A Second Order Lagrangian Model for Irregular Ocean Waves

2004 ◽  
Author(s):  
Se´bastien Fouques ◽  
Harald E. Krogstad ◽  
Dag Myrhaug
Keyword(s):  
Specular Reflection ◽  
Fluid Velocity ◽  
Equations Of Motion ◽  
Wave Theory ◽  
Ocean Waves ◽  
Second Order ◽  
Lagrangian Model ◽  
Lagrangian Description ◽  
Linear Wave ◽  
Irregular Solution

Synthetic Aperture Radar (SAR) imaging of ocean waves involves both the geometry and the kinematics of the sea surface. However, the traditional linear wave theory fails to describe steep waves, which are likely to bring about specular reflection of the radar beam, and it may overestimate the surface fluid velocity that causes the so-called velocity bunching effect. Recently, the interest for a Lagrangian description of ocean gravity waves has increased. Such an approach considers the motion of individual labeled fluid particles and the free surface elevation is derived from the surface particles positions. The first order regular solution to the Lagrangian equations of motion for an inviscid and incompressible fluid is the so-called Gerstner wave. It shows realistic features such as sharper crests and broader troughs as the wave steepness increases. This paper proposes a second order irregular solution to these equations. The general features of the first and second order waves are described, and some statistical properties of various surface parameters such as the orbital velocity, the slope and the mean curvature are studied.

A Second Order Lagrangian Model for Irregular Ocean Waves

2005 ◽  
Vol 128 (3) ◽  
pp. 177-183 ◽  
Author(s):  
Sébastien Fouques ◽  
Harald E. Krogstad ◽  
Dag Myrhaug
Keyword(s):  
Specular Reflection ◽  
Fluid Velocity ◽  
Equations Of Motion ◽  
Wave Theory ◽  
Ocean Waves ◽  
Second Order ◽  
Lagrangian Model ◽  
Lagrangian Description ◽  
Linear Wave ◽  
Irregular Solution

Synthetic aperture radar (SAR) imaging of ocean waves involves both the geometry and the kinematics of the sea surface. However, the traditional linear wave theory fails to describe steep waves, which are likely to bring about specular reflection of the radar beam, and it may overestimate the surface fluid velocity that causes the so-called velocity bunching effect. Recently, the interest for a Lagrangian description of ocean gravity waves has increased. Such an approach considers the motion of individual labeled fluid particles and the free surface elevation is derived from the surface particles positions. The first order regular solution to the Lagrangian equations of motion for an inviscid and incompressible fluid is the so-called Gerstner wave. It shows realistic features such as sharper crests and broader troughs as the wave steepness increases. This paper proposes a second order irregular solution to these equations. The general features of the first and second order waves are described, and some statistical properties of various surface parameters such as the orbital velocity, slope, and mean curvature are studied.

A Note on the Second-Order Contribution to Extreme Waves Generated During Hurricanes

2019 ◽  
Vol 141 (4) ◽  
Author(s):  
Mark L. McAllister ◽  
Thomas A. A. Adcock ◽  
Paul H. Taylor ◽  
Ton S. van den Bremer
Keyword(s):  
Wind Waves ◽  
Low Frequency ◽  
Nonlinear Behavior ◽  
Wave Theory ◽  
Linear Range ◽  
Second Order ◽  
Linear Wave ◽  
Extreme Waves ◽  
Order Difference ◽  
Hurricane Wind

High wind speeds generated during hurricanes result in the formation of extreme waves. Extreme waves by nature are steep meaning that linear wave theory alone is insufficient in understanding and predicting their occurrence. The complex, highly transient nature of the direction of wind and hence of waves generated during hurricanes affects this nonlinear behavior. Herein, we examine how this directionality can affect the second-order nonlinearity of extreme waves generated during hurricanes. This is achieved through both deterministic calculations and experiments based on the observations of Young (2006, “Directional Spectra of Hurricane Wind Waves,” J. Geophys. Res. Oceans, 111(C8), epub). Our calculations show that interactions between the tail and peak of the spectrum can become significant when they travel in different directions, resulting in second-order difference components that exist in the linear range of frequencies. These calculations are generally supported by experimental observations, but we note the difficulty of generating and focusing the high-frequency tail of the spectrum experimentally. Bound second-order difference components or subharmonics typically exist as low frequency infra-gravity waves. Components that exist in the linear range of frequencies may be missed by conventional methods of processing field data where low-pass filtering is used and hence overlooked. In this note, we show that in idealized directional spreading conditions representative of a hurricane, failing to account for second-order difference components may lead to underestimation of extreme wave height.

