Step-flux residual circulation in a stratified channel (${\chi }^{\ast }(z)=-\frac{1}{2}{A}_0\mathrm{sgn}(z)e^{-\kappa |z|}$)

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.