 Steady quasi-geostrophic response to localized heating

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 $\beta \ne 0$, 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.