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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.