Problem 2: Storms

In class we discussed that in the most simple way we can think of the Jetstream as a vertically sheared flow on a rotating disc. An analysis of the potential vorticity equation then showed that any perturbation on this background flow will have to obey: \[ \phi_{zz} - \frac{N^2 k^2}{f^2} \phi = 0 \] the boundary conditions being no normal flow at the bottom (i.e., the ground) and top of the atmosphere (with fixed height \(D\)).

(1) What is the most general solution for the perturbation?

(2) How are the boundary conditions expressed in \(\phi\)?
(Hint: evaluate the potential vorticity equation at the boundaries)

(3) Under what conditions are the perturbations growing?

(4) Plot the growth rate as a function of wavelength using appropriate software, and determine the fastest growing mode.


1 This problem set contains 6 problems, and your final score will be based on the 4 problems on which you earned the highest scores.