## 1. Introduction

^{−1}in less than 5 min and lasted about 20 min, while the significant wave height increased sixfold. To model such events there have been no adequate tools. At present all wave modeling is based on the numerical integration of the kinetic (Hasselmann) equation for the energy spectra of wave fields. The equation takes into account wind input, dissipation, and interaction between waves of different scales and directions and describes the slow evolution of wind-wave spectra in time and space (Komen et al. 1994; Janssen 2004). In terms of wave action density

*t*are “slow” spatial and temporal variables, it can be written as

*O*(

*ε*

^{−4}) time scale, which allows wind forcing to vary only on the same or larger time scale. Therefore, strictly speaking, the Hasselmann equation is not applicable to situations with rapid changes of wind. Because of the lack of alternatives, this difficulty is usually ignored, and the Hasselmann theory (Hasselmann 1962) is currently being used beyond the domain of its applicability. In the detailed study by Young and van Agthoven (1998), the Hasselmann equation was used to model the response to a sharp increase or decrease of wind. It is not clear to what extent these results can be trusted since such an increase clearly violates the equation conditions of validity. It is worth noting that both sea (van Vledder and Holthuijsen 1993) and wave tank observations (Autard 1995; Waseda et al. 2001) and direct numerical simulations (Annenkov and Shrira 2009) show a faster response of a wave field to an instant perturbation, which suggests a discrepancy with the strict

*O*(

*ε*

^{−4}) evolution time scale embedded into the Hasselmann equation. One of the motivations for this work is to quantify applicability of the Hasselmann equation to changing winds. We specifically address the spectral response of a wind-wave field to a squall, defined as a sharp increase of wind speed, which lasts for a short time period before the wind speed returns to near its previous value. Our aim is to perform the modeling of wave field under squall with the generalized kinetic equation (GKE), which was derived without the quasi-stationarity assumption (Annenkov and Shrira 2006). The GKE differs from the Hasselmann equation by the form of

*O*(10

^{2}) characteristic wave periods. We find that during the squall, the spectral evolution obtained with the GKE is close to that obtained with the Hasselmann equation. However, after the end of the squall, the GKE shows that a transient spectrum is formed, which has a considerably narrower peak than the Hasselmann equation predicts. This transient spectrum predicted by the GKE exists for a few hundred characteristic wave periods after the squall.

## 2. Theoretical background

*t*is time. Statistical description of the wave field is sought in terms of correlators of complex amplitude

*g*is gravity,

*δ*is the Dirac delta function,

**k**plane. The compact notation used designates the arguments by indices, for example,

## 3. Numerical algorithm

Although the GKE [Eq. (2)] is nonlocal in time, it can be solved iteratively by specifying the current value of ^{9}. Time stepping is performed by the Runge–Kutta–Fehlberg algorithm with absolute tolerance 10^{−10} and the time step limited from above by the approximately ⅓ characteristic wave period. A typical computation takes 1–3 days on 64 computational cores. Initial conditions were specified as the Donelan et al. (1985) spectra for *c* is the phase speed of the spectral peak, and *U*_{10} is wind at 10 m; wind forcing is according to Hsiao and Shemdin (1983). The specific choice of the wind generation model is of little importance here, since our focus is on the nonlinear interactions. The spectral evolution was traced with the same initial value of

Simulations of the Hasselmann Eq. (1) were performed using the Gurbo Quad 5 set of subroutines based on the Webb–Resio–Tracy (WRT) algorithm (van Vledder 2006), kindly provided by Gerbrant van Vledder. The same computational grid was used with the same maximum time step and absolute tolerance (10^{−6}) and the same initial conditions. To perform a more detailed and accurate comparison of the simulations with the GKE and Hasselmann equation, in a number of runs the spectra obtained with the GKE were used as initial conditions for the Hasselmann equation.

## 4. Results

First, the new algorithm for the numerical integration of the GKE was thoroughly validated on a number of model situations without forcing or dissipation, with the analysis of the invariants conservation properties and the large time asymptotics of the spectral evolution. The results of these preliminary simulations are not reported in this paper. Here, for the algorithm validation, we consider the development of the wave spectrum under the action of constant wind and compare the results with the simulations of the Hasselmann equation for the same forcing and dissipation and the same initial conditions. In Fig. 1a, results of the simulations for the GKE and Hasselmann equation are shown for a rather high wind (initially

Our primary interest is in the effects of the squall, here understood as an instant increase of wind (start of squall) followed by its sharp decrease back to the initial value after *O*(10^{2}) characteristic wave periods (end of squall). Since the GKE is nonlocal in time, its numerical modeling is performed as one continuous computation under changing wind conditions. Numerical simulation of the Hasselmann equation can be performed in the same way, or, to facilitate comparison between the two equations, the spectra obtained with the GKE at the start and the end of the squall can be used as initial conditions for the Hasselmann equations. In this study, we use both approaches.

