Experimental and numerical evidence have shown that nonuniform bathymetry may alter significantly... more 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.
At the Helmholtz Zentrum Geesthacht (HZG), Germany an oceanic monitoring system, based on a Doppl... more At the Helmholtz Zentrum Geesthacht (HZG), Germany an oceanic monitoring system, based on a Dopplerized microwave radar was developed. Within this paper we present the capabilities of this system to measure ocean surface wave properties such as significant wave height, peak wave period, peak wave lengths as well as other spectral wave parameters. In addition, the system is able to detect wave breaking, shear stress of the wind field, near-surface currents, and the bathymetry in shallow water areas. Because of the diversity of measurement capabilities, the system particularly well suited for monitoring and investigation of hydrographic and oceanographic process.
The authors discuss the spatio-temporal domain, here referred to as the predictable zone, in whic... more The authors discuss the spatio-temporal domain, here referred to as the predictable zone, in which waves can be predicted deterministically based on an observation in a limited spatial or temporal domain. A key issue is whether the group or phase speed of the observed waves governs the extent of the predictable zone. The authors have addressed this issue again using linear wave theory on both computer-generated synthetic wave fields and laboratory experimental observations. The authors find that the group speed adequately indicates the predictable zone for forecasting horizons relevant for offshore and maritime applications.
Resonant interaction of wave trains on the sea surface has been recognized as a fundamental mode ... more Resonant interaction of wave trains on the sea surface has been recognized as a fundamental mode of energy interchange between waves of different lengths, directions and frequencies. To predict the development of the entire sea-wave spectrum, it is further necessary to understand the mutual influence of short and long waves. Such understanding is also useful for interpreting remote sensing data by microwave radar. In the high frequency end of the spectrum, corresponding to microwaves in the X-band, resonance of three trains of short gravity—capillary McGoldrick waves has been examined by McGoldrick (1965), Simmons (1969), Meiss and Watson (1978) and others. It is natural to ask how this resonance behavior might change in the presence of a much longer wave. Our purpose here is to study the evolution of three resonating gravity—capillary waves propagating in different directions on a long gravity wave. This topic has been considered in the closely related paper Trulsen and Mei (1995) which we subsequently refer to as TM
There has been much interest in freak or rogue waves in recent years, especially after the Draupn... more There has been much interest in freak or rogue waves in recent years, especially after the Draupner “New Year Wave” that occurred in the central North Sea on January 1st 1995. From the beginning there have been two main research directions, deterministic and statistical. The deterministic approach has concentrated on focusing mechanisms and modulational instabilities, these are explained in Chap. 3 and some examples are also given in Chap. 4. A problem with many of these deterministic theories is that they require initial conditions that are just as unlikely as the freak wave itself, or they require idealized instabilities such as Benjamin-Feir instability to act over unrealistically long distances or times. For this reason the deterministic theories alone are not very useful for understanding how exceptional the freak waves are. On the other hand, a purely statistical approach based on data analysis is difficult due to the unusual character of these waves. Recently a third research direction has proved promising, stochastic analysis based on Monte-Carlo simulations with phase-resolving deterministic models. This approach accounts for all possible mechanisms for generating freak waves, within a sea state that is hopefully as realistic as possible. This chapter presents several different modified nonlinear Schrodinger (MNLS) equations as candidates for simplified phase-resolving models, followed by an introduction to some essential elements of stochastic analysis. The material is aimed at readers with some background in nonlinear wave modeling, but little background in stochastic modeling. Despite their simplicity, the MNLS equations capture remarkably non-trivial physics of the sea surface such as the establishment of a quasi-stationary spectrum with ω−4 power law for the high-frequency tail, and nonlinear probability distributions for extreme waves. In the end we will suggest how often one should expect a “New Year Wave” within the sea state in which it occurred.
21st International Conference on Offshore Mechanics and Arctic Engineering, Volume 1, 2002
A detailed description of a fully nonlinear numerical method for computing wave interaction effec... more A detailed description of a fully nonlinear numerical method for computing wave interaction effects around arbitrary marine structures is presented. The paper highlights application to one specific geometry: A single, fixed vertical truncated cylinder. The fully nonlinear computations are compared with linear and second-order nonlinear results obtained with the perturbation approach, as well as with experiments.
