Evolution of kurtosis for wind waves

Transcription

1 GEOPHYSICAL RESEARCH LETTERS, VOL. 36, L13603, doi: /2009gl038613, 2009 Evolution of kurtosis for wind waves S. Y. Annenkov 1 and V. I. Shrira 1 Received 14 April 2009; revised 21 May 2009; accepted 27 May 2009; published 3 July [1] Long-term evolution of random wind waves is studied by direct numerical simulation within the framework of the akharov equation. The emphasis is on kurtosis as a single characteristics of field departure from Gaussianity. For generic wave fields generated by a steady or changing wind, kurtosis is found to be almost entirely due to bound harmonics. This observation enables one to predict the departure of evolving wave fields from Gaussianity, capitalizing on the already existing capability of wave spectra forecasting. Kurtosis rapidly adjusts to a sharp increase of wind and slowly decreases after a drop of wind. Typically kurtosis is in the range , which implies a tangible increase of freak wave probability compared to the Rayleigh distribution. Evolution of narrow-banded fields is qualitatively different from the generic case of wind waves: statistics is essentially non-gaussian, which confirms that in this special case the standard kinetic equation paradigm is inapplicable. Citation: Annenkov, S. Y., and V. I. Shrira (2009), Evolution of kurtosis for wind waves, Geophys. Res. Lett., 36, L13603, doi: /2009gl Introduction [2] The necessity to predict the wave height probability distribution at any given place and time becomes more and more acute with the growth of society reliance on shipping and offshore activities. Theoretical description began with the natural idea of a narrow-banded linear wave field obeying stationary Gaussian statistics; these assumptions lead to the Rayleigh distribution for wave heights [Longuet-Higgins, 1952]. The account of weak nonlinearity through bound harmonics leads to a departure from a Gaussianity. Numerous further developments of varying degree of rigour incorporating a finite spectral width and bound harmonic nonlinearity produced a number of wave height distributions [Tayfun and Fedele, 2007; Fedele, 2008]. The crucial question is the behaviour of the distribution tails: the slightest difference in the distribution shape could result in huge disparities in the probability of freak waves and, correspondingly, significant scatter of the highest wave type estimates required by industry. The fundamental assumption of all such models is that the probability distributions are stationary and completely determined by the energy spectrum. [3] The first major step towards understanding the evolution of wave distributions beyond spectra was made by Janssen [2003], who extended the kinetic description 1 Department of Mathematics, EPSAM, Keele University, Keele, UK. Copyright 2009 by the American Geophysical Union /09/2009GL employed in the derivation of the Hasselmann (kinetic) equation to describe evolution of fourth moment of surface elevation. Within the framework of this approach a wave field is considered in terms of canonical variables representing nonlinear normal modes with the bound harmonics eliminated by the canonical transformation; then the field statistics is Gaussian unless interactions between the modes due to cubic nonlinearity are taken into account. These interactions lead to wave spectrum evolution and, at the same time, to a departure from Gaussianity. Kurtosis is a convenient measure of this departure and, hence, of freak wave probability. We will refer to this kurtosis as dynamic and denote it as C (d) 4. Janssen [2003] described evolution of C (d) 4 in terms of wave energy spectrum. Note that dependence (d) of C 4 on spectra is non-local in time, i.e. depends on history of spectral evolution, unless the large-time asymptotics is taken. The focus of further developments was upon narrow-banded fields at the limits (and beyond) of the theory applicability [e.g., Mori and Janssen, 2006; Onorato et al., 2006]. A version of this approach is now used for operational forecasting of freak waves [Janssen and Bidlot, 2009]. [4] The contribution to kurtosis due to bound harmonics, which we denote as C 4, can be derived from the canonical transformation. Under the assumption of quasi- Gaussianity Janssen [2008] derived a local in time expression for C 4 in terms of wave spectrum, that is C 4 (in contrast to C (d) 4 ) is determined by the current spectrum and does not depend on its history. At present, the magnitude of C (d) 4 for a naturally evolving broad-band (d) spectra is not