Wave forecasting and extreme events

Transcription

1 Wave forecasting and extreme events Peter Janssen European Centre for Medium-Range Weather Forecasts 1.

2 INTRODUCTION Interest in wave prediction started during the Second World War because of the practical need for knowledge of the sea state during landing operations. The first operational predictions were based on the work of Sverdrup and Munk (1947), who introduced a parametrical description of the sea state, consisting of wind sea and swell. Manual techniques based on this approach have been used by operational forecasters for many years (Groen and Dorrestein, 1976). 2.

3 z U 0 (z) a Air(ρ a ) x g λ Water(ρ w ) Steepness s = ka =2πa/λ Figure 1: Schematic of the problem in two dimensions. Dispersion relation: ω = gk, phase speed c = ω/k = g/ω. Here ω = 2π f is the angular frequency of the wave. 3.

4 The work of Sverdrup and Munk introduced the appropriate framework to analyze the growth of waves by wind. Dimensionless parameters were introduced using acceleration of gravity g and wind speed U 10. Specifically, with fetch X, significant wave height H S and peak phase speed c p = g/ω p, dimensionless parameters such as gh S, U 2 10 gx U 2 10, and c p U 10 = g U 10 ω p (the wave age!) were introduced which gave a considerable simplification in the analysis of observations of ocean wave growth. 4.

5 ε Bothnian Sea Lake Ontario Lake Ontario, orthog ν X10 X 10 Figure 2: Dimensionless energy ε 10 = g 2 E/U 4 10 (with E = H2 S /16) and peak frequency ν 10 = U 10 /c p as function of dimensionless fetch X 10 = gx/u

6 Programme of the talk WHAT HAPPENED NEXT? Sketch the further developments: Wave Spectrum, Energy balance equation, Theories of wind-wave generation and nonlinear interactions. JONSWAP WAVE MODELLING AND SATELLITE DATA The 1980s gave the first supercomputers and the promise of the wealth of data on the ocean surface from remote-sensing instruments on board of new satellites. Wave forecasting based on the energy balance equation became feasible. WAVE PHYSICS: THE SOURCE FUNCTIONS Windinput is based on critical layer theory, while Hasselmann s four-wave interactions represent the nonlinear transfer. Dissipation by white capping is based on scaling arguments. 6.

7 NEW APPLICATIONS A number of new applications have emerged. Air-sea momentum and heat transfer, sea surface albedo, radar backscatter from an Altimeter, Freak waves. Wave dissipation determines the mixing of momentum and heat in the upper ocean. THE FUTURE: ONE MODEL FOR THE GEOSPHERE Ocean waves play a role in air-sea momentum transfer and in ocean mixing. Also, momentum transfer and the sea state is affected by surface currents, hence it makes sense to introduce a three-way coupling between atmosphere, ocean circulation and surface waves. The end result is one model for the geosphere. At ECMWF, a rudimentary version of such a model is used in our seasonal forecasting system. 7.

8 The following is a list of landmarks: A BRIEF HISTORY the wave spectrum (Pierson et al, 1955) the energy balance equation (Gelci et al., 1957). wind-wave generation theories by Phillips (1957) and Miles (1957) and nonlinear transfer by (Hasselmann, 1962). Understanding of wave evolution was greatly improved because of the in-situ observations from the JOint North Sea Wave Project ( JONSWAP, 1973) 8.

9 m 2 /Hz Hz Figure 3: Evolution of wave spectra with fetch for offshore winds (11-12 h, Sept. 15, 1968). The spectra are labelled with the fetch in kilometres. (From Hasselmann et al, 1973.) 9.

10 WAVE MODELLING, BUOY AND SATELLITE DATA In the 1980s the first supercomputers and the promise of the wealth of data from new satellites such as ERS-1 and Topex-Poseidon provided a significant stimulus to the development of modern ocean wave prediction systems. For the first time the energy balance equation, including the nonlinear interactions, could be used in the practice of wave forecasting and results on the sea state could be verified on a global scale. Combined with significant progress in weather forecasts, resulting in high-quality surface wind fields, this led to unprecedented improvements in the quality of the sea state forecast. 10.

