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

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.