Karsten Trulsen

• Trulsen, Karsten; Stansberg, Carl Trygve & Dysthe, Kristian (2025). Experimental evidence of transverse modulation and frequency downshift of uniform waves in a long tank. Journal of Fluid Mechanics. ISSN 0022-1120. 1024, p. 1–30. doi: 10.1017/jfm.2025.10822. Full text in Research Archive Show summary

  We report experiments in a long tank showing that transverse Benjamin–Feir instability of Stokes waves can lead to a significant energy transfer into oscillations across the tank. We observe frequency downshift in the long-term evolution of Stokes waves essentially when significant energy is transferred to narrow-banded transverse modes. Experiments with Stokes waves are often carried out with wavelengths that are not long compared with the width of the tank, permitting transverse instabilities to be excited. With insufficient resolution of measurements across the tank, transfer of energy to transverse modes can be misinterpreted as dissipation. Our experiments suggest that the frequency downshift depends as much on energy-preserving transverse modulations of type I as it does on damping or wave breaking. Broad-banded unstable modulations of type II do not imply downshift.

• Samseth, Karen & Trulsen, Karsten (2025). Extreme statistics of wave forces on a horizontal cylinder behind a shoal. Journal of Fluid Mechanics. ISSN 0022-1120. 1007. doi: 10.1017/jfm.2025.124. Full text in Research Archive Show summary

  We report laboratory experiments of long-crested irregular water surface waves propagating over a shoal, with attention to the region over the down-slope behind the shoal. We measure the surface elevation field, the horizontal velocity field in the water, and the resulting forces on a horizontal submerged cylinder placed over the down-slope of the shoal. In addition, we calculate the horizontal acceleration field. From this, we find that the presence of the shoal can modify the wave field such that the resulting forces on the submerged cylinder can be enhanced with thicker extreme tails and increased values of skewness and kurtosis depending on the location of the cylinder. The spatial dependence of the statistics of forces is different from the spatial dependence of the statistics of horizontal velocity, horizontal acceleration and surface elevation.

• Bitner-Gregersen, E. M.; Gramstad, Odin; Trulsen, Karsten; Magnusson, Anne Karin; Støle-Hentschel, Susanne & Aarnes, Ole Johan [Show all 7 contributors for this article] (2024). Rogue waves: Results of the ExWaMar project. Ocean Engineering. ISSN 0029-8018. 292. doi: 10.1016/j.oceaneng.2023.116543. Full text in Research Archive Show summary

  Rogue waves, also known as freak waves, have received much attention in the scientific community as well as in media and the marine industry in the past two decades. Forecasts of extreme weather events have always been welcomed by the marine industry. Therefore, the development of warning criteria for rogue waves, being scientifically challenging forecast products, has also been encouraged. Such criteria would help mariners avoid sea states where rogue waves occur. The paper summarizes the main findings of the Norwegian research project ExWaMar dedicated to development of improved warning criteria for extreme and rogue waves. Three approaches for warning criteria for extreme and rogue waves based on information provided by the weather forecast are proposed. They include use of integrated wave parameters, coupling of a phase-averaged wave spectral model and phase-resolving wave model and application of Machine Learning methodology. Challenges related to development of warning criteria for rogue waves are discussed.

• Vanem, Erik; Gramstad, Odin; Babanin, Alexander; Bin, Riccardo De & Trulsen, Karsten (2024). On the distribution of ocean wave crest heights in varying wave conditions. Journal of Ocean Engineering and Marine Energy. ISSN 2198-6444. 10(4), p. 797–815. doi: 10.1007/s40722-024-00350-0. Full text in Research Archive Show summary

  This paper addresses the statistical distribution of wave crest heights in ocean environments. This is much needed in several engineering applications including risk and reliability assessment of marine and coastal structures and is an important input for design of ocean structures. However, even though crest height distributions have received a lot of attention within both academia and the industry, current state-of-the-art models have notable shortcomings and do not describe the distribution of crest heights well in all sea states. In particular, commonly used models do not accurately describe the upper tail in certain unstable wave conditions. This will be addressed in this paper, which investigates in what sea-state conditions, established crest height distributions describe the data well and under what conditions, these models fail to fit well to the data. A large dataset of crest heights observed in different sea states is analyzed, and various goodness-of-fit tests are applied to determine when the different statistical models are appropriate. Then rejection rates of the various statistical models will be linked to selected sea-state parameters to identify wave conditions where alternative statistical models are needed. Results from this study indicate that the well-known Forristall distribution, which is known to capture second-order effects well, fails to perform well in some sea states. Different variants of the Tayfun–Fedele distribution are also analyzed, and results suggest that this represent a slight improvement. Somewhat surprisingly, sea-state parameters such as significant wave height (Hs), significant wave steepness (S1), and the Benjamin–Feir index (BFI) do not seem to be very influential on the probability of rejecting these distributions, even though empirical skewness and kurtosis play a significant role. The implications of this for describing the distribution of crest heights well in certain sea-state conditions are discussed.

• Lawrence, Christopher; Trulsen, Karsten & Gramstad, Odin (2022). Extreme wave statistics of surface elevation and velocity field of gravity waves over a two-dimensional bathymetry. Journal of Fluid Mechanics. ISSN 0022-1120. 939, p. 1–18. doi: 10.1017/jfm.2022.227. Full text in Research Archive
• Trulsen, Karsten (2022). Ekstrembølger i havet - hva er det, hvordan oppstår de og hvilken betydning kan de ha for virksomhet til havs. Årbok / Det norske videnskaps-akademi. ISSN 0332-6209. Årbok 2021, p. 147–156. Full text in Research Archive
• Lawrence, Christopher; Trulsen, Karsten & Gramstad, Odin (2021). Statistical properties of wave kinematics in long-crested irregular waves propagating over non-uniform bathymetry. Physics of Fluids. ISSN 1070-6631. 33. doi: 10.1063/5.0047643. Full text in Research Archive Show summary

  Experimental and numerical evidence have shown that nonuniform bathymetry may alter significantly the statistical properties of surface elevation in irregular wave fields. The probability of “rogue” waves is increased near the edge of the upslope as long-crested irregular waves propagate into shallower water. The present paper studies the statistics of wave kinematics in long-crested irregular waves propagating over a shoal with a Monte Carlo approach. High order spectral method is employed as wave propagation model, and variational Boussinesq model is employed to calculate wave kinematics. The statistics of horizontal fluid velocity can be different from statistics in surface elevation as the waves propagate over uneven bathymetry. We notice strongly non-Gaussian statistics when the depth changes abruptly in sufficiently shallow water. We find an increase in kurtosis in the horizontal velocity around the downslope area. Furthermore, the effects of the bottom slope with different incoming waves are discussed in terms of kurtosis and skewness. Finally, we investigate the evolution of kurtosis and skewness of the horizontal velocity over a sloping bottom in a deeper regime. The vertical variation of these statistical quantities is also presented.

