“Fast” Nonlinear Evolution in Wave Turbulence

Modeling of nonlinear random wave fields in nature (and, in particular, their most common example—wind waves in the ocean) is one of the fundamental open problems of natural sciences. The existing theoretical approaches based on the kinetic equation paradigm assume a proximity to stationarity and homogeneity. In reality this assumption is often violated and how a wave field evolves is not known. We show by direct numerical simulation that after a strong perturbation the wave field evolves on the much faster $O({\epsilon }^{-2})$ “dynamic” [rather then $O({\epsilon }^{-4})$ “kinetic”] time scale; here $\epsilon$ is the characteristic wave steepness ($\epsilon \ll 1$). The phenomenon of fast evolution is universal, and it must occur whenever there is a strong external perturbation.

Figure 1

An example of DNS of random wave field evolution under an abrupt increase of wind (Run 3). (a) Time evolution of wave action spectrum, in steps of approximately 400 periods of the spectral peak (intermediate curves are dashed). Regions of forcing and dissipation are delimited by dotted vertical lines. Equilibrium spectra $N_0(k)$ and $N_f(k)$ are shown by shading. (b) Evolution of wave steepness $\epsilon$. Equilibrium values of steepness before and after the increase of forcing (which occurs at $t=T_0$) are marked by dashed horizontal lines. (c) Evolution of spectral peak $k_p$. Theoretical (according to the large-time kinetic equation asymptotics) downshift rate $k_p\sim t^{-6/11}$ is shown by the dashed line.

Figure 2

Response of the wave action spectrum to the sharp increase of forcing (run 3). (a) Response of sample spectral bands (integrated from $k_0$ to $1.1k_0$). Values of the integrated spectrum $N(k)$ are normalized by the equilibrium values $N_0(k)$ and $N_f(k)$. (b) Maximal and average growth rates ($M(k)$ and $m(k)$ respectively, defined in the text) as functions of the wave number.

Figure 3

Integrated (from $k=0.5$ to $k=1.0$) maximal growth rates $\overline{M}$ and mean growth rates $\overline{m}$ versus wave steepness ${\epsilon }_2$, on a logarithmic scale, with least-squares fits (dashed lines) for the scaling ${\epsilon }^{\lambda }$ ($\lambda =3.5$ for $\overline{M}$ and $\lambda =4.3$ for $\overline{m}$). Theoretical dynamic ${\epsilon }^4$ and kinetic ${\epsilon }^6$ scalings are also shown (dotted lines). Numbers identify numerical runs (Table ).