Two really seminal papers were published in the area of physical oceanography about half a century ago: the article by and the work of . Both of them concentrated on the same problem: what is going on with ocean waves growing under the influence of wind?

O. Phillips suggested that this growth is arrested by wave breaking, in other words, by the formation of “white caps” (or “white horses”). Wave breaking is the main mechanism of energy dissipation. In “white caps”, mechanical energy of waves transforms to turbulence on small scales, then to heat. This is a strongly nonlinear phenomenon that cannot be studied by perturbative methods. An analytical theory of wave breaking has not been developed yet. The Phillips' assumption on dominance of wave-breaking effects allowed him to offer, on the basis of dimensional considerations, a universal ω-5 energy spectrum of wind waves (Phillips' spectrum), whereas a more elaborate explanation of the underlying physics was put forward by .

K. Hasselmann developed a completely different theory. He noticed that a typical ensemble of ocean waves possesses a small parameter: the average steepness (slope) of the surface μ. One can define it, for instance, as μ=〈|∇η|2〉, where η is the shape of the surface. This definition has the clearest geometrical meaning. Another definition estimates it as the average amplitude A of the waves multiplied by the wave vector of the spectral peak kp. This definition is easy to use for experimental observations. Later, we shall consider another definition, which involves the spectral distribution function. All these definitions give close values of the average steepness. Typically, μ ≃ 0.1. This fact allows one to assume that the wind-driven sea is an ensemble of weakly interacting waves, which can be described statistically by the use of expansion in powers of average steepness μ. This is a tedious procedure because one has to expand up to order μ4. However, K. Hasselmann coped with this hard work and derived his famous Hasselmann kinetic equation for a wave energy spectrum.

Then, it was found that the Hasselmann equation has exact stationary solutions (Kolmogorov–Zakharov or KZ spectra) decaying at a high-frequency range as ω-4, i.e., slower than the Phillips spectrum. A question arises as to which theory is closer to reality. The answer is given by experiment.

Numerous measurements (e.g., ) performed in large lakes and the ocean demonstrated that the spectrum is the weak-turbulent KZ one E(ω) ∼ ω-4, at least for the range of scales ωp < ω < 4ωp; here, ωp is the frequency of the spectral peak. At the same time, the Phillips spectrum was observed for rough seas for the tail of the spectrum. This fact has the following explanation. Wave turbulence in the wind-driven sea is a mixture of weak and strong turbulence. This is a question of phase correlation. Weakly nonlinear interaction provides correlations of waves phases on large scales – hundreds of characteristic wavelengths. At the same time, the wave breaking and whitecapping is the phenomenon localized in space. As a result, looking at the wind-driven sea, one observes the formation of short-lived localized wave breaking events embedded into a homogeneous weakly nonlinear background .

A similar situation, the coexistence of weak turbulence and localized coherent structures, is typical of wave turbulence in general. It takes place, for instance, in nonlinear optics , where optical turbulence coexists with self-focusing or in isotropic plasma where weak Langmuir turbulence coexists with Langmuir collapses . Similar phenomena were observed recently in numerical simulations for gravity waves . In all these situations, the usual course of action is to augment corresponding kinetic equations of weak turbulence by the introduction of an additional term, describing dissipation of energy in the coherent structures (corresponding to whitecapping onsets and wave breaking) . The question arises whether it is still reasonable to use a kinetic equation as an adequate model of wave interaction in the presence of coherent structures. Maybe, the first attempt to answer this question was the work by on spectral dissipation due to whitecapping. In this work, the concept of a “weakly nonlinear on average” sea was introduced, which justifies the weakly nonlinear description, resulting in the Hasselmann equation. This hypothesis was not confirmed numerically because the required numerical simulation of primordial dynamical equations was not feasible at that time.

In this paper, we show that at least some of the assumptions under which the Hasselmann kinetic equation was derived still hold in the presence of a number of whitecapping events. To do this, we perform massive numerical simulations of primordial weakly nonlinear dynamical equations (Hamiltonian equations with a Hamiltonian truncated at four-wave interaction terms) describing potential flow of Euler equations of an ideal fluid with a free surface. It should be noted that we only mimic energy dissipation due to whitecapping through the regularization of the equations by relatively close dissipation regions (approximately one decade from the pumping scale). As a result, although the whitecapping cannot be described rigorously in the weakly nonlinear model, one can dissipate part of the energy of the wave involved in the whitecapping by the proposed mechanism. The statement that we dissipate the right part of the energy is a hypothesis, although we give some explanations showing that it looks like quite a reasonable claim.

In our numerical experiments, we study Fourier spectra of space–time correlation functions of normal canonical variables describing the surface dynamics. Our goal was to determine the “shape of the line”, i.e., the frequency spectra of certain spatial Fourier harmonics. According to the theory of weak turbulence, these spectra must be narrow and concentrated near the frequency given by the dispersion relation for waves of small amplitude. We obtained the following important result: in a broad range of wave numbers, the spectra remain narrow even in the presence of whitecapping dissipation events, when the spatial spectrum obeys the Phillips law. This justifies the construction of an appropriately adjusted dissipation term that can be used thereafter in wave forecasting.