• Støle-Hentschel, Susanne; Borge, Jose Carlos Nieto & Trulsen, Karsten (2021). The deconvolution as a method to deal with gaps in ocean wave measurements. Ocean Engineering. ISSN 0029-8018. 219. doi: 10.1016/j.oceaneng.2020.108373. Full text in Research Archive Show summary

  This work introduces the deconvolution as a technique to reconstruct missing information in data. While the method was originally developed for ocean waves, it will be useful in a wider range of applications where gaps in data may alter the statistics or spikes have to be eliminated without removing extreme values. For the application to ocean waves, it is estimated that gaps as long as half of the peak period may be reconstructed well. It is possible to reconstruct data of longer gaps, however, in total the amount of missing points should be less than 50% of all points and the missing data should not be clustered.

• Trulsen, Karsten; Raustøl, Anne; Jorde, Stian & Rye, Lisa Bæverfjord (2020). Extreme wave statistics of long-crested irregular waves over a shoal. Journal of Fluid Mechanics. ISSN 0022-1120. 882. doi: 10.1017/jfm.2019.861. Full text in Research Archive Show summary

  We report laboratory experiments of long-crested irregular water surface waves propagating over a shoal. For a sufficiently shallow shoal we find that the surface elevation can have a local maximum of skewness and kurtosis above the shallower part of the shoal close to the edge on the incoming side, and a local minimum of skewness over the downward slope on the lee side of the shoal. We find that the horizontal fluid velocity can have a local maximum and minimum of skewness at the same locations as those for the surface elevation. However, the kurtosis of the horizontal fluid velocity can have a local maximum over the downward slope on the lee side of the shoal, different from the location of the maximum of kurtosis of the surface elevation.

• Støle-Hentschel, Susanne; Trulsen, Karsten; Borge, José Carlos Nieto & Olluri, Shkurte (2020). Extreme Wave Statistics in Combined and Partitioned Windsea and Swell. Water Waves. ISSN 2523-367X. 2, p. 169–184. doi: 10.1007/s42286-020-00026-w. Full text in Research Archive Show summary

  Abstract We investigate how the extreme wave statistics of a combined windsea and swell appears to be different from the extreme wave statistics of the corresponding windsea and swell partitions. We consider the situation of following long-crested windsea and swell in laboratory experiments and in simulations using a high-order spectral method (HOSM). We also consider the situation of short-crested windsea and swell crossing at nearly right angle, corresponding to the sea state when the Prestige accident happened, in hindcast simulations combined with HOSM. For cases when the two wave systems do not interact much, the combined wave system appears to be more Gaussian than the corresponding partitioned wave systems, consistent with the central limit theorem. This result is found for kurtosis and exceedance probability of envelope and crest height.

• Lawrence, Christopher; Gramstad, Odin & Trulsen, Karsten (2020). Variational Boussinesq model for kinematics calculation of surface gravity waves over bathymetry. Wave motion. ISSN 0165-2125. 100. doi: 10.1016/j.wavemoti.2020.102665. Full text in Research Archive Show summary

  Many of the widely used models for description of nonlinear surface gravity waves, in deep or shallow water, such as High Order Spectral Method (HOSM) and Boussinesq-type equations, rely on the elimination of the vertical coordinate from the basic three-dimensional Euler equations. From a numerical point of view such models are often computationally efficient, which is one of the main reasons that many such models are frequently used in studies on nonlinear surface waves. While surface-based models provide the time-evolution of surface quantities, typically the surface elevation η and velocity potential at the surface φ , they do not directly provide the water particle kinematics in the fluid interior. However, in many practical applications information about the water-particle kinematics is crucial. The present paper presents a new method for the calculation of water-particle kinematics, from information about surface quantities. The presented methodology is a non-perturbative approach based on the fully nonlinear Variational Boussinesq model, and can be applied to wave propagation over both constant and variable water depth. The proposed method is validated on several cases, including Stokes waves, a solitary wave, and irregular waves over flat bottom. We have carried out new laboratory experiments of regular waves over a shoal with measurements of the horizontal velocity specifically taken for validation of the method. We also employ recent laboratory experiments for validation of statistical properties of wave kinematics of long crested irregular waves propagating over a shoal.

• Magnusson, Anne Karin; Trulsen, Karsten; Bitner-Gregersen, Elzbieta M.; Aarnes, Ole Johan & Malila, Mika Petteri (2019). “Three Sisters” Measured as a Triple Rogue Wave Group. In ASME, . (Eds.), ASME 2019 38th International Conference on Ocean, Offshore and Arctic Engineering. Volume 3: Structures, Safety, and Reliability. The American Society of Mechanical Engineers (ASME). ISSN 9780791858783. doi: 10.1115/OMAE2019-96837. Full text in Research Archive Show summary

  On November 30, 2018, our attention was caught when analyzing wave profile time series measured by a platform mounted wave sensor (a SAAB REX radar) at Ekofisk, central North Sea. The 20-minute time series had not only one, but three consecutive waves with individual heights that all were more than twice the significant wave height, the two last of them being almost equally high with a factor 2.35 to the significant wave height of 4m (from 4σ(η), over 20 minutes). Counting three rogue waves in one sequence seems to be very rare. In this study we analyze how the shape is evolving in space and time using linear and non-linear propagation methods developed by Mark Donelan [1,2] and Karsten Trulsen [3,4]. Weather conditions and characteristics of the sea state with the ‘Three Sisters’ (named the “Justine Three Sisters”) are presented. It is found that the Three Sisters occurred in a crossing sea condition, with wind sea and swell coming from directions 60 degrees apart, with about same frequency, but very different energy.

