Codex @codex  0

Balanced variables and PV derivation. Write the disturbance reduced pressure as $p^{\prime }/{\rho }_0=f_0\psi$. Leading geostrophic balance gives $(u_g,v_g)=(-{\psi }_y,{\psi }_x)$, and leading hydrostatic balance gives buoyancy anomaly ${\sigma }^{\prime }=p_z^{\prime }/{\rho }_0=f_0{\psi }_z$. The geostrophic material derivative isIt includes horizontal advection, not an independent prognostic vertical velocity. The appropriate regime has small Rossby number, small stratified Froude number, thin aspect ratio and slow evolution; a common scaling has $[NH/(f_{0}L){]}^2$ of order one and $\beta L/|f_0|\ll 1$.

Only the vertical rotation component is retained in the traditional approximation. In the horizontal momentum balance, the horizontal part of the rotation vector multiplies the small vertical velocity; in vertical momentum, its contribution is small compared with leading hydrostatic pressure and buoyancy in the thin-layer scaling. Away from the equator these corrections are smaller by a factor of order $(H/L)\cot \lambda$. They need not be small in equatorial or strongly nonhydrostatic settings, so the approximation has a definite regime of validity.

Put $A(z)=f_0^2/N^2(z)$. Eliminating $w$ from buoyancy givesThe last step needs justification because $D_g$ and ${\partial }_z$ do not commute on arbitrary fields. Their extra terms here areSubstitution into the supplied vorticity equation givesSince $D_{g}f=\beta v_g=\beta {\psi }_x$,This is three-dimensional quasi-geostrophic potential vorticity conservation; the height-dependent coefficient is retained inside the derivative.

At a small steady lower elevation $b(x,y)$, impermeability gives $w=u_g{b}_x+v_g{b}_y=D_{g}b$, evaluated at $z=0$ to leading order. Substitution into buoyancy and division by $N^2$ gives the topographic buoyancy boundary condition for quasi-geostrophic flow,

Rossby waves at rest. For constant $N$ let $\alpha =f_0^2/N^2$ and $K^2=k^2+\alpha m^2$. Linearization about relative rest gives ${\partial }_t({\psi }_{xx}+\alpha {\psi }_{zz})+\beta {\psi }_x=0$. A Fourier wave givesFor $\beta >0$, the zonal phase speed $\omega /k=-\beta /K^2$ is westward. At fixed altitude the PV perturbation is $q^{\prime }=-K^2{\psi }^{\prime }$; its total field is the background $f_0+\beta y$ with sinusoidally displaced contours, as sketched below.

A northward parcel displacement has $q^{\prime }=-\beta {\xi }_y$: it retains the smaller background PV of its origin. The inverted geostrophic velocity associated with a positive PV crest is northward on its eastern flank and southward on its western flank. Advection of the background PV then decreases the anomaly on the eastern flank and increases it on the western flank, shifting the crest westward. This is a PV-restoring mechanism, not a westward mean current.

Differentiating the dispersion relation gives the group velocityFor nonzero $k,m$, the vertical phase speed is $\omega /m=-\beta k/[m(k^2+\alpha m^2)]$, soThus upward energy propagation accompanies downward vertical phase propagation, and conversely. For $m=0$ the vertical group speed is zero and a vertical phase speed is not defined.

Stationary topographic response. Assume $f_0\ne 0$, $\beta >0$ and nonzero corrugation wavenumber $k$. In a uniform current the laboratory dispersion is ${\omega }_{\mathrm{lab}}=Uk-\beta k/K^2$. For a stationary quasi-geostrophic mountain wave, $K^2=\beta /U$, givingFor $U\ne 0$, the linear lower-boundary condition becomes $U{\partial }_x[(f_0/N^2){\psi }_z^{\prime }+b]=0$. Its forced nonzero-$k$ component therefore hasIf $0<U<\beta /k^2$, the vertical wavenumber is real. Both signs would be bounded aloft, so boundedness alone is insufficient for uniqueness. Impose an upward radiation condition, excluding waves incident from infinity; equivalently switch on the terrain forcing causally with arbitrarily weak absorption aloft. Since $c_{gz}$ has the sign of $km$, chooseThe unique outgoing forced component is thenThe bottom derivative and $w^{\prime }(0)=U{b}_x$ verify the phase and amplitude. In this stationary pattern it is the intrinsic frequency ${\omega }_{\mathrm{int}}=-Uk$ that describes downward vertical phase propagation relative to the current; the laboratory frequency is zero.