We solve numerically weakly nonlinear Euler equations for dynamics of incompressible deep fluids with free surfaces in the presence of gravity by the pseudo-spectral code described in . The code was verified in our previous papers . The equations are written for surface elevation η(x, y, t) and hydrodynamic velocity potential on the surface ψ(x, y, t). Equations are the result of weakly nonlinear expansion of the Hamiltonian up to the fourth-order terms in steepness η˙=k^ψ-(∇(η∇ψ))-k^[ηk^ψ]+k^(ηk^[ηk^ψ])+12Δη2k^ψ+12k^η2Δψ-F^-1γkηk, ψ˙=-gη-12(∇ψ)2-(k^ψ)2-[k^ψ]k^[ηk^ψ]-[ηk^ψ]Δψ-F^-1γkψk+F^-1Pk. Here, the dot means the time derivative, Δ = ∇2 the Laplace operator in the r = (x, y)T plane, k^ = -Δ, F^-1 is an inverse Fourier transform, γk is a dissipation rate depending on the scale (according to recent work by , it has to be included in both equations), which corresponds to viscosity on small scales and, if needed, “artificial” damping on large scales. Pk is the driving term that simulates pumping on relatively large scales. Here and further on, we use the following formula for coefficients of Fourier series ηk=1(2π)2∫ηre-ikrd2r. The specific form of pumping is not important in our simulations. We have chosen it in the following form: Pk=fkeiRk(t),fk=4F0k-kp1kp2-kkp2-kp12,0-ifk<kp1ork>kp2. Here, kp1 = 28, kp2 = 32 and F0 = 1.5 × 10-5; Rk(t) was a uniformly distributed random number in the interval (0, 2π] for each k and t. The initial condition was a low-amplitude noise in all harmonics. The time step was Δt = 6.7 × 10-4. The dissipation function is γk = γk(1) + γk(2).

Artificial viscosity γk(1) was the same in all experiments: γk(1)=γ0k-kd2,k>kd,0,ifk≤kd, where kd = 256 and γ0 = 0.97 × 102.

γk(2) is the dissipation concentrated in small wave numbers. It was zero only in one of three experiments. It is described separately for every experiment below.

The linear dispersion relation for gravity waves was ωk = gk, with g the gravity acceleration, which was taken as g = 1. Equation (2) was solved in a 2π × 2π periodic box. The number of modes was 1024 × 1024.

Let us consider explicitly discrete Fourier transforms in the case of the 2π × 2π periodic box: ηk=1(2π)2∫02π∫02πηreikrd2r,ηr=∑-Nx/2Nx/2-1∑-Ny/2Ny/2-1ηke-ikr≃∫ηke-ikrd2k. Here, we took into account the fact that, for the 2π × 2π region, Δk = 1.

The same for ψr and ψk. Normal canonical variables are ak=ωk2kηk+ik2ωkψkηk=k2ωkak+a-k*,ψk=ωk2kak-a*-k. Inside, the box spectra were practically isotropic; thus, one can put |ηk|2≃k2ωknk+n-k,nk=|ak|2. For average squared steepness, we obtain μ2=〈|∇η(r)|2〉=1(2π)2∫|∇η(r)|2d2r=(2π)2(2π)2∫k2|ηk|2d2k. Using Eq. (), one can get |ηk|2=k2ωk|ak|2+|a-k|2+ak*a-k*+aka-k. If we consider cases of random phases (as in our simulation), in the isotropic case, 〈ak*a-k*〉ϕ=〈aka-k〉ϕ=0, where 〈⋅〉ϕ means averaging over the azimuthal angle 〈f(k)〉ϕ=12π∫02πf(k,ϕ)dϕ.

Using Eq.  (), we can simplify the expression for μ2: ∫k2|ηk|2d2k=∫0+∞k2kωk∫02π|ak|2dϕkdk. For isotropic cases, it is natural to introduce a spectrum averaged by an angle (this is what we usually measure in our numerical experiments): nk=〈nk〉ϕ=12π∫02π|ak|2dϕ. Then, we have μ2=∫k2|ηk|2d2k=2π∫0+∞k2kωknkkdk. For surface gravity waves over the fluid of infinite depth, ωk = gk. In our simulations, g = 1, so from Eq. (), we get μtotal2=2π∫0+∞k7/2nkdk. We would like to emphasize that, for the isotropic case definition, Eq. () coincides with Eq. ().

In order to understand which scales are important for the steepness, we can introduce μ2(k) as follows: μ2(k)=2π∫0kk7/2nkdk. Essentially, we introduced a steepness μ(k) accumulated by the scales with wave numbers less than k as a square root of Eq. ().

Here, we use angle-averaged nk (Eq. ), and summation is replaced by integration over k = |k|. Average steepness can be obtained as a limit: μtotal=lim⁡k→+∞μk. This definition gives a value of average steepness that differs from the geometrical definition μ = 〈|∇η|2〉; in our experiments, the discrepancy does not exceed 0.005, which is in most cases just a few percent of the characteristic average steepness. This discrepancy arises from the fact that, close to the origin of the k plane, we have a small number of harmonics for the angle averaging. As a result, terms in Eq. () do not vanish completely, because the summation over a coarse discrete grid of wave numbers is too crude an approximation of the integration.

