Proceedings of OMAE2006 25th International Conference on Offshore Mechanics and Arctic Engineering June 4-9, 2006, Hamburg, Germany

OMAE2006-92491

KINEMATICS UNDER EXTREME WAVES

Carl Trygve Stansberg MARINTEK Trondheim, Norway

Ove T. Gudmestad Statoil Stavanger, Norway

ABSTRACT Four different methods for prediction of wave-zone particle velocities under steep crests in random seas are compared. The study includes linear prediction, a second-order random wave model, Wheeler’s method, and a new method proposed by Grue et al. (2003). Comparison to laboratory data is also made. The purpose is to observe and evaluate differences in predictions for high and extreme waves, and how well they agree with measurements. The whole range from below still water level up to the free surface is considered. It is found that the second-order random wave model works best at all levels of the water column under a steep crest in deep water, and is therefore recommended. Grue’s method works reasonably well in many cases for z > 0, i.e. above the calm water level, but it overpredicts the velocities for z < 0. Wheeler’s method, when used with a measured or a secondorder input elevation record, predicts fairly well the velocities at the free surface z=ηmax, but it underpredicts around z=0 as well as at lower levels. The relative magnitude of this underprediction is slightly lower than the local steepness kA0 and can be quite significant in extreme waves. If Wheeler’s method is used with a linear input, the same error occurs also at the free surface.

INTRODUCTION A need has been identified to assess methods that take into account nonlinear effects in wave zone particle kinematics up to the free surface in random seas. Traditionally, in addition to linear modeling, Wheeler’s method (1971) has been frequently used. In a review study on different methods, Gudmestad (1993) found that Wheeler’s method under-predicts the velocities under steep irregular waves, and he identified a need for further research. Another method is the second-order

Sverre K. Haver Statoil Stavanger, Norway

random wave modeling described in Stansberg (1994), Stansberg & Gudmestad (1996), using the formulation by Marthinsen & Winterstein (1992). This method has not yet been very widely in use. Recently, a new method has been suggested by Grue et. al (2003). These three methods, plus the linear model, are compared in the present study. In addition, one part of the work also includes comparison to selected experimental data from the NHL-LDV study by Skjelbreia et al. (1991). It is necessary to use the most correct method for wave kinematics in time series simulations. A general feeling of uncertainties related to the correctness of using the traditional Wheeler method for calculating kinematics under extreme waves, combined with the interesting report of Grue et. al (2003), have been the main reasons for revisiting the present subject. Three different types of data are used: − Purely numerical data, including regular, bi-chromatic and irregular waves − Use of measured elevation records from MARINTEK Ocean Basin laboratory data − Use of measured elevation records from NHL – LDV data, including comparisons to measured velocities Within the scope of the present study, selected events are considered only, while a more systematic analysis is recommended for future studies.

DESCRIPTION OF METHODS Four different methods are compared, for the prediction of wave zone particle velocities u up to the free surface: − Linear

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− Second-order − Grue’s method − Wheeler’s method In this comparison, we focus on the horizontal velocity ux under the peaks Amax ≡ ηmax of selected wave crests. In particular, extreme (steep) waves in irregular wave trains on deep or almost deep water are considered, but regular and bichromatic waves are also addressed. Unidirectional waves are assumed Linear model Horizontal velocities are modeled up to the calm water surface, z=0, by use of commonly known linear theory for arbitrary water depth (see e.g. Dean & Dalrymple, 1991). In the following, u(1)(z) shall denote the linear velocity amplitude at a level z ≤ 0, while u0 ≡ u(1)(0) shall denote the linear horizontal velocity amplitude at z=0. Second-order model Time series of horizontal velocities are consistently modeled up to the linear free surface wave elevation A0 ≡ ηlinear,max by use of a second-order irregular wave model (Stansberg, 1994; Stansberg & Gudmestad, 1996) based on the formulation by Martinsen and Winterstein (1992). A linear input wave record is used, either purely numerical, or extracted from measurements. Arbitrary water depth is taken into account in the present model. Full storm durations can be modeled. The horizontal velocity amplitude at a level z under a crest is formulated as:

