[width=10cm]k2500_ampDetail.eps

Figures 7 and 8 show time evolution of the amplitude and the (interaction representation) phase respectively for . The phase evolution seen in figure 8 consists of the time intervals when it oscillates quasiperiodically (with amplitude less than ) which are interleaved by sudden ``phase runs'',  fast monotonic phase changes by values which can significantly exceed . By juxtaposing figures 7 and 8, one can see that when the phase runs happen then the amplitude is close to zero. We zoom in at the amplitude graph in a time interval characterized by low amplitudes (and therefore phase runs) in figure 9. One can see largeamplitude quasiperiodic oscillations on ,  it changes in value severalfold over a time comparable to the linear wave period. This indicates that such phase run intervals mark the places where the WT assumption of weak nonlinearity breaks down.
This behavior, together with the twopeak character of the timeFourier spectrum, suggest that at each wavenumber there are two modes:
Note that such observed behavior of the phase is very dependent on the wave variables: it occurs in natural variables such as the surface height and velocity but is absent for the waveaction variable after the canonical transformation leading to the Zakharov equation. Thus, the phase runs can say nothing about the dynamics or statistics of the waveaction amplitude used in WT, but they mark short time intervals when nonlinearity fails to be weak for a particular mode, and this will be used as a diagnostic tool in our simulations.
[width=10cm]phase_runs.eps

In figure 11 we show comparison of the mean rate of the phase change during the upward phase runs and the second peak's frequency at different wavenumbers. A significant coincidence of these two curves supports the proposed above twomode explanation.
[width=10cm]HighFrequencyPeak.eps

A word of caution is due about the simple twomode explanation of the phase runs. Indeed, according to this picture the phase should always run to higher values whereas in figure 8 we see both upward and downward runs. A possible explanation of this is that phase runs may be triggered not only by sharp peaks but also by broadband distributions with frequencies less than the linear one. Such broadband distributions could be made of modes which are strong (and therefore can produce phase runs) but whose duration in time is short and sporadic and at different frequencies (hence a broad spectrum). This picture is supported by the fact that the amount of total wave energy in the frequency range below the linear frequency is similar (and for low wavenumbers even greater) than the amount of energy in the range above the linear frequencies, see figure 13. Evidence of relation of the sublinear modes and the downward phase runs is shown in figure 12 where the mean rate of the phase change during the downward phase runs and the frequency of the highest sublinear peak. Again, we can see agreement of these two curves although with a greater level of fluctuations due to the fact the sublinear modes are very spread over different frequencies.
[width=10cm]LowFrequencyPeak.eps

[width=10cm]power.eps
