 Solution

For constant density $\rho$ and constant angular velocity $\mathbf{\Omega }$, the Euler equations for an inviscid fluid in the rotating frame isThe last two forces are gravity and centrifugal acceleration; $2\mathbf{\Omega }\times \mathbf{u}$ is the Coriolis force term on the left. Incorporate both conservative forces into the modified pressureso the right side is $-\nabla \Pi$.

At latitude $\lambda$ on a spherical Earth of radius $R$, the outward local vertical component of centrifugal acceleration is ${\Omega }_E^{2}R{\cos }^2\lambda$. Its ratio to gravity is at most ${\Omega }_E^{2}R/g\simeq 3.5\times 10^{-3}$, using ${\Omega }_E\simeq 7.3\times 10^{-5}{\mathrm{s}}^{-1}$, $R\simeq 6.4\times 10^6\mathrm{m}$, and $g\simeq 9.8{\mathrm{m}\mathrm{s}}^{-2}$. Thus this component is small compared with gravity everywhere on Earth.

For a velocity scale $U$ and length scale $\ell$, nonlinear advection scales as $U^2/\ell$, whereas the Coriolis term scales as $2\Omega U$. Their ratio is the Rossby number $U/(2\Omega \ell )$. If it is small, for example by increasing rotation at fixed $U,\ell$, neglect nonlinear advection in perturbations about a fluid at rest. The conservative equilibrium forces have already been absorbed into pressure; with a rotation-timescale unsteady motion the leading equations areThese are the linear equations governing inertial waves.