Grue’s method A new method has been proposed in Grue et al. (2003), based on observations from fully nonlinear wave simulations and from experimental results in a wave flume. Steep deepwater transient waves and similar events are considered. The method is phenomenological but has some physical basis in third-order Stoke’s regular wave theory. It is intended for use on individual waves one by one, from observed crest heights and wave periods only. It is rather simple and is therefore potentially an interesting method. The hypothesis is that the vertical profile of the horizontal velocity is simply given by: ux(z) = u0’exp(k’z) with the normalized reference velocity at z=0 defined as: u0’ = ε’ √(g/k’)

(3)

Here k’ is the actual (nonlinear) angular wave number 2π/L, L is the wave length, and ε’ is a steepness parameter. In Grue et al. (2003) ε’ is found implicitly from measurements, see eqs. (4,5) below. (Invoking third-order Stokes theory, we identify it as k’A0 , where A0 is the linear crest height, but neither A0 nor k’A0 are explicitly expressed in the original reference). If linear theory is valid, eq.(3) reduces to u0’ = ω’A0. The nonlinear wave number k’ and the steepness ε’ used in this formulation are given by the following third-order Stokes regular wave formulation: k’ηmax = ε’ + ½ ε’2 + ½ ε’3

z≤0 ⎫ ⎬(1) utot(z) = u0 +(∂ u(1)/∂z│z=0)·z + u(2,sum)(0) + u(2,diff)(0); z>0 ⎭ utot(z) = u(1)(z) + u(2 sum)(z) + u(2, diff)(z);

where u(2,sum) and u(2, diff) are the contributions from the sumand difference-frequency potentials, respectively. For more details, we refer to the above references. Roughly speaking, in deep water this model represents a linear extrapolation of the linear velocity gradient for z > 0, plus a second-order difference-frequency potential term which is generally negative under energetic wave groups. Note that in deep water the sum-frequency velocity potential is zero, and for regular waves also the differencefrequency contribution is zero then. In finite waters this is modified, but in almost deep water the modifications are small. An essential item in the model is the choice of the low-pass filter in the tail of the linear spectrum, which is needed in order to assure consistency in the perturbation to second order. Here we use the cut-off criterion proposed in Stansberg (1998) for deep-water waves: ωhigh ≡ √(khigh g) , where khigh = 2/Hs. Comparisons to experimental data in Stansberg & Gudmestad (1996) indicate that this works reasonably well. A discussion of this criterion was also made by Brodtkorb (2003).

(2)

ω’2/gk’ = 1 + ε’2

(4) (5)

Following the procedure in Grue et al. (2003), we find the wave frequency ω’ from the trough-to-trough period TTT observed from the wave time series. This differs from the definition chosen for the other methods. We use a zero-crossing criterion here, unless otherwise stated. (Notice that in a broadbanded spectrum, estimates from the zero-crossing method may differ from that using a “local” minimum identification method). In the procedure described, also the crest height ηmax is found from the measurement. The method is based on a regular wave theory and thus does not take into account any low-frequency differencefrequency contributions. Comment on definitions and notations - Grue’s method The estimated wave frequency ω’ and wave number k’ defined above are generally different from those used elsewhere in this study. This is partly because the local wave period definitions differ, and in order to distinguish the

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parameters from the others we have therefore identified the present ones by use of the mark ’ . Furthermore, the wave number k’ is nonlinear and decreases with increasing steepness, while a linear wave number definition is used in the other methods. In this study, we also find it convenient to define a “linear” wave number k0’ found directly from TTT : k0’≡ ω’2/g ;

k’ = k0’/( 1 + ε’2)

(6)

