Miles–Howard theorem

Remarks

Normal modes with stream function amplitude $\varphi(z)$ satisfy the Taylor–Goldstein equation $$(U - c)(\varphi'' - \alpha^2 \varphi) - U'' \varphi + \frac{N^2}{U - c} \varphi = 0, \qquad \varphi(z_1) = \varphi(z_2) = 0.$$ Howard's proof substitutes $\varphi = (U - c)^{1/2} G$, multiplies by $\bar G$ and integrates. The imaginary part of the result is $$c_i \left[ \int \left( \abs{G'}^2 + \alpha^2 \abs{G}^2 \right) dz + \int \frac{N^2 - U'^2/4}{\abs{U - c}^2} \abs{G}^2 \, dz \right] = 0,$$ with $c = c_r + i c_i$, and when $N^2 \ge U'^2/4$ the bracket is positive, so $c_i = 0$.

The theorem was proved by Miles and by Howard in 1961. The condition is sufficient for stability, not necessary: a flow with $\mathrm{Ri} < 1/4$ somewhere may still be stable. With $N = 0$ the condition fails wherever $U' \neq 0$; the unstratified case of the Euler equations is covered by Rayleigh's inflection point criterion instead. The theorem and its proof extend to unbounded layers with disturbances that decay at infinity. Example: for the stratified mixing layer $U = \tanh z$, $N^2 = J \operatorname{sech}^2 z$ on the whole line, one has $\mathrm{Ri}(z) = J \cosh^2 z \ge J$, so the flow is stable when $J \ge 1/4$.

Going deeper

For a student

The theorem is the stratified counterpart of the inviscid criteria of hydrodynamic stability for parallel flows. The threshold $1/4$ is widely used in oceanography and meteorology as a rule of thumb for when shear-driven turbulence and Kelvin–Helmholtz billows can occur. Its proof is the same energy-type identity as in Rayleigh's criterion, after Howard's substitution. The same substitution gives Howard's semicircle theorem, which bounds the complex wave speed of unstable modes.