= Steady quasi-geostrophic response to localized heating
{title2=$\psi=-\dfrac{f_0}{\beta N^2}\int_x^\infty R_z(X,y,z)\,dX$}
In the weak-forcing steady response about rest, <diabatically forced quasi-geostrophic potential vorticity> gives $\beta\psi_x=(f_0/N^2)R_z$. For $\beta\ne0$, an eastern zero-flow condition fixes the horizontal circulation through $\psi=-(f_0/\beta N^2)\int_x^\infty R_z\,dX$, up to a horizontally uniform pressure gauge. The thermodynamic equation gives $w=R/N^2$. A western zonal wake can remain even though the heating and vertical velocity vanish there. This is a formal steady interior solution: specified initial boundary buoyancy can prevent it from being the attained steady state. For example, under a rigid lid where the heating vanishes, initially uniform boundary buoyancy remains uniform at linear order.
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