STRUCTURES GENERATED BY MODEL NONLINEAR HAMILTONIANS BASED ON OCEAN WAVES

Abstract

When surface gravity waves in deep water are driven to a balance between energy input and dissipation, it has been observed that the spectrum of wave-height correlations exhibits power-law behavior consistent with the interpretation that over a range of length scales the surface is a self-affine fractal. Zakharov and L’vov long ago invoked a Kolmogorov cascade hypothesis to describe the nonlinear modal transfer of energy which might explain this observation, even in the absence of driving and dissipation. If their hypothesis is correct, a purely autonomous Hamiltonian system would be capable of driving a surface to fractal roughness, in analogy to dynamical system chaos. In this report I summarize a body of recent work aimed at testing the Z-L hypothesis through numerical integration of Zakharov’s Hamiltonian equations. I investigate both scaling and the existence of postulated emergent dynamical times which scale with modenumber. The results are mixed, in ways not easily abstracted. They lead to hard questions regarding the roles of driving and dissipation, and the process of equilibration of the waveheight statistical distribution, which cannot be answered at our present state of knowledge. But I hope that the form of the questions will prove stimulating.