Transformed Eulerian mean (TEM)

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.

The local meridional eddy buoyancy flux is the covariance of disturbance meridional velocity with disturbance buoyancy. In a Boussinesq approximation, $b^{\prime }=-g{\rho }^{\prime }/{\rho }_0$. This is a local transport density, distinct from a plume's cross-section-integrated buoyancy flux. In the transformed Eulerian mean, division by the background buoyancy frequency squared gives an eddy-induced meridional overturning streamfunction.

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.

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.

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.

Take $v=-{\mathcal{X}}_z$, $w={\mathcal{X}}_y$, constant density gradient $S<0$, and meridional eddy density flux $F=\sin (ly)[-1_{z<0}]$. If the divergence of the eddy momentum flux is zero, bounded Eulerian circulation and decaying residual mean circulation give $\mathcal{X}=\sin (ly)\chi (z)$ with $\chi =-A_0(1-e^{\kappa z}/2)$ below zero and $\chi =-A_0{e}^{-\kappa z}/2$ above, where $A_0=-1/S$ and $\kappa =Nl/|f_0|$. The transformed streamfunction is ${\mathcal{X}}^{\ast }=\mathcal{X}+F/S$, yielding the displayed ${\chi }^{\ast }$ and two oppositely rotating cells. Its jump represents a horizontal delta-function transport at the discontinuous flux termination. A smooth flux cutoff replaces that sheet by a thin regular return flow. Density flux alone does not determine either circulation if the momentum-flux divergence is left unspecified.

For a sinusoidal Eliassen–Palm flux whose vertical divergence is constant in $-D<z<D$ and zero elsewhere, the residual mean circulation is determined by a forced one-dimensional modified Helmholtz equation. With $v^{\ast }=C^{\prime }(z)\sin (\pi y/L)$ and $w^{\ast }=-(\pi /L)C(z)\cos (\pi y/L)$, a forcing proportional to $-G_0\sin (\pi y/L)/(2D)$ givesThe response is uniquely selected by decay of the residual circulation at both vertical infinities. An Eulerian mean vertical flow can persist in the undamped wave region even where this residual response vanishes.

In the localized wave-drag response in a stratified channel, the full vertical integral of zonal acceleration equals the deposited wave momentum, $-G_0\sin (\pi y/L)$. The fraction inside the directly forced layer is $1-(1-e^{-2s})/(2s)$, where $s=\pi ND/(|f_0|L)$. A small $s$ spreads most acceleration outside the dissipation layer; a large $s$ confines most of it inside.