Eliassen–Palm flux (EP flux)

The Eliassen–Palm flux is a wave-activity flux whose divergence forces the transformed mean flow. Its vertical component combines vertical eddy momentum flux with a correction from meridional buoyancy transport.

For zonal averages, $u^{\prime }=-{\psi }_y^{\prime }$, $v^{\prime }={\psi }_x^{\prime }$, and $Q^{\prime }={\nabla }_H^2{\psi }^{\prime }+{\partial }_z(S{\psi }_z^{\prime })$, with $S=f_0^2/N^2$ depending only on height, define $F=-\overline{u^{\prime }{v}^{\prime }}$ and $G=S\overline{{\psi }_x^{\prime }{\psi }_z^{\prime }}$. Integrating zonal derivatives by parts gives $\overline{v^{\prime }{Q}^{\prime }}={\partial }_y\overline{{\psi }_x^{\prime }{\psi }_y^{\prime }}+{\partial }_z(S\overline{{\psi }_x^{\prime }{\psi }_z^{\prime }})$. This identity is exact within quasi-geostrophic potential vorticity, without assuming small disturbance amplitudes. Periodicity or appropriate vanishing zonal boundary terms is essential.

Steady conservative small-amplitude waves have divergence-free Eliassen–Palm flux by wave activity conservation. In a balanced mean state with homogeneous impermeable boundaries and no independently forced residual mean circulation, this gives zero residual circulation and zero mean acceleration. The Eulerian mean circulation can nevertheless be nonzero because the residual circulation includes an eddy-induced correction. Wave dissipation, time-dependent wave activity or boundary forcing can invalidate the conclusion.

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.

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.