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12 ENERGY BALANCE EQUATION From the previous picture we see how complicated the sea state is. Therefore, try to predict average sea state in a grid box. Fundamental quantity is the action density spectrum N(k,t), because it is an adiabatic invariant. Given the action N the energy spectrum F(k,t) is given by F(k,t) = ω(k) N(k,t) From first principles one finds the following evolution equation t N + x (ẋn)+ k ( kn) = S = S in + S nl + S ds, where ẋ = ω/ k, k = ω/ x, and the source functions S represent the physics of wind-wave generation, dissipation by wave breaking and nonlinear four-wave interactions. 12.

13 T = 2h T = 4h T = 8h T = 12h T = 24h F(f) (m 2 /Hz) f (Hz) Figure 4: Evolution in time of the one-dimensional frequency spectrum for a wind speed of 18 m/s. 13.

14 Sin Snl Sds S(f) frequency (Hz) Figure 5: The energy balance of wind-generated ocean waves for a duration of 3 hrs, and a wind speed of 18 m/s. 14.

15 70 N 50 N 30 N 10 N S 20 S 20 S 30 S 40 S 40 S 50 S 60 S 60 S 70 S Tuesday 14 March UTC ECMWF Forecast t+36 VT: Wednesday 15 March UTC Surface: significant wave height 20 E 60 N 60 N 40 N 40 N 20 N 20 N 20 E 40 E 40 E 60 E 60 E 80 E 80 E 100 E 100 E 120 E 120 E 140 E 140 E 160 E 160 E W 160 W 140 W 140 W 120 W 120 W 100 W 100 W 80 W 80 W 60 W 60 W 40 W 40 W 20 W 20 W 70 N 50 N 30 N 10 N 10 S 30 S 50 S 70 S

16 SATELLITE DATA In 1991 the European Space Agency (ESA) launched a satellite which was completely devoted to observing properties of the ocean surface. The main instruments on board were an Altimeter measuring wave height, wind speed and the mean sea level along the satellite track a Scatterometer measuring wind speed and direction in a wide swath. a Synthetic Aperture Radar (SAR) measuring the spectrum of the long waves. The availability of wind and wave observations on a global scale has provided a significant stimulus to the development of a realistic wave prediction system. Following the succes of ERS-1 and Topex-Poseidon, numerous Satellites have been launched: ERS-2, Jason 1 and 2, Envisat, GFO, Cryosat

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18 ENVISAT WAVEHEIGHTS (M). Surface gravity waves Entries STATISTICS ENTRIES MEAN WAM MEAN ENVISAT BIAS (ENVISAT - WAM) STANDARD DEVIATION SCATTER INDEX CORRELATION SYMMETRIC SLOPE REGR. COEFFICIENT REGR. CONSTANT ( ) ( ) ( ) WAM WAVEHEIGHTS (M) Figure 22.Comparison between ENVISAT Altimeter Ku-Band and WAM (first guess) significant wave heights for December 2011 (Global) 18.

19 WAVE PHYSICS In the context of wave forecasting the physics of waves is all contained in the source functions S in, S nl and S ds. Although in the last decade a great effort has been spent in understanding the physics of wave breaking/white capping, the dissipation source function S ds may still be regarded as the Cindarella of the source functions. Therefore, this source function is based on a combination of scaling arguments and some rudimentary knowledge of white capping. The wind input source function and the nonlinear transfer are founded on a more solid basis. 19.

20 DISSIPATION The dissipation source function is quasi-linear, i.e. it is the product of a damping factor which is a functional of the entire spectrum times the spectrum at the wavenumber of interest: S ds = γ d N, with γ d = β ω ( [ k 2 ) m k m 0 k + a ( ) k ], k where β = 1.6, m = 2 and a = 0.5 are tunable constants. The other parameters are related to moments of the spectrum, such as the zero moment m 0 and the mean frequency ω : m 0 = dk F(k), ω = dk ωf(k)/m 0 while < k >=< ω > 2 /g. In accordance with one s intuition the dissipation increases with increasing steepness parameter k 2 m

21 W(%)^(1/3) U10N (m/s) 21.

22 WAVE GROWTH BY WIND and AIR-SEA INTERACTION The main mechanism for wave growth was suggested by Miles who considered the simplified problem where effects of turbulence on the gravity wave motion were ignored. Critical layer mechanism : resonant interaction of airflow at critical height z c U 0 (z = z c ) = c = ω k with surface wave with wavenumber k and angular frequency ω. In fact, it turns out that the growth rate of the waves by wind is proportional to the curvature in the wind profile at the critical height. The wind input source function becomes of the form S in = γn where γ is the growth rate of the energy of the waves which is given by γ = εωβ(u /c) 2 cos 2 (θ φ), θ φ < π/2. Here, ε the air-water density ratio, which is a small number typically of the order of 10 3, and β is the so-called Miles parameter. The strength of the wind is expressed 22.

