(a) Show that for an incompressible fluid, $\nabla \times \boldsymbol{\omega}=-\nabla^{2} \mathbf{u}$, where $\boldsymbol{\omega}$ is the flow vorticity,

(b) State the equation of motion for an inviscid flow of constant density in a rotating frame subject to gravity. Show that, on Earth, the local vertical component of the centrifugal force is small compared to gravity. Present a scaling argument to justify the linearisation of the Euler equations for sufficiently large rotation rates, and hence deduce the linearised version of the Euler equations in a rapidly rotating frame.

(c) Denoting the rotation rate of the frame as $\boldsymbol{\Omega}=\Omega \mathbf{e}_{z}$, show that the linearised Euler equations may be manipulated to obtain an equation for the velocity field $\mathbf{u}$ in the form

$$\frac{\partial^{2} \nabla^{2} \mathbf{u}}{\partial t^{2}}+4 \Omega^{2} \frac{\partial^{2} \mathbf{u}}{\partial z^{2}}=\mathbf{0}$$

(d) Assume that there exist solutions of the form $\mathbf{u}=\mathbf{U}_{0} \exp [i(\mathbf{k} \cdot \mathbf{x}-\omega t)]$. Show that $\omega=\pm 2 \Omega \cos \theta$ where the angle $\theta$ is to be determined.