Wave-induced longitudinal-vortex instability in shear flows

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Abstract
The instability of two-dimensional periodic flows to spanwise-periodic ‘longitudinal- vortex’ modes is examined. The undisturbed state comprises a parallel shear flow and a two-dimensional O(ε) wave field as encountered in, say, water-wave or hydrodynamic-stability theories.When the mean shear is weak, of order ε2, the present theory reduces to that of Craik (1977) and Leibovich (1977b, 1980). For stronger but still weak shear, of order ε, it is established that the Craik-Leibovich instability mechanism is essentially unchanged, apart from scaling factors. For strong O(1) shear flows, the governing equations are derived by using, in part, a generalized Lagrangian-mean formulation. The resultant eigenvalue problem for the longitudinal-vortex instability is then more complex, but simplifies in the case of small spanwise spacing of the vortices, in the inviscid limit. An example is given of flows that exhibit instability in this limiting case. Such instability seems likely to occur for a wide class of periodic shear flows. Complementary physical interpretations of the instability mechanism are discussed.