On the Generalized Kinetic Equation for Surface Gravity Waves, Blow-Up and Its Restraint

Abstract

This article is concerned with the non-linear interaction of homogeneous random ocean surface waves. Under this umbrella, numerous kinetic equations have been derived to study the evolution of the spectral action density, each employing slightly different assumptions. Using analytical and numerical tools, and providing exact formulas, we demonstrate that the recently derived generalized kinetic equation exhibits blow up in finite time for certain degenerate quartets of waves. This is discussed in light of the assumptions made in the derivation, and this equation is contrasted with other kinetic equations for the spectral action density.

Strong solutions for the Alber equation and stability of unidirectional wave spectra

“…e.g. [2,33], and the results of this paper could motivate analogous advances for those equations as well.…”

Nonlinear Wave Interaction in Coastal and Open Seas: Deterministic and Stochastic Theory

“…2.19] generalized this to include frequency correction terms. The same generalized equation was recently studied by Andrade et al [6], and found to exhibit finite-time blow-up for some degenerate quartets. When performing Monte-Carlo simulations, Annenkov & Shrira [7,Sec.…”

“…A number of generalizations of the kinetic equation (27) [6], and found to exhibit finite-time blow-up for some degenerate quartets. When performing Monte-Carlo simulations, Annenkov & Shrira [7,Sec.…”

“…φ tt + gφ z = 0 on z = 0, (6) φ z + φ x h x + φ y h y = 0 on z = −h(x, y), (7) where the surface boundary conditions have been combined to eliminate η from the problem. Equation (7) simplifies to φ z = 0 on z = −h for constant depth, resulting in the Laplace equation on a horizontal strip, or, for infinite depth, a half space.…”

Kinetic equations in a third-generation spectral wave model

“…All these limitations of our GKE algorithm would practically inhibit its large-scale, realistic applications. Andrade, Stuhlmeier & Stiassnie (2019) reported that when applied to degenerate quartets, the GKE may exhibit blow-up under certain conditions, further implying a possible inherent deficiency of the GKE. Such strange behaviour, however, is not observed in our simulations for wave spectra with multiple wave modes.…”