Past exam of the physics of atmospheres and oceans course of the University of Oxford / 2017 / 5 / b / Solution

First check the printed equations literally. Set $v=0$ and use ${\rho }_0^{\prime }/{\rho }_0=-1/H$. Differentiating continuity in time and using horizontal momentum givesVertical momentum and buoyancy give $w_{tt}+N^{2}w=-p_{zt}/{\rho }_0$. Combining them therefore yieldswhich differs from the requested equation. The factor two arises when differentiating both ${\rho }_0$ and $w_z-w/H$. Keeping variable density in continuity while treating it as constant during that differentiation would be inconsistent.

For the intended anelastic reduced-pressure Kelvin wave, make the explicit correction $w_t-b+{\Phi }_z=0$, while $u_t+{\Phi }_x=0$ and $\Phi =p^{\prime }/{\rho }_0$. Continuity then gives ${\Phi }_{xx}=w_{zt}-w_t/H$, and vertical momentum gives $w_{tt}+N^{2}w=-{\Phi }_{zt}$. Cross differentiation now producesThis proves the advertised equation for the corrected model, without claiming it follows from the incompatible printed set.

A concrete counterexample makes the distinction testable. In nondimensional units $H=N=k=1$, take $m=0$, $\omega =2/\sqrt{5}$ and $w=e^{z/2}{e}^{i(x-\omega t)}$, multiplied by its Gaussian meridional envelope. This satisfies the last boxed wave equation, but substitution into the literal printed-system wave operator gives $-2w/5\ne 0$. No choice of accompanying pressure and horizontal velocity can then make it a solution of all five printed equations. Conversely, the literal operator has vertical factors $e^{z/H+imz}$ and dispersion ${\omega }^2=N^2{k}^2/(k^2+m^2)$; its scaling is different from the requested half-scale-height factor.