Past exam of the mathematics course of the University of Cambridge / 2024 / iii / Paper 333 / 4 / e

Define the Eulerian-mean streamfunction byWiththe transformation in part a givesHence, up to an irrelevant additive constant,For the step-profile flux,

The residual streamfunction ${\overline{\chi }}_a^{\ast }$ vanishes on the rigid boundaries, decays away from the absorption level, and has opposite-signed values immediately below and above $z=H$. Its vertical derivative gives a zonal acceleration ${\overline{u}}_t$ concentrated around $H$ and largest near the channel center. Its meridional derivative gives a dipolar density tendency: ${\overline{\rho }}_t$ changes sign across the channel center and reverses vertical structure across the absorption level.

The eddy term in ${\overline{\chi }}_a$ has a compensating downward jump at $H$, so the Eulerian-mean streamfunction is continuous even though ${\overline{\chi }}_a^{\ast }$ jumps in the idealized step limit. Below the critical layer, the Eulerian view contains a broad circulation associated with the eddy density flux; the transformed view subtracts that reversible eddy-induced motion and isolates the residual circulation forced where the waves dissipate.

In the Eulerian density budget, vertical advection by ${\overline{w}}_a$ and the divergence of $\overline{{\rho }^{\prime }{v}^{\prime }}$ can be individually large and largely cancel. The transformed Eulerian mean combines them into advection by ${\overline{w}}_a^{\ast }$, making the irreversible mean response to wave-activity deposition much clearer.