• Trulsen, Karsten (2018). Rogue Waves in the Ocean, the Role of Modulational Instability, and Abrupt Changes of Environmental Conditions that Can Provoke Non Equilibrium Wave Dynamics. In Velarde, Manuel G.; Tarakanov, Roman Yu. & Marchenko, Alexey V. (Ed.), The Ocean in Motion. Springer. ISSN 9783319719344. p. 239–247. doi: 10.1007/978-3-319-71934-4_17. Full text in Research Archive Show summary

  Modulational instability is an efficient mechanism for the generation of rogue waves in the limit of narrow-banded and long-crested wave fields. While such wave fields are easily achieved in laboratories, there appears to be lacking evidence that known occurrences of rogue waves in the ocean (e.g. Draupner “New Year” wave, Andrea wave) or ship accidents that could have been provoked by rogue waves (e.g. the Prestige accident) actually happened in sea states favorable for the modulational instability to have played an important role. The absence of modulational instability does not mean that nonlinear interactions are unimportant. Here we point out recent results that suggest large deviations from Gaussian statistics can happen due to nonlinearity in the absence of modulational instability, the key ingredient seems to be that the wave field is brought into a state of non-equilibrium.

• Simanesew, Abushet; Krogstad, Harald E; Trulsen, Karsten & Borge, Jose Carlos Nieto (2018). Bimodality of directional distributions in ocean wave spectra: A comparison of data-adaptive estimation techniques. Journal of Atmospheric and Oceanic Technology. ISSN 0739-0572. 35(2), p. 365–384. doi: 10.1175/JTECH-D-17-0007.1. Full text in Research Archive Show summary

  The properties of directional distributions in ocean wave spectra are studied, with an emphasis on sea states with bimodal directional distributions in the high-frequency tails of single-peaked wave systems. A peak-splitting tendency has been a challenge in the interpretation of results from some data-adaptive estimation methods. After a survey of the theory, mathematical and numerical explanations are presented regarding domains of uni- and bimodality for symmetric Burg and Shannon maximum entropy methods. The study finds that both the Burg and Shannon maximum entropy methods have a tendency to split peaks, and that the domains of uni- and bimodality for these two methods depend on the Fourier coefficients input into the algorithms. Comparisons of data-adaptive methods based on data collected near the Ekofisk oil field in the North Sea and from nonlinear wave simulations are presented. The maximum likelihood (ML) method, the iterative maximum likelihood (IML) method, and the Burg and Shannon maximum entropy methods are applied. A large fraction of the directional wave spectra from Ekofisk shows bimodal features for distributions above the spectral peak for all of the abovementioned methods. In particular, strong similarity in bimodal features between the iterative maximum likelihood and the Burg maximum entropy methods are found. In general, the bimodality is consistent with previous observations, and it seems to be associated with wave and spectral development owing to nonlinear wave–wave interactions rather than being associated with the peak-splitting tendency in the estimates from any of the algorithms. The bimodal directional distributions were sometimes persistent and sometimes formed or decayed within the order of hours.

• Gramstad, Odin; Bitner-Gregersen, Elzbieta; Trulsen, Karsten & Borge, Jose Carlos Nieto (2018). Modulational instability and rogue waves in crossing sea states. Journal of Physical Oceanography. ISSN 0022-3670. 48(6), p. 1317–1331. doi: 10.1175/JPO-D-18-0006.1. Full text in Research Archive Show summary

  Wave statistical properties and occurrence of extreme and rogue waves in crossing sea states are investigated. Compared to previous studies a more extensive set of crossing sea states are investigated, both with respect to spectral shape of the individual wave systems and with respect to the crossing angle and separation in peak frequency of the two wave systems. It is shown that, because of the effects described by Piterbarg, for a linear sea state the expected maximum crest elevation over a given surface area depends on the crossing angle so that the expected maximum crest elevation is largest when two wave systems propagate with a crossing angle close to 90°. It is further shown by nonlinear phase-resolving numerical simulations that nonlinear effects have an opposite effect, such that maximum sea surface kurtosis is expected for relatively large and small crossing angles, with a minimum around 90°, and that the expected maximum crest height is almost independent of the crossing angle. The numerical results are accompanied by analysis of the modulational instability of two crossing Stokes waves, which is studied using the Zakharov equation so that, different from previous studies, results are valid for arbitrary-bandwidth perturbations. It is shown that there is a positive correlation between the value of kurtosis in the numerical simulations and the maximum unstable growth rate of two crossing Stokes waves, even for realistic broadband crossing sea states.

• Støle-Hentschel, Susanne; Trulsen, Karsten; Rye, Lisa Bæverfjord & Raustøl, Anne (2018). Extreme wave statistics of counter-propagating, irregular, long-crested sea states. Physics of Fluids. ISSN 1070-6631. 30(6). doi: 10.1063/1.5034212. Full text in Research Archive Show summary

  Extreme wave statistics of unidirectional and counter-propagating seas are investigated, for the special case of long-crested irregular waves, with laboratory experiments and with numerical simulations using the higher order spectral method. Both the kurtosis of the surface elevation and the exceedance probability of the crest height are larger in unidirectional seas than in counter-propagating seas. Numerical simulations show that even a small amount of wave energy travelling against an essentially unidirectional wave system can significantly reduce the kurtosis of the surface elevation. This research was originally published in Physics of Fluids. © 2018 AIP Publishing

• Støle-Hentschel, Susanne; Seemann, Jörg; Borge, Jose Carlos Nieto & Trulsen, Karsten (2018). Consistency between Sea Surface Reconstructions from Nautical X-Band Radar Doppler and Amplitude Measurements. Journal of Atmospheric and Oceanic Technology. ISSN 0739-0572. 35, p. 1201–1220. doi: 10.1175/JTECH-D-17-0145.1. Full text in Research Archive Show summary

