Localized wave-drag response in a stratified channel ($m=\pi N/(|f_0|L)$)

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.