A Compressible Miles Instability for Wind-Wave Generation in a Neutral Atmosphere

Abstract

We study the normal-mode stability of a flat interface between compressible Euler flow in the air and incompressible Euler flow in the water, including gravity and surface tension. The base state is quiescent water beneath a hydrostatic, constant-entropy shear flow $U(z)$ with $U(0)=0$, so the atmosphere is neutrally stratified. For a small ratio $\epsilon$ of the air density at the interface to the water density, and for uniformly subsonic wave-relative flow, we establish a compressible analog of the critical-layer instability of Miles, extending the rigorous incompressible theory of Bühler, Shatah, Walsh, and Zeng. For a real phase speed $c_r$, the singular coefficient at a regular critical level $U(z_c)=c_r$ has a residue proportional to $Q=U^{\prime \prime }+g{U}^{\prime }/a^2={\bar{\rho }}_0(U^{\prime }/{\bar{\rho }}_0{)}^{\prime }$, with gravitational acceleration $g$, sound speed $a$, and normalized base-state air density ${\bar{\rho }}_0$. Thus the gradient of specific vorticity replaces the curvature $U^{\prime \prime }$ of the incompressible theory. We prove a limiting absorption principle at regular, non-resonant phase speeds and a residue formula for the imaginary part of the boundary value $Z_{\mathrm{a}}^{+}(k,c_r)$ of the air-side pressure response at wavenumber $k>0$. Let $c_k>0$ be the phase speed of the free gravity-capillary wave. If $\mathrm{Im}{Z}_{\mathrm{a}}^{+}(k,c_k)>0$, then, for sufficiently small $\epsilon$, the dispersion relation has exactly one unstable root $c_{\epsilon }$ in a neighborhood of $c_k$ that does not depend on $\epsilon$. A sufficient condition is $Q\le 0$ at every critical level and $Q<0$ at the highest one. As the Mach number $M\to 0$, $Z_{\mathrm{a}}^{+}$ converges to its incompressible counterpart at rate $O(M^2)$. Under the corresponding incompressible sign condition, the unstable root satisfies $|c_{\epsilon }(M)-c_{\epsilon }(0)|\le C\epsilon M^2$ with $C$ independent of $\epsilon$ and $M$.