  This study comprises the analysis and the interpretation of the coherent and the noncoherent parts of a coherent-on-receive microwave radar at grazing incidence conditions. The Doppler measurement is an extension of standard civil marine radar technology. While intensity images require interpretation based on understanding the underlying imaging mechanism, the Doppler signal measures the motion of an area of sea surface and is therefore closely related to the wave physics. Both the measured Doppler signal and the backscatter intensity signal are suitable for surface inversion and give almost identical surface elevations. A statistical comparison with a nearby buoy showed good correlation for the significant wave height and the peak period. By comparing the Doppler signal and the amplitude in the backscatter, the study amends the understanding of imaging mechanisms in marine radars at grazing incidence. This research was originally published in Journal of Atmospheric and Oceanic Technology. © 2018 American Meteorological Society

• Simanesew, Abushet; Trulsen, Karsten; Krogstad, Harald E & Borge, José Carlos Nieto (2017). Surface wave predictions in weakly nonlinear directional seas. Applied Ocean Research. ISSN 0141-1187. 65, p. 79–89. doi: 10.1016/j.apor.2017.03.009. Full text in Research Archive Show summary

  We have employed laboratory and numerical experiments in order to investigate propagation of waves in both long and short-crested wave fields in deep water. For long-crested waves with steepness, ϵ = kcac = 0.1 (a fairly extreme case), reliable prediction can be performed with the modified nonlinear Schrödinger equation up to about 40 characteristic wavelengths. For short-crested waves the accuracy of prediction is strongly reduced with increasing directional spread.

• Taklo, Tore Magnus Arnesen; Trulsen, Karsten; Krogstad, Harald E & Borge, Jose Carlos Nieto (2017). On dispersion of directional surface gravity waves. Journal of Fluid Mechanics. ISSN 0022-1120. 812, p. 681–697. doi: 10.1017/jfm.2016.817. Full text in Research Archive Show summary

  Using a nonlinear evolution equation we examine the dependence of the dispersion of directional surface gravity waves on the Benjamin–Feir index (BFI) and crest length. A parameter for describing the deviation between the dispersion of simulated waves and the theoretical linear dispersion relation is proposed. We find that for short crests the magnitude of the deviation parameter is low while for long crests the magnitude is high and depends on the BFI. In the present paper we also consider laboratory data of directional waves from the Marine Research Institute of the Netherlands (MARIN). The MARIN data confirm the simulations for three cases of BFI and crest length.

• Simanesew, Abushet; Krogstad, Harald E; Trulsen, Karsten & Borge, Jose Carlos Nieto (2016). Development of frequency-dependent ocean wave directional distributions. Applied Ocean Research. ISSN 0141-1187. 59, p. 304–312. doi: 10.1016/j.apor.2016.06.011. Full text in Research Archive Show summary

  The paper discusses the development of a frequency dependent directional spread from an initial condition of frequency-independence. The study applies basin directional measurements from the Maritime Research Institute Netherlands (MARIN), simulated data from a nonlinear wave equation, and field measurements from the Ekofisk field. The basin experiments and numerical simulations are initialized with a JONSWAP spectrum with frequency-independent directional distributions. In both cases we observe the development of a strong frequency-dependence of the directional spread. The numerical simulations suggest that static nonlinear contributions to the surface elevation partially explain the behavior below the spectral peak in accordance with [1]. There are also dynamic nonlinear contributions on both sides of the spectral peak.

• Trulsen, Karsten; Borge, Jose Carlos Nieto; Gramstad, Odin; Aouf, Lotfi & Lefevre, Jean-Michel (2015). Crossing sea state and rogue wave probability during the Prestige accident. Journal of Geophysical Research (JGR): Oceans. ISSN 2169-9275. 120(10), p. 7113–7136. doi: 10.1002/2015JC011161. Full text in Research Archive Show summary

  We discuss the crossing sea state and the probability of rogue waves during the accident of the tanker Prestige on 13 November 2002. We present newly computed hindcast spectra for every hour during that day at nearby locations, showing the development of a bimodal sea state with two wave systems crossing at nearly right angle. We employ four different nonlinear models capable of computing the phase-resolved sea surface from the hindcast spectra, allowing us to estimate statistics for the occurrence of rogue waves. At the location and moment of the accident, the models give expected values for the kurtosis κ = 3.0119 ± 0.0078. The models coincide that the maximum crest elevation was about 5–6% larger than the expected maximum crest elevation in a Gaussian sea at the moment of the accident. We also conclude that the possible nonlinear interaction between the two crossing wave systems practically did not modify neither the kurtosis nor the largest crest elevation. This research was originally published in the Journal of Geophysical Research: Oceans. © American Geophysical Union.

• Taklo, Tore Magnus Arnesen; Trulsen, Karsten; Gramstad, Odin; Krogstad, Harald E & Jensen, Atle (2015). Measurement of the dispersion relation for random surface gravity waves. Journal of Fluid Mechanics. ISSN 0022-1120. 766, p. 326–336. doi: 10.1017/jfm.2015.25. Full text in Research Archive Show summary

  We report laboratory experiments and numerical simulations of the Zakharov equation, designed to have sufficient resolution in space and time to measure the dispersion relation for random surface gravity waves. The experiments and simulations are carried out for a JONSWAP spectrum and Gaussian spectra of various bandwidths on deep water. It is found that the measured dispersion relation deviates from the linear dispersion relation above the spectral peak when the bandwidth is sufficiently narrow.

