Inertia-gravity wave

An inertia-gravity wave is restored jointly by buoyancy or free-surface gravity and the Coriolis force.

With downward buoyancy $b=g\rho \prime /{\rho }_0$ in a uniform rotating Boussinesq approximation, $q=\zeta -f{b}_z/N^2$ is time-independent. The linear pressure obeys ${\nabla }^2{p}_{tt}+f^2{p}_{zz}+N^2{\nabla }_H^{2}p={\rho }_{0}f{N}^{2}q$. The initial potential vorticity supplies the steady pressure source; nonzero-frequency inertia-gravity waves have zero perturbation PV.

For a plane inertia-gravity wave on a resting layer, $\eta ={\eta }_0\cos (kx-\omega t)$, the horizontal velocities are $u=U\cos \theta$ and $v=V\sin \theta$, where $U=\omega {\eta }_0/(k{h}_0)$ and $V=f{\eta }_0/(k{h}_0)$. First-order integration at a fixed equilibrium coordinate gives $\xi =-U\sin \theta /\omega$, $\upsilon =V\cos \theta /\omega$. The particle trajectory is an ellipse with axis ratio $|f|/\omega$. For positive $f$, the orbit is clockwise in the $(x,y)$ plane.

An external gravity wave displaces a fluid's free surface and moves nearly uniformly through the depth in the long-wave limit. In a layer of depth $H$, its leading shallow-water approximation speed is $\sqrt{gH}$.

On an f-plane, linear shallow-water plane waves have one zero-frequency geostrophic mode and two inertia-gravity modes with ${\omega }^2=f^2+c^2(k^2+l^2)$. The crossover scale $c/|f|$ is the barotropic deformation radius.