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 ε 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 cr, the singular coefficient at a regular critical level U(zc)=cr has a residue proportional to Q=U''+gU'/a2=ρ̅0(U'/ρ̅0)', with gravitational acceleration g, sound speed a, and normalized base-state air density ρ̅0. Thus the gradient of specific vorticity replaces the curvature U'' 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 Za+(k,cr) of the air-side pressure response at wavenumber k>0. Let ck>0 be the phase speed of the free gravity-capillary wave. If Im Za+(k,ck)>0, then, for sufficiently small ε, the dispersion relation has exactly one unstable root cε in a neighborhood of ck that does not depend on ε. A sufficient condition is Q≤0 at every critical level and Q<0 at the highest one. As the Mach number M→0, Za+ converges to its incompressible counterpart at rate O(M2). Under the corresponding incompressible sign condition, the unstable root satisfies |cε(M)−cε(0)|≤ Cε M2 with C independent of ε and M.
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BibTeX
@misc{li2026compressible,
title = {{A Compressible Miles Instability for Wind-Wave Generation in a Neutral Atmosphere}},
author = {Tianyi Li},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.02628},
eprint = {2610.02628},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.02628},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - A Compressible Miles Instability for Wind-Wave Generation in a Neutral Atmosphere
AU - Tianyi Li
PY - 2026
DA - 2026/10/02
PB - arXiv
DO - 10.48550/arxiv.2610.02628
UR - https://arxiv.org/abs/2610.02628
AB - 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 ε 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''+gU'/a^2=ρ̅_0(U'/ρ̅_0)', with gravitational acceleration g, sound speed a, and normalized base-state air density ρ̅_0. Thus the gradient of specific vorticity replaces the curvature U'' 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_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 Im Z_a^+(k,c_k)>0, then, for sufficiently small ε, the dispersion relation has exactly one unstable root c_ε in a neighborhood of c_k that does not depend on ε. A sufficient condition is Q≤0 at every critical level and Q<0 at the highest one. As the Mach number M→0, Z_a^+ converges to its incompressible counterpart at rate O(M^2). Under the corresponding incompressible sign condition, the unstable root satisfies |c_ε(M)−c_ε(0)|≤ Cε M^2 with C independent of ε and M.
ER -