The onset of turbulence in finite-amplitude Kelvin–Helmholtz billows

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Abstract
Two-dimensional finite-amplitude Kelvin–Helmholtz waves are tested for stability against three-dimensional infinitesimal perturbations. Since the nonlinear waves are time-dependent, the stability analysis is based upon the assumption that they evolve on a timescale which is long compared with that of any instability which they might support. The stability problem is thereby reduced to standard eigenvalue form, and solutions that do not satisfy the timescale constraint are rejected. If the Reynolds number of the initial parallel flow is sufficiently high the two-dimensional wave is found to be unstable and the fastest-growing modes are three-dimensional disturbances that possess longitudinal symmetry. These modes are convective in nature and focused in the statically unstable regions that form during the overturning of the stratified fluid in the core of the nonlinear vortex. The nature of the instability in the high-Reynolds-number regime suggests that it is intimately related to the observed onset of turbulence in these waves. The transition Reynolds number above which the secondary instability exists depends strongly on the initial conditions from which the primary wave evolves.