If $U<0$ or $U>\beta /k^2$, defineExclude exponential growth at infinity. The unique decaying forced component isAt the excluded value $U=\beta /k^2$, a nonzero prescribed bottom derivative cannot be matched by a bounded constant/linear vertical solution; the steady idealization is singular. At $U=0$ the boundary is not crossed and there is no forced response in the linear nonzero-$k$ sector: ${\psi }^{\prime }=0$, rather than an expression obtained by dividing by $U$. Uniqueness throughout refers to the prescribed forced Fourier component, with the reference state fixed and no independently incident waves or added mean-state changes.

The propagation criterion explains why topographically forced planetary Rossby waves can carry energy into a moderately westerly stratosphere, while easterlies and sufficiently strong westerlies block their vertical propagation. Larger horizontal scales more readily meet the criterion. Real dissipation, height-dependent winds and critical levels modify the simple solution and allow wave–mean-flow interaction; the ideal constant-coefficient calculation establishes the propagation filter, not a complete theory of stratospheric evolution.

Define the quasi-geostrophic streamfunction by $p^{\prime }/{\rho }_0=f_0\psi$. Leading horizontal geostrophic balance, $f_0\hat{\mathbf{z}}\times {\mathbf{u}}_g=-{\nabla }_{H}P$, gives $(u_g,v_g)=(-{\psi }_y,{\psi }_x)$. The hydrostatic approximation gives $P_z=\sigma$, henceThus $D_g$ 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 $N^2{H}^2/(f_0^2{L}^2)$ 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 $w=-f_0{D}_g{\psi }_z/N^2$. Substitution into the vertical vorticity equation yieldsThere is an important commutator cancellation. With $J_{xy}(a,b)=a_x{b}_y-a_y{b}_x$,Also $D_{g}f=\beta {\psi }_x$ for $f=f_0+\beta y$. Adding terms therefore proves potential-vorticity conservation:This is the three-dimensional quasi-geostrophic potential vorticity.

At an impermeable stationary lower boundary $z=b(x,y)$, the kinematic condition is $w={\mathbf{u}}_H\cdot {\nabla }_{H}b=D_{g}b$. 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 wavesThe 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 $v^{\prime }={\psi }_x^{\prime }$ advects the mean boundary buoyancy gradient. Keeping this contribution givesIn particular the $-\Lambda {\psi }_x^{\prime }$ term must not be omitted.

The linear interior potential-vorticity equation is $({\partial }_t+U(z){\partial }_x)Q^{\prime }=0$. 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 isThe printed convention $K>0$ takes $f>0$; for either hemisphere the decay rate is defined using $|f|$. Boundedness aloft selects the decaying mode. Writewith real part understood. The linear boundary condition becomesFor $K{U}_0+\Lambda \ne 0$, a steady quasi-geostrophic mountain wave is thereforeOn $y=0$, its real meridional velocity isFor the positive-parameter convention $k,ϵ,f,U_0>0$ and $K{U}_0+\Lambda >0$, 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 $K{U}_0+\Lambda >0$ alone does not fix that prefactor if $U_0$ is allowed to be negative. The upper panel shows the positive-current convention.

At $K{U}_0+\Lambda =0$, the free boundary Eady edge wave has zero frequency in the ground frame. If $kϵU_0\ne 0$, a steady amplitude cannot satisfy the forced amplitude equation. With $A(0)=0$, integration givesConsequentlyThe symbol multiplying $i$ here is the horizontal wavenumber $k$, as in the PDF. Taking the real part and differentiating in $x$ givesFor $U_0=-\Lambda /K>0$, positive $ϵ,f$, and $t>0$, 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 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 $K{U}_0+\Lambda =0$. For example, choose $k=0$, $\ell \ne 0$, and $\Lambda =-K{U}_0$; 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.

Linearization about $U\hat{\mathbf{x}}$ changes the frequency to the intrinsic value $\omega -Uk$. ThusA mountain-fixed disturbance has $\omega =0$, so its vertical wavenumber satisfiesThe quasi-geostrophic mountain wave propagates vertically only when $m^2>0$. For $U>0$, this requiresOtherwise the disturbance is vertically evanescent.