The Focusing of Uni-Directional Gaussian Wave-Groups in Finite Depth: An Approximate NLSE Based Approach

2010 ◽  
Author(s):  
Thomas A. A. Adcock ◽  
Shiqiang Yan
Keyword(s):  
Analytical Model ◽  
Deep Water ◽  
Finite Depth ◽  
Full Potential ◽  
Wave Group ◽  
Freak Waves ◽  
Wave Groups ◽  
Non Linear ◽  
Cubic Nonlinear Schrödinger Equation ◽  
Linear Interactions

The non-linear changes to a NewWave type wave-group are helpful in developing our understanding of the non-linear interactions which can lead to the formation of freak waves. In addition, Gaussian wave-groups are used in model tests where it is useful to have a simple model for their non-linear dynamics. This paper derives a simple analytical model to describe the nonlinear changes to a wave-group as it focuses. This paper is an extension to finite depth of the theory developed for deep water in Adcock & Taylor (2009) (Proc. Roy. Soc. A 465(2110)). The model is derived using the conserved quantities of the cubic nonlinear Schrodinger equation (NLSE). In deep water there are substantial changes to the group shape and spectrum as the wave-group focuses, and the characteristics of these changes are governed by the Benjamin-Feir Index. However, in finite depth the characteristics of the non-linear interactions change, reducing the non-linear changes to the group shape. The analytical model is validated against simulations using the NLSE and against full potential flow solutions using a QALE-FEM numerical scheme. We also compare its predictions against experiments in a physical wavetank. We find that the NLSE, and thus analytical theories derived from it, capture the dominant physics in the evolution of narrowbanded wave-groups.

Ocean Wave Non-Linearity and Wind Input in Directional Seas: Energy Input During Wave-Group Focussing

2018 ◽  
Author(s):  
Thomas A. A. Adcock ◽  
Paul H. Taylor
Keyword(s):  
Energy Input ◽  
Wind Waves ◽  
Rogue Waves ◽  
Wave Group ◽  
Spectral Evolution ◽  
Wave Groups ◽  
Small Influence ◽  
Non Linear ◽  
Linear Effects ◽  
Wind Input

There has been speculation that energy input (wind) can play an important role in the formation of rogue waves in the open ocean. Here we examine the role energy input can play by adding energy to the modified non-linear Schrödinger equation. We consider NewWave type wave-groups with spectra which are realistic for wind waves. We examine the case where energy input is added to the group as the wave-group focuses. We consider whether this energy input can cause significant non-linear effects to the subsequent spatial and spectral evolution. For the parameters considered here we find this to have only a small influence.

Wave breaking onset and strength for two-dimensional deep-water wave groups

2007 ◽  
Vol 585 ◽  
pp. 93-115 ◽  
Author(s):  
MICHAEL L. BANNER ◽  
WILLIAM L. PEIRSON
Keyword(s):  
Growth Rate ◽  
Energy Density ◽  
Deep Water ◽  
Wave Energy ◽  
Water Wave ◽  
Wave Group ◽  
Two Dimensional ◽  
Carrier Wave ◽  
Wave Groups ◽  
Deep Water Wave