23 in terms of the friction velocity u = τ 1/2 where the kinematic stress τ gives the total loss of momentum from air to water. Finally, θ is the direction in which the waves propagate and φ is the wind direction. The Miles parameter follows from the solution of the Rayleigh equation. An approximate solution can also be found. It is of the form β = β m κ 2 µ log4 (µ), µ < 1, where µ = kz c is the dimensionless critical height. For a logarithmic wind profile, U 0 (z) = u κ log(z/z 0), with z 0 the roughness length, µ becomes µ = ( u ) 2 Ωm exp(κ/x), κc with Ω m = gκ 2 z 0 /u

24 . Surface gravity waves β c/u* γ/f u * /c

25 However, a definite answer for the growth of the waves requires knowledge of z 0! The roughness as felt by the air flow is to a large extent determined by the waves themselves. The growing waves slow down the air flow. This gives a strong coupling between the wind and the waves. Air-sea interaction is governed by momentum conservation. In the steady state: τ = τ w (z)+τ turb (z), with τ w (z) the wave induced stress profile with surface value τ w = P t = dωdθ ks in. wind This then results in a dimensionless roughness length, or Charnock parameter, as given by 25.

26 z 0 = gz 0 u 2 = α 1 τ w τ, α 0.01 and depends on the ratio of wave-induced stress τ w to total stress τ. Using the Charnock relation z 0 = z 0 u2 /g, the neutral drag coefficient is given by C D (10) = u 2 U N (10) 2 = ( ) κ 2 log(10/z 0 ) As the coupled system results in a sea state dependent Charnock parameter, the drag over the ocean is sea state dependent as well. This is illustrated below where observed Charnock parameter is plotted against the inverse of the wave age parameter c p /u. The wave age parameter measures the stage of development of windsea. 26.

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28 The graph of Charnock parameter versus inverse wave age shows two regimes: for extreme young windseas roughness increases with wave age (occurs in Hurricane conditions), while for larger wave ages but still young windseas the roughness decreases with wave age. The first regime hardly ever occurs, so let us give some results for the normal regime of young windseas. Check on statistical properties of the ECMWF coupled system: compare average drag as function of windspeed with most recent observations. Impact on forecast skill. 28.

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30 100 % FORECAST VERIFICATION HEIGHT OF WAVES SURFACE LEVEL ANOMALY CORRELATION FORECAST AREA=N.HEM TIME=00 MEAN OVER 82 CASES DATE1= /... DATE2= /... two-way one-way 100 % FORECAST VERIFICATION 500 hpa GEOPOTENTIAL ANOMALY CORRELATION FORECAST AREA=N.HEM TIME=00 MEAN OVER 82 CASES DATE1= /... DATE2= /... two-way one-way MAGICS 6.12 aurora - dax Tue May 4 13:23: Verify SCOCOM Forecast Day MAGICS 6.12 aurora - dax Tue May 4 13:25: Verify SCOCOM Forecast Day FORECAST VERIFICATION HEIGHT OF WAVES SURFACE LEVEL STANDARD DEVIATION OF ERROR FORECAST AREA=N.HEM TIME=00 MEAN OVER 82 CASES DATE1= /... DATE2= / MAGICS 6.12 aurora - dax Tue May 4 13:23: Verify SCOCOM * 1 ERROR(S) FOUND * Forecast Day two-way one-way 100 M FORECAST VERIFICATION 500 hpa GEOPOTENTIAL ROOT MEAN SQUARE ERROR FORECAST AREA=N.HEM TIME=00 MEAN OVER 82 CASES DATE1= /... DATE2= / MAGICS 6.12 aurora - dax Tue May 4 13:25: Verify SCOCOM Forecast Day two-way one-way 30.

31 NONLINEAR INTERACTIONS Ocean waves may be regarded most of the time as weakly nonlinear, dispersive waves. Hence, there is a small parameter present which permits to study the effect of nonlinearity on wave evolution by means of a perturbation expansion with starting point linear, freely propagating ocean waves. It should be pointed out that the subject of nonlinear ocean waves has conceptually much in common with nonlinear wave phenomena arising in diverse fields such as optics and plasma physics. Because of the common denominater we have seen a relatively rapid progress in the field of nonlinear ocean waves. Starting point for the development is the Hamiltonian formulation of the water wave problem. A small steepness expansion then gives the Zakharov Equation which describes the evolution of the free waves. This deterministic evolution equation is then used as a starting point for deriving the evolution of the ensemble mean of the sea state. 31.

