Nonlinearly active modes and cascade ``avalanches''

Component with the shifted frequency $\omega^*$ is clearly a nonlinear effect (there is no frequency shift in linear dynamics). Thus, the relative strength of $c_2$ and $c_1$ can be used as a measure of nonlinearity. Particularly, the phase runs mark the events when nonlinearity becomes strong. Figure 10 shows locations of the phase runs in the 2D wavenumber space which happened at $t=500$. Note that at that time ZF steady spectrum has already formed. In the energy cascade range, we see that the phase run density is increasing toward high $k$'s, which is in agreement with the WT prediction that the nonlinearity grows as one cascades down-scale [9,10]. Curiously, we also observe high density of the phase runs within th circle $k<6$, which is, perhaps, manifestation of a waveaction accumulation via an inverse cascade process. However, this range is too small for any meaningful conclusions to be made about the inverse cascade properties.

The energy cascade from the forcing region toward the high wavenumber region proceeds in a non-uniform in time fashion somewhat resembling sporadic sandpile avalanches. This arises due to the $k$-grid discreteness effects which tend to block the resonant wave interaction when the wave intensities are small. This situation resembles ``frozen turbulence'' of [14]. Thus, the wave energy does not cascade to high wavenumbers and it tends to accumulate near the forcing scales until the wave intensity is strong enough to restore the resonant interaction via the nonlinear resonance broadening. At this moment the energy cascade toward high wavenumbers sets in, and this leads to depletion of energy at the forcing scale, - ``sandpile tips over''. In turn, depletion of energy at the forcing scale leads to blocking of the energy cascade, and the process continues in a repetitive manner. As a result system oscillates between the state of ``frozen turbulence'' and the state of ``avalanche cascade''. This behavior is illustrated in figure 14 which shows percentage of modes experiencing phase runs in two different wavenumber ranges $13 < k < 29$ and $30 < k < 45$. One can see that the shapes of these two curves bear a great degree of similarity up to a certain time delay and a vertical shift in the second curve with respect to the first one. The vertical shift reflects the fact that the energy cascade gets stronger as it proceeds to large wavenumbers. The time delay, on the other hand indicates the direction and the character of the sporadic energy cascade. It shows that a higher (lower) nonlinear activity at low $k$'s after a finite delay causes a higher (lower) activity at higher $k$'s, which could be compared with propagation of an avalanche (quenching) down a sandpile.

Figure 14: Percentage of modes experiencing phase runs in the range $13 < \vert k\vert < 29$ (lower curve) and in the range $30 < \vert k\vert < 45$ (upper curve).

[width=10cm]2regions_runs.eps