The numerical study of J. Song & M. L. Banner (J. Phys. Oceanogr. vol. 32, 2002, p. 254) proposed a generic threshold parameter for predicting the onset of breaking within two-dimensional groups of deep-water gravity waves. Their parameter provides a non-dimensional measure of the wave energy convergence rate and geometrical steepening at the maximum of an evolving nonlinear wave group. They also suggested that this parameter might control the strength of breaking events. The present paper presents the results of a detailed laboratory observational study aimed at validating their proposals.For the breaking onset phase of this study, wave potential energy was measured at successive local envelope maxima of nonlinear deep-water wave groups propagating along a laboratory wave tank. These local maxima correspond alternately to wave group geometries with the group maximum occurring at an extreme carrier wave crest elevation, followed by an extreme carrier wave trough depression. As the nonlinearity increases, these crest and trough maxima can have markedly different local energy densities owing to the strong crest–trough asymmetry. The local total energy density was reconstituted from the potential energy measurements, and made dimensionless using the square of the local (carrier wave) wavenumber. A mean non-dimensional growth rate reflecting the rate of focusing of wave energy at the envelope maximum was obtained by smoothing the local fluctuations.For the cases of idealized nonlinear wave groups investigated, the observations confirmed the evolutionary trends of the modelling results of Song & Banner (2002) with regard to predicting breaking onset. The measurements confirmed the proposed common breaking threshold growth rate of 0.0014±0.0001, as well as the predicted key evolution times: the time taken to reach the energy maximum for recurrence cases; and the time to reach the breaking threshold and then breaking onset, for breaking cases.After the initiation and subsequent cessation of breaking, the measured wave packet mean energy losses and loss rates associated with breaking produced an unexpected finding: the post-breaking mean wave energy did not decrease to the mean energy level corresponding to maximum recurrence, but remained significantly higher. Therefore, pre-breaking absolute wave energy or mean steepness do not appear to be the most fundamental determinants of post-breaking wave packet energy density.However, the dependence of the fractional breaking energy loss of wave packets on the parametric growth rate just before breaking onset proposed by Song & Banner (2002) was found to provide a plausible collapse to our laboratory data sets, within the experimental uncertainties. Further, when the results for the energy loss rate per unit width of breaking front were expressed in terms of a breaker strength parameter b multiplying the fifth power of the wave speed, it is found that b was also strongly correlated with the parametric growth rate just before breaking. Measured values of b obtained in this investigation ranged systematically from 8 × 10−5 to 1.2 × 10−3. These are comparable with open ocean estimates reported in recent field studies.

scholarly journals MNLS simulations of surface wave groups with directional spreading in deep and finite depth waters

2021 ◽  
Author(s):  
Dylan Barratt ◽  
Ton Stefan van den Bremer ◽  
Thomas Alan Adcock Adcock
Keyword(s):  
Potential Flow ◽  
Finite Depth ◽  
Spectral Peak ◽  
Wave Group ◽  
Carrier Wave ◽  
Linear Dispersion ◽  
Wave Groups ◽  
Fully Nonlinear ◽  
Nonlinear Potential ◽  
The Impact

AbstractWe simulate focusing surface gravity wave groups with directional spreading using the modified nonlinear Schrödinger (MNLS) equation and compare the results with a fully-nonlinear potential flow code, OceanWave3D. We alter the direction and characteristic wavenumber of the MNLS carrier wave, to assess the impact on the simulation results. Both a truncated (fifth-order) and exact version of the linear dispersion operator are used for the MNLS equation. The wave groups are based on the theory of quasi-determinism and a narrow-banded Gaussian spectrum. We find that the truncated and exact dispersion operators both perform well if: (1) the direction of the carrier wave aligns with the direction of wave group propagation; (2) the characteristic wavenumber of the carrier wave coincides with the initial spectral peak. However, the MNLS simulations based on the exact linear dispersion operator perform significantly better if the direction of the carrier wave does not align with the wave group direction or if the characteristic wavenumber does not coincide with the initial spectral peak. We also perform finite-depth simulations with the MNLS equation for dimensionless depths ($$k_{\text {p}}d$$ k p d ) between 1.36 and 5.59, incorporating depth into the boundary conditions as well as the dispersion operator, and compare the results with those of fully-nonlinear potential flow code to assess the finite-depth limitations of the MNLS.

scholarly journals On the Rapid Spectral Evolution of Steep Wave Groups with Directional Spreading at Intermediate Depths

2020 ◽  
Author(s):  
Dylan Barratt ◽  
Harry B. Bingham ◽  
Paul H. Taylor ◽  
Ton S. van den Bremer ◽  
Thomas A. A. Adcock
Keyword(s):  
Wave Train ◽  
Three Dimensional ◽  
Finite Depth ◽  
Wave Group ◽  
Spectral Evolution ◽  
Wave Groups ◽  
Return Current ◽  
Resonant Interactions ◽  
Energy Transfers ◽  
Steep Wave

