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Eliassen–Palm flux and E–P flux divergence

Also called: E-P flux, EP flux, Eliassen-Palm flux, E-P flux divergence, EP flux divergence, wave driving

The Eliassen–Palm (E–P) flux is a vector in the latitude–height plane that shows where atmospheric waves carry their activity, and its divergence is the force those waves exert on the zonal-mean westerlies. It is the standard diagnostic of wave–mean-flow interaction: of how baroclinic eddies maintain the jet stream, and of how planetary waves rising out of the troposphere decelerate the stratospheric polar vortex before a sudden warming.

The equations

In the transformed Eulerian mean (TEM) form of the zonal-mean momentum equation, the eddies enter only through the divergence of the E–P flux $\mathbf{F}$:

$$\frac{\partial \bar u}{\partial t} - f\,\bar v^{*} = \frac{\nabla\cdot\mathbf{F}}{\rho_0\,a\cos\phi} + \bar X, \qquad \nabla\cdot\mathbf{F} = \frac{1}{a\cos\phi}\frac{\partial}{\partial\phi}\!\left(F^{(\phi)}\cos\phi\right) + \frac{\partial F^{(z)}}{\partial z}$$

where $\bar v^{*}$ is the residual meridional circulation and $\bar X$ is any other force, such as gravity-wave drag. In the quasi-geostrophic approximation the two components are

$$F^{(\phi)} = -\rho_0\,a\cos\phi\;\overline{u'v'}, \qquad F^{(z)} = \rho_0\,a\cos\phi\;f\,\frac{\overline{v'\theta'}}{\partial\bar\theta/\partial z}$$

The meridional component is the eddy momentum flux with its sign reversed; the vertical component is proportional to the eddy heat flux. Waves that tilt westward with height carry heat poleward and therefore carry E–P flux upward.

How to read it

• The arrows point where wave activity is going, along the waves' group velocity. Upward arrows at the tropopause in winter are planetary waves entering the stratosphere; equatorward arrows in the upper troposphere are baroclinic waves propagating toward their critical lines in the subtropics.
• Convergence ($\nabla\cdot\mathbf{F}<0$) is an easterly force: the waves decelerate the westerlies there. Sustained convergence in the polar stratosphere is what weakens, and in a sudden warming reverses, the polar vortex. The same wave drag, integrated, drives the Brewer–Dobson circulation.
• Divergence ($\nabla\cdot\mathbf{F}>0$) is a westerly force.
• Only the longest planetary waves (zonal wavenumbers 1–3) can propagate into the winter stratospheric westerlies (Charney and Drazin, 1961), which is why stratospheric E–P diagnostics are usually restricted to them.

The live forecast

The loop below is the E–P flux for zonal wavenumbers 1–3 and its divergence, computed from every member of the ECMWF AIFS ensemble (the control and 25 perturbed members, 14 pressure levels) and stepped through the 15-day forecast. The flux is quadratic in the wave amplitude, so it is computed for each member and then averaged; the flux of the ensemble-mean fields would fade with lead time as the members' wave phases drift apart. The quasi-geostrophic form is used with a global-mean static stability, and latitudes poleward of 82° are masked. It is recomputed every 00 and 12 UTC cycle.