• Naaijen, Peter; Trulsen, Karsten & Blondel-Couprie, Elise (2014). Limits to the extent of the spatio-temporal domain for deterministic wave prediction. International Shipbuilding Progress. ISSN 0020-868X. 61(3-4), p. 203–223. doi: 10.3233/ISP-140113. Full text in Research Archive
• Gramstad, Odin; Zeng, Huiming; Trulsen, Karsten & Pedersen, Geir Kleivstul (2013). Freak waves in weakly nonlinear unidirectional wave trains over a sloping bottom in shallow water. Physics of Fluids. ISSN 1070-6631. 25(12). doi: 10.1063/1.4847035. Full text in Research Archive
• Zeng, Huiming & Trulsen, Karsten (2012). Evolution of skewness and kurtosis of weakly nonlinear unidirectional waves over a sloping bottom. Natural Hazards and Earth System Sciences. ISSN 1561-8633. 12(3), p. 631–638. doi: 10.5194/nhess-12-631-2012. Full text in Research Archive
• Trulsen, Karsten; Zeng, Huiming & Gramstad, Odin (2012). Laboratory evidence of freak waves provoked by non-uniform bathymetry. Physics of Fluids. ISSN 1070-6631. 24(9). doi: 10.1063/1.4748346. Full text in Research Archive Show summary

  We show experimental evidence that as relatively long unidirectional waves propagate over a sloping bottom, from a deeper to a shallower domain, there can be a local maximum of kurtosis and skewness close to the shallower side of the slope. We also show evidence that the probability of large wave envelope has a local maximum near the shallower side of the slope. We therefore anticipate that the probability of freak waves can have a local maximum near the shallower side of a slope for relatively long unidirectional waves. Copyright 2012 American Institute of Physics. This article may be downloaded for personal use only. Any other use requires prior permission of the author and the American Institute of Physics. The following article appeared in Phys. Fluids 24, 097101 (2012)

• Gramstad, Odin & Trulsen, Karsten (2011). Hamiltonian form of the modified nonlinear Schrödinger equation for gravity waves on arbitrary depth. Journal of Fluid Mechanics. ISSN 0022-1120. 670, p. 404–426. doi: 10.1017/S0022112010005355. Full text in Research Archive
• Gramstad, Odin & Trulsen, Karsten (2011). Fourth-order coupled nonlinear Schrödinger equations for gravity waves on deep water. Physics of Fluids. ISSN 1070-6631. 23(6). doi: 10.1063/1.3598316. Full text in Research Archive
• Gramstad, Odin & Trulsen, Karsten (2010). Can swell increase the number of freak waves in a wind sea? Journal of Fluid Mechanics. ISSN 0022-1120. 650, p. 57–79. doi: 10.1017/S0022112009993491. Full text in Research Archive
• Toffoli, Alessandro; Gramstad, Odin; Trulsen, Karsten; Monbaliu, Jaak; Bitner-Gregersen, Elzbieta & Onorato, Miguel (2010). Evolution of weakly nonlinear random directional waves: laboratory experiments and numerical simulations. Journal of Fluid Mechanics. ISSN 0022-1120. 664, p. 313–336. doi: 10.1017/S002211201000385X. Full text in Research Archive
• Krogstad, Harald E & Trulsen, Karsten (2010). Interpretations and observations of ocean wave spectra. Ocean Dynamics. ISSN 1616-7341. 60(4), p. 973–991. doi: 10.1007/s10236-010-0293-3. Full text in Research Archive
• Hjelmervik, Karina Bakkeløkken & Trulsen, Karsten (2009). Freak wave statistics on collinear currents. Journal of Fluid Mechanics. ISSN 0022-1120. 637, p. 267–284. doi: 10.1017/S0022112009990607. Full text in Research Archive
• Onorato, M; Waseda, T; Toffoli, A; Cavaleri, L; Gramstad, Odin & Janssen, PAEM [Show all 14 contributors for this article] (2009). Statistical Properties of Directional Ocean Waves: The Role of the Modulational Instability in the Formation of Extreme Events. Physical Review Letters. ISSN 0031-9007. 102(11). doi: 10.1103/PhysRevLett.102.114502. Full text in Research Archive
• Onorato, M; Cavaleri, L; Fouques, S; Gramstad, Odin; Janssen, PAEM & Monbaliu, J [Show all 12 contributors for this article] (2009). Statistical properties of mechanically generated surface gravity waves: a laboratory experiment in a three-dimensional wave basin. Journal of Fluid Mechanics. ISSN 0022-1120. 627, p. 235–257. doi: 10.1017/S002211200900603X. Full text in Research Archive
• Gramstad, Odin & Trulsen, Karsten (2007). Influence of crest and group length on the occurrence of freak waves. Journal of Fluid Mechanics. ISSN 0022-1120. 582, p. 463–472. doi: 10.1017/S0022112007006507. Full text in Research Archive
• Trulsen, Karsten (2007). Weakly nonlinear sea surface waves - freak waves and deterministic forecasting. In Hasle, Geir; Lie, Knut-Andreas & Quak, Ewald Gunther (Ed.), Geometric Modelling, Numerical Simulation, and Optimization: Applied Mathematics at SINTEF. Springer Publishing Company. ISSN 9783540687825. p. 191–209. Full text in Research Archive
• Trulsen, Karsten (2006). Weakly nonlinear and stochastic properties of ocean wave fields. Application to an extreme wave event. In Grue, John & Trulsen, Karsten (Ed.), Waves in Geophysical Fluids Tsunamis, Rogue Waves, Internal Waves and Internal Tides. Springer Science+Business Media B.V.. ISSN 9783211374603. p. 49–106. Full text in Research Archive
• Liu, Jingdong; Krogstad, Harald E; Trulsen, Karsten; Dysthe, Kristian & Socquet-Juglard, Herve (2005). The statistical distribution of a nonlinear ocean surface. International Journal of Offshore and Polar Engineering. ISSN 1053-5381. 15(3), p. 168–174. Full text in Research Archive
• Socquet-Juglard, Herve; Dysthe, Kristian; Trulsen, Karsten; Krogstad, Harald E. & Liu, Jingdong (2005). Probability distributions of surface gravity waves during spectral changes. Journal of Fluid Mechanics. ISSN 0022-1120. 542, p. 195–216. doi: 10.1017/S0022112005006312. Full text in Research Archive
• Trulsen, Karsten (2005). Spatial evolution of water surface waves, Fifth International Symposium on Ocean Wave Measurement and Analysis. WAVES 2005. IAHR Secretariat. ISSN 9789078046028. Full text in Research Archive Show summary

  Weakly nonlinear phase-resolving modeling of the spatial evolution of water surface wave trains is capable of predicting the deterministic evolution of long-crested waves over a substantial number of wavelengths. The extension of this capability to directionally distributed waves is discussed. Application to long wave tanks are given.