<p>We have performed numerical simulations of steep three-dimensional wave groups, formed by dispersive focusing, using the fully-nonlinear potential flow solver <em>OceanWave3D</em>. We find that third-order resonant interactions result in directional energy transfers to higher-wavenumber components, forming steep wave groups with augmented kinematics and a prolonged lifespan. If the wave group is initially narrow banded, <em>quasi-degenerate interactions</em> resembling the instability band of a regular wave train arise, characterised by unidirectional energy transfers and energy transfers along the resonance angle, ±35.26°, of the Phillips ‘figure-of-eight’ loop. Spectral broadening due to the quasi-degenerate interactions eventually facilitates <em>non-degenerate interactions</em>, which dominate the spectral evolution of the wave group after focus. The non-degenerate interactions manifest primarily as a high-wavenumber sidelobe, which forms at an angle of ±55° to the spectral peak. We consider finite-depth effects in the range of deep to intermediate waters (5.592 ≥ <em>k<sub>p</sub>d</em> ≥ 1.363), based on the characteristic wavenumber (<em>k<sub>p</sub></em>) and the domain depth (<em>d</em>), and find that all forms of spectral evolution are suppressed by depth. However, the quasi-degenerate interactions exhibit a greater sensitivity to depth, suggesting suppression of the modulation instability by the return current, consistent with previous studies. We also observe sensitivity to depth for <em>k<sub>p</sub>d</em> values commonly considered "deep", indicating that the length scales of the wave group and return current may be better indicators of dimensionless depth than the length scale of any individual wave component. The non-degenerate interactions appear to be depth resilient with persistent evidence of a ±55° spectral sidelobe at a depth of <em>k<sub>p</sub>d</em> =1.363. Although the quasi-degenerate interactions are significantly suppressed by depth, the interactions do not entirely disappear for <em>k<sub>p</sub>d</em> =1.363 and show signs of biasing towards oblique, rather than unidirectional, wave components at intermediate depths. The contraction of the wavenumber spectrum in the <em>k<sub>y</sub></em>-direction has also proved to be resilient to depth, suggesting that lateral expansion of the wave group and the "wall of water" effect of Gibbs & Taylor (2005) may persist at intermediate depths.</p>

scholarly journals WAVE GROUP ANATOMY OF OCEAN WAVE SPECTRA

1984 ◽  
Vol 1 (19) ◽  
pp. 45 ◽  
Author(s):  
Warren C. Thompson ◽  
Arthur R. Nelson ◽  
Dean G. Sedivy
Keyword(s):  
Statistical Analysis ◽  
Deep Water ◽  
Wave Spectrum ◽  
Ocean Wave ◽  
Group Model ◽  
Wave Spectra ◽  
Wave Group ◽  
Wave Groups

This paper inquires into the questions of how wave groups are related to the wave spectrum, and how they differ in sea versus swell. Some results are presented in the form of a wave group model for sea spectra and for swell spectra. The models were developed from statistical analysis of a large number of wave records and apply to deep water only.

Focusing of unidirectional wave groups on deep water: an approximate nonlinear Schrödinger equation-based model

2009 ◽  
Vol 465 (2110) ◽  
pp. 3083-3102 ◽  
Author(s):  
T. A. A. Adcock ◽  
P. H. Taylor
Keyword(s):  
Schrödinger Equation ◽  
Nonlinear Schrödinger Equation ◽  
Deep Water ◽  
Schrodinger Equation ◽  
Nonlinear Schrodinger Equation ◽  
Wave Groups ◽  
Flow Solver ◽  
Nonlinear Schrödinger ◽  
Nonlinear Potential ◽  
Cubic Nonlinear Schrödinger Equation

This paper sets out an approximate analytical model describing the nonlinear evolution of a Gaussian wave group in deep water. The model is derived using the conserved quantities of the cubic nonlinear Schrödinger equation (NLSE). The key parameter for describing the evolution is the amplitude-to-wavenumber bandwidth ratio, a quantity analogous to the Benjamin–Feir index for random sea-states. For smaller values of this parameter, the group is wholly dispersive, whereas for more nonlinear cases, solitons are formed. Our model predicts the characteristics and the evolution of the groups in both regimes. These predictions are found to be in good agreement with numerical simulations using the NLSE and are in qualitative agreement with numerical results from a fully nonlinear potential flow solver and experimental results.

