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Nonlinear internal gravity wave beams. (English) Zbl 1057.76010

Summary: Based on linear inviscid theory, a two-dimensional source oscillating with frequency \(\omega_{0}\) in a uniformly stratified (constant Brunt-Väisälä frequency \(N_{0}\)) Boussinesq fluid induces a steady-state wave pattern, also known as St Andrew’s Cross, that features four straight wave beams stretching radially outwards from the source at angles \(\pm\cos^{-1}(\omega_{0}/N_{0})\) relative to the vertical. Similar wave beams are generated by oscillatory stratified flow over topography and also appear in simulations of thunderstorm-generated gravity waves in the atmosphere. Uniform plane-wave beams of infinite extent are in fact exact solutions of the nonlinear inviscid equations of motion, and this property is used here to study the propagation of finite-amplitude wave beams taking into account weak viscous and refraction effects. Oblique beams (\(\omega_0<N_0\)) are considered first and an amplitude-evolution equation is derived assuming slow modulations along the beam direction. Remarkably, the leading-order nonlinear terms cancel out in this evolution equation and, as a result, the steady-state similarity solution of N. H. Thomas and T. N. Stevenson [J. Fluid Mech. 54, 495–506 (1972; Zbl 0247.76095)] for linear viscous beams is also valid in the nonlinear régime. Moreover, for the same reason, nonlinear effects are found to be relatively unimportant for two-dimensional and axisymmetric beams that propagate nearly vertically (\(\omega_{0}\approx N_0\)) in a Boussinesq fluid. Owing to the fact that the group velocity vanishes when \(\omega_0=N_0\), however, the transient evolution of nearly vertical beams takes place on a slower time scale than that of oblique beams; this is shown to account for the discrepancies between the steady-state similarity solution of D. Gordon and T. N. Stevenson [J. Fluid Mech. 56, 629–639 (1972; Zbl 0257.76015)] and their experimental observations. Finally, the present asymptotic theory is used to study the refraction of nearly vertical nonlinear beams in the presence of background shear and variations in the Brunt-Väisälä frequency.

MSC:

76B55 Internal waves for incompressible inviscid fluids
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