Effects of finite non-Gaussianity on evolution of a random wind wave field.

We examine the long-term evolution of a random wind wave field generated by constant forcing, by comparing numerical simulations of the kinetic equation and direct numerical simulations (DNS) of the dynamical equations. While the integral characteristics of the spectra are in reasonably good agreement, the spectral shapes differ considerably at large times, the DNS spectral shape being in much better agreement with field observations. Varying the number of resonant and approximately resonant wave interactions in the DNS numerical scheme, we show that when the ratio of nonlinear and linear parts of the Hamiltonian tends to zero, the DNS spectral shape approaches the shape predicted by the kinetic equation. We attribute the discrepancies between the kinetic equation modeling, on one side, and the DNS and observations, on the other, to the neglect of non-Gaussianity in the derivation of the kinetic equation

Spectral evolution of weakly nonlinear random waves: kinetic description versus direct numerical simulations

Kinetic equations are widely used in many branches of science to describe the evolution of random wave spectra. To examine the validity of these equations, we study numerically the long-term evolution of water wave spectra without wind input using three different models. The first model is the classical kinetic (Hasselmann) equation (KE). The second model is the generalised kinetic equation (gKE), derived employing the same statistical closure as the KE but without the assumption of quasistationarity. The third model, which we refer to as the DNS-ZE, is a direct numerical simulation algorithm based on the Zakharov integrodifferential equation, which plays the role of the primitive equation for a weakly nonlinear wave field. It does not employ any statistical assumptions. We perform a comparison of the spectral evolution of the same initial distributions without forcing, with/without a statistical closure and with/without the quasistationarity assumption. For the initial conditions, we choose two narrow-banded spectra with the same frequency distribution and different degrees of directionality. The short-term evolution ( wave periods) of both spectra has been previously thoroughly studied experimentally and numerically using a variety of approaches. Our DNS-ZE results are validated both with existing short-term DNS by other methods and with available laboratory observations of higher-order moment (kurtosis) evolution. All three models demonstrate very close evolution of integral characteristics of the spectra, approaching with time the theoretical asymptotes of the self-similar stage of evolution. Both kinetic equations give almost identical spectral evolution, unless the spectrum is initially too narrow in angle. However, there are major differences between the DNS-ZE and gKE/KE predictions. First, the rate of angular broadening of initially narrow angular distributions is much larger for the gKE and KE than for the DNS-ZE, although the angular width does appear to tend to the same universal value at large times. Second, the shapes of the frequency spectra differ substantially (even when the nonlinearity is decreased), the DNS-ZE spectra being wider than the KE/gKE ones and having much lower spectral peaks. Third, the maximal rates of change of the spectra obtained with the DNS-ZE scale as the fourth power of nonlinearity, which corresponds to the dynamical time scale of evolution, rather than the sixth power of nonlinearity typical of the kinetic time scale exhibited by the KE. The gKE predictions fall in between. While the long-term DNS show excellent agreement with the KE predictions for integral characteristics of evolving wave spectra, the striking systematic discrepancies for a number of specific spectral characteristics call for revision of the fundamentals of the wave kinetic description

Vortical and Wave Modes in 3D Rotating Stratified Flows: Random Large Scale Forcing

Utilizing an eigenfunction decomposition, we study the growth and spectra of energy in the vortical and wave modes of a 3D rotating stratified fluid as a function of $\epsilon = f/N$. Working in regimes characterized by moderate Burger numbers, i.e. $Bu = 1/\epsilon^2 < 1$ or $Bu \ge 1$, our results indicate profound change in the character of vortical and wave mode interactions with respect to $Bu = 1$. As with the reference state of $\epsilon=1$, for $\epsilon < 1$ the wave mode energy saturates quite quickly and the ensuing forward cascade continues to act as an efficient means of dissipating ageostrophic energy. Further, these saturated spectra steepen as $\epsilon$ decreases: we see a shift from $k^{-1}$ to $k^{-5/3}$ scaling for $k_f < k < k_d$ (where $k_f$ and $k_d$ are the forcing and dissipation scales, respectively). On the other hand, when $\epsilon > 1$ the wave mode energy never saturates and comes to dominate the total energy in the system. In fact, in a sense the wave modes behave in an asymmetric manner about $\epsilon = 1$. With regard to the vortical modes, for $\epsilon \le 1$, the signatures of 3D quasigeostrophy are clearly evident. Specifically, we see a $k^{-3}$ scaling for $k_f < k < k_d$ and, in accord with an inverse transfer of energy, the vortical mode energy never saturates but rather increases for all $k < k_f$. In contrast, for $\epsilon > 1$ and increasing, the vortical modes contain a progressively smaller fraction of the total energy indicating that the 3D quasigeostrophic subsystem plays an energetically smaller role in the overall dynamics.Comment: 18 pages, 6 figs. (abbreviated abstract

Nonlinear wave interaction in coastal and open seas -- deterministic and stochastic theory

We review the theory of wave interaction in finite and infinite depth. Both of these strands of water-wave research begin with the deterministic governing equations for water waves, from which simplified equations can be derived to model situations of interest, such as the mild slope and modified mild slope equations, the Zakharov equation, or the nonlinear Schr\"odinger equation. These deterministic equations yield accompanying stochastic equations for averaged quantities of the sea-state, like the spectrum or bispectrum. We discuss several of these in depth, touching on recent results about the stability of open ocean spectra to inhomogeneous disturbances, as well as new stochastic equations for the nearshore