known. The role of C 4 and C 4 is not established, so it is not known whether and when the wave height distribution can be inferred just from the spectrum. Nothing is known on how C 4 and C (d) 4 react to rapid changes of environment (e.g., a gust of wind). [5] In this Letter we address these questions by direct numerical simulation (DNS) within the framework of the akharov equation. This approach allows us, for the first time, to simulate the kurtosis evolution directly. The adopted approach enables us to consider separately dynamical and bound harmonics parts of kurtosis. We study the kurtosis evolution both with and without wind forcing, under constant and rapidly changing wind. The key finding is that only in very special circumstances (very narrowbanded spectra, initial stages of wind wave development) the dynamical part of kurtosis C (d) 4 is dominant. Then and only then the excursions of kurtosis could be of order one, which implies essential non-gaussianity of the statistics. In generic situations (developed waves) the kurtosis is almost entirely due to bound harmonics. This enables one to predict the departure from Gaussianity of evolving wind wave L of5

2 fields, capitalizing on the existing capability of wave spectra forecasting. 2. Theoretical Background [6] We consider gravity waves on the surface of deep ideal fluid governed by the akharov equation [Krasitskii, 1994] and then the dynamic contribution to kurtosis C 4 (d) is C ðdþ 4 ¼ m 4 =m 2 2 3; where m 2 ¼ w 0 n 0 dk 0 : The bound harmonics contribution C 4 is obtained using the canonical transformation equation (2) and ensemble averaging, ð5þ ¼ ðw 0 þ ig 0 Þb 0 þ T 0123 b 1 *b 2 b 3 d 0þ1 2 3 dk 123 : ð1þ C ðþ b 4 ¼ 12 m 2 2 F 012 w 0 w 1 w 2 n 0 n 1 n 2 dk 012 : ð6þ Here, b(k) is a canonical complex variable in Fourier space, k is the wavevector, k = jkj, w(k) =(gk) 1/2 is the linear dispersion relation, ig(k) is the small imaginary correction to frequency due to forcing or dissipation, gravity g is normalized to unity, and integration in equation (1) is performed over the entire k-plane. The compact notation used designates the arguments by indices, e.g., T 0123 = T(k, k 1, k 2, k 3 ), d = d(k + k 1 k 2 k 3 ), asterisk means complex conjugation, and t is time. The canonical variable b(k) is linked to the Fourier-transformed primitive physical variables z(k, t) and 8(k, t) (position of the free surface and the velocity potential at the surface respectively) through an integral-power series [Krasitskii, 1994] ( bðkþ ¼ 1 rffiffiffiffiffiffiffiffiffiffi sffiffiffiffiffiffiffiffiffiffi ) wðkþ k pffiffi zðkþþ i 8ðkÞ þ OðÞ: e 2 k wðkþ Derivation of equation (1) assumes that wave slopes are O(e) small, and includes expansion in powers of e [Krasitskii, 1994]. [7] We are interested in the evolution of statistical characteristics of wave field, usually studied in terms of correlators of b(k, t). The classical derivation [e.g., akharov et al., 1992] uses equation (1) as the starting point and leads to the kinetic equation for the second statistical moment ¼ 4p T f 0123d 0þ1 2 3 Dw dk 123 þ S f ; where n 0 is the second-order correlator, hb* 0 b 1 i = n 0 d 0 1, angular brackets mean ensemble averaging, f 0123 = n 2 n 3 (n 0 + n 1 ) n 0 n 1 (n 2 + n 3 ), Dw = d(w 0 + w 1 w 2 w 3 ), and S f is the forcing/dissipation term. The derivation of equation (3) assumes a random wave field proximity to Gaussianity and stationarity, which makes equation (3) inapplicable for wave fields far from equilibrium, e.g., subjected to a rapid change of wind [Annenkov and Shrira, 2009]. Here, we study evolution of wave fields by solving equation (1) by DNS without such assumptions. [8] The fourth moment m 4 in terms of b(k) is calculated directly by integrating equation (1) and averaging over realizations m 4 ¼ 3 ðw 0 w 1 w 2 w 3 Þ 1=2 hb 0 *b 1 *b 2 b 3 idk 0123 þ c:c:; ð4þ 4 ð2þ ð3þ The lengthy coefficient F 012 is derived by Janssen [2008]. This derivation assumes quasi-gaussianity of a wave field in b(k). Although this is not always the case for the simulations considered below, we use equation (6) as an estimate for C Numerical Algorithm [9] We perform DNS of equation (1) using an efficient algorithm tested to give good agreement with the kinetic equation where it is applicable [Annenkov and Shrira, 2006a, 2006b], and used for the study of the wave field fast evolution, when the standard statistical theory cannot be applied [Annenkov and Shrira, 2009]. We build in Fourier space a grid consisting of wave packets, coupled through exact and approximate