Solution
= Solution
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.