This “linear” wave number is generally different from the one defined for the other methods (k0), due to the wave period definition mentioned above. Also the present reference velocity u0’ is defined differently from a corresponding parameter used in the other methods. Further details on the definitions are given in a later section describing the actual data. Wheeler’s method This method, proposed in 1970 by Wheeler (1971), is widely in use since it is simple and it takes into account an observed reduction from linear predictions around z=0. At the same time, when a measured record is used as input, it also predicts reasonable free-surface velocities. The basic principle is that from a given elevation record, one computes the velocity for each frequency component using linear theory, assuming each component to be freely propagating (although they are in reality nonlinear in the higher frequency tail of the spectrum). Then, for each time step in the time series, the vertical (z) coordinate is “stretched” from the original level z to a level z’: z’ = (z - η) / (1 + η/d)

(8)

If, on the other hand, a nonlinear (e.g. a measured or a second-order) elevation record is used as input, nonlinear components will add as if they were independent and “free” near the free surface. It can be shown analytically that for a deep-water regular wave, a purely second-order elevation input will give exactly the same free-surface velocity as the consistent second-order model: The consistent second-order model gives, from eq.(1): utot(z=A0) = u0 + (∂u/∂z│z=0)·z = u0 + ω k0(A0)2

ufree surface = u0 + 2ωA(2) = u0 + 2ω[0.5 k0(A0)2] = u0 + ω k0(A0)2

(10)

where we have used the fact that the second-order elevation component is A(2) = 0.5 k0(A0)2 (from Stokes theory). The above effect is strongly reduced or vanished around z=0 and below, in which region the Wheeler method will still predict approximately those reduced velocities as with a linear input. With a measured input record, a low-pass filter is needed to avoid too high frequencies leading to excessive free-surface velocity estimates. A reasonable filter choice can be to try to include linear plus second-order contributions only, since that will lead to a free-surface velocity reasonably close to that of the consistent second-order model (see the paragraph above). In practice, fcut ≈ 4 fpeak , where fcut is the cut-off frequency and fpeak is the spectral peak frequency, often leads to a useful result.

DATA SETS Numerical simulations Linear and second-order numerical deep-water wave simulations are made by use of MARINTEK’s in-house software for modeling of second-order wave kinematics, based on the formulations in Stansberg (1994, 1998) and Marthinsen & Winterstein (1992). Four different cases are run:

(7)

where η is the elevation and d is the water depth. Thus for deep water it implies simply a time-varying vertical shift following the elevation. It should be noted that if a linear input elevation record is used, the method transforms (stretches) the linear fluid velocities up and down according to the elevation. Thus there may be a significant velocity reduction under high crests relative to the linear model throughout the whole water column in the wave zone. The relative magnitude of this reduction is approximately equal to the normalized vertical shift k0ηmax:

∆u/u0 = exp(k0ηmax ) – 1 ≈ k0ηmax (in deep water)

while Wheeler’s method with a second-order elevation input gives:

− Test 1: Regular wave H=7.6m, T=12.0s (k0A0 = 0.10) − Test 3: Regular wave H=21.6m, T=12.0s (k0A0 = 0.30) − Test 23: Bi-chromatic wave H1=11.9m, T1=15.0s H2=5.6m, T2=9.0s (Max. local steepness k0A0 = 0.40) − Test 100: Irregular wave Hs=16m Tp=14s JONSWAP Gamma=2.5 (Event 1: Local steepness k0A0 = 0.517; Event 2: k0A0 = 0.378) In the above definitions, the local steepness k0A0 is defined from actual crest parameters as described in the section “Normalization of velocity data” below. Laboratory data – elevation measurements only Selected measured elevation records from an earlier experiment with extreme waves in MARINTEK’s Ocean Basin are used as input to wave-zone kinematics estimation. The data are described as follows: − Model scale: 1:55. − Irregular waves, Hs=20m, Tp=20s, Torsethaugen spectrum. − Two different events are selected, two from test run 8525 and one from run 8526. One of these events correspond

(9)