scholarly journals ENGINEERING DESIGN IN THE PRESENCE OF WAVE GROUPS

2011 ◽  
Vol 1 (32) ◽  
pp. 68
Author(s):  
Thomas D. Shand ◽  
William L. Peirson ◽  
Ronald J. Cox
Keyword(s):  
Water Level ◽  
Deep Water ◽  
Group Development ◽  
Design Guidelines ◽  
Wave Group ◽  
Non Linear ◽  
Present Design ◽  
Wave Trains ◽  
Monochromatic Waves ◽  
Group Effects

Determining the largest wave height, H which can occur in water of depth, d without breaking is a fundamental reference quantity for the design of coastal structures. Current design guidelines, used to predict the ratio of breaking height to depth (Hb/d), also known as the breaker index, are based on investigations which predominantly used monochromatic waves, thereby implicitly neglecting group effects. Groupiness or height modulation in wave trains is an inherent characteristic of freely propagating waves in deep water and has been shown within previous studies to induce breaker indices substantially exceeding those predicted by design guidelines. Additionally, the raw data upon which present design guidelines have been based exhibit considerable scatter. This scatter is surprising given the monochromatic and uniform nature of the laboratory waves. A physical investigation at the Water Research Laboratory using new techniques for data extraction and visualisation has yielded new insights into the shoaling and breaking processes of regular and grouped waves and revealed deficiencies in the present design techniques. Monochromatic waves trains were found to develop amplitude modulation with distance along the flume due to non-linear instabilities. These instabilities are well recognized in deep-water waves and contribute to group development and the occurrence of low-probability extreme waves. These modulations induced variation in breaking wave heights, locations and derived breaker indices. Such modulation of initially regular wave trains is proposed as a possible cause of the scatter observed in raw laboratory breaker index data. Wave group testing has revealed evolutionary cycles in local energy density during deep water propagation and that the spatial phasing of this evolution with the initiation of shoaling yielded considerably different shoaling and breaking regimes. Critically, smaller waves within the group, particularly those occurring at the front of the wave group were, at times, able to propagate into shallower water before breaking than is presently predicted by existing design guides. Causes for this discrepancy, including differences in definitions of water level and depth are investigated. However, discrepancies between observed and predicted values are found to remain. Revision to present design guidelines to directly incorporate non-linear group effects and group-induced water level variation are presented.

scholarly journals On Determining the Onset and Strength of Breaking for Deep Water Waves. Part II: Influence of Wind Forcing and Surface Shear

2002 ◽  
Vol 32 (9) ◽  
pp. 2559-2570 ◽  
Author(s):  
Michael L. Banner ◽  
Jin-Bao Song
Keyword(s):  
Growth Rate ◽  
Group Dynamics ◽  
Deep Water ◽  
Water Waves ◽  
Numerical Study ◽  
Wind Forcing ◽  
Vertical Shear ◽  
Wave Group ◽  
Wave Groups ◽  
Deep Water Waves

Abstract Part I of this study describes the authors' findings on a robust threshold variable that determines the onset of breaking for unforced, irrotational deep water waves and proposes a means of predicting the strength of breaking if the breaking threshold is exceeded. Those results were based on a numerical study of the unforced evolution of fully nonlinear, two-dimensional inviscid wave trains and highlight the fundamental role played by the nonlinear wave group dynamics. In Part II the scope of these calculations is extended to investigate the additional influence of wind forcing and background shear on the evolution to breaking. Using the methodology described in Part I, the present study focuses on the influence of wind forcing and vertical shear on long-term evolution toward breaking or recurrence of the maximum of the local energy density within a wave group. It investigates the behavior of a dimensionless local growth rate parameter that reflects the mean energy flux to the energy maximum in the wave group and provides a clearer physical interpretation of the evolution toward recurrence or breaking. Typically, the addition of the wind forcing and surface layer shear results in only small departures from the irrotational, unforced cases reported in Part I. This indicates that nonlinear hydrodynamic energy fluxes within wave groups still dominate the evolution to recurrence or breaking even in the presence of these other mechanisms. Further, the calculations confirm that the breaking threshold for this growth rate found for unforced irrotational wave groups in Part I is also applicable for cases with wind forcing and shear typical of open ocean conditions. Overall, this approach provides an earlier and more decisive indicator for the onset of breaking than previously proposed breaking thresholds and suggests a foundation for predicting the strength of breaking events.

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