= Solution
For constant density $\rho$ and constant angular velocity $\boldsymbol\Omega$, the <Euler equations for an inviscid fluid> in the rotating frame is
$$
\partial_t\mathbf u+(\mathbf u\cdot\nabla)\mathbf u+2\boldsymbol\Omega\times\mathbf u=-\frac1\rho\nabla p-\nabla\Phi_g-\boldsymbol\Omega\times(\boldsymbol\Omega\times\mathbf r),\qquad\nabla\cdot\mathbf u=0.
$$
The last two forces are gravity and <centrifugal acceleration>; $2\boldsymbol\Omega\times\mathbf u$ is the <Coriolis force> term on the left. Incorporate both conservative forces into the modified pressure
$$
\Pi=p/\rho+\Phi_g-\tfrac12|\boldsymbol\Omega\times\mathbf r|^2,
$$
so 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 $\Omega_E^2R\cos^2\lambda$. Its ratio to gravity is at most $\Omega_E^2R/g\simeq3.5\times10^{-3}$, using $\Omega_E\simeq7.3\times10^{-5}\,\mathrm{s}^{-1}$, $R\simeq6.4\times10^6\,\mathrm m$, and $g\simeq9.8\,\mathrm{m\,s}^{-2}$. 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 are
$$
\boxed{\partial_t\mathbf u+2\boldsymbol\Omega\times\mathbf u=-\nabla\Pi',\qquad\nabla\cdot\mathbf u=0.}
$$
These are the linear equations governing <inertial waves>.
Back to article page
