= Solution
For $\mathbf u_0=-\tfrac32\Omega x\mathbf e_y$, advection and viscosity vanish, while its <Coriolis acceleration> cancels $-\nabla\Phi_t$; constant pressure and $\theta=0$ complete the equilibrium. For axisymmetric perturbations,
$$
\partial_tu_x'-2\Omega u_y'=-\rho_0^{-1}\partial_xP'-N^2\theta'+\nu\nabla^2u_x',
$$
$$
\partial_tu_y'+\frac12\Omega u_x'=\nu\nabla^2u_y',\qquad
\partial_tu_z'=-\rho_0^{-1}\partial_zP'+\nu\nabla^2u_z',
$$
$$
\partial_t\theta'=u_x',\qquad
\partial_xu_x'+\partial_zu_z'=0.
$$
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