resonant interactions. The grid is logarithmic in the wavenumber k (161 points within a span 0.13 < k <2.12m 1 ) and regular in the angle q (31 point within p/3 q p/3. A quartet of grid points is assumed to be in approximate resonance if its wavenumber and frequency mismatch satisfies a pair of conditions Dw/w min < l w, Dk/k min < l k w/w min, where Dw and Dk are the frequency and wavenumber mismatch in the quartet, w min and k min are the minimum values of frequency and wavenumber in the quartet, w is the mean frequency, and l w and l k are detuning parameters, chosen to ensure that the total number of resonances is O(N 2 ), where N is the number of grid points. In this study, N = 4991, and l w = l k = 0.01; results were verified to be non-dependent on specific values in a wide range of l w, l k. For simulations of wind-generated waves the specific choice of forcing proved to produce no qualitative difference. The reported results were obtained with dissipation applied to k >1.62m 1,and forcing prescribed by an empirical formula [Hsiao and Shemdin, 1983] and confined to the range 1.0 < k < 1.29 m 1. Initial phases of waves are chosen randomly, averaging is over 30 realizations. For simulations of initially narrow-banded spectra, only a part of the grid is used: p/6 q p/6; N = 2737, there is no forcing, dissipation is applied to k >1.57m 1, averaging is over 600 realizations. 4. Results 4.1. Evolution of Narrow-Banded Initial Spectra [10] First, to verify the code and relate to earlier numerical simulations and laboratory experiments, we study the evolution without wind forcing of initially narrow-banded spectrum (Figure 1). The initial energy spectrum is onedimensional, nonzero and constant in the vicinity of the 2of5

3 Figure 1. Initially one-dimensional narrow spectrum without forcing, with characteristic steepness 0.2 and BFI = 1.4.(a) Evolution of the spectrum, in steps of 64 characteristic periods. Dissipation, shown by an arrow, is applied to the highfrequency part of the spectrum (w > 1.25). Evolution of kurtosis in time, measured in characteristic wave periods. Dynamic part C (d) 4, bound harmonic part C (d) 4 and their sum C tot 4 are shown. central frequency w 0 = 1, with relative spectral width d w = Dw/w 0 = 0.2. The characteristic steepness e p is taken to be 0.2, so that the Benjamin-Feir index BFI = e ffiffiffi 2 /dw 1.4. [11] Note that the evolution of the statistical characteristics of such a wave field cannot be addressed by the kinetic equation or its generalizations because of the role played by coherent patterns quasisolitons. Eventually, due to slow angular spreading, the wave field ceases to be onedimensional, and at large times the Kolmogorov-akharov spectra are formed. [12] In the one-dimensional model, the evolution of a narrow spectrum and of the related kurtosis was extensively studied by Janssen [2003]. Here, we consider the evolution of a one-dimensional spectrum within a two-dimensional DNS. Evolution of the energy spectrum E(w) is shown in Figure 1a, evolution of the corresponding kurtosis (dynamical C (d) 4 and bound harmonics C 4 parts) - in Figure 1b. During the first few hundred characteristic periods, the spectrum remains nearly one-dimensional and quite irregular, while C (d) 4 is positive and large, reaching a maximum of 0.65 and then decreasing. At this stage the results agree with the available one-dimensional simulations [Janssen, 2003; Shemer and Sergeeva, 2009]. Then, the spectrum approaches a quasi-equilibrium state, with subsequent slow angular spreading. [13] In the simulations of initially two-dimensional narrow-banded spectra order one excursions of kurtosis were again found, but in contrast to the one-dimensional initial conditions, these excursions are negative for fully two-dimensional fields. These large excursions of kurtosis demonstrate the key features of evolution of narrow-banded fields. First, the evolution is non-gaussian, as manifested by the values of kurtosis. Second, the kurtosis evolution is dominated by the dynamical part C (d) 4, which is not totally determined by the spectrum, but depends on the evolution history. The narrow-banded fields got the lion s share of attention in the literature; their study is beyond the scope of the present work. Here we just stress their very atypical character Evolution of Wind Generated Waves [14] In a series of numerical experiments, we consider the evolution of broad banded wave fields generated by realistic winds. Forcing is applied to the high-frequency part of the spectrum only, which means