= Solution
Define the <quasi-geostrophic streamfunction> by $p'/\rho_0=f_0\psi$. Leading horizontal <geostrophic balance>, $f_0\hat{\boldsymbol z}\times\boldsymbol u_g=-\nabla_HP$, gives $(u_g,v_g)=(-\psi_y,\psi_x)$. The <hydrostatic approximation> gives $P_z=\sigma$, hence
$$
\boxed{\sigma=f_0\psi_z,\qquad D_g=\partial_t-\psi_y\partial_x+\psi_x\partial_y.}
$$
Thus $D_g$ is the <material derivative> along the leading <geostrophic flow>, and the vertical derivative of <pressure> determines the <buoyancy perturbation>. This also gives <thermal wind>: $u_{gz}=-\sigma_y/f_0$ and $v_{gz}=\sigma_x/f_0$.
The balanced regime is favored by small <Rossby number>, a shallow aspect ratio, and a <Burger number> $N^2H^2/(f_0^2L^2)$ of order one. Equivalently the stratified <Froude number> $U/(NH)=\mathrm{Ro}/\sqrt{\mathrm{Bu}}$ is small. On a <beta plane>, require $\beta L/|f_0|=O(\mathrm{Ro})$, and take slow, initially balanced motion away from the equator. Under the <traditional approximation>, the horizontal <Coriolis force> generated by vertical <velocity> is small because vertical <velocity> is small, while the vertical rotational force is small compared with the leading <hydrostatic pressure> and gravity terms. Strong stratification and small aspect ratio justify retaining only the vertical component of Earth's rotation in this regime; this argument is not uniform near the equator.
Put $A(z)=f_0^2/N^2(z)$. The <buoyancy> equation gives $w=-f_0D_g\psi_z/N^2$. Substitution into the vertical <vorticity equation> yields
$$
D_g\nabla_H^2\psi+\beta\psi_x=-\partial_z(A D_g\psi_z).
$$
There is an important commutator cancellation. With $J_{xy}(a,b)=a_xb_y-a_yb_x$,
$$
\partial_z(A D_g\psi_z)=D_g\partial_z(A\psi_z)+A J_{xy}(\psi_z,\psi_z)=D_g\partial_z(A\psi_z).
$$
Also $D_gf=\beta\psi_x$ for $f=f_0+\beta y$. Adding terms therefore proves <potential-vorticity conservation>:
$$
\boxed{D_gQ=0,\qquad Q=f+\nabla_H^2\psi+\partial_z\left(\frac{f_0^2}{N^2}\psi_z\right).}
$$
This is the <three-dimensional quasi-geostrophic potential vorticity>.
At an impermeable stationary lower boundary $z=b(x,y)$, the kinematic condition is $w=\boldsymbol u_H\cdot\nabla_Hb=D_gb$. For small elevation and slope, evaluate this at the reference level $z=0$ to leading order. Combining it with the <buoyancy> equation gives the <rigid-boundary buoyancy condition for quasi-geostrophic waves>
$$
\boxed{D_g\left(\frac{f_0}{N^2}\psi_z+b\right)=0\quad\text{at }z=0.}
$$
The quantity in parentheses is the boundary displacement plus its buoyancy-equivalent displacement.
Now take constant $N,f$ and the mean <quasi-geostrophic streamfunction> $\bar\psi=-(U_0+\Lambda z)y$. Its mean <buoyancy perturbation> is $-f\Lambda y$, consistent with <thermal wind>, and its interior <potential vorticity> is constant. In the <boundary condition>, the perturbation <velocity> $v'=\psi'_x$ advects the mean boundary <buoyancy> gradient. Keeping this contribution gives
$$
\boxed{(\partial_t+U_0\partial_x)\psi'_z-\Lambda\psi'_x+\frac{N^2U_0}{f}b_x=0\quad\text{at }z=0.}
$$
In particular the $-\Lambda\psi'_x$ term must not be omitted.
