The force on the mean flow and the wave-activity tendency obey
$$
-\partial_z\overline{u'w'}
=\partial_z\mathcal F_{\mathcal A}
=-\partial_t\mathcal A.
$$
Hence 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 $\widehat\omega=-kU$, but upward propagation now selects $km<0$. For $k>0$ one takes $m<0$, and therefore
$$
\overline{u'w'}=-\frac12mkU^2h_0^2>0.
$$
The 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.

Solution

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

 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2021/iii/paper-333/4/vi/solution