32 Hamiltonian formulation The total energy of the fluid is given by E = 1 2 dx η D 0 dz( φ) 2 + g 2 dx η 2. (1) and this is a conserved quantity. By choosing η and ψ = φ(z = η) as canonical variables, Zakharov (1968) realized that the kinematic boundary condition and Bernoulli s equation then follow from the Hamilton equations, η t = δe δψ, ψ = δe t δη, (2) where δe/δψ is the functional derivative of E with respect to ψ = φ(z = η), etc. 32.

33 Inside the fluid the potential φ satisfies Laplace s equation, 2 φ = 0 (3) with boundary conditions φ(x,z = η) = ψ (4) and, with D 0 the water depth, φ(x,z) z = 0, z = D 0, (5) By solving the potential problem, φ may be expressed in terms of the canonical variables η and ψ. Then the energy E may be evaluated in terms of the canonical variables, and the evolution in time of η and ψ follows at once from Hamilton s equations (Eq.(2)). In particular for small steepness ε the potential problem (3-5) may be solved in an iterative fashion. 33.

34 Introduce the Fourier transforms of η and ψ, for example η = dk ˆη(k)e ik.x (6) where ˆη and ˆψ are the Fourier transforms of η and ψ. Here, k is the wavenumber vector, and k its absolute value. In order to proceed, introduce the linear dispersion relation for surface gravity waves ω 2 = gkt 0, T 0 = tanhkd 0. (7) Next, anticipating the fact we have two oscillation modes, introduce the following relation between the Fourier transform of η and ψ and the action density variable A(k,t) ˆη = ω g 2g (A(k)+A ( k)), ˆψ = i 2ω (A(k) A ( k)). (8) 34.

35 In terms of the action variable the energy of the fluid is given by a series expansion which up to fourth order in amplitude involves quadratic (linear theory), cubic and quartic terms. It should now be realized that surface gravity waves do not enjoy resonant three wave interactions. For example, the resonance conditions of the type k 1 + k 2 = k 3 and ω 1 + ω 2 = ω 3 cannot be satisfied when the dispersion relation is given by the one for gravity waves. A simple graphical construction shows this. Therefore, there is a canonical transformation of the type A = A(a,a ) (9) which removes the non-resonant third and fourth order terms as much as possible. Here, a is the amplitude of the free waves, while the canonical transformation generates the contributions by bound waves (i.e. second harmonics). 35.

36 Application of Krasitskii s transformation, A = A(a,a ), results in a considerable simplification of the Hamiltonian E(a,a ): E = dk 1 ω 1 a 1a dk 1,2,3,4 T 1,2,3,4 a 1a 2a 3 a 4 δ , where the interaction matrix T is given by Krasitskii (1994), and reflects contributions from bound waves and direct interactions. Most important symmetry is T 1,2,3,4 = T 3,4,1,2 as this condition implies conservation of E. The evolution equation for the amplitudes a follows then from Hamilton s equations i a/ t = δe/δa, giving the well-known Zakharov equation t a 1 + iω 1 a 1 = i dk 2,3,4 T 1,2,3,4 a 2a 3 a 4 δ (10) 36.

37 The Zakharov Equation is an evolution equation that describes four-wave interactions. Note the special case of resonant four-wave interactions: k 1 + k 2 = k 3 + k 4,ω 1 + ω 2 = ω 3 + ω 4. Phillips has shown that for gravity waves these resonance conditions can be satisfied, while at the same time he showed that resonant three wave interactions are impossible. Many properties of the Zakharov equation have been studied in the past. An example is the instability of a narrow-band wave train ( the Benjamin-Feir instability ) which may result in nonlinear focusing which helps in the generation of wave groups. This process is thought to play an important role in the formation of extreme events such as Freak Waves. 37.

38 1 Detail of surface elevation time series NLS with fixed initial phase 0.5 η Time NLS with random phase η Time 38.