A Modified Linear Lagrangian Model for Irregular Long-Crested Waves

2009 ◽  
Author(s):  
Se´bastien Fouques ◽  
Carl Trygve Stansberg
Keyword(s):  
Mass Conservation ◽  
Second Order ◽  
Lagrangian Model ◽  
Wave Crest ◽  
Lagrangian Description ◽  
Wave Steepness ◽  
Linear Solution ◽  
Fluid Particles ◽  
Wave Properties ◽  
Crest Height

Wave crest height and steepness are crucial parameters for the design of ships and offshore structures. For irregular sea states, they are commonly predicted by using linear wave theory and a Eulerian description of the fluid motion. This theory only applies when the wave steepness is small and it fails to capture extreme wave events. Such linear solutions can also be extended by including second-order terms in order to provide more realistic wave properties. The paper describes a model for irregular long-crested waves that is based on a modified linear solution derived from a Lagrangian description of the fluid, i.e. by considering the motion of individual fluid particles. Lagrangian solutions have the advantage of showing realistic wave profiles with sharp crests and broad troughs already at the first order, whereas these features only appear at the second order when using the Eulerian approach. Still, a severe drawback with the former is that the mass conservation is not fulfilled exactly. The aim of the modification in the present Lagrangian model is to ensure that the mass conservation is always fulfilled in the solution. This is done by using the second-order residual in the continuity equation to lift up the fluid particles vertically. Comparative investigations of wave properties such as the crest height and the wave steepness are further carried out by making use of both numerical case studies and wave tank recordings. The wave models used in the comparisons include linear and second-order Eulerian solutions as well as the modified linear Lagrangian one.

scholarly journals Phase-Resolved Reconstruction Algorithm and Deterministic Prediction of Nonlinear Ocean Waves From Spatio-Temporal Optical Measurements

2018 ◽  
Author(s):  
Nicolas Desmars ◽  
Yves Pérignon ◽  
Guillaume Ducrozet ◽  
Charles-Antoine Guérin ◽  
Stephan T. Grilli ◽  
...  
Keyword(s):  
Free Surface ◽  
Wave Model ◽  
Reconstruction Algorithm ◽  
Wave Theory ◽  
Ocean Waves ◽  
Optical Measurements ◽  
Linear Wave ◽  
Offshore Structure ◽  
Highly Nonlinear ◽  
Spatio Temporal

We investigate a nonlinear phase-resolved reconstruction algorithm and models for the deterministic prediction of ocean waves based on a large number of spatio-temporal optical measurements of surface elevations. We consider a single sensor (e.g., LIDAR, stereo-video, etc.) mounted on a fixed offshore structure and remotely measuring fields of free surface elevations. Assuming a uniform distribution of measurement points over the sensor aperture angles, the density of free surface observation points geometrically decreases with the distance from the sensor. Additionally, wave shadowing effects occur, which become more important at small viewing angles (i.e., grazing incidence on the surface). These effects result in observations of surface elevation that are sparsely distributed. Here, based on earlier work by [1], we present and discuss the characteristics of an algorithm, aimed at assimilating such sparse data and able to deterministically reconstruct and propagate ocean surface elevations for their prediction in time and space. This algorithm could assist in the automatic steering and control of a variety of surface vehicles. Specifically, we compare prediction results using linear wave theory and the weakly nonlinear Choppy Wave Model [2, 3], extended here to an “improved” second order formulation. The latter model is based on an efficient Lagrangian formulation of the free surface and was shown to be able to model wave properties that are important to the proper representation of nonlinear free surfaces, namely wave shape and celerity. Synthetic datasets from highly nonlinear High Order Spectral simulations are used as reference oceanic surfaces. Predicted results are analyzed over an area that evolves in time, using the theoretical amount of information assimilated during the reconstruction of the wave field. For typical horizons of prediction, we discuss the capabilities of our assimilation process for each wave model considered.

scholarly journals COUPLING STOKES AND CNOIDAL WAVE THEORIES IN A NONLINEAR REFRACTION MODEL

1988 ◽  
Vol 1 (21) ◽  
pp. 42
Author(s):  
Thomas A. Hardy ◽  
Nicholas C. Kraus
Keyword(s):  
Nonlinear Refraction ◽  
Wave Model ◽  
Wave Theory ◽  
Quantitative Agreement ◽  
Wave Mode ◽  
Finite Amplitude ◽  
Second Order ◽  
Stokes Wave ◽  
Linear Wave ◽  
Cnoidal Wave

An efficient numerical model is presented for calculating the refraction and shoaling of finite-amplitude waves over an irregular sea bottom. The model uses third-order Stokes wave theory in relatively deep water and second-order cnoidal wave theory in relatively shallow water. It can also be run using combinations of lower-order wave theories, including a pure linear wave mode. The problem of the connection of Stokes and cnoidal theories is investigated, and it is found that the use of second-order rather than first-order cnoidal theory greatly reduces the connection discontinuity. Calculations are compared with physical model measurements of the height and direction of waves passing over an elliptical shoal. The finite-amplitude wave model gives better qualitative and quantitative agreement with the measurements than the linear model.

