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 $\varepsilon$ 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 $Q=U&#39;&#39;+gU&#39;/a<sup>2=\barρ_0(U&#39;/\barρ_0)&#39;$, with gravitational acceleration $g$, sound speed $a$, and normalized base-state air density $\barρ<em>0$. Thus the gradient of specific vorticity replaces the curvature $U&#39;&#39;$ 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 $Z</em>{\mathrm a}<sup>+(k,c_r)$ of the air-side pressure response at wavenumber $k&gt;0$. Let $c_k&gt;0$ be the phase speed of the free gravity-capillary wave. If $\mathrm{Im}\,Z_{\mathrm a}<sup>+(k,c_k)&gt;0$, then, for sufficiently small $\varepsilon$, the dispersion relation has exactly one unstable root $c_\varepsilon$ in a neighborhood of $c_k$ that does not depend on $\varepsilon$. A sufficient condition is $Q\le0$ at every critical level and $Q&lt;0$ at the highest one. As the Mach number $M\to0$, $Z_{\mathrm a}<sup>+$ converges to its incompressible counterpart at rate $O(M<sup>2)$. Under the corresponding incompressible sign condition, the unstable root satisfies $$|c_\varepsilon(M)-c_\varepsilon(0)|\le C\varepsilon M<sup>2$$ with $C$ independent of $\varepsilon$ and $M$.