= Solution
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> by
$$
(u_g,v_g)=(-\psi_y,\psi_x),\qquad b'=f_0\psi_z,\qquad D_g=\partial_t+J(\psi,\cdot),\qquad J(a,b)=a_xb_y-a_yb_x.
$$
The <buoyancy> and vertical <vorticity> equations at the retained order are
$$
f_0D_g\psi_z+N_0^2w=R,\qquad D_g\nabla_h^2\psi+\beta\psi_x=f_0w_z.
$$
Differentiate the first in $z$. Since $J(\psi_z,\psi_z)=0$, its material-derivative term becomes $f_0D_g\psi_{zz}$. Elimination of $w_z$ gives <diabatically forced quasi-geostrophic potential vorticity>:
$$
\boxed{D_gq=\frac{f_0}{N_0^2}R_z,\qquad q=\nabla_h^2\psi+\frac{f_0^2}{N_0^2}\psi_{zz}+\beta y.}
$$
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 $\beta\ne0$. The <quasi-geostrophic potential-vorticity equation> reduces to $\beta\psi_x=f_0R_z/N_0^2$. Choosing the horizontally uniform pressure gauge so that $\psi\to0$ in the east gives the <steady quasi-geostrophic response to localized heating>
$$
\psi=-\frac{f_0}{\beta N_0^2}\int_x^\infty R_z(X,y,z)\,dX.
$$
Set
$$
I(x)=\int_x^\infty e^{-X^2/L^2}\,dX=\frac{\sqrt\pi L}{2}\operatorname{erfc}(x/L),\qquad F(z)=(1+z/H)e^{z/H},\qquad K=\frac{f_0R_0}{\beta N_0^2H}.
$$
Using the <error function> to integrate the <Gaussian function> gives
$$
\boxed{\psi=KF(z)e^{-y^2/L^2}I(x),\qquad u_g=\frac{2y}{L^2}\psi,\qquad v_g=-KF(z)e^{-(x^2+y^2)/L^2}.}
$$
These 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 $\beta v_g=f_0R_z/N_0^2<0$, 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:
$$
f_0\nabla_h^2\psi_{zt}+f_0\nabla_h^2J(\psi,\psi_z)+N_0^2\nabla_h^2w=\nabla_h^2R,
$$
$$
f_0\nabla_h^2\psi_{zt}+f_0\partial_zJ(\psi,\zeta)+f_0\beta\psi_{xz}=f_0^2w_{zz}.
$$
Subtracting eliminates the pressure tendency and yields
$$
\boxed{N_0^2\nabla_h^2w+f_0^2w_{zz}=\nabla_h^2R+f_0\beta\psi_{xz}+f_0\left[\partial_zJ(\psi,\zeta)-\nabla_h^2J(\psi,\psi_z)\right].}
$$
The beta term is linear and must remain when the nonlinear terms are neglected. For the steady weak-forcing solution, $f_0\beta\psi_{xz}=f_0^2R_{zz}/N_0^2$, so
$$
\boxed{w=\frac{R}{N_0^2}=\frac{-R_0z}{N_0^2H}e^{-(x^2+y^2)/L^2+z/H}.}
$$
This 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.
\Image[/past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2018/iii/paper-333-heating-circulation.png]
{title=Thermally driven circulation and its boundary layers}
{description=Left: surface streamlines and velocity direction for positive $f_0,\beta,R_0$. Right: meridional and vertical arrows at $x=0$, with components separately scaled to show direction; dashed line marks the reversal depth $z=-H$. The vertical slice is a projection of three-dimensional flow.}
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 $b'(0)=f_0\psi_z(0)$. 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.
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