Eliassen equation for residual circulation

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.