Consider the Rossby-wave equation

$$\frac{\partial}{\partial t}\left(\frac{\partial^{2}}{\partial x^{2}}-\ell^{2}\right) \varphi+\beta \frac{\partial \varphi}{\partial x}=0,$$

where $\ell>0$ and $\beta>0$ are real constants. Find and sketch the dispersion relation for waves with wavenumber $k$ and frequency $\omega(k)$. Find and sketch the phase velocity $c(k)$ and the group velocity $c_{g}(k)$, and identify in which direction(s) the wave crests travel, and the corresponding direction(s) of the group velocity.

Write down the solution with initial value

$$\varphi(x, 0)=\int_{-\infty}^{\infty} A(k) e^{i k x} d k$$

where $A(k)$ is real and $A(-k)=A(k)$. Use the method of stationary phase to obtain leading-order approximations to $\varphi(x, t)$ for large $t$, with $x / t$ having the constant value $V$, for

(i) $0<V<\beta / 8 \ell^{2}$,

(ii) $-\beta / \ell^{2}<V \leqslant 0$,

where the solutions for the stationary points should be left in implicit form. [It is helpful to note that $\omega(-k)=-\omega(k)$.]

Briefly discuss the nature of the solution for $V>\beta / 8 \ell^{2}$ and $V<-\beta / \ell^{2}$. [Detailed calculations are not required.]

[Hint: You may assume that

$$\int_{-\infty}^{\infty} e^{\pm i \gamma u^{2}} d u=\left(\frac{\pi}{\gamma}\right)^{\frac{1}{2}} e^{\pm i \pi / 4}$$

for $\gamma>0 .]$