that for dominant waves wind input is exactly balanced by dissipation. Then, the most energetic part of wave field evolves entirely due to nonlinear interactions, which is typical of developed seas. [15] We start with low-intensity white noise, and then run the model for several thousand characteristic wave periods, to allow the spectrum to develop under the constant wind, so that on the slope the wave spectrum is nearly stationary and in local equilibrium with forcing. Then, the forcing is instantaneously increased or decreased, and the wave field evolves towards a new quasi-equilibrium state. [16] Evolution of spectrum in case of an instant increase of wind is shown in Figure 2a, the corresponding evolution of kurtosis - in Figure 2b. Under steady wind, the evolution of the developed spectrum is slow and is described by largetime asymptotics of the kinetic equation. The dynamical part of kurtosis C (d) 4 is very small (10 2 ), negative and nearly constant, the bound harmonics part C 4 is much larger and positive, slowly growing with the powerlike increase of the total energy. The sharp increase in wind at t = T 0 abruptly drives the wave system out of the local equilibrium with forcing. Then the evolution of a wave field occurs on O(e 2 ) timescale, rather than the O(e 4 ) timescale predicted by the kinetic equation [Annenkov and Shrira, 2009]. The wave energy and, in particular, characteristic wave steepness rise on a fast timescale, the downshift of the spectral peak stops for a few hundred wave periods, and then resumes at an increased rate, gradually slowing down back to the kinetic equation asymptotics. On this background C (d) 4 abruptly decreases, reaching a new negative equilibrium value. At the same time, the dominant contribution to the kurtosis C 4 rises sharply over a few hundred wave periods, then returning to the slow growth at the new equilibrium with the increased forcing. 3of5

4 Figure 2. Waves generated by constant wind with an instant increase of wind speed at t = T 0 = 6500 periods from 10 to 16 m/s. (a) Evolution of the spectrum, in steps of 400 characteristic periods. Equilibrium spectra before and after the change of forcing (N 0 (k) and N f (k) respectively) are shown by thick curves with shading. Evolution of kurtosis in time. [17] Figures 3a and 3b show a sample of field evolution for the case of a sharp drop of wind. Qualitatively, the process of adjustment to a new forcing is similar: after a drop of wind, both the spectrum and kurtosis tend to their equilibrium values with the new forcing. However, the timescales are much slower, about a few thousand characteristic periods. 5. Concluding Remarks [18] Here we summarize and discuss the main results of our study. The use of the DNS within the framework of the akharov equation enabled us, for the first time, to simulate the kurtosis evolution directly, without the assumptions used in the statistical theory and to examine separately the dynamic and bound harmonic contributions to kurtosis. [19] Our study of a model problem of the evolution of a narrow-banded spectrum without forcing demonstrated that initially one-dimensional wave fields with BFI > 1 exhibit strong departures from Gaussianity. This result agrees with earlier works, which reported, in numerical and tank observations, a strong increase in the probability of large amplitude events (freak waves) due to nonlinearity [Janssen, 2003; Socquet-Juglard et al., 2005; Onorato et al., 2006; Shemer and Sergeeva, 2009]. However, such a behaviour of the kurtosis is confined to these special initial conditions. Our results suggest that one-dimensional models, as well as laboratory observations in long narrow tanks, could demonstrate behaviour that is qualitatively different from that of generic two-dimensional situations. Note that this special behaviour of narrow-banded spectra, characterized by significant (positive or negative) excursions of the kurtosis, renders the kinetic description inapplicable, since the required proximity to Gaussianity does not hold. [20] For generic broad-banded wave fields generated by a steady or rapidly changing wind, our simulations show that kurtosis is always positive and almost entirely due to bound harmonics. Thus, at any instant the kurtosis is determined by the spectrum at the same moment. This observation enables one to predict departure from Gaussianity of evolving wave fields utilizing the already existing capability of wave spectra forecasting. That is, if n(k, t) is known, either from the kinetic equation (3) or its generalization [Annenkov and Shrira, 2006b] for wave fields not in equilibrium with forcing, the kurtosis C 4 (t) can be found from equation (6), while the dynamical contribution C 4 (d) (t) can be neglected. [21] At the philosophical level, the idea that the bound harmonics contribute and even determine the departure from Gaussianity is not new [e.g., Tayfun, 1980]. Recently it has gained extra support from analysis of a representative data Figure 3. Same as Figure 2, but with an instant decrease of wind speed at t = T 0 = 4000 periods from 16 to 10 m/s. 4of5

