Reflection of internal gravity waves in the form of quasi-axisymmetric beams
Résumé
Preservation of the angle of reflection when an internal gravity wave hits a sloping boundary generates a focusing mechanism if the angle between the direction of propagation of the incident wave and the horizontal is close to the slope inclination (near-critical reflection). We establish a rigorous analysis of this phenomenon in two space dimensions, by providing an explicit description of the form of the leading approximation of the unique Leray solution to the near-critical reflection of internal waves from a slope within a certain (nonlinear) time-scale. More precisely, we construct a consistent and Lyapunov stable approximate solution, $L^2$ -close to the Leray solution, in the form of a beam wave.
Besides, beams being physically more meaningful than plane waves, their spatial localization plays a key role in improving the previous mathematical results from a twofold viewpoint: 1) our beam wave approximate solution is the sum of a finite number of terms, each of them is a consistent solution to the system and there is no artificial/non-physical corrector; 2) thanks to 1) and the special structure of the nonlinear term, we can improve the expansion of our solution up to two next orders, so providing a more accurate approximation leading to a longer consistency time-scale.
Finally, our results provide a set of initial conditions localized on rays, for which the Leray solution maintains approximately in $L^2$ the same localization.
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