Geometric Modelling, Numerical Simulation, and Optimization
The material contained here is to a large extent motivated by the so-called Draupner “New Year Wa... more The material contained here is to a large extent motivated by the so-called Draupner “New Year Wave”, an extreme wave event that was recorded at the Draupner E platform in the central North Sea on January 1st 1995 [4], [5]. This location has an essentially uniform depth of 70 m. The platform is of jacket type and is not expected to modify the wave field in any significant way. The platform had been given a foundation of a novel type, and for this reason was instrumented with a large number of sensors measuring environmental data, structural and foundation response. We are particularly interested in measurements taken by a down looking laser-based wave sensor, recording surface elevation at a speed of 2.1333 Hz during 20 minutes of every hour. The full 20 minute time series recorded starting at 1520 GMT is shown in Figure 1 and a close-up of the extreme wave event is shown in Figure 2. To remove any doubt that the measurements are of good quality, Figure 3 shows an even finer close-up with the individual measurements indicated. It is clear that the extreme wave is not an isolated erroneous measurement. The minimum distance between the sensor and the water surface was 7.4 m.
We report laboratory experiments and numerical simulations of the Zakharov equation, designed to ... more 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.
ABSTRACT Nonlinear problems of wave-packet propagation along the interface between two fluids of ... more ABSTRACT Nonlinear problems of wave-packet propagation along the interface between two fluids of different densities taking into account the surface tension are investigated. Two problems are considered, one for two half-spaces, the other for the layer over a half-space. Asymptotic solutions are developed on the basis of the method of multiple scale expansions. Unlike previous investigations dealing with only three approximations in this paper four asymptotic approximations, have been developed by using symbolic algebra. The evolution equations are obtained in the form of nonlinear higher-order Schrödinger equations. The stability of solutions is investigated. As a result, the new region of stability for capillary waves and the new region of instability for gravity waves have been discovered in the case of the layer of finite thickness unlike the case of two fluid half-spaces.
Extreme value analysis of measured and simulated ocean waves has a long history, and the one-dime... more Extreme value analysis of measured and simulated ocean waves has a long history, and the one-dimensional theory is well developed. Today, we are starting to obtain spatial measurements of waves by remote sensing techniques (Schulz-Stellen ‡eth and Lehner, 2004), and discussions about spatial and even spatio/temporal extremes arise. For rogue waves the question naturally comes up whether such a wave is truly exceptional, or whether it may actually happen somewhere or sometimes in a storm sea with considerable temporal and spatial extent. Another question is the spatial shapes of the large waves.
It is well-known that a uniform train of surface gravity waves is unstable to the so- called Benj... more It is well-known that a uniform train of surface gravity waves is unstable to the so- called Benjamin-Feir (BF) instability. The stability of a narrow band of Gaussian random wavetrains was considered by Alber and Saffman (1978), Alber (1978) and Crawford et al. (1980). For waves on deep water they found that the BF instability persists provided that the relative
ABSTRACT The linear dispersion relation for water surface waves is often taken for granted for th... more ABSTRACT The linear dispersion relation for water surface waves is often taken for granted for the interpretation of wave measurements. High-resolution spatiotemporal measurements suitable for direct validation of the linear dispersion relation are on the other hand rarely available. While the imaging of the ocean surface with nautical radar does provide the desired spatiotemporal coverage, the interpretation of the radar images currently depends on the linear dispersion relation as a prerequisite, (Nieto Borge et al., 2004). Krogstad & Trulsen (2010) carried out numerical simulations with the nonlinear Schrödinger equation and its generalizations demonstrating that the nonlinear evolution of wave fields may render the linear dispersion relation inadequate for proper interpretation of observations, the reason being that the necessary domain of simultaneous coverage in space and time would allow significant nonlinear evolution. They found that components above the spectral peak can have larger phase and group velocities than anticipated by linear theory, and that the spectrum does not maintain a thin dispersion surface. We have run laboratory experiments and accurate numerical simulations designed to have sufficient resolution in space and time to deduce the dispersion relation directly. For a JONSWAP spectrum we find that the linear dispersion relation can be appropriate for the interpretation of spatiotemporal measurements. For a Gaussian spectrum with narrower bandwidth we find that the dynamic nonlinear evolution in space and time causes the directly measured dispersion relation to deviate from the linear dispersion surface in good agreement with our previous numerical predictions. This work has been supported by RCN grant 214556/F20. Krogstad, H. E. & Trulsen, K. (2010) Interpretations and observations of ocean wave spectra. Ocean Dynamics 60:973-991. Nieto Borge, J. C., Rodríguez, G., Hessner, K., Izquierdo, P. (2004) Inversion of marine radar images for surface wave analysis. J. Atmos. Ocean. Tech. 21:1291-1300.
ABSTRACT Evolution of statistics of weakly nonlinear unidirectional waves over a sloping bottom i... more ABSTRACT Evolution of statistics of weakly nonlinear unidirectional waves over a sloping bottom in shallow water
Experimental and numerical evidence have shown that nonuniform bathymetry may alter significantly... more 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.
