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

Begin with the Boussinesq approximation primitive equations on a beta plane, decompose every field into a zonal mean and a disturbance, and average over longitude. The zonal momentum equation then contains the divergence of the eddy momentum flux $\overline{u^{\prime }{v}^{\prime }}$, while the mean density equation contains the divergence of the eddy density flux $\overline{{\rho }^{\prime }{v}^{\prime }}$. At small Rossby number, use geostrophic balance, hydrostatic pressure, thermal-wind balance, and the leading eddy equations to combine those fluxes.

Defineand introduce the residual mean circulationThe added eddy-induced velocity is nondivergent, soIt absorbs the eddy density-flux divergence into advection by the transformed circulation, giving

The mean zonal momentum equation becomeswhere the zonally averaged Eliassen–Palm flux in the meridional-vertical plane isThus the transformed Eulerian mean gathers the wave forcing into one flux divergence and makes density evolve under one residual circulation.

Define the residual-circulation stream function byThis satisfies residual mass conservation identically. LetThe momentum equation is

Differentiate geostrophic balance vertically and hydrostatic pressure meridionally to obtain thermal-wind balanceAfter a time derivative, the transformed density equation givesOn the other hand, the $z$ derivative of momentum givesEliminating ${\overline{u}}_{tz}$ yields the Eliassen equation for residual circulation

Put $l=\pi /L$. For the specified quasi-geostrophic streamfunction, the complex amplitudes of the disturbance fields areThe product of these two amplitudes is purely imaginary after one is conjugated, so its zonal mean vanishes:Consequently

Geostrophic balance and hydrostatic pressure giveThereforeSince $d{\rho }_s/dz=-{\rho }_0{N}^2/g$, the vertical Eliassen–Palm flux isThus it has the stated formwith

Letwhere $\mathcal{H}$ is the Heaviside step function. Thenso the Eliassen equation for residual circulation isTake no normal residual flow at $y=0,L$ and $z=0$, decay as $z\to \infty$, and choose the streamfunction constant on the connected rigid boundary to be zero. Thus

The required Fourier series in sine modes isDefineFor each mode, the vertical equation isThe Dirac delta function requiresThe solution satisfying the boundary conditions is thereforewhere

The momentum equation givesThe jump of ${\overline{\chi }}_a^{\ast }$ supplies an equal positive delta function in $f_0{\overline{\chi }}_{a,z}^{\ast }$, so the singular terms cancel. The regular acceleration iswhereFinally, the transformed density equation givesThese exponentially decaying modes are the balanced mean response to wave-activity deposition at $z=H$.

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.

Let the wave-forcing layer have vertical scale $D$ and meridional scale $L$. In the Eliassen equation for residual circulation, the two restoring terms scale asTheir ratio is controlled by

In the shallow-forcing limit $\Lambda \ll 1$, the vertical derivative term dominates:After one vertical integration,The Coriolis force on the residual mean circulation therefore balances most of the wave forcing, andis small at leading order. The response is primarily an overturning circulation with an associated density tendency.

In the deep-forcing limit $\Lambda \gg 1$, the meridional term dominates:Nowso the residual circulation is too weak to balance the forcing. The dominant momentum response is direct zonal acceleration,with a comparatively weak overturning and density response.