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to very rare extreme waves, picked out from a largenumber of different 3-hour storm realizations. − Waves were measured without any structure in the basin. − Water depth 335m (full scale), i.e. deep water − Only elevation was measured, not kinematics From the measurements, the linear (“free”) wave component is extracted using a second-order filtering technique followed by the low-pass filtering procedure described in Section 3.2 above. Laboratory data including kinematics measurements Selected measurements from an earlier LDV (LaserDoppler-Velocimetry) experiment (Skjelbreia et al., 1991), made in a wave flume at NHL, Trondheim, are considered. The data are described as follows: − Model scale: 1:1. − Irregular waves, Hs=0.22m, Tp=1.8s, JONSWAP spectrum. Test case i18. − Two different events are selected among the largest in a run with approximately 500 waves. − Waves were measured without any structure in the basin. − Water depth 1.306m, i.e. deep water for wave periods T< 1.3s. In addition to surface elevations, particle velocities were measured at fixed vertical (z-) level with 0.05m intervals in the wave zone. The experiment was repeated for each z- level. Normalization of velocity data For each selected event, the velocity parameters are normalized in the following way, including regular, bichromatic as well as irregular wave records: First, velocities are divided by the linear estimate uref ≡ u0 at z=0, under the crest peak of the actual event. For purely numerical simulations, u0 is found directly from the linear input wave elevation record by use of the linear velocity transfer function (ref. e.g. Dalrymple & Dean, 1991). For numerical reconstructions of measured elevation records, u0 is found from the estimated linear wave component, in the same manner. Then, the vertical level z is multiplied by the linear angular wave number k0 estimated locally for the actual wave crest, found from the linear wave component in the following way: k0 ≡ ω02 /g

(11)

where the local angular wave frequency ω0 is defined from the linear wave component time series, by using the approximation of regular wave theory:

ω0 ≡ u0 / A0

(12)

and g is the acceleration of gravity, A0 is the linear crest height.

Special note on Grue’s method Note that the above linear wave number k0 is generally different from the wave number k’ estimated as a part of the method by Grue et al. (2003), see the method description above. This is due to two facts: First, k’ is a nonlinear wave number that changes (decreases) with the nonlinear dispersion effect in steep waves, and secondly, the frequency ω’ estimated directly from the troughs in the time series may differ from ω0 . Thus graphs from Grue’s method will in general appear shrinked (or sometimes stretched) on the plots relative to what they would have looked like using their definitions. Furthermore, the normalizing velocity unit in Grue’s method is given in eq. (3): u0’ = ε’ √(g/k’) , which can be written in terms of the parameters ω0 , ω’, u0 , A0 , and k’ above: u0’ = u0 (ω’/ω0 ) / √[1 + (k’A0)2]

(13)

(to third order). This means that the normalized velocity at z=0 in general differs from 1, and in most cases be somewhat lower.

RESULTS Elevation records of selected events Plots of elevation time series for all the selected events are shown in Figures 1-8. Elevations are shown in full scale except from the NHL-LDV experimental data in Figs. 7-8. The figures are as follows: Fig. 1: Fig. 2: Figs. 3-4: Figs. 5-6: Figs. 7-8:

Regular waves (numerical) Bi-chromatic waves (numerical) Irregular wave (numerical) Irregular wave; based on measured elevation Irregular wave with comparison to measured kinematics.

For each case except the regular waves, two plots are shown: A time series sample including an extreme wave (upper) and a zoomed-in plot of the actual wave (lower). Velocity profile comparisons Vertical profiles of estimated horizontal velocities under crests are shown in Figures 9 – 16. All velocities are normalized according to the procedure described in previous sections, and shown with velocities along the horizontal axis; vertical level z along the vertical axis. The following methods are compared: − Linear − Second-order − Grue’s method − Wheeler’s method, based on linear elevation input − Wheeler’s method, based on second-order elevation input, or measured if available

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Fig. 1. Regular wave elevation, numerical simulation. Steepness kA0=0.10 (upper) and 0.30 (lower).

Fig. 3. Irregular wave, numerical simulation, event 1. Maximum local steepness kA0=0.52.

Fig. 2. Bi-chromatic wave elevation, numerical simulation. Maximum local steepness kA0=0.40.

