Past exam of the advanced fluid dynamics course of the University of Oxford / 2018 / 1 / a / Solution

Let $\hat{\mathbf{u}}$ and $\hat{p}$ be the complex plane wave amplitudes and set $k_x=k_{\perp }$, $k_z=k_{\parallel }$. The linearized Euler equations in the rotating frame areHere $p$ is pressure per unit mass density, including the centrifugal potential. The last line is incompressibility, not an additional equation of state.

Taking the curl eliminates pressure. For constant $\mathbf{\Omega }$ and a divergence-free vector field, $\nabla \times (2\mathbf{u}\times \mathbf{\Omega })=2(\mathbf{\Omega }\cdot \nabla )\mathbf{u}$. Thus the inertial-wave vorticity equation givesApply $\mathbf{k}\times$ once more and use $\mathbf{k}\times (\mathbf{k}\times \hat{\mathbf{u}})=-k^2\hat{\mathbf{u}}$. For a nonzero oscillatory amplitude,so the inertial-wave dispersion relation isThe restoring influence is the Coriolis acceleration. In particular $|\omega |\le 2|\Omega |$. For $k_{\parallel }=0$, the projected linear dynamics have zero frequency; these are steady balanced disturbances rather than the nonzero-frequency branch used in the division above. The spatially uniform $k=0$ mode must be treated separately from this dispersion relation.