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For constant Coriolis parameter $f_0$ and buoyancy frequency $N$, thermal-wind balance and transformed mean-flow equations givewhere ${\chi }^{\ast }$ is the residual-circulation streamfunction and $\mathbf{F}$ is the Eliassen–Palm flux.

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.

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

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

For linear perturbations about a uniform zonal flow, multiplying the forced quasi-geostrophic potential-vorticity equation by the PV anomaly gives a local conservation law for quadratic wave activity. Its flux is the Eliassen–Palm flux, and forcing or damping appears as a wave-activity source or sink.

The transformed Eulerian mean combines the Eulerian-mean meridional circulation with an eddy-induced circulation. This moves eddy buoyancy transport out of the mean thermodynamic equation and gathers wave forcing into the divergence of an Eliassen–Palm flux.

When waves dissipate or encounter a critical layer, their Eliassen–Palm flux changes with position. Its divergence transfers wave pseudomomentum to the mean flow and produces wave drag.