Fig. 4. Irregular wave, numerical simulation, event 2. Maximum local steepness kA0=0.38

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Fig. 5. Irregular wave, MARINTEK experiment, event 1. Maximum local steepness kA0=0.28.

Fig. 7. Irregular wave, NHL-LDV experiment, event 1. Maximum local steepness kA0=0.26.

Fig. 6. Irregular wave, MARINTEK experiment, event 2. Maximum local steepness kA0=0.38.

Fig. 8. Irregular wave, NHL-LDV experiment, event 2. Maximum local steepness kA0=0.40.

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The velocity profile figures are grouped as follows, with references given to the corresponding elevation plot numbers:

Linear (up to z=0) Second-order Grue's method Wheeler (from linear) Wheeler (from 2nd ord)

Test 23 - Bichr W #1 (k0A0=0.4) 0,5

k0z

Fig. 9: Regular waves (numerical) (re: fig. 1) Fig. 10: Bichromatic wave (numerical) (re: fig. 2) Figs. 11-12: Irregular wave (numerical) (re: fig. 3-4) Figs. 13-14: Irregular wave; based on measured elevation (re: fig. 5-6) Figs. 15-16: Irregular wave with comparison to measured kinematics (re: fig. 7-8).

0

-0,5 0

0,5

1

1,5

Ux/Uref0

Fig. 10. As Fig. 9 but bi-chromatic wave, kA0=0.4 Linear (up to z=0) Second-order Grue's method Wheeler (from linear) Wheeler (from 2nd ord)

Test 1 - Reg W k0A0=0.10

Test 100 - Irr W - event #1 k0A0=0.517

0,5

Linear (up to z=0) Second-order Grue's method Wheeler (from linear) Wheeler (from 2nd ord)

0

k 0z

k 0z

0,5

0

-0,5

-0,5

0

0,5

1

1,5

0

0,5

Ux/Uref0

1

1,5

Ux/Uref0 Linear (up to z=0) Second-order Grue's method Wheeler (from linear) Wheeler (from 2nd ord)

Test 3 - Reg W k0A0=0.30

Test 100 - Irr W - event #1 (a) k0A0=0.517

0,5

Linear (up to z=0) Second-order Grue's method Wheeler (from linear) Wheeler (from 2nd ord)

0

k 0z

k0z

0,5

0

-0,5 0

0,5

1

1,5

-0,5

Ux/Uref0

Fig. 9. Vertical profile of horizontal velocity, regular wave, Normalized values. Upper: steepness kA0=0.1. Lower: kA0=0.3.

0

0,5

1

1,5

Ux/Uref0

Fig. 11. As Fig. 9 but irregular wave event, kA0=0.52. Upper and lower: different definitions of wave period.

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Linear (up to z=0) Second-order Grue's method Wheeler (from linear) Wheeler (from 2nd ord)

Test 100 - Irr W - event #2 k0A0=0.378

Test 1826 - Irr W - 98s k0A0=0.353

0,5

0,5

0

k0z

k0z

Linear Second-order Grue's method Wheeler (from linear) Wheeler (from measur) LDV experiment

-0,5 0

0,5

1

0

-0,5

1,5

0

0,5

Ux/Uref0

1

1,5

Ux/Uref0

Fig. 12. As Fig. 11 (upper) but different event, kA0=0.38

Fig. 15. As Fig. 13 but event from different experiment including comparison to measured velocities , kA0=0.35

Linear (up to z=0) Second-order Grue's method Wheeler (from linear) Wheeler (from measured)

Test 8525 - Irr W - event #1 k0A0=0.283

Linear Second-order Grue's method Wheeler (from linear) Wheeler (from measur) LDV experiment

Test 1826 - Irr W - 721s k0A0=0.395

0,5

0

k0z

k0z

0,5

0

-0,5 0

0,5

1

1,5

-0,5

Ux/Uref0

0

1

1,5

Fig. 16. As Fig. 15 but different event, kA0=0.40

Linear Second-order Grue's method Wheeler (from linear) Wheeler (from measured)

