3.4 The energy balance in a growing wind sea
Having discussed the parametrization of the physics source terms we now proceed with studying the impact of wind input, nonlinear interaction and whitecap dissipation on the evolution of the wave spectrum for the simple case of a duration-limited wind sea. To this end we numerically solved equation (2.24) for infinite depth and a constant wind of approximately 18 m/s, neglecting currents and advection. Typical results are shown in Fig. 3.4 for a young wind sea (
) and in Fig. 3.5 for an old wind sea (
). In either case the directional averages of 
, 
and 
are shown as functions of frequency. First of all we observe that, as expected from our previous discussions, the wind input is always positive, and the dissipation is always negative, while the nonlinear interactions show a three lobe structure of different signs. Thus, the intermediate frequencies receive energy from the airflow which is transported by the nonlinear interactions towards the low and high frequencies.
Figure 3.4 The energy balance for young duration-limited wind sea.
Figure 3.5 The energy balance for old wind sea.
Concentrating for the moment on the case of young wind sea, we immediately conclude that the one-dimensional frequency spectrum in the `high'-frequency range must be close to 
, because the nonlinear source term is quite small (see the discussion in § II.3.10 of Komen et al. (1994) on the energy cascade caused by the four-wave interactions and the associated equilibrium shape of the spectrum). We emphasize, however, that because of the smallness of 
it cannot be concluded that the nonlinear interactions do not control the shape of the spectrum in this range. On the contrary, a small deviation from the equilibrium shape would give rise to a large nonlinear source term which will drive the spectrum back to its equilibrium shape. The role of wind input and dissipation in this relaxation process can only be secondary because these source terms are approximately linear in the wave spectrum. The combined effect of wind input and dissipation is more of a global nature in that they constrain the magnitude of the energy flow through the spectrum (which is caused by the four-wave interactions).
At low frequencies we observe from Fig. 3.4 that the nonlinear interactions maintain an `inverse' energy cascade by transferring energy from the region just beyond the location of the spectral peak (at 
) to the region just below the spectral peak, thereby shifting the peak of the spectrum towards lower frequencies. This frequency downshift is, however, to a large extent, determined by the shape and magnitude of the spectral peak itself. For young wind sea, having a narrow peak with a considerable peak enhancement, the rate of downshifting is significant while for old wind sea this is much less so. During the course of time the peak of the spectrum gradually shifts towards lower frequencies until the peak of the spectrum no longer receives input from the wind because these waves are running faster than the wind. Under these circumstances the waves around the spectral peak are subject to a considerable dissipation so that their wave steepness becomes reduced. Consequently, because the nonlinear interactions depend on the wave steepness, the nonlinear transfer is reduced as well. The peak of the positive low-frequency lobe of the nonlinear transfer remains below the peak of the spectrum, where it compensates the dissipation. As a result, a quasi-equilibrium spectrum emerges. The corresponding balance of old wind sea is shown in Fig. 3.5 . The nature of this balance depends on details of the directional distribution (see Komen et al., 1984 for additional details). The question of whether an exact equilibrium exists appears of little practical relevance. For old wind sea the timescale of downshifting becomes much larger than the typical duration of a storm. Thus, although from the present knowledge of wave dynamics it cannot be shown that wind-generated waves evolve towards a steady state, for all practical purposes they do!
This concludes our discussion of the parametrization of the physics source terms. Before presenting a discussion of the numerical scheme we have used to solve the action balance equation we shall first describe the data assimilation scheme.
