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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 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.

Define the transformed Eulerian mean circulation byThe added terms form a nondivergent eddy-induced circulation, so ${\partial }_y{\overline{v}}_a^{\ast }+{\partial }_z{\overline{w}}_a^{\ast }=0$. The buoyancy equation becomesSubstituting ${\overline{v}}_a={\overline{v}}_a^{\ast }+N^{-2}{\partial }_z\overline{v^{\prime }{\sigma }^{\prime }}$ into the momentum equation giveswhere the components of the Eliassen–Palm flux are

The residual mean circulation is the meridional and vertical circulation used by the transformed Eulerian mean. It combines the Eulerian-mean circulation with an eddy-induced contribution so that mean tracer evolution is advected by a single nondivergent flow.