Test 8526 - Irr W - event #1 k0A0=0.379

0,5

Ux/Uref0

Fig. 13. As Fig. 9 but irregular wave event, kA0=0.28

k0z

0,5

0

-0,5 0

0,5

1

1,5

Ux/Uref0

Fig. 14. As Fig. 13 but different wave event, kA0=0.38

DISCUSSION In the lower wave zone z < 0, all nonlinear models generally predict lower velocities than the linear model for irregular waves. This is also confirmed by the measured velocity data. For z > 0, linear data cannot be defined (and should be assumed to be constant - equal to u0), while all the other models show increased velocities with increasing z, in agreement with the measurements. At the free surface, predicted velocities are up to 1.3 - 1.5 times u0 in the steepest events. The “performances” of each of the different nonlinear models are discussed in the following. The second-order model is the one that compares best with the LDV measurements of test 1826, both with respect to the

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gradient for z > 0 and with respect to the reduction (relative to linear) at z < 0. The difference observed at the highest measurement level in the 98s event may be due partly to higher-order effects, but the possible measuring uncertainty at such measurement points with a very short fluid measurement duration should also be kept in mind. Furthermore, all examples (numerical and experimental) show that for z > 0 the predicted gradient of the normalized velocity agrees reasonably well with k0z, or is slightly higher. This is a helpful result, indicating that the regular wave approximation in the estimation of the local k0 works quite well. The choice of the low-pass filter for the input spectra, based on a criterion taking into account the perturbation problem, seems to have been satisfactory. It is, however, recommended to address this further, e.g. through sensitivity studies. Another possible topic to study further is the consequencues from the use of such a nonlinear kinematic model on statistical properties of velocities and possible forces and responses. Grue’s method compares fairly well both with the measurements and with the second-order model in the zone z > 0. It is, however, not always quite robust with respect to the trough-to-trough estimation of the wave period T’, since for irregular waves there may sometimes be alternative and equally logical choices for this estimation, except for narrow-banded spectra. It is not clear from the definition in the original reference whether a zero-crossing or a local minimum trough criterion is used in their work. In our work, we have basically used zero-crossing, but our experience also indicates that a local minimum criterion may in fact work better. More systematic studies should be made to clarify this. Furthermore, the method basically neglects the negative return current effect for z < 0 under energetic wave groups, although a minor effect is in most cases still apparently predicted near z=0 due to the higher-order wave length elongation reducing the reference velocity u0’. Some overprediction at levels below z=0 can also be identified in the original results, Grue et al. (2003), which qualitatively confirms our finding. Wheeler’s method, if based upon a linear wave elevation record only, significantly underpredicts the velocities at all depth levels. The relative error is in the range 0.5k0A0 – 0.75k0A0, where k0A0 is the local steepness. If Wheeler’s method is based upon a nonlinear record, e.g. upon a second-order or a measured time series, the velocity predicted near the surface is clearly improved compared to the linear input case above. However, it decreases rapidly with z, and at z=0 and below the relative underprediction is typically 0.5k0A0 – 0.75k0A0s as with a linear input record. The rapid decrease is strongest for the steepest waves – hence the method may be a fairly reasonable choice for low-steepness waves. For a typical individual design wave in the North Sea with H=29m, T=15s, k0A0 is approximately 0.26. Assuming a second-order input record then, use of Wheeler’s method will predict reasonably well at the free surface, while it will be expected to underpredict by approximately 10% – 20% in the region around z=0 and below. Another item, which makes the method

less robust, is the need to define a low-pass filter for a measured input wave signal, to avoid excessive high-frequency contributions. In a way this problem may appear to be similar to the filtering problem in the second-order model, but the difference is that in the Wheeler method case there seems to be no clear physical criterion behind the filter choice. One reasonable criterion, however, might be to try to filter away all contributions of order higher than 2, since use of a pure 2nd order model as input gives a free-surface velocity reasonably close to a consistent 2nd order model (but it still predicts too low velocities further down in the fluid). In this paper, we have made limited selections from large data sets. In the underlying work, comparisons for more examples were made, which generally support the findings shown here. Still, there is a need to include a wider variety of data for a more systematic study. A limitation has also been made in the selection of methods. Further work with a broader scope is therefore suggested.