In this way, we consider the development of the wind-wave field under moderate constant wind forcing (

Comparison of the GKE (blue curves) and Hasselmann equation (red curves) solutions during the squall. Wind forcing is instantly increased from

Citation: Journal of Physical Oceanography 45, 3; 10.1175/JPO-D-14-0182.1

Comparison of the GKE (blue curves) and Hasselmann equation (red curves) solutions during the squall. Wind forcing is instantly increased from

Citation: Journal of Physical Oceanography 45, 3; 10.1175/JPO-D-14-0182.1

Comparison of the GKE (blue curves) and Hasselmann equation (red curves) solutions during the squall. Wind forcing is instantly increased from

Citation: Journal of Physical Oceanography 45, 3; 10.1175/JPO-D-14-0182.1

After approximately 100 characteristic periods, the wind drops to its value before the squall. The subsequent evolution obtained with both equations is plotted in Fig. 3. Both equations demonstrate that the spectrum changes its form, developing a wide and flat “gap” on the slope at slightly higher frequencies than the frequency of the spectral peak. However, the GKE spectrum develops a considerably narrower peak. Here, the width of the peak is understood as the width of the spectrum at half the peak amplitude. As shown in Fig. 4a, the squall causes the peak of the spectrum to become narrower, this effect being captured by both equations. However, immediately after the end of the squall the peak of the spectrum obtained with the GKE continues to narrow down, while within the Hasselmann equation the peak widens considerably. This discrepancy between the two equations persists for about a hundred characteristic wave periods.

Evolution of energy spectrum

Citation: Journal of Physical Oceanography 45, 3; 10.1175/JPO-D-14-0182.1

Evolution of energy spectrum

Citation: Journal of Physical Oceanography 45, 3; 10.1175/JPO-D-14-0182.1

Evolution of energy spectrum

Citation: Journal of Physical Oceanography 45, 3; 10.1175/JPO-D-14-0182.1

(a) Evolution of spectral width during and after the squall, with the GKE (blue) and the Hasselmann equation (red). Initial condition in both cases corresponds to the GKE solution at the start of the squall. (b) Evolution of peakedness parameter *γ* with the GKE (blue) and the Hasselmann equation (red).

Citation: Journal of Physical Oceanography 45, 3; 10.1175/JPO-D-14-0182.1

(a) Evolution of spectral width during and after the squall, with the GKE (blue) and the Hasselmann equation (red). Initial condition in both cases corresponds to the GKE solution at the start of the squall. (b) Evolution of peakedness parameter *γ* with the GKE (blue) and the Hasselmann equation (red).

Citation: Journal of Physical Oceanography 45, 3; 10.1175/JPO-D-14-0182.1

(a) Evolution of spectral width during and after the squall, with the GKE (blue) and the Hasselmann equation (red). Initial condition in both cases corresponds to the GKE solution at the start of the squall. (b) Evolution of peakedness parameter *γ* with the GKE (blue) and the Hasselmann equation (red).

Citation: Journal of Physical Oceanography 45, 3; 10.1175/JPO-D-14-0182.1

To compare the form of the spectrum obtained with different equations, it is instructive to perform a fit of the obtained spectrum to the JONSWAP spectral parameters *α* and *γ*. The dependence of *γ* on time during the evolution of the spectra obtained with both equations is shown in Fig. 4b. The squall causes the increase of *γ* in both cases, and after the end of the squall, *γ* continues to increase for a few dozen wave periods. However, this increase is much sharper for the spectrum obtained with the GKE. The angular spectra, prescribed at the initial moment according to Donelan et al. (1985) by the

## 5. Concluding remarks

In this paper, we report the key first step toward the modeling of short-lived transient sea states. It has been demonstrated that by employing the GKE, which allows the fast dynamic of wave fields and takes into account nonresonant interactions, it is possible to simulate their evolution quite effectively. The simulations aimed at modeling wave field dynamics during an idealized squall event, apart from being the first simulations of squall, brought in unexpected results. First, both the field evolution under a moderate wind and the rapid growth of the wave spectrum during the squall as modeled with the GKE proved to be nearly identical to the predictions made with the Hasselmann equation. While such an agreement was expected for steady wind situations, the negligible discrepancy found during the rapid growth contradicts earlier direct numerical simulations (Annenkov and Shrira 2009). A direct comparison for the same initial conditions and spectral resolution is not possible at present, and hence there is a fundamental open question that requires a dedicated study. At the rapid growth stage we also observed a peculiar shape of the spectra with a dip on the spectral slope not reported in the literature. Second, counterintuitively, the predictions of the GKE and the Hasselmann equation diverge qualitatively just *after* the squall, that is, when the wind drops to its presquall level; the GKE predicts noticeably more narrow spectra characterized by a substantially higher peakedness parameter *γ*. At present there are no observations allowing a detailed quantitative test of the simulations. In the future the most perspective way to test the simulations against reality is to examine specially designed tank observations in the spirit of Autard (1995) and Waseda et al. (2001); however, a quantitative comparison is not straightforward and a sustained effort from two sides is required to bridge the gap between the simulations and observations.

The results suggest, and this is one of the main unexpected implications of the work, that the Hasselmann equation has a much wider range of applicability than follows from the assumptions adopted for its derivation. This adds solidity to the already existing simulations with the Hasselmann equation carried out outside the range of its formal validity, for example, in Young and van Agthoven (1998). Probably the most far-reaching implications stem from the demonstrated efficiency of the proposed highly parallel algorithm for solving the GKE. We have got the tool not only for studying a variety of short-lived transient processes; in the long term, it has a potential to replace the Hasselmann equation as the basis of wave modeling. Since the GKE algorithm is highly parallel, and has a lot of room for efficiency improvement, with a sufficient number of parallel processors it could be even made much faster than the existing codes for the Hasselmann equation.

## Acknowledgments

We are grateful to G. van Vledder for providing his code for the Hasselmann equation. The work was made possible by U.K. NERC Grant NE/I01229X/1; it was also supported by NERC Grant NE/M016269/1 and EU FP7 612610. The access to the ECMWF supercomputing facility under special project SPGBVSSA is gratefully acknowledged.

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