Past exam of the mathematics course of the University of Cambridge / 2022 / iii / Paper 333 / 4

Put $A=\gamma -i\omega$ and $B=\alpha -i\omega$. For $\hat{v}=0$,Continuity gives the damped Equatorial Kelvin wave dispersion relationOnly roots satisfying $\mathrm{Re}(i\beta k/A)<0$ are trapped; for weak damping this is the eastward branch $k\mathrm{Re}\omega >0$. The other algebraic root makes the Gaussian grow with $|y|$ and must be rejected.

Elimination of $\hat{u}$ and $\hat{\varphi }$ givesChoose ${\sigma }^2=(\beta /c)(B/A{)}^{1/2}$ with $\mathrm{Re}{\sigma }^2>0$ and set $Y=\sigma y$. The Hermite eigenvalue condition gives, for $n=1,2,\dots$,Squaring the eigenvalue condition introduces an extraneous branch, so one must also impose $ikc/\sqrt{AB}=2n+1$ and $\mathrm{Re}{\sigma }^2>0$. For real $k$, trapped propagating roots exist whenand the accepted root propagates westward, $k\mathrm{Re}\omega <0$. The other root has an outward-growing meridional structure.

At $\omega =0$, the Kelvin relation gives $k=\pm i\sqrt{\alpha \gamma }/c$. Meridional trapping accepts onlywhich is the response east of the forcing and has zonal decay length $c/\sqrt{\alpha \gamma }$. The Rossby relations giveTrapping now accepts onlyso the response west of the forcing is a sum of Equatorial Rossby waves; its longest zonal scale is $c/(3\sqrt{\alpha \gamma })$ for $n=1$.

Both families haveThus the damped equatorial Kelvin and Rossby response extends farther east than west. Increasing either damping rate shortens the zonal tails, while their ratio controls the common meridional trapping width.