Past exam of the mathematics course of the University of Cambridge / 2016 / ib / Paper 4 / 18C (Fluid Dynamics)

By definition the vorticity is $\mathbf{\omega }=\nabla \times \mathbf{u}$. The curl of the curl identity yieldsFor incompressible flow, $\nabla \cdot \mathbf{u}=0$, so

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.

Take the curl of the linear momentum equation. For constant $\mathbf{\Omega }$ and zero divergence, $\nabla \times (\mathbf{\Omega }\times \mathbf{u})=-(\mathbf{\Omega }\cdot \nabla )\mathbf{u}$. With $\mathbf{\Omega }=\Omega {\mathbf{e}}_z$, this givesTake another curl and use part (a): $-{\partial }_t{\nabla }^2\mathbf{u}=2\Omega {\partial }_z\mathbf{\omega }$. Differentiate in time and substitute the previous equation:ConsequentlyThe two curl operations remove the pressure gradient while retaining the rotational restoring force.

For a nonzero wavevector $\mathbf{k}$, substitute the plane wave into part (c). Each time derivative contributes $-i\omega$, each Laplacian contributes $-|\mathbf{k}{|}^2$, and each second $z$ derivative contributes $-k_z^2$. A nonzero amplitude therefore requiresIf $\theta$ is the angle between the wavevector and the rotation axis, $\cos \theta =k_z/|\mathbf{k}|$, andThis is the inertial wave dispersion relation. Incompressibility also requires $\mathbf{k}\cdot {\mathbf{U}}_0=0$; the original first-order equations select the corresponding transverse polarizations. A horizontal wavevector gives the zero-frequency geostrophic case.