On nonlinear wave groups and crest statistics

2009 ◽  
Vol 620 ◽  
pp. 221-239 ◽  
Author(s):  
FRANCESCO FEDELE ◽  
M. AZIZ TAYFUN
Keyword(s):  
Nonlinear Effects ◽  
Theoretical Expression ◽  
Large Wave ◽  
Weakly Nonlinear ◽  
The North ◽  
Wave Measurements ◽  
Surface Displacements ◽  
Skewness Coefficient ◽  
Broadband Waves ◽  
Generalized Distribution

We present a second-order stochastic model of weakly nonlinear waves and develop theoretical expressions for the expected shape of large surface displacements. The model also leads to an exact theoretical expression for the statistical distribution of large wave crests in a form that generalizes the Tayfun distribution (Tayfun, J. Geophys. Res., vol. 85, 1980, p. 1548). The generalized distribution depends on a steepness parameter given by μ = λ3/3, where λ3 represents the skewness coefficient of surface displacements. It converges to the Tayfun distribution in narrowband waves, where both distributions describe the crests of all waves well. In broadband waves, the generalized distribution represents the crests of large waves just as well whereas the Tayfun distribution appears as an upper bound and tends to overestimate them. However, the theoretical nature of the generalized distribution presents practical difficulties in oceanic applications. We circumvent these by adopting an appropriate approximation for the steepness parameter. Comparisons with wind-wave measurements from the North Sea suggest that this approximation allows both distributions to assume an identical form with which we can describe the distribution of large wave crests fairly accurately. The same comparisons also show that third-order nonlinear effects do not appear to have any discernable effect on the statistics of large surface displacements or wave crests.

Envelope and Phase Statistics of Large Waves

2008 ◽  
Author(s):  
M. Aziz Tayfun ◽  
Francesco Fedele
Keyword(s):  
North Sea ◽  
Phase Distribution ◽  
Theoretical Expression ◽  
Large Wave ◽  
Wave Phase ◽  
The North ◽  
Wave Measurements ◽  
Surface Displacements ◽  
The North Sea ◽  
Simple Limit

A theoretical expression derived previously for describing the joint distribution of the envelope and phase of nonlinear waves is verified with wind-wave measurements collected in the North Sea. The same distribution is explored further to derive the marginal and conditional distributions of wave envelopes and phases. The nature and implications of these are examined with emphasis on the occurrence of large waves and associated phases. It is shown that the wave-phase distribution assumes two distinct forms depending on whether if envelope elevations exceed the significant envelope height or not. For envelope elevations sufficiently larger than this threshold, the wave-phase distribution approaches a simple limit form, indicating that large surface displacements can occur only above the mean sea level. Comparisons with the North Sea data confirm these theoretical results and also suggest that large surface displacements and thus large wave heights arise from the constructive interference of spectral components with different amplitudes and phases. Further, large waves with high and sharp crests do not display any secondary maxima and minima. They appear more regular or narrow-banded than relatively low waves, and their heights and crests do not violate the Miche-Stokes type upper limits.

scholarly journals Distributions of Envelope and Phase in Wind Waves

2008 ◽  
Vol 38 (12) ◽  
pp. 2784-2800 ◽  
Author(s):  
M. Aziz Tayfun
Keyword(s):  
North Sea ◽  
Phase Distribution ◽  
Wind Waves ◽  
Second Order ◽  
Theoretical Expression ◽  
Wave Phase ◽  
The North ◽  
Wave Measurements ◽  
Surface Displacements ◽  
The North Sea

Abstract A theoretical expression derived previously for describing the joint distribution of the envelope and phase of second-order nonlinear waves is verified with wind wave measurements gathered in the North Sea. The same distribution is explored further to obtain the marginal and conditional distributions of wave envelopes and phases. The nature and implications of these are examined, with emphasis on the occurrence of large waves and associated phases. It is shown that the wave phase distribution assumes two distinct forms depending on whether envelope heights exceed the significant envelope height. For envelope heights sufficiently larger than this threshold, the wave phase distribution approaches a simple limit form, indicating that large surface displacements can occur only above the mean sea level. Comparisons with the North Sea data confirm these theoretical results and indicate that large surface displacements and thus large waves result from the random superposition of elementary spectral components enhanced by second-order nonlinear interactions. Further, large waves with higher and sharper crests do not display any secondary maxima or minima. They appear more regular or “narrow banded” than relatively low waves, and their heights and crests do not violate the Miche–Stokes-type upper limits. The results also suggest that third-order nonlinearities do not affect the surface statistics in any discernable way.