At the Helmholtz Zentrum Geesthacht (HZG), Germany an oceanic monitoring system, based on a Doppl... more At the Helmholtz Zentrum Geesthacht (HZG), Germany an oceanic monitoring system, based on a Dopplerized microwave radar was developed. Within this paper we present the capabilities of this system to measure ocean surface wave properties such as significant wave height, peak wave period, peak wave lengths as well as other spectral wave parameters. In addition, the system is able to detect wave breaking, shear stress of the wind field, near-surface currents, and the bathymetry in shallow water areas. Because of the diversity of measurement capabilities, the system particularly well suited for monitoring and investigation of hydrographic and oceanographic process.
The authors discuss the spatio-temporal domain, here referred to as the predictable zone, in whic... more The authors discuss the spatio-temporal domain, here referred to as the predictable zone, in which waves can be predicted deterministically based on an observation in a limited spatial or temporal domain. A key issue is whether the group or phase speed of the observed waves governs the extent of the predictable zone. The authors have addressed this issue again using linear wave theory on both computer-generated synthetic wave fields and laboratory experimental observations. The authors find that the group speed adequately indicates the predictable zone for forecasting horizons relevant for offshore and maritime applications.
Resonant interaction of wave trains on the sea surface has been recognized as a fundamental mode ... more Resonant interaction of wave trains on the sea surface has been recognized as a fundamental mode of energy interchange between waves of different lengths, directions and frequencies. To predict the development of the entire sea-wave spectrum, it is further necessary to understand the mutual influence of short and long waves. Such understanding is also useful for interpreting remote sensing data by microwave radar. In the high frequency end of the spectrum, corresponding to microwaves in the X-band, resonance of three trains of short gravity—capillary McGoldrick waves has been examined by McGoldrick (1965), Simmons (1969), Meiss and Watson (1978) and others. It is natural to ask how this resonance behavior might change in the presence of a much longer wave. Our purpose here is to study the evolution of three resonating gravity—capillary waves propagating in different directions on a long gravity wave. This topic has been considered in the closely related paper Trulsen and Mei (1995) which we subsequently refer to as TM
There has been much interest in freak or rogue waves in recent years, especially after the Draupn... more There has been much interest in freak or rogue waves in recent years, especially after the Draupner “New Year Wave” that occurred in the central North Sea on January 1st 1995. From the beginning there have been two main research directions, deterministic and statistical. The deterministic approach has concentrated on focusing mechanisms and modulational instabilities, these are explained in Chap. 3 and some examples are also given in Chap. 4. A problem with many of these deterministic theories is that they require initial conditions that are just as unlikely as the freak wave itself, or they require idealized instabilities such as Benjamin-Feir instability to act over unrealistically long distances or times. For this reason the deterministic theories alone are not very useful for understanding how exceptional the freak waves are. On the other hand, a purely statistical approach based on data analysis is difficult due to the unusual character of these waves. Recently a third research direction has proved promising, stochastic analysis based on Monte-Carlo simulations with phase-resolving deterministic models. This approach accounts for all possible mechanisms for generating freak waves, within a sea state that is hopefully as realistic as possible. This chapter presents several different modified nonlinear Schrodinger (MNLS) equations as candidates for simplified phase-resolving models, followed by an introduction to some essential elements of stochastic analysis. The material is aimed at readers with some background in nonlinear wave modeling, but little background in stochastic modeling. Despite their simplicity, the MNLS equations capture remarkably non-trivial physics of the sea surface such as the establishment of a quasi-stationary spectrum with ω−4 power law for the high-frequency tail, and nonlinear probability distributions for extreme waves. In the end we will suggest how often one should expect a “New Year Wave” within the sea state in which it occurred.
21st International Conference on Offshore Mechanics and Arctic Engineering, Volume 1, 2002
A detailed description of a fully nonlinear numerical method for computing wave interaction effec... more A detailed description of a fully nonlinear numerical method for computing wave interaction effects around arbitrary marine structures is presented. The paper highlights application to one specific geometry: A single, fixed vertical truncated cylinder. The fully nonlinear computations are compared with linear and second-order nonlinear results obtained with the perturbation approach, as well as with experiments.