5 set of 9h storm (characterized by the broadband spectra) from the North Sea Tern platform [Fedele, 2008; Fedele and Tayfun, 2009]. In this Letter, we have shown that in generic case the bound harmonic contribution dominates, and found its magnitude by DNS. We have also revealed the picture of the kurtosis adjustment to rapid changes of wind: an instant increase of wind results in a quite steep increase of kurtosis, while a sharp drop of wind leads to a more gradual (10 3 periods) transition towards the equilibrium with forcing. An analysis of applications for specific scenarios of spectral evolution goes beyond the scope of this work and will be considered elsewhere. [22] Finally, we note that our observation that in generic situations kurtosis is always positive implies an increase of probability of large amplitude events compared to the Rayleigh distribution. The obtained values of kurtosis can be easily translated into the wave envelope probability density function [Janssen and Bidlot, 2009]. For example, in a wave field with kurtosis C 4 > 0, the probability of wave exceeding twice the significant wave height increases by the factor (1 + 24C 4 ), compared to the Rayleigh distribution. [23] Acknowledgments. This work was supported by INTAS grant We are grateful to P. Janssen and L. Shemer for helpful discussions. References Annenkov, S. Y., and V. I. Shrira (2006a), Direct numerical simulation of downshift and inverse cascade for water wave turbulence, Phys. Rev. Lett., 96, , doi: /physrevlett Annenkov, S. Y., and V. I. Shrira (2006b), Role of non-resonant interactions in the evolution of nonlinear random water wave fields, J. Fluid Mech., 561, Annenkov, S. Y., and V. I. Shrira (2009), Fast nonlinear evolution in wave turbulence, Phys. Rev. Lett., 102, , doi: /physrevlett Fedele, F. (2008), Rogue waves in oceanic turbulence, Physica D, 237, Fedele, F., and M. A. Tayfun (2009), On nonlinear wave groups and crest statistics, J. Fluid Mech., 620, Hsiao, S. V., and O. H. Shemdin (1983), Measurements of wind velocity and pressure with a wave follower during MARSEN, J. Geophys. Res., 88, Janssen, P. (2003), Nonlinear four-wave interactions and freak waves, J. Phys. Oceanogr., 33, Janssen, P. (2008), On some consequences of the canonical transformation in the Hamiltonian theory of water waves, ECMWF Tech. Memo. 579, Eur. Cent. for Medium-Range Weather Forecasts, Reading, UK. Janssen, P., and J. Bidlot (2009), On an extension of the freak wave warning system and its verification, ECMWF Tech. Memo. 588, Eur. Cent. for Medium-Range Weather Forecasts, Reading, UK. Krasitskii, V. P. (1994), On reduced Hamiltonian equations in the nonlinear theory of water surface waves, J. Fluid Mech., 272, Longuet-Higgins, M. S. (1952), The statistical distribution of the heights of sea waves, J. Mar. Res., 11, Mori, N., and P. Janssen (2006), On kurtosis and occurrence probability of freak waves, J. Phys. Oceanogr., 36, Onorato, M., A. R. Osborne, M. Serio, L. Cavaleri, C. Brandini, and C. T. Stansberg (2006), Extreme waves, modulational instability and second order theory: Wave flume experiments on irregular waves, Eur. J. Mech. B, 25, Shemer, L., and A. Sergeeva (2009), An experimental study of spatial evolution of statistical parameters in a unidirectional narrow-banded random wavefield, J. Geophys. Res., 114, C01015, doi: / 2008JC Socquet-Juglard, H., K. Dysthe, K. Trulsen, H. E. Krogstad, and J. Liu (2005), Probability distributions of surface gravity waves during spectral changes, J. Fluid Mech., 542, Tayfun, M. A. (1980), Narrow-band nonlinear sea waves, J. Geophys. Res., 85, Tayfun, M. A., and F. Fedele (2007), Wave-height distributions and nonlinear effects, Ocean Eng., 34, akharov, V. E., V. S. L vov, and G. Falkovich (1992), Kolmogorov Spectra of Turbulence I: Wave Turbulence, Springer, Berlin. S. Y. Annenkov and V. I. Shrira, Department of Mathematics, EPSAM, Keele University, Keele ST5 5BG, UK. (s.annenkov@maths.keele.ac.uk; v.i.shrira@maths.keele.ac.uk) 5of5