• Krogstad, Harald E; Liu, Jingdong; Socquet-Juglar, Herve; Dysthe, Kristian & Trulsen, Karsten (2004). Spatial Extreme Value Analysis of Nonlinear Simulations of Random Surface Waves. In ASME, . (Eds.), 23rd International Conference on Offshore Mechanics and Arctic Engineering, Volume 2. The American Society of Mechanical Engineers (ASME). ISSN 9780791837443. doi: 10.1115/OMAE2004-51336. Full text in Research Archive Show summary

  The paper first recalls the Slepian Model Representation and a theorem of V. I. Piterbarg as generic tools for analyzing the spatial characteristic of ocean waves. We then consider numerical simulations of random surface gravity waves carried out in space and time by means of the modified nonlinear Schrödinger equation. It is shown that the extreme waves in the simulations are steeper and more asymmetric than predicted by the Gaussian theory. Moreover, the reconstructed wave fields shows extreme crest heights well in excess of what is expected from the Gaussian theory.

• Selezov, Igor; Avramenko, Olga; Kharif, Christian & Trulsen, Karsten (2003). High-order evolution equation for nonlinear wave-packet propagation with surface tension accounting. Comptes rendus de l'Académie des sciences. Série IIb, Mécanique. ISSN 1620-7742. 331, p. 197–201. doi: 10.1016/S1631-0721(03)00043-3. Full text in Research Archive
• Trulsen, Karsten (2003). Real time deterministic ocean wave forecasting using weakly nonlinear wave evolution theory, MekIT'03, Second national conference on Computational Mechanics, Trondheim 8-9 May 2003. Tapir Akademisk Forlag. ISSN 9788251918688. p. 371–390. Full text in Research Archive
• Dysthe, Kristian; Trulsen, Karsten; Krogstad, Harald E & Socquet-Juglard, Herve (2003). Evolution of a narrow-band spectrum of random surface gravity waves. Journal of Fluid Mechanics. ISSN 0022-1120. 478, p. 1–10. doi: 10.1017/S0022112002002616. Full text in Research Archive
• Trulsen, Karsten & Teigen, Per Steinar (2002). Wave scattering around a vertical cylinder: Fully nonlinear potential flow calculations compared with low order perturbation results and experiment, 21ST International Conference on Offshore Mechanics and Arctic Engineering. American Society of Mechanical Engineers (ASME). ISSN 9780791835999. Full text in Research Archive
• Morozov, Eugene G.; Trulsen, Karsten; Velarde, Manuel G. & Vlasenko, Vasily I. (2002). Internal tides in the Strait of Gibraltar. Journal of Physical Oceanography. ISSN 0022-3670. 32, p. 3193–3206. doi: 10.1175/ITITSO%253E2.0.CO;2. Full text in Research Archive
• Trulsen, Karsten (2001). Simulating the spatial evolution of a measured time series of a freak wave. In Olagnon, Michel & Athanassoulis, Gerassimos (Ed.), Rogue waves 2000. Editions Ifremer. ISSN 9782844330635. p. 265–273. Full text in Research Archive Show summary

  Proceedings of a Workshop organized by Ifremer and held in Brest, France 29-30 November 2000 within the Brest SeaTechWeek 2000.

• Teigen, Per Steinar & Trulsen, Karsten (2001). Numerical investigation of nonlinear wave effects around multiple cylinders. In Chung, Jin S.; Prevosto, Marc & Mizutani, Norimi (Ed.), The Proceedings of The Eleventh (2001) International Offshore and Polar Engineering Conference. International Society of Offshore and Polar Engineers (ISOPE). ISSN 9781880653548. p. 369–378. Full text in Research Archive
• Trulsen, Karsten & Stansberg, Carl Trygve (2001). Spatial evolution of water surface waves: Numerical simulation and experiment of bichromatic waves. In Chung, Jin S.; Prevosto, Marc & Mizutani, Norimi (Ed.), The Proceedings of The Eleventh (2001) International Offshore and Polar Engineering Conference. International Society of Offshore and Polar Engineers (ISOPE). ISSN 9781880653548. p. 71–77. Full text in Research Archive Show summary

  The modified nonlinear Schrödinger (MNLS) equation for spatial evolution of weakly nonlinear water surface waves is shown to yield good comparisons with experimental measurements of bichromatic waves in a long tank. While linear theory does not predict neither the phase velocity nor the evolution of the envelope well, the cubic nonlinear Schrödinger (NLS) equation improves the prediction of the phase velocity but not the modulation of the envelope. The MNLS equation predicts both the evolution of individual wave crests and the modulation of the envelope over longer fetch, and thus permits accurate forecasting of individual ocean wave crests over a fetch of several tens of wavelengths.

• Trulsen, Karsten; Gudmestad, Ove Tobias & Velarde, Manuel G. (2001). The nonlinear Schrödinger method for water wave kinematics on finite depth. Wave motion. ISSN 0165-2125. 33(4), p. 379–395. doi: 10.1016/S0165-2125(00)00077-9. Full text in Research Archive
• Trulsen, Karsten; Kliakhandler, Igor L; Dysthe, Kristian & Velarde, Manuel G. (2000). On weakly nonlinear modulation of waves on deep water. Physics of Fluids. ISSN 1070-6631. 12(10), p. 2432–2437. doi: 10.1063/1.1287856. Full text in Research Archive Show summary

  We propose a new approach for modeling weakly nonlinear waves, based on enhancing truncated amplitude equations with exact linear dispersion. Our example is based on the nonlinear Schrödinger (NLS) equation for deep-water waves. The enhanced NLS equation reproduces exactly the conditions for nonlinear four-wave resonance (the "figure 8" of Phillips) even for bandwidths greater than unity. Sideband instability for uniform Stokes waves is limited to finite bandwidths only, and agrees well with exact results of McLean; therefore, sideband instability cannot produce energy leakage to high-wave-number modes for the enhanced equation, as reported previously for the NLS equation. The new equation is extractable from the Zakharov integral equation, and can be regarded as an intermediate between the latter and the NLS equation. Being solvable numerically at no additional cost in comparison with the NLS equation, the new model is physically and numerically attractive for investigation of wave evolution.

