Kelvin–Helmholtz instability

Remarks

The normal modes satisfy the dispersion relation $$\frac{\omega}{k} = \frac{\rho_1 U_1 + \rho_2 U_2}{\rho_1 + \rho_2} \pm \left( \frac{g}{k} \frac{\rho_1 - \rho_2}{\rho_1 + \rho_2} + \frac{\sigma k}{\rho_1 + \rho_2} - \frac{\rho_1 \rho_2 (U_1 - U_2)^2}{(\rho_1 + \rho_2)^2} \right)^{1/2}.$$ So the wavenumber $k$ is unstable exactly when $$\rho_1 \rho_2 (U_1 - U_2)^2 > (\rho_1 + \rho_2) \left( \frac{g (\rho_1 - \rho_2)}{k} + \sigma k \right).$$

Example: with $g = 0$ and $\sigma = 0$ the condition holds for every $k > 0$, and the growth rate $\operatorname{Im} \omega = k \sqrt{\rho_1 \rho_2} \abs{U_1 - U_2} / (\rho_1 + \rho_2)$ is unbounded as $k \to \infty$. Non-example: with $\rho_1 > \rho_2$, $\sigma > 0$, $g > 0$ and $(U_1 - U_2)^2 \le 2 (\rho_1 + \rho_2) \sqrt{g \sigma (\rho_1 - \rho_2)} / (\rho_1 \rho_2)$, no wavenumber is unstable, since the right side of the condition has minimum $2 (\rho_1 + \rho_2) \sqrt{g \sigma (\rho_1 - \rho_2)}$ over $k > 0$.

The name is also used for the instability of continuous shear layers, such as $U(z) = \tanh z$, which roll up into vortices in the same way.