CONCLUSIONS At the free surface of steep crests, z=ηmax, all the three nonlinear methods predict the maximum velocity reasonably well, except when Wheeler’s method is used with linear input in which case a significant under-prediction is observed. The maximum velocities are typically 30 - 40% higher than linear predictions. At lower levels in the wave zone, there are larger discrepancies between the models. An overall conclusion is that among the methods investigated here, the second-order random wave model works best at all levels of the water column under a steep crest in deep water, and is therefore recommended. Grue’s method works reasonably well in most cases for z>0, i.e. above the calm water level, while it generally over-predicts the velocities for z<0. Wheeler’s method, when used with a measured or a second-order input elevation record, predicts fairly well the velocities at the free surface, but it underpredicts around z=0 as well as at lower levels. The relative magnitude of this under-prediction is slightly lower than the local steepness kA0 and can be quite significant in extreme waves. If Wheeler’s method is used with a linear input, the same error occurs also at the free surface. Further work is recommended to compare the different methods in a more systematic way and also in a broader range of sea states, not only selected events in extreme conditions as focused on here. Comparisons to related results in the literature are also recommended. Furthermore, a sensitivity analysis of the low-pass filter needed in the second-order random wave model is needed to establish it as a robust model. Finally, the consequences from the present findings on the resulting drag loads on slender structures, and their statistical properties, should be investigated.

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ACKNOWLEDGMENTS This study has been financed by Statoil. The authors gratefully acknowledge the permission to publish this paper. Experimental kinematics data were obtained from the SINTEF NHL Wave Kinematics Experiment (1988-1991), which was financed by: NTNF, Statoil, Amoco, Conoco, Exxon and Mobil.

REFERENCES Brodtkorb, P.A. (2004), The Probability of Occurrence of Dangerous Wave Situations at Sea, Dr. ing. Thesis, NTNU, Trondheim, Norway. Dean, R.G. and Dalrymple, R.A. (1991), Water Wave Mechanics for Engineers and Scientists, World Scientific, Singapore. (Chapter 4). Grue, J., Clamond, D., Huseby, M. and Jensen, A. (2003), “Kinematics of Extreme Waves in Deep Water”, Applied Ocean Research, Vol. 25, pp 355-366. Gudmestad, O.T. (1993), “Measured and Predicted Deep Water Wave Kinematics in Regular and Irregular Seas”, Marine Structures, Vol. 6, pp. 1-73.

Marthinsen, T. and Winterstein, S. (1992), “On the Skewness of Random Surface Waves”, Proc., Vol.3, 2nd ISOPE Conf., San Francisco, Cal., USA, pp. 472-478. Skjelbreia, J.E., Berek, G., Bolen, Z.K., Gudmestad, O.T., Heideman, J.C., Ohmart, R.D., Spidsøe, N. and Tørum, A. (1991), “Wave Kinematics in Irregular Waves”, Proc., Vol. 1A, the 10th OMAE Conf., Stavanger, Norway, pp. 223-228. Stansberg, C. T. (1994), "Second-Order Effects in Random Wave Modeling", Proc., Vol. 2, International Symposium on Waves - Physical and Numerical Modeling, Vancouver, Canada, pp. 793-802. Stansberg, C.T. (1998), “Non-Gaussian Extremes in Numerically Generated Second-Order Random Waves on Deep Water”, Proc., Vol. III, the 8th ISOPE Conference, Montreal, Canada, pp. 103-110. Stansberg, C.T. and Gudmestad, O.T. (1996), “Nonlinear Random Wave Kinematics Models Verified Against Measurements in Steep Waves”, Proc. Vol. IA, the 15th OMAE Conf., Florence, Italy, (1996), pp. 15-24. Wheeler, J.D.E. (1970), “Method for Calculating Forces Produced by Irregular Waves”, Journal of Petroleum Tech., Vol. 249, pp. 359-367.

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