It is worth noting that the usual definition of the steepness through the average squared amplitude and the wave number of the spectral peak takes into account mostly regions of the spectral peak, while our computations of function μ(k) show that the contribution from the tail of the spectrum is a significant part of total steepness μ and cannot be neglected.

For the spatial harmonic with wave vector k, we defined the time Fourier transform as follows: a(k,ω)=1T∫0Ta(k,t)e-iωtdt. Here, T = 2π/ωk is the period of the chosen wave. Thereafter, I(k, ω) = |a(k, ω)|2, which is a spatio-temporal spectrum of the wave field.

We performed three series of experiments, choosing different functions γk(2) describing damping in the area of small wave numbers.

The picture we observed is typical of wave turbulence and was described theoretically in detail by . Firstly, we get fast growth of waves corresponding to the pumping scale. Then, a direct cascade of energy results in the relatively fast formation of the spectrum tail from the pumping region to the high wave numbers until it reaches the dissipation scale kd. The wave turbulence theory predicts the KZ tail for the direct cascade 〈|ak|2〉 ∼ k-4. The Phillips spectrum corresponds to 〈|ak|2〉 ∼ k-9/2. We observed both spectra.

There is a simultaneous, although much slower, process of formation of the inverse cascade, corresponding to the flux of wave action from the pumping scale into the larger scales (smaller wave numbers). After very slow development of the inverse cascade spectrum, formation of a strong long wave background starts. This strong long wave background is often called the condensate (the name comes from some similarity to the phenomenon of Bose–Einstein condensation). Formation of the condensate is due to the fact that, at large scales (small wave numbers), the discrete grid of wave vectors becomes too coarse for resonant interaction of waves. In this range of scales, we switch from normal weak turbulence through the mesoscopic regime as described in into regimes without nonlinear resonant interaction. As a result, we have an influx of waves' action from small scales to the large scales due to the inverse cascade, which is arrested by the discreteness of the wave number grid at large scales. We get an accumulation of waves on the border of the scales where at least mesoscopic wave turbulence is still possible. These accumulated waves form the intense long wave background, which we call the condensate. It should be emphasized that a discrete grid of wave vectors appears not only in numerical simulations, but also in all wave tank experiments or other finite-size systems. The formation of the condensate in large lakes or the sea can be neglected due to the fact that downshift of the spectral maximum becomes slower the further it is shifted in large scales. As a result, the free path time of the wave is not enough to reach the scale where resolution of the grid of wave numbers becomes significant.

We believe that our experiments make possible a “Marriage of Heaven and Hell” in the spirit of William Blake. Both outstanding scientists – O. Phillips and K. Hasselmann – were right. If the local steepness μ is small (μ ≤ 0.1), the Hasselmann kinetic equation is valid without any augmentation by additional dissipation terms, at least for not too broad a spectrum, similar to what we have in the experiments. For μ noticeably higher than 0.1, in the area of very short waves, the kinetic equation is not applicable in its “pure” form. It is impossible to separate “free” and “slave” harmonics in this area. This part of the ocean spectrum cannot be described analytically by the use of perturbative methods. Nevertheless, the Phillips spectrum in this area is still applicable. The only theoretical reason for this statement is dimensional considerations supported by experimental data. At the same time, there is also a “grey area”, which is especially interesting because it generates a significant part of the steepness where whitecapping has equal footing with weakly nonlinear resonant wave interaction. As a result of the balance of nonlinear flux and dissipation, the spectrum slope can change , as has been demonstrated in our simulations.

We expect that spectra in this range could be described by an “augmented” Hasselmann equation, including an additional term describing dissipation energy due to wave breaking. Similar additional “dissipation terms” Sdiss are widely used in well-developed operational wave forecasting models. But, they are introduced “out of the blue” and are not supported either by theoretical considerations or, in many cases, by experimental observations. Moreover, it is not clear a priori that one can use the Hasselmann kinetic equation in a situation where wave-breaking events are frequent enough. We hope that our experiments showed that this is likely possible. A necessary condition of the applicability of the Hasselmann equation is narrowness of the spectral line (we should stress that, in this paper, the “dynamical” effects, like “gusty” pumping, described in a recent work by and confirmed numerically in , are not considered). In the present article, we assert that, in the “grey area” with a frequency range in one half of the decade, the frequency spectra of harmonics are still narrow lines. It means that the Hasselmann equation is applicable there in the sense that it is a good approximation for the description of nonlinear wave interaction. Of course, it must be augmented by a proper dissipative term. The existing and widely used dissipative terms can be improved. One possible and hopefully plausible variant was recently suggested by , another by . The dependence of the dissipation term on the average steepness was recently measured directly from the numerical experiment; the preliminary results can be found in . In our next paper, we shall analyze what information about possible shapes of the dissipative term can be extracted from massive numerical experiments.