• Trulsen, Karsten; Stansberg, Carl Trygve & Velarde, Manuel G. (1999). Laboratory evidence of three-dimensional frequency downshift of waves in a long tank. Physics of Fluids. ISSN 1070-6631. 11, p. 235–237. doi: 10.1063/1.869915. Full text in Research Archive
• Dysthe, Kristian & Trulsen, Karsten (1999). Note on breather type solutions of the NLS as models for freak-waves. Physica Scripta. ISSN 0031-8949. T82, p. 48–52. doi: 10.1238/Physica.Topical.082a00048. Full text in Research Archive
• Trulsen, Karsten (1999). Wave Kinematics Computed With the Nonlinear Schrödinger Method for Deep Water. Journal of Offshore Mechanics and Arctic Engineering. ISSN 0892-7219. 121, p. 126–130. doi: 10.1115/1.2830077. Full text in Research Archive
• Trulsen, Karsten (1998). Crest pairing predicted by modulation theory. Journal of Geophysical Research (JGR). ISSN 0148-0227. 103(C2), p. 3143–3147. doi: 10.1029/97JC03284. Full text in Research Archive Show summary

  Crest pairing of water surface waves is a consequence of linear dispe rsion alone and is predicted by the linear Schrodinger equation for t he slow evolution of a wave train. Linear theory also explains why cr est pairing is observed in front of a wave group. The role of nonline arity is to redistribute sufficient energy to the sidebands such that the linear dispersive mechanism can become effective. There is also quantitative agreement with experimental observations for the propaga tion of a wave group faster than the linear group velocity.

• Trulsen, Karsten; Temmos, Jean-Marc; Dysthe, Kristian; Gudmestad, Ove Tobias & Velarde, Manuel G. (1998). The Schrödinger method for water wave kinematics. In Zhang, Jun (Eds.), Ocean wave kinematics, dynamics and loads on structures. American Society of Civil Engineers. ISSN 9780784403365. p. 76–83. Full text in Research Archive Show summary

  Proceedings of the 1998 international OTRC symposium, April 30-May 1, 1998, Houston, Texas.

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• Grue, John & Trulsen, Karsten (2006). Waves in Geophysical Fluids Tsunamis, Rogue Waves, Internal Waves and Internal Tides. Springer Science+Business Media B.V.. ISBN 9783211374603. 344 p. Full text in Research Archive

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• Torgersen, Eivind; Trulsen, Karsten; Samseth, Karen & Berner, Eva Charlotte (2022). Oppfordrer til skjerpet sikkerhet etter eksperimenter med ekstrembølger under vann. [Journal]. https://titan.uio.no/. Full text in Research Archive
• Torgersen, Eivind; Trulsen, Karsten; Samseth, Karen & Berner, Eva Charlotte (2022). Oppfordrer til skjerpet sikkerhet etter eksperimenter med ekstrembølger under vann. [Journal]. https://www.tu.no/. Full text in Research Archive
• Trulsen, Karsten (2021). Ekstrembølger i havet – hva er det, hvordan oppstår de og hvilken betydning kan de ha for virksomhet til havs. Full text in Research Archive
• Magnusson, Anne Karin; Trulsen, Karsten; Bitner-Gregersen, Elzbieta M.; Aarnes, Ole Johan & Malila, Mika Petteri (2019). “Three Sisters” Measured as a Triple Rogue Wave Group. Full text in Research Archive
• Seemann, J.; Carrasco, R.; Stresser, M.; Horstmann, J.; Støle-Hentschel, Susanne & Trulsen, Karsten [Show all 7 contributors for this article] (2017). An Operational Wave Monitoring System Based on a Dopplerized Marine Radar. Full text in Research Archive
• Haakstad, Espen & Trulsen, Karsten (2016). Kan en monsterbølge ha forårsaket dette? [Journal]. https://titan.uio.no. Full text in Research Archive
• Støle-Hentschel, Susanne; Rye, Lisa Bæverfjord; Raustøl, Anne & Trulsen, Karsten (2016). Extreme seas and non-linear waves. Full text in Research Archive
• Haakstad, Espen & Trulsen, Karsten (2016). Did a giant wave provoke an environmental disaster? [Journal]. https://titan.uio.no. Full text in Research Archive
• Marstrander, Christian & Trulsen, Karsten (2016). Bølgeforsker kaster nytt lys over oljekatastrofe. [Radio]. NRK Radio - Ekko. Full text in Research Archive
• Vogt, Yngve & Trulsen, Karsten (2016). Norsk matematiker løste gåten bak oljekatastrofe. [Journal]. Apollon. Full text in Research Archive
• Simanesew, Abushet; Krogstad, Harald E; Trulsen, Karsten & Borge, Jose Carlos Nieto (2015). Frequency dependence of directional characteristics of ocean wave spectra. Full text in Research Archive
• Trulsen, Karsten; Raustøl, Anne & Rye, Lisa Bæverfjord (2015). Freak waves in nonlinear unidirectional wave trains over a sloping bottom. Full text in Research Archive
• Trulsen, Karsten; Raustøl, Anne & Rye, Lisa Bæverfjord (2015). Freak waves in nonlinear unidirectional wave trains over a sloping bottom. Full text in Research Archive
• Taklo, Tore Magnus Arnesen; Trulsen, Karsten; Krogstad, Harald E & Borge, Jose Carlos Nieto (2014). Direct measurement of dispersion relation for directional random surface gravity waves. Full text in Research Archive
• Taklo, Tore Magnus Arnesen; Trulsen, Karsten; Krogstad, Harald E; Gramstad, Odin; Borge, Jose Carlos Nieto & Jensen, Atle (2013). Direct measurement of nonlinear dispersion relation for water surface waves. Full text in Research Archive
• Taklo, Tore Magnus Arnesen; Trulsen, Karsten; Gramstad, Odin; Krogstad, Harald E & Jensen, Atle (2013). Direct measurement of dispersion relation for random surface gravity waves on deep water. Full text in Research Archive
• Trulsen, Karsten (2013). Bølgebeistene. [Journal]. Aftenposten Junior. Full text in Research Archive
• Trulsen, Karsten; Taklo, Tore Magnus Arnesen; Borge, Jose Carlos Nieto & Krogstad, Harald E (2012). Experimental validation of the linear dispersion relation, application to the interpretation of radar images. Full text in Research Archive
• Trulsen, Karsten (2012). Når bølgebeistene skyller inn. [Journal]. Dagsavisen. Full text in Research Archive
• Trulsen, Karsten (2012). Bølgebeistene. [Internet]. abcnyheter. Full text in Research Archive
• Gramstad, Odin; Lefevre, Jean-Michel & Trulsen, Karsten (2011). Model and case study for freak waves in crossing swell and wind sea. Full text in Research Archive
• Zeng, Huiming; Trulsen, Karsten & Gramstad, Odin (2011). Variation of kurtosis over non-uniform bathymetry. Full text in Research Archive
• Gramstad, Odin; Zeng, Huiming & Trulsen, Karsten (2010). Finite depth and freak waves: Hamiltonian form of the modified nonlinear Schrödinger equation. Can non-uniform bathymetry provoke freak waves? Full text in Research Archive
• Hjelmervik, Karina Bakkeløkken & Trulsen, Karsten (2008). Freak waves on collinear current jets. Full text in Research Archive
• Gramstad, Odin & Trulsen, Karsten (2008). Can swell increase the number of freak waves in a wind-sea? Full text in Research Archive
• Hjelmervik, Karina Bakkeløkken & Trulsen, Karsten (2008). Extreme waves on current jets - are they freak? Full text in Research Archive
• Gramstad, Odin; Cavaleri, L.; Janssen, P.; Monbaliu, J.; Onorato, M. & Osborne, A. [Show all 10 contributors for this article] (2008). Statistical properties of directional surface gravity waves: A laboratory experiment with mechanically generated waves. Full text in Research Archive
• Gramstad, Odin & Trulsen, Karsten (2007). Influence of crest and group length on the occurrence of freak waves. Full text in Research Archive
• Trulsen, Karsten (2005). Spatial evolution of directionally distributed water surface waves: numerical simulation and experiment. Full text in Research Archive Show summary

