(i) In a reference frame rotating with constant angular velocity $\boldsymbol{\Omega}$ the equations of motion for an inviscid, incompressible fluid of density $\rho$ in a gravitational field $g=-\nabla \Phi$ are

$$\rho \frac{D \mathbf{u}}{D t}+2 \rho \boldsymbol{\Omega} \wedge \mathbf{u}=-\nabla p+\rho \mathbf{g}, \quad \nabla \cdot \mathbf{u}=0$$

Define the Rossby number and explain what is meant by geostrophic flow.

Derive the vorticity equation

$$\frac{D \boldsymbol{\omega}}{D t}=(\boldsymbol{\omega}+2 \boldsymbol{\Omega}) \cdot \nabla \mathbf{u}+\frac{\nabla \rho \wedge \nabla p}{\rho^{2}}$$

$\left[\right.$ Recall that $\mathbf{u} \cdot \nabla \mathbf{u}=\nabla\left(\frac{1}{2} \mathbf{u}^{2}\right)-\mathbf{u} \wedge(\nabla \wedge \mathbf{u})$.]

Give a physical interpretation for the term $(\boldsymbol{\omega}+2 \boldsymbol{\Omega}) \cdot \nabla \mathbf{u}$.

(ii) Consider the rotating fluid of part (i), but now let $\rho$ be constant and absorb the effects of gravity into a modified pressure $P=p-\rho \mathbf{g} \cdot \mathbf{x}$. State the linearized equations of motion and the linearized vorticity equation for small-amplitude motions (inertial waves).

Use the linearized equations of motion to show that

$$\nabla^{2} P=2 \rho \boldsymbol{\Omega} \cdot \boldsymbol{\omega} .$$

Calculate the time derivative of the curl of the linearized vorticity equation. Hence show that

$$\frac{\partial^{2}}{\partial t^{2}}\left(\nabla^{2} \mathbf{u}\right)=-(2 \mathbf{\Omega} \cdot \nabla)^{2} \mathbf{u}$$

Deduce the dispersion relation for waves proportional to $\exp [i(\mathbf{k} \cdot \mathbf{x}-n t)]$. Show that $|n| \leq 2 \Omega$. Show further that if $n=2 \Omega$ then $P=0$.