Past exam of the atmospheric physics course of the University of Cambridge / 2014 / Paper 1 / 3 / d

Use a uniform westerly Barotropic Rossby wave model on a local beta plane. Its dispersion is $\omega =Uk-\beta k/(k^2+l^2)$. A zonal wavenumber-two pattern at latitude $\varphi$ has $k_2=2/(a\cos \varphi )$, so stationarity requiresFor the zonally varying, meridionally uniform idealization $l=0$, this is $U=\Omega a{\cos }^3\varphi /2$. Meridional structure, deformation radius, mean-flow shear and full spherical geometry change the condition; the latitude and zonal wavenumber alone do not fix a unique observed jet speed.

For a shorter zonal perturbation $n>2$ with the same $l=0$ assumptions,Its phase travels eastward relative to the stationary long wave, but westward relative to the air at speed $4U/n^2$; its packet propagates downstream faster than the mean flow in this idealization. For example $n=6$ gives $c_n=8U/9$ and $c_g=10U/9$.

Within the meandering flow, poleward parcel motion lowers relative vorticity and equatorward motion raises it through $D\zeta /Dt\simeq -\beta v$. Synoptic anomalies can reinforce or oppose the background ridges/troughs according to phase, exchange momentum with it, and transfer energy downstream. Linear superposition alone does not cause mutual growth; stronger interactions require shear, baroclinicity or nonlinear PV advection. No unique amplification rate follows from the supplied information.