39 Nonlinear focusing Basically a problem concerning the balance between dispersion of the waves and its nonlinearity (giving focusing). In the narrow-band approximation and 1D, this balance may be expressed by the so-called Benjamin-Feir Index BFI = ε 2/δ ω, where δ ω = σ ω /ω 0 is the relative width of the frequency spectrum and ε = (k 2 0 < η2 >) 1 2 is an integral measure of wave steepness (with < η 2 > the average surface elevation variance and k 0 the peak wave number). In 2D, an additional parameter plays a role, namely the ratio of directional width δ θ and frequency width δ ω, R = 1 2 δ 2 θ δ 2. ω 39.

40 Stochastic approach In wave forecasting we are interested in predicting quantities such as the second moment B 1,2 =< a 1 a 2 >, where angle brackets denote an ensemble average. Following methods employed in Statistical Mechanics (Liouville Boltzmann) one obtains from the deterministic Zakharov equation an equation for the action density N. Two assumptions are made: Because of nonlinearity second moment is coupled to fourth moment, etc. Closure achieved by the assumption that the sea state is close to a Gaussian, because nonlinearity is weak. For example, the fourth moment is < a j a k a l a m > = B j,l B k,m + B j,m B k,l + D j,k,l,m, where D is the so-called fourth cumulant, which vanishes for a Gaussian sea state. For weakly nonlinear waves D is small, but finite, and this enables one to close the hierarchy of equations. 40.

41 The first two members of the BBGKY hierarchy look like: [ ] t + i(ω i ω j ) B i, j = i dk 2,3,4 [T i,2,3,4 < a ja 2a 3 a 4 > δ i c.c.(i j)], Similarly, the equation for the fourth moment involves the sixth moment. It becomes [ ] t + i(ω i + ω j ω k ω l ) < a i a j a k a l >= i + i dk 2,3,4 [T i,2,3,4 < a 2a k a l a 3a 4 a j > δ i (i j)] dk 2,3,4 [T k,2,3,4 < a 3a 4a l a 2a i a j > δ k (k l)]. 41.

42 The second assumption we make is that of a homogeneous wave field. For this the two point correlation function < η(x 1 )η(x 2 ) > depends only on the distance x 1 x 2. As a consequence, the second moment becomes B i, j = N i δ(k i k j ), where N i is the spectral action density. Applying the random phase approximation to the sixth moment gives for the fourth cumulant D, subject to the initial value D(t = 0) = 0, D i, j,k,l = 2T i, j,k,l δ i+ j k l G( ω,t)[n i N j (N k + N l ) (N i + N j )N k N l ] where ω = ω i + ω j ω k ω l. Requires extensive use of the symmetries of T. In addition, the action density N is assumed to evolve on the slow time scale. The function G is defined as G( ω,t) = i t 0 dτe i ω(τ t) = R r ( ω,t)+ir i ( ω,t). 42.

43 Knowledge of the fourth cumulant is essential for evolution of N caused by four-wave interactions determination of deviations from normality. Hence, use of D in second moment equation gives t N 4 = 4 dk 1,2,3 T 2 1,2,3,4 δ(k 1 + k 2 k 3 k 4 )R i ( ω,t) [N 1 N 2 (N 3 + N 4 ) N 3 N 4 (N 1 + N 2 )], where ω = ω 1 + ω 2 ω 3 ω 4. The Boltzmann equation! Note there are now two timescales implied by R i ( ω,t) = sin( ωt)/ ω short times: lim t 0 R i ( ω,t) = t, hence T NL = O(1/ε 2 ω 0 ), the Benjamin-Feir timescale, corresponding to non-resonant interactions. large times: lim t R i ( ω,t) = πδ( ω), corresponding to resonant wave-wave interactions, hence T NL = O(1/ε 4 ω 0 ) (Hasselmann, 1962). 43.

44 Nonlinearity (both bound waves and dynamics) induce deviations from Gaussian statistics. When these deviations are small the pdf of e.g. the surface elevation η will follow the Gram-Charlier expansion, i.e. with x = η/ < η 2 > 1/2 the normalized surface elevation, p(x) = ( 1+ C 3 6 d 3 dx 3 + C 4 8 where f 0 is given by the normal distribution f 0 = 1 2π exp ( x2 d 4 ) dx Deviations from Normality are therefore most conveniently expressed by means of skewness C 3 and kurtosis C 4. However, in practice users are most interested in the wave height which is the distance between crest and trough of a wave. Therefore, kurtosis is of most interest. It is here defined as 2 ), C 4 =< η 4 > /3 < η 2 > 2 1. f 0, 44.