  Weakly nonlinear phase-resolving modeling of the spatial evolution of water surface wave trains is capable of predicting the deterministic evolution of long-crested waves over a substantial number of wavelengths. The extension of this capability to directionally distributed waves is discussed. Application to long wave tanks are given.

• Dysthe, Kristian B.; Socquet-Juglard, Herve; Trulsen, Karsten; Krogstad, Harald E. & Liu, Jingdong (2005). "Freak" waves and large-scale simulations of surface gravity waves. Full text in Research Archive
• Trulsen, Karsten; Jensen, Atle & Grue, John (2005). Water wave kinematics of steep irregular waves - systematic perturbation approach, empirical law, PIV measurements and engineering practice. Full text in Research Archive
• Jenkins, Alastair David; Magnusson, Anne Karin; Niedermeier, Andreas; Monbaliu, Jaak; Toffoli, Alessandro & Trulsen, Karsten (2004). Rogue waves and extreme events in measured time-series. Full text in Research Archive Show summary

  We report on the wave time-series data used in the MAXWAVE project, with sample time series and statistics of wave height, crest height, and trough depth. Data are included from the Ekofisk and Draupner oil fields, and from Belgian coastal waters. As well as extreme value analysis, wavelet analysis has been were performed. Studies were also made of buoy time series transformed by quasi-Lagrangian correction of the buoy motion, and of laser time series propagated to other locations with up to 220 m separation from the original measurement location using linear and nonlinear techniques.

• Socquet-Juglard, Herve; Dysthe, Kristian; Trulsen, Karsten; Krogstad, Harald E & Liu, Jingdong (2004). Distribution of extreme waves by simulations. Full text in Research Archive
• Socquet-Juglard, Herve; Dysthe, Kristian; Trulsen, Karsten; Fouques, Sébastien; Liu, Jingdong & Krogstad, Harald E (2004). Spatial Extremes, Shapes of Large Waves, and Lagrangian Models. Full text in Research Archive
• Trulsen, Karsten (2004). Transversal crest and group modulation of extreme waves. Full text in Research Archive
• Krogstad, Harald E.; Socquet-Juglard, Herve; Liu, Jingdong; Dysthe, Kristian B. & Trulsen, Karsten (2004). Spatial extreme value analysis of nonlinear simulations of random surface waves. Full text in Research Archive
• Krogstad, Harald E; Dysthe, Kristian B.; Trulsen, Karsten; Socquet-Juglard, Hervé & Liu, Jingdong (2003). Spectral evolution and extreme value analysis of non-linear numerical simulations of narrow band random surface gravity waves. Full text in Research Archive
• Socquet-Juglard, Herve; Dysthe, Kristian B.; Trulsen, Karsten & Krogstad, Harald E. (2003). Evolution of a directional spectrum of random surface graviy waves. Full text in Research Archive
• Socquet-Juglard, Herve; Dysthe, Kristian B.; Trulsen, Karsten; Liu, Jingdong & Krogstad, Harald E. (2003). Spectral evolution and extreme value analysis of non-linear numerical simulations of narrow band random surface gravity waves. Full text in Research Archive
• Dysthe, Kristian & Trulsen, Karsten (2001). The evolution of an evolution equation. Full text in Research Archive
• Trulsen, Karsten & Stansberg, Carl Trygve (1998). Laboratory evidence of frequency downshift in three-imensional waves in a wave tank. Full text in Research Archive
• Trulsen, Karsten; Temmos, Jean-Marc; Dysthe, Kristian; Gudmestad, Ove Tobias & Velarde, Manuel G. (1998). The Schrödinger method for water wave kinematics. Full text in Research Archive Show summary

  The Schrödinger method is developed for computation of water wave kinematics on finite depth, and is applied to laboratory experimental data for irregular waves.

• Hjelmervik, Karina Bakkeløkken & Trulsen, Karsten (2009). New current modified Schrödinger equations. Universitetet i Oslo. Full text in Research Archive Show summary

  New current modified Schrödinger equations are derived suited to study waves on both potential and non-potential inhomogeneous currents. Split-step schemes of first, second, and fourth order are discussed. Different results are presented regarding the current terms and the model setup. This paper mainly serve as background information for Hjelmervik & Trulsen (2009), but the current modified Schrödinger equations and model setup presented here are expected to have an even larger range of application possibilities.

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