Geometric Modelling, Numerical Simulation, and Optimization
The material contained here is to a large extent motivated by the so-called Draupner “New Year Wa... more The material contained here is to a large extent motivated by the so-called Draupner “New Year Wave”, an extreme wave event that was recorded at the Draupner E platform in the central North Sea on January 1st 1995 [4], [5]. This location has an essentially uniform depth of 70 m. The platform is of jacket type and is not expected to modify the wave field in any significant way. The platform had been given a foundation of a novel type, and for this reason was instrumented with a large number of sensors measuring environmental data, structural and foundation response. We are particularly interested in measurements taken by a down looking laser-based wave sensor, recording surface elevation at a speed of 2.1333 Hz during 20 minutes of every hour. The full 20 minute time series recorded starting at 1520 GMT is shown in Figure 1 and a close-up of the extreme wave event is shown in Figure 2. To remove any doubt that the measurements are of good quality, Figure 3 shows an even finer close-up with the individual measurements indicated. It is clear that the extreme wave is not an isolated erroneous measurement. The minimum distance between the sensor and the water surface was 7.4 m.
We report laboratory experiments and numerical simulations of the Zakharov equation, designed to ... more 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.
ABSTRACT Nonlinear problems of wave-packet propagation along the interface between two fluids of ... more ABSTRACT Nonlinear problems of wave-packet propagation along the interface between two fluids of different densities taking into account the surface tension are investigated. Two problems are considered, one for two half-spaces, the other for the layer over a half-space. Asymptotic solutions are developed on the basis of the method of multiple scale expansions. Unlike previous investigations dealing with only three approximations in this paper four asymptotic approximations, have been developed by using symbolic algebra. The evolution equations are obtained in the form of nonlinear higher-order Schrödinger equations. The stability of solutions is investigated. As a result, the new region of stability for capillary waves and the new region of instability for gravity waves have been discovered in the case of the layer of finite thickness unlike the case of two fluid half-spaces.
Extreme value analysis of measured and simulated ocean waves has a long history, and the one-dime... more Extreme value analysis of measured and simulated ocean waves has a long history, and the one-dimensional theory is well developed. Today, we are starting to obtain spatial measurements of waves by remote sensing techniques (Schulz-Stellen ‡eth and Lehner, 2004), and discussions about spatial and even spatio/temporal extremes arise. For rogue waves the question naturally comes up whether such a wave is truly exceptional, or whether it may actually happen somewhere or sometimes in a storm sea with considerable temporal and spatial extent. Another question is the spatial shapes of the large waves.
It is well-known that a uniform train of surface gravity waves is unstable to the so- called Benj... more It is well-known that a uniform train of surface gravity waves is unstable to the so- called Benjamin-Feir (BF) instability. The stability of a narrow band of Gaussian random wavetrains was considered by Alber and Saffman (1978), Alber (1978) and Crawford et al. (1980). For waves on deep water they found that the BF instability persists provided that the relative
ABSTRACT The linear dispersion relation for water surface waves is often taken for granted for th... more ABSTRACT The linear dispersion relation for water surface waves is often taken for granted for the interpretation of wave measurements. High-resolution spatiotemporal measurements suitable for direct validation of the linear dispersion relation are on the other hand rarely available. While the imaging of the ocean surface with nautical radar does provide the desired spatiotemporal coverage, the interpretation of the radar images currently depends on the linear dispersion relation as a prerequisite, (Nieto Borge et al., 2004). Krogstad & Trulsen (2010) carried out numerical simulations with the nonlinear Schrödinger equation and its generalizations demonstrating that the nonlinear evolution of wave fields may render the linear dispersion relation inadequate for proper interpretation of observations, the reason being that the necessary domain of simultaneous coverage in space and time would allow significant nonlinear evolution. They found that components above the spectral peak can have larger phase and group velocities than anticipated by linear theory, and that the spectrum does not maintain a thin dispersion surface. We have run laboratory experiments and accurate numerical simulations designed to have sufficient resolution in space and time to deduce the dispersion relation directly. For a JONSWAP spectrum we find that the linear dispersion relation can be appropriate for the interpretation of spatiotemporal measurements. For a Gaussian spectrum with narrower bandwidth we find that the dynamic nonlinear evolution in space and time causes the directly measured dispersion relation to deviate from the linear dispersion surface in good agreement with our previous numerical predictions. This work has been supported by RCN grant 214556/F20. Krogstad, H. E. & Trulsen, K. (2010) Interpretations and observations of ocean wave spectra. Ocean Dynamics 60:973-991. Nieto Borge, J. C., Rodríguez, G., Hessner, K., Izquierdo, P. (2004) Inversion of marine radar images for surface wave analysis. J. Atmos. Ocean. Tech. 21:1291-1300.
ABSTRACT Evolution of statistics of weakly nonlinear unidirectional waves over a sloping bottom i... more ABSTRACT Evolution of statistics of weakly nonlinear unidirectional waves over a sloping bottom in shallow water
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