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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.

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.