Wave Statistics for Intermediate Depth Water—NewWaves and Symmetry

2004 ◽  
Vol 126 (1) ◽  
pp. 54-59 ◽  
Author(s):  
P. H. Taylor ◽  
B. A. Williams
Keyword(s):  
Linear Part ◽  
Nonlinear Behavior ◽  
Wave Theory ◽  
Order Theory ◽  
Small Degree ◽  
Second Order ◽  
Linear Wave ◽  
Intermediate Water ◽  
Wave Statistics ◽  
The Hilbert Transform

A study has been made into the average shape of large crests and troughs during several storms using wave elevation data from the WACSIS measurement program. The analysis techniques adopted were data-driven at all times, in order to test whether second-order wave theory could reproduce important features in the field data. The sea surface displayed obvious nonlinear behavior, reflected in the fact that the shapes of crests were always sharper and larger than their trough equivalents. Assuming that the dominant nonlinear correction is second order in the wave steepness (but without a knowledge of the detailed form of second-order theory), the average shapes of maxima in the underlying linear wave components were shown to match NewWave. This NewWave is the scaled auto-correlation function for a linear random process with the same power spectrum as the measured waves. Thus, NewWave was shown to be an acceptable model for the linear part of large waves on intermediate water depth (here ∼17 m). Assuming that NewWave is a good model for the linear part of large crests and troughs, a value for the second-order coefficient required to estimate crest elevation statistics was derived from the measured data for several storms. This coefficient was in good agreement with the results of the second-order random simulations of Forristall and Prevosto [1]. As well as studying vertical asymmetry, required for crest and trough statistics, horizontal asymmetry was examined using the Hilbert transform. Compared to a large amount of vertical asymmetry, the analysis showed that there was virtually no horizontal asymmetry for the bulk of the waves in the records. However, there is a very small degree of horizontal asymmetry exhibited in the largest waves in the records. Thus, given a surface elevation record, it is difficult to distinguish the direction of the time axis, again consistent with most of the nonlinearity being due to simple second-order bound waves.

Higher Order Wave Loading on Fixed, Slender, Surface-Piercing, Rigid Cylinders

1991 ◽  
Vol 113 (1) ◽  
pp. 23-29
Author(s):  
K. Thiagarajan ◽  
R. E. Baddour
Keyword(s):  
Dynamic Pressure ◽  
Wave Theory ◽  
Offshore Structures ◽  
Second Order ◽  
Pressure Effects ◽  
Inertia Force ◽  
Wave Forces ◽  
Linear Wave ◽  
Wave Loading ◽  
Theoretical Evaluation

The use of Morison’s equation together with the linear wave theory is considered a first approximation to evaluate the inline wave forces on a surface-piercing cylinder. Significant second-order forces are expected to arise from the waterline and dynamic pressure effects, even when a wave is described by the linear theory. Experiments have been carried out at the MUN (Memorial University of Newfoundland) wave tank facility to identify these second-order forces for various wave frequencies and for various cylinder diameters. A strain gage force transducer has been used for this purpose. First and second-order force components have been identified using a Fast Fourier Transform. Theoretical evaluation of wave forces involved computing components from Morison’s equation using second-order Stokes theory. The waterline forces and convective acceleration forces which contribute toward the total second-order force have also been evaluated. First-order results are in acceptance with previously established data. Theoretical considerations for second order are satisfactory. Scatter in second-order experimental results were observed. Different approaches to the second-order inertia force are compared. It is expected that the inclusion of second-order forces will lead to a better representation of wave loading on offshore structures.

Wave Statistics for Intermediate Depth Water: New Waves and Symmetry

2002 ◽  
Author(s):  
P. H. Taylor ◽  
B. A. Williams
Keyword(s):  
Linear Part ◽  
Wave Theory ◽  
Order Theory ◽  
Small Degree ◽  
Second Order ◽  
Linear Wave ◽  
Intermediate Water ◽  
Wave Statistics ◽  
Non Linear ◽  
The Hilbert Transform

A study has been made into the average shape of large crests and troughs during several storms using wave elevation data from the WACSIS measurement programme. The analysis techniques adopted were data-driven at all times, in order to test whether 2nd order wave theory could reproduce important features in the field data. The sea surface displayed obvious non-linear behaviour, reflected in the fact that the shapes of crests were always sharper and larger than their trough equivalents. Assuming that the dominant non-linear correction is second order in the wave steepness (but without a knowledge of the detailed form of 2nd order theory), the average shapes of maxima in the underlying linear wave components were shown to match NewWave. This NewWave is the scaled auto-correlation function for a linear random process with the same power spectrum as the measured waves. Thus, NewWave was shown to be an acceptable model for the linear part of large waves on intermediate water depth (here ∼17m). Assuming that NewWave is a good model for the linear part of large crests and troughs, a value for the second order coefficient required to estimate crest elevation statistics was derived from the measured data for several storms. This coefficient was in good agreement with the results of the 2nd order random simulations of Forristall and Prevosto. As well as studying vertical asymmetry, required for crest and trough statistics, horizontal asymmetry was examined using the Hilbert transform. Compared to a large amount of vertical asymmetry, the analysis showed that there was virtually no horizontal asymmetry for the bulk of the waves in the records. However, there is a very small degree of horizontal asymmetry exhibited in the largest waves in the records. Thus, given a surface elevation record, it is difficult to distinguish the direction of the time axis, again consistent with most of the non-linearity being due to simple 2nd order bound waves.

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