Past exam of the mathematics course of the University of Cambridge / 2021 / iii / Paper 333 / 4

Let every perturbation be proportional to $\exp [i(kx+mz-\omega t)]$ and define the intrinsic frequencyEliminating pressure with incompressibility, and then eliminating buoyancy, gives the internal gravity wave dispersion relationThe vertical group velocity isThus energy propagates upward when $\hat{\omega }m<0$ and downward when $\hat{\omega }m>0$. Equivalently, on the positive-intrinsic-frequency branch upward propagation requires $m<0$, while on the negative branch it requires $m>0$.

The exact no-penetration condition on the stationary boundary is that the velocity be tangent to it. Linearized at $z=0$, it isThe terrain is stationary, so $\omega =0$ and $\hat{\omega }=-kU$. The dispersion relation givesA vertically propagating wave therefore exists if and only ifThe upward radiation condition $\hat{\omega }m<0$ selects $km>0$ for $U>0$. Taking $k>0$ gives $m>0$. Lines of constant phase satisfyso they tilt upstream as height increases. The discarded sign of $m$ gives the mirror-image phase tilt and downward energy propagation.

For $k>0$, choose the upward-radiating rootThe boundary condition gives a vertical-velocity amplitude of magnitude $Uk{h}_0$. Incompressibility gives $u^{\prime }=-(m/k)w^{\prime }$, so horizontal averaging over one wavelength yields the wave momentum fluxIf the wave is completely absorbed in a remote dissipation layer, this negative flux increases to zero across the layer. The integrated force on the mean flow is thereforeIt acts opposite to the positive basic flow.

The sloping boundary exerts the pressure dragon the fluid. At generation this equals ${\rho }_0\overline{u^{\prime }{w}^{\prime }}$: the mountain supplies a negative vertical flux of horizontal momentum to the wave. Where the wave dissipates, convergence of that flux transfers the same westward momentum to the mean flow. A forcing that maintains the prescribed uniform basic current must supply the compensating eastward momentum.

Define the perturbation vorticity componentTaking the curl of the linear momentum equation and using the buoyancy equation givesMultiply the first equation by $b^{\prime }$, the second by ${\zeta }^{\prime }$, and horizontally average. Periodicity makes the averaged $x$ derivatives vanish, whileby incompressibility and integration by parts. It follows thatTherefore the wave activity conservation law is

The force on the mean flow and the wave-activity tendency obeyHence growth of positive wave activity produces a negative mean-flow force, while decay deposits negative momentum into the mean flow. Since part iii found a negative force for a growing, upward-radiating wave with $U>0$, its wave activity is positive.

If $U<0$, stationarity still gives $\hat{\omega }=-kU$, but upward propagation now selects $km<0$. For $k>0$ one takes $m<0$, and thereforeThe integrated force reverses and accelerates the mean flow in the positive $x$ direction. The activity of a growing wave correspondingly has the opposite sign and is negative.