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.
Highlights
The dominant non-linear effect on the propagation of surface gravity waves in deep water is due to the resonant and almost resonant four-wave interactions discovered by Phillips [1]
This process of four-wave interaction is succinctly described by the Zakharov equation [2] which is a deterministic model accurate to third order in the wave steepness
On the other hand we show that solutions of the JKE always remain bounded, even in cases when there is blow-up in the generalized kinetic equation (GKE)
Summary
The dominant non-linear effect on the propagation of surface gravity waves in deep water is due to the resonant and almost resonant four-wave interactions discovered by Phillips [1]. This process of four-wave interaction is succinctly described by the Zakharov equation [2] which is a deterministic model accurate to third order in the wave steepness. Elsewhere, subscripts denote a wavenumber component, so Bi = B(ki , t). The wavenumber resonance condition for quartets of waves is evident in the delta function δ0+1−2−3 = δ(k0 + k1 − k2 − k3 ), while the frequency detuning ∆2,3
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