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In a log-pressure coordinate, quasi-geostrophic potential vorticity uses density-weighted vertical stretching. For stratification $N^2=RS/H$ and geostrophic flow ${\mathbf{u}}_g=(-{\psi }_y,{\psi }_x,0)$, the inviscid equation is $D_{g}q/Dt=0$. Combining horizontal vorticity balance with the thermal equation eliminates ageostrophic flow and gives this conservation law.

Adopt the Boussinesq approximation, hydrostatic approximation, traditional approximation, and inviscid quasi-geostrophic approximation. Assume stable constant $N_0^2>0$, nonzero Coriolis parameter $f_0$, a beta plane, small Rossby number, $\beta L/|f_0|$ of that small order, and small displacement of the background stratification. Vertical advection of the buoyancy perturbation is then higher order. Define the quasi-geostrophic streamfunction byThe buoyancy and vertical vorticity equations at the retained order areDifferentiate the first in $z$. Since $J({\psi }_z,{\psi }_z)=0$, its material-derivative term becomes $f_0{D}_g{\psi }_{zz}$. Elimination of $w_z$ gives diabatically forced quasi-geostrophic potential vorticity:Thus it is the height gradient of heating, rather than heating alone, that produces potential vorticity at this order.

For the steady response linearized about rest, assume $\beta \ne 0$. The quasi-geostrophic potential-vorticity equation reduces to $\beta {\psi }_x=f_0{R}_z/N_0^2$. Choosing the horizontally uniform pressure gauge so that $\psi \to 0$ in the east gives the steady quasi-geostrophic response to localized heatingSetUsing the error function to integrate the Gaussian function givesThese velocities vanish as $x\to +\infty$. A depth-dependent but horizontally uniform addition to $\psi$ is dynamically irrelevant for the requested horizontal velocity. At $z=0$, streamlines are contours of $e^{-y^2/L^2}I(x)$. For $f_0,\beta ,R_0>0$, flow approaches the heating region from the west on its northern side, turns southward, and returns westward on its southern side. The contours are open, not closed gyres: $I$ tends to a nonzero constant in the west.

For $-H<z<0$, $R_z<0$ and the forcing decreases potential vorticity; steady balance requires $\beta v_g=f_0{R}_z/N_0^2<0$, giving southward flow. Below $z=-H$, $R_z>0$ and the meridional flow reverses. A constant heating rate independent of $z$ would have no such interior potential vorticity source.

For the nonlinear diagnostic quasi-geostrophic omega equation, write $\zeta ={\nabla }_h^2\psi$. Apply ${\nabla }_h^2$ to buoyancy evolution and $f_0{\partial }_z$ to vorticity evolution:Subtracting eliminates the pressure tendency and yieldsThe beta term is linear and must remain when the nonlinear terms are neglected. For the steady weak-forcing solution, $f_0\beta {\psi }_{xz}=f_0^2{R}_{zz}/N_0^2$, soThis also follows immediately from the steady linear buoyancy equation. It satisfies $w=0$ at $z=0$, decay at depth and horizontal infinity, and the linear quasi-geostrophic omega equation; with these homogeneous conditions the elliptic problem has no extra decaying homogeneous solution. The rigid-lid interpretation is an additional boundary approximation for this diagnostic response.

At $x=0$, the displayed $v_g$ and $w$ provide the requested meridional/vertical arrows. For positive heating there is upwelling concentrated near $y=0$, maximal in depth at $z=-H$, with northward flow below that depth and southward flow above. The vertical circulation tends to zero far from the forcing. A western zonal wake persists, but its meridional and vertical components vanish as $x\to -\infty$. Horizontal ageostrophic flow supplies the divergence associated with the upwelling; the projected $(v_g,w)$ slice is not itself a two-dimensional incompressible flow and need not have closed streamlines.

The steady calculation fixes an interior circulation, not its attainability from every possible initial and boundary state. For example, a rigid lid with initially uniform boundary buoyancy conserves that buoyancy at linear order because $R(0)=w(0)=0$. The formal steady profile instead generally has nonuniform $b^{\prime }(0)=f_0{\psi }_z(0)$. Such additional initial boundary data require a time-dependent adjustment and cannot simply be imposed on this particular steady solution. No boundary buoyancy data are supplied for the requested steady problem.