Wave-Height Distributions and Nonlinear Effects

2006 ◽  
Author(s):  
M. Aziz Tayfun ◽  
Francesco Fedele
Keyword(s):  
Nonlinear Waves ◽  
Deterministic Model ◽  
Nonlinear Effects ◽  
Type Model ◽  
Large Wave ◽  
The North ◽  
Linear Waves ◽  
Wave Heights ◽  
Nonlinear Extensions ◽  
Fifth Order

Theoretical distributions for describing the crest-to-trough heights of linear waves are reviewed briefly. To explore the effects of nonlinearities, these and two approximations that follow from the Tayfun [28] model are generalized to nonlinear waves via the second-order quasi-deterministic model of Fedele and Arena [7]. The potential utility of Gram-Charlier type approximations [17, 18, 29, 31] in representing the statistics of nonlinear wave heights is also explored. All models and a fifth-order Stokes-Rayleigh type model recently proposed by Dawson [5] are compared with linear and nonlinear waves simulated from the JONSWAP spectrum representative of long-crested extreme seas, and also with observational data gathered during two severe storms in the North Sea. The results indicate first that nonlinearities do not appear to affect the crest-to-trough wave heights significantly. Most models and their nonlinear extensions yield similar and reasonable predictions of the data trends observed. The present comparisons do not confirm the efficacy of Gram-Charlier type approximations in modeling the statistics of unusually large wave heights.

scholarly journals Focusing of long waves with finite crest over constant depth

2013 ◽  
Vol 469 (2153) ◽  
pp. 20130015 ◽  
Author(s):  
Utku Kânoğlu ◽  
Vasily V. Titov ◽  
Baran Aydın ◽  
Christopher Moore ◽  
Themistoklis S. Stefanakis ◽  
...  
Keyword(s):  
Source Region ◽  
Wave Theory ◽  
Long Waves ◽  
Constant Depth ◽  
Large Wave ◽  
Weakly Nonlinear ◽  
Scaling Relationships ◽  
2011 Japan Tsunami ◽  
Wave Heights ◽  
Run Up

Tsunamis are long waves that evolve substantially, through spatial and temporal spreading from their source region. Here, we introduce a new analytical solution to study the propagation of a finite strip source over constant depth using linear shallow-water wave theory. This solution is not only exact, but also general and allows the use of realistic initial waveforms such as N -waves. We show the existence of focusing points for N -wave-type initial displacements, i.e. points where unexpectedly large wave heights may be observed. We explain the effect of focusing from a strip source analytically, and explore it numerically. We observe focusing points using linear non-dispersive and linear dispersive theories, analytically; and nonlinear non-dispersive and weakly nonlinear weakly dispersive theories, numerically. We discuss geophysical implications of our solutions using the 17 July 1998 Papua New Guinea and the 17 July 2006 Java tsunamis as examples. Our results may also help to explain high run-up values observed during the 11 March 2011 Japan tsunami, which are otherwise not consistent with existing scaling relationships. We conclude that N -waves generated by tectonic displacements feature focusing points, which may significantly amplify run-up beyond what is often assumed from widely used scaling relationships.

scholarly journals “Extraordinary” modulation instability in optics and hydrodynamics

2021 ◽  
Vol 118 (14) ◽  
pp. e2019348118
Author(s):  
Guillaume Vanderhaegen ◽  
Corentin Naveau ◽  
Pascal Szriftgiser ◽  
Alexandre Kudlinski ◽  
Matteo Conforti ◽  
...  
Keyword(s):  
Wave Propagation ◽  
Stability Analysis ◽  
Linear Stability ◽  
Nonlinear Theory ◽  
Linear Stability Analysis ◽  
Water Waves ◽  
Modulation Instability ◽  
Nonlinear Effects ◽  
Weakly Nonlinear ◽  
Weakly Nonlinear Theory

The classical theory of modulation instability (MI) attributed to Bespalov–Talanov in optics and Benjamin–Feir for water waves is just a linear approximation of nonlinear effects and has limitations that have been corrected using the exact weakly nonlinear theory of wave propagation. We report results of experiments in both optics and hydrodynamics, which are in excellent agreement with nonlinear theory. These observations clearly demonstrate that MI has a wider band of unstable frequencies than predicted by the linear stability analysis. The range of areas where the nonlinear theory of MI can be applied is actually much larger than considered here.

scholarly journals Precessional instability of a fluid cylinder

2011 ◽  
