Spectrum

Measuring the spectrum has by far dominated WT studies because this is the most basic and robust theoretical object and because this quantity is easier to observe experimentally that more subtle statistical quantities. For the surface gravity waves, the WT prediction about the energy cascade spectrum have been confirmed in several recent numerical studies [2,3,1]. Here, we also start by presenting the spectrum. Figure 2 shows the spectrum at $t=300 T_{p}$ at $t=2000T_{p}$, where $T_{p}$ is period of the slowest mode ($k=10$). The obtained spectrum is in agreement with the $k^{-4}$ shape predicted by WT theory in the inertial range, and this serves as a validation of our code. Our subsequent statistical measurements will be made at the time where this steady state spectrum has already got established.

Figure 2: Waveaction spectrum of waves. Dashed and solid lines show spectra after 300 and 2000 periods of mode $k=10$. The straight line corresponds to the ZF spectrum ($-4$ slope).

[width=10cm]1Dspectrum.eps