45 Using the expression for the fourth cumulant the contribution to kurtosis by means of four-wave interactions is found to be C dyn 4 = 4 g 2 m 2 dk1,2,3,4 T 1,2,3,4 δ (ω 1 ω 2 ω 3 ω 4 ) 1 2 R r ( ω,t)n 1 N 2 N 3, 0 which connects deviations from Gaussianity to the mean state. This expression is too involved in an operational context. However, for Gaussian-shaped spectra in the narrow band approximation the kurtosis can be shown to have a particularly simple form: C dyn 4 = π R BFI 2, R 0 = R/R 0 4π 3, hence the kurtosis depends on the square of the BF index and on the ratio of directional width and frequency width R. In addition, the bound waves are represented through the canonical transformation and they give the contribution C can 4 = 6ε 2 so that the total kurtosis is C 4 = C dyn 4 + 6ε

46 Comparison with Monte Carlo Simulations As the formation of freak waves depends in a sensitive manner on the initial phases, we perform Monte Carlo simulations with the evolution equations for water waves: Amplitudes are drawn from the wave spectrum, while phases are drawn randomly. Concentrate on the one-dimensional case! Main process that causes change are the nonlinear interactions. The next slides show for the 1-D case: spectral shape for BFI = 1.4 for Zakharov Equation impact on pdf of surface elevation dependence of kurtosis on BFI while for the 2-D case: kurtosis as function of BFI and R for 2D NLS 46.

47 0.020 Wavenumber Spectrum Zakharov Eq.; BFI=1.4; n_modes = Initial Spectrum Final Spectrum(MCFW) Final Spectrum(Theory) F(k) k Initial and final time wave number spectrum using the Zakharov equation. Error bars give 95% confidence limits. Results from theory are shown as well. 47.

48 0.0 Pdf of surface elevation Zakharov Eq; BFI=1.4; n_modes= log(pdf) 8.0 Gaussian MCFW Theory x/sqrt(m_0) Log of PDF for surface elevation (BFI=1.4). For reference the Gaussian distribution is shown as well. Freak waves correspond to a normalized height of 4 or larger. 48.

49 0.8 Kurtosis versus final time BF Index n_modes = 41 (focussing) C MCFW (NLS) Theory(NLS) MCFW(Zakh Eq) Theory (Zakh Eq) BFI(t=15) Normalized Kurtosis as function of the BF Index. Shown are results for focussing from simulations with NLS and with the Zakharov equation, and corresponding theoretical results. 49.

50 Empirical κ BFI σ y

51 NEW APPLICATIONS Nowadays, a number of new applications have emerged. Traditionally, ocean wave forecasts are used for guidance in shipping, fisheries, offshore operations and coastal protection. Ocean waves control to a large extent the air-sea momentum and heat transfer. Spectral information is important for the sea surface albedo, the Altimeter Radar back scatter. The spectral shape is known to determine the probabilty of extreme events, while wave dissipation plays an important role in the mixing of heat and momentum in the upper ocean. 51.

52 FREAK WAVES Recently evidence for the existence of freak waves has been found. Even modern ship are not designed to always withstand these exceptional circumstances, hence it is important to predict the probability that freak waves occur. On the open ocean, freak wave generation is based on nonlinear wave -wave interactions: A wave may borrow temporarily energy from its neighbours thus becoming huge for a while after which it returns the energy back. Developed probability theory for maximum wave height including nonlinear effects (kurtosis). 52.

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55 . Surface gravity waves Evolution of surface elevation in space and time from the big wave tank in Trondheim (from Onorato et al, 2004). The formation of Freak Waves is clearly seen. 55

56 Thursday 14 April UTC ECMWF Forecast t+36 VT: Friday 15 April UTC Surface: mean sea level pressure/surf: 10 mtr v Thursday 14 April UTC ECMWF Forecast t+36 VT: Friday 15 April UTC Surface: Wave spectral kurtosis W 40 W m/s W W

57 1 buoys model, random draw, 100 min linear model, random draw, 100 min log10 (pdf) Hmax/Hs 57.

58 CONCLUSIONS The work of Sverdrup and Munk, Miles, Phillips and Hasselmann was instrumental in a rapid development of ocean wave forecasting. Ocean wave forecasting is still an exciting field, drawing on clues from observations from buoys and satellites on the one hand and obtaining inspiration from theoretical developments. Research on Freak Waves has also triggered interesting developments in other fields of physics, e.g. in nonlinear optics. Finally, ocean waves also play an important role in air-sea interaction and the momentum and heat exchange in atmosphere and ocean. 58.