.
$$
Thus the <Bottom Ekman layer> has
$$
\boxed{\delta=\left(\frac{2\nu}{f}\right)^{1/2},quad
u=U[1-e^{-z/\delta}\cos(z/\delta)],quad
v=Ue^{-z/\delta}\sin(z/\delta).}
$$
Its integrated anomalous transport is
$$
\boxed{\mathbf u_T=\frac\delta2(-\mathbf U_g+\widehat{\mathbf z}\times\mathbf U_g).}
$$
It is the transport required by the vertically integrated momentum balance between Coriolis acceleration and bottom stress. For slowly varying geostrophic flow, $\nabla\cdot\mathbf U_g=0$ and $\nabla\cdot(\widehat z\times\mathbf U_g)=-\zeta_g$, so
$$
\nabla\cdot\mathbf u_T=-\frac\delta2\zeta_g.
$$
Mass conservation therefore gives the interior <Ekman pumping> condition
$$
\boxed{w(0^+)=\frac\delta2\zeta_g=\frac\delta2\nabla_h^2\psi.}
$$

Solution

Past exam of the mathematics course of the University of Cambridge / 2022 / iii / Paper 333 / 2 / a / Solution

 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2022/iii/paper-333/2/a/solution