When $N=0$, the factorized relation is $s(D^2+\varpi^2)=0$, so
$$
\boxed{s=0,\qquad s=-\nu k^2\pm i\varpi}.
$$
The oscillatory roots are <inertial waves> restored by rotation; viscosity leaves their leading oscillation frequency unchanged and damps their amplitude at rate $\nu k^2$. The zero root is the decoupled buoyancy-variable mode.

Solution

Past exam of the mathematics course of the University of Cambridge / 2025 / iii / Paper 321 / 3 / d / Solution

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An inertial wave is a rotating-fluid wave restored by the Coriolis force. Its frequency depends on the angle between its wavevector and the rotation axis.


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