Problem 3: Kelvin-Helmholtz Instability

For a Kelvin–Helmholtz instability in a continuously stratified ocean, obtain a globally integrated energy equation in the form: \[ \frac{1}{2} \frac{d}{dt} \int \left( u^2 + w^2 + \frac{g^2 \rho^2}{\rho_0^2 N^2} \right) dV = - \int uw \frac{\partial U}{\partial z} dV. \] Discuss the physical meaning of each term and the mechanism of instability.

ControlVolume

The integration in \(x\) takes place over an integer number of wavelengths.


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.