Let $D_g={\partial }_t+{\mathbf{u}}_g\cdot {\nabla }_h$, $\psi =\tilde{\Phi }/f_0$, and ${\mathbf{u}}_g=(-{\psi }_y,{\psi }_x,0)$. The hydrostatic and thermal equations combine to giveTaking the vertical curl of the horizontal momentum equations to leading quasi-geostrophic order givesThe log-pressure coordinate continuity equation gives ${\nabla }_h\cdot {\mathbf{u}}_{a,h}=-e^{z/H}{\partial }_z(e^{-z/H}w)$. Substitute $w=-f_0{D}_g{\psi }_z/N^2$. In commuting the vertical derivative with $D_g$, the extra term is ${\mathbf{u}}_{g,z}\cdot {\nabla }_h{\psi }_z=(-{\psi }_{yz},{\psi }_{xz})\cdot ({\psi }_{xz},{\psi }_{yz})=0$. Therefore the log-pressure quasi-geostrophic potential vorticity equation is

The required small parameters have distinct roles. The Rossby number ${ϵ}_R=U/(|f_0|L)$ makes inertial acceleration small relative to the leading geostrophic balance. The parameter ${ϵ}_{\beta }=\beta L/|f_0|$ lets that leading balance use a single $f_0$ while retaining the beta effect in the slower vorticity evolution. Finally, $\psi \sim UL$ and the thermal equation imply $w\sim |f_0|U^2/(N^{2}D)$. Density-weighted continuity has vertical derivative scale $1/\delta$, where $\delta =\min (D,H)$, so $|{\mathbf{u}}_{a,h}|\sim Lw/\delta$. The vertical-advection consistency criterion for log-pressure quasi-geostrophy is thusIt also controls the omitted vertical-advection ratio $wL/(UD)\sim {ϵ}_a\delta /D$. Small Rossby number alone would not ensure small ageostrophic flow for every choice of stratification and vertical scales.

For constant $N$, linearization about rest givesPutting $\psi =\mathrm{Re}[{\hat{\psi }}_c{e}^{z/(2H)}{e}^{i(kx+mz-\omega t)}]$ replaces the vertical operator by $-m^2-1/(4H^2)$. Consequently the Rossby wave in a log-pressure atmosphere has

For the forced half-space, substitution of the specified ansatz givesAssume $\beta >0$ and $k>0$, and define ${\omega }_{\ast }=-\beta k/[k^2+f_0^2/(4N^2{H}^2)]<0$.

If $\omega >0$ or $\omega <{\omega }_{\ast }$, then $m^2<0$. Boundedness excludes the growing solution and uniquely selects $\boxed{\hat{\chi }={\Psi }_b{e}^{-\sqrt{-m^2}z}}$. If $\omega ={\omega }_{\ast }$, boundedness excludes the linear-in-$z$ solution, leaving $\hat{\chi }={\Psi }_b$; this is the zero-vertical-wavenumber threshold.

If ${\omega }_{\ast }<\omega <0$, both oscillatory solutions are bounded. Boundedness alone does not give a unique propagating solution. With forcing at the bottom and no incident wave from above, impose an upward radiation condition. Sincepositive $m$ has upward group velocity. The outgoing solution is $\boxed{\hat{\chi }={\Psi }_b{e}^{imz},m=\sqrt{m^2}>0}$. The physical streamfunction includes the prescribed $e^{z/(2H)}$ density-weighting factor; it is the amplitude $\hat{\chi }$ that must remain bounded.

Thus the strict range for vertically propagating waves isAt $\omega =0$, the unforced steady interior equation requires $\beta {\psi }_x=0$, so a nonzero $k$ bottom streamfunction cannot have a time-independent solution of this ideal rest-state problem without additional dynamics.

For scales $U,L,D$, thermal balance estimates $w\sim f_0{U}^2/(N^{2}D)$. Density-weighted continuity gives horizontal ageostrophic flow of order $Lw/\min (D,H)$. It is small relative to $U$ only ifThis condition supplements small Rossby number and small relative variation $\beta L/|f_0|$ of the Coriolis parameter.