Past exam of the mathematics course of the University of Cambridge / 2015 / iii / Paper 79 / 4

Use $(\bar{u},\bar{v})=(-{\bar{\psi }}_y,{\bar{\psi }}_x)$ and the hydrostatic thermal-wind balance convention $\bar{b}=f_0{\bar{\psi }}_z$. Because the basic state is independent of $x$, $\bar{v}=0$. The thermal-wind relation isIntegration from the resting lower boundary givesA compatible quasi-geostrophic streamfunction iswhere $C$ is a spatial constant. Its horizontal Laplacian is $z{A}^{\prime \prime }(y)/f_0$ and its second vertical derivative is $B^{\prime }(z)/f_0$. Substituting into the given three-dimensional quasi-geostrophic potential vorticity yieldsIn particular, the horizontal-curvature term has a plus sign: $\bar{\zeta }=-{\bar{u}}_y=z{A}^{\prime \prime }/f_0$.

The reference-buoyancy convention in quasi-geostrophic potential vorticity should be stated. If $\bar{b}$ includes the full reference profile $N^{2}z$, the formula above includes its constant contribution $f_0$. If instead the quasi-geostrophic streamfunction represents pressure relative to that reference hydrostatic state, use $\bar{b}-N^{2}z=f_0{\bar{\psi }}_z$. Then the quoted $\bar{q}$ is reduced by the constant $f_0$. Both conventions give exactly the same velocity and potential-vorticity gradients, and therefore the same subsequent dynamics. The coefficient $N$ in the stated linear model is a prescribed reference buoyancy frequency; if it is also required to equal the full physical buoyancy frequency everywhere, then $B^{\prime }(z)=N^2$, rather than an arbitrary function. The formal expression above retains the requested arbitrary $B$ with the stated constant coefficient.

Put $S=M^2/f_0$. The thermal wind is $\bar{U}(z)=-Sz$, and the basic potential vorticity is spatially constant: $\bar{q}=2f_0$ when the full reference buoyancy is included, or $f_0$ when that reference contribution has been subtracted. Either way ${\nabla }_h\bar{q}=0$.

For a smooth normal mode, the linear quasi-geostrophic potential-vorticity equation becomesTake $k\ne 0$ and $K>0$, as required for a nontrivial wave with the printed phase-speed parameterization. Except possibly at an isolated critical level of a shear-flow wave, this implies $\hat{q}=0$; regularity then extends that condition through the isolated level. HenceDecay at infinity excludes the growing exponential, giving the decaying vertical structure of a semi-infinite Eady edge waveThe amplitude $C$ is arbitrary. The semi-infinite Eady model supports a wave trapped at its lower boundary with penetration depth $m^{-1}$. The zero-horizontal-wavenumber case has no nonzero decaying solution of this homogeneous vertical equation. Singular neutral interior potential vorticity sheets belong to a different continuous-spectrum class; they are not the smooth decaying edge-wave eigenfunction requested here.

The PDF contains the request for this vertical structure and uses the phase factor $e^{i(kx+ly-kct)}$. The TeX aid omits that request and also inserts a duplicate factor without $c$. The solution uses the original PDF form throughout.

The interior potential-vorticity equation alone does not determine the frequency. Use the rigid-boundary buoyancy condition for quasi-geostrophic waves. Linear adiabatic buoyancy conservation about the basic state givesAt $z=0$, the impermeability condition sets $w^{\prime }=0$ and the basic velocity is zero. ThereforeSubstitution of ${\hat{\psi }}^{\prime }(0)=-m\hat{\psi }(0)$ yields the dispersion relation of a semi-infinite Eady edge waveFor $f_0>0$, this is $c=-M^2/[N\sqrt{k^2+l^2}]$. The sign is fixed by both the thermal-wind relation and the lower-boundary buoyancy budget.

The steering height of a semi-infinite Eady edge wave is obtained by cancelling the phase-speed term against the basic advection in the wave-following frame:Thus the phase velocity equals the basic thermal-wind velocity at one penetration depth. For nonzero $M$ they have the same sign in the laboratory frame. If “exactly opposes the thermal wind” is read literally as $c=-\bar{U}(z)$, it instead gives $z=-1/m$, outside the fluid: there is no positive physical height with opposite laboratory velocities. The positive height above is the physically meaningful critical level, where the $-c$ and $\bar{U}$ contributions to the wave-frame material derivative oppose and cancel. This distinction resolves the ambiguous printed wording without reversing the derived wave speed. If $M=0$, the shear and $c$ vanish, so there is no distinguished steering height.

For an exponentially growing normal mode with $k\ne 0$, $\mathrm{Im}(kc)>0$. Such a complex $c$ cannot equal the real basic velocity anywhere, so the interior equation forces $\hat{q}=0$ at every height. Decay then fixes the unique vertical structure $C{e}^{-mz}$, and the lower-boundary condition fixes $c=-S/m$, which is real. This contradiction proves the absence of exponential instability in the semi-infinite Eady model:For $k=0$, there is no advective frequency in the linear interior equation, so an exponentially growing mode would again have $q^{\prime }=0$. The wall buoyancy equation then forces ${\psi }_z^{\prime }(0)=0$, which the nonzero decaying exponential cannot satisfy. Thus these modes do not provide a missing growing branch. This conclusion rules out exponential eigenmode growth, not every possible transient response or singular neutral potential vorticity disturbance.

With a second rigid boundary at $z=D$, the same uniform-shear flow is the Eady model. Both boundaries carry buoyancy perturbations and support Boundary Rossby waves. Their intrinsic propagation directions oppose one another relative to their respective local basic flows. At suitable wavelengths their interacting fields can phase-lock and release basic-state potential energy, producing baroclinic instability. The single-boundary system lacks that second interacting boundary wave.

For a direct comparison, write $\Lambda ={\bar{U}}_z=-S$ and $m=NK/|f_0|$. The vertical solution between the two boundaries is $A\cosh (mz)+B\sinh (mz)$, and their buoyancy conditions areEliminating $A,B$ gives the finite-depth Eady dispersion relationThe bracket is negative for $0<mD<2.399\dots$, so those modes with $k\ne 0$ have a growing and a decaying member. At short wavelengths the two edge waves interact weakly and the frequencies are real. In the semi-infinite limit, the lower-wave root tends to $c=\Lambda /m=-S/m$, recovering the neutral result above.