Reflection of internal gravity waves in the form of quasi-axisymmetric beams

Abstract

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. Contents 1 3 , ω ∼ ± sin γ. 3.2. Proof of Proposition 3.1 3.3. Linear analysis in different regimes 4. The weakly nonlinear system 4.1. Interactions of type (a) 4.2. Interactions of type (b) 4.3. Proof of Proposition 4.1 5. Higher-order Approximation 5.1. Proof of Theorem 2.10 Appendix A. Proof of Lemma 2.4 Appendix B. A useful lemma