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V.V. Voronovich, D.E. Pelinovsky, and V.I. Shrira

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On internal wave-shear flow resonance in shallow water

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J. Fluid Mech. 354, 209-237 (1998)

**Abstract:**

The work is concerned with long nonlinear internal waves
interacting with a shear flow localized near the sea surface.
The study is focused on the most intense resonant
interaction occurring when the phase velocity of internal
waves matches the flow velocity at the surface. The
perturbations of the shear flow are considered as 'vorticity
waves', which enables us to treat the wave-flow resonance
as the resonant wave-wave interaction between an internal
gravity mode and the vorticity mode. Within the
weakly nonlinear long-wave approximation a system of
evolution equations governing the nonlinear dynamics
of the waves in resonance is derived and an asymptotic
solution to the basic equations is constructed. At
resonance the nonlinearity of the internal wave dynamics
is due to the interaction with the vorticity mode, while the
wave's own nonlinearity proves to be negligible. The
equations derived are found to possess solitary wave
solutions of different polarities propagating slightly faster or
slower than the surface velocity of the shear flow.
The amplitudes of the 'fast' solitary waves are
limited from above; the crest of the limiting wave
forms a sharp corner.
The solitary waves of amplitude smaller than a certain
threshold are shown to be stable; 'subcritical' localized
pulses tend to such solutions. The localized pulses of
amplitude exceeding this threshold form infinite slopes
in finite time, which indicates wave breaking.

**Keywords:**

SOLITARY WAVES, INSTABILITY, SOLITONS