Past exam of the mathematics course of the University of Cambridge / 2025 / iii / Paper 333 / 4

Define the transformed Eulerian mean circulation byThe added terms form a nondivergent eddy-induced circulation, so ${\partial }_y{\overline{v}}_a^{\ast }+{\partial }_z{\overline{w}}_a^{\ast }=0$. The buoyancy equation becomesSubstituting ${\overline{v}}_a={\overline{v}}_a^{\ast }+N^{-2}{\partial }_z\overline{v^{\prime }{\sigma }^{\prime }}$ into the momentum equation giveswhere the components of the Eliassen–Palm flux are

For an Equatorial Kelvin wave, set $\hat{v}=0$. The zonal and meridional momentum equations givesoHydrostatic balance and buoyancy evolution give $\hat{\sigma }=im\hat{\varphi }$ and $\hat{w}=-\omega m\hat{\varphi }/N^2$. Continuity then givesBecause the mode must decay as $|y|\to \infty$, one needs $k/\omega >0$. With the stipulated $\omega >0$, the acceptable branch is thereforeThe branch $\omega =-Nk/|m|$ would require $k<0$ when $\omega >0$, making the Gaussian exponent positive and the mode unbounded; it is not an equatorially trapped solution.

For real $k,m,\omega$,Upward group propagation has $m<0$, while a Kelvin wave has $k>0$, so $\overline{u^{\prime }{w}^{\prime }}>0$. Dissipation makes this eastward momentum flux decrease with height; therefore $-{\partial }_z\overline{u^{\prime }{w}^{\prime }}>0$ and the wave exerts an eastward force on the mean flow.

Choose the phase so that the trapped meridional structure $\hat{v}(y)$ is real. The linear wave equations then make $\hat{\varphi }$ and $\hat{u}$ purely imaginary relative to $\hat{v}$, soFor every equatorial mode, zonal momentum, hydrostatic balance, and buoyancy evolution giveSubstitution into the transformed flux shows that the terms involving $\hat{v}{\hat{\varphi }}^{\ast }$ cancel between momentum and buoyancy transport. Hence

For upward-propagating waves $m<0$, so $F^{(z)}$ has the opposite sign to $k$. Where a wave dissipates, its flux tends toward zero with height: eastward modes with $k>0$ have $F^{(z)}<0$ and exert an eastward force, while westward modes with $k<0$ have $F^{(z)}>0$ and exert a westward force. Thus dissipation deposits each wave's signed zonal pseudomomentum in the stratospheric mean flow. The formula also shows why $F^{(z)}$ is a wave-activity or pseudomomentum flux rather than merely $-\overline{u^{\prime }{w}^{\prime }}$: the eddy buoyancy transport supplies the correction required by the transformed circulation.