The linear interior <potential-vorticity equation> is $(\partial_t+U(z)\partial_x)Q'=0$. Starting with zero interior perturbation <potential vorticity>, and with forcing only through the boundary, it remains zero. For a steady solution this is also the regular zero-interior-PV choice, continued through any isolated zero of $U$. The inversion equation for a horizontal mode is
$$
\psi'_{zz}-K^2\psi'=0,\qquad K=\frac{N\sqrt{k^2+\ell^2}}{f}>0.
$$
The printed convention $K>0$ takes $f>0$; for either hemisphere the decay rate is defined using $|f|$. Boundedness aloft selects the decaying mode. Write
$$
\psi'=A(t)\cos(\ell y)e^{ikx}e^{-Kz}
$$
with real part understood. The linear <boundary condition> becomes
$$
\boxed{A'+ik\left(U_0+\frac{\Lambda}{K}\right)A=\frac{ik\epsilon N^2U_0}{Kf}.}
$$
For $KU_0+\Lambda\ne0$, a steady <quasi-geostrophic mountain wave> is therefore
$$
\boxed{\psi'_{\mathrm s}=\frac{\epsilon N^2U_0}{f(KU_0+\Lambda)}\cos(\ell y)e^{-Kz}e^{ikx}.}
$$
On $y=0$, its real meridional <velocity> is
$$
v'_{\mathrm s}=-\frac{k\epsilon N^2U_0}{f(KU_0+\Lambda)}e^{-Kz}\sin(kx),\qquad b(x,0)=\epsilon\cos(kx).
$$
For the positive-parameter convention $k,\epsilon,f,U_0>0$ and $KU_0+\Lambda>0$, the <velocity> minimum lies one quarter <wavelength> downstream of each topographic maximum; its maximum lies one quarter <wavelength> downstream of each topographic minimum. If the <velocity> prefactor changes sign, maxima and minima interchange. The condition $KU_0+\Lambda>0$ alone does not fix that prefactor if $U_0$ is allowed to be negative. The upper panel shows the positive-current convention.
\Image[/past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2007/iii/paper-77-topographic-phases.png]
{title=Meridional velocity relative to topography for a steady mountain wave and a resonantly growing edge wave}
At $KU_0+\Lambda=0$, the free boundary <Eady edge wave> has zero <frequency> in the ground frame. If $k\epsilon U_0\ne0$, a steady amplitude cannot satisfy the forced amplitude equation. With $A(0)=0$, integration gives
$$
\boxed{A(t)=\frac{ik\epsilon N^2U_0}{Kf}t,\qquad \psi'(t)=\frac{ik\epsilon N^2U_0}{Kf}t\cos(\ell y)e^{-Kz}e^{ikx}.}
$$
Consequently
$$
\boxed{\psi'_t=\frac{ik\epsilon N^2U_0}{Kf}\cos(\ell y)e^{-Kz}e^{ikx}.}
$$
The symbol multiplying $i$ here is the horizontal <wavenumber> $k$, as in the PDF. Taking the real part and differentiating in $x$ gives
$$
\psi'_{\mathrm{real}}=-\frac{k\epsilon N^2U_0t}{Kf}\cos(\ell y)e^{-Kz}\sin(kx),\qquad v'=-\frac{k^2\epsilon N^2U_0t}{Kf}\cos(\ell y)e^{-Kz}\cos(kx).
$$
For $U_0=-\Lambda/K>0$, positive $\epsilon,f$, and $t>0$, \b[the meridional-velocity minima coincide with topographic maxima, and its maxima with topographic minima]. This is the lower panel. The change from quadrature to opposition is the <phase of a resonantly forced topographic edge wave>.
The stationary relief continuously excites a boundary <Rossby wave> whose intrinsic propagation exactly cancels mean <advection>. The forcing therefore remains coherent with that wave; with no damping or radiation loss for this trapped mode, its linear amplitude grows secularly. This is a <resonant topographic quasi-geostrophic wave>. Unlimited growth is a conclusion of the linear idealization: eventually the small-disturbance approximation fails.
The assertion that resonance must be unsteady needs \b[nonzero forcing]. If $k=0$, then $b_x=0$ and the boundary current does not cross the relief; if $U_0=0$, the forcing also vanishes. In either case the zero disturbance is a steady solution even if $KU_0+\Lambda=0$. For example, choose $k=0$, $\ell\ne0$, and $\Lambda=-KU_0$; this is a concrete exception to an unconditional resonance claim. When both horizontal <wavenumbers> vanish, $K=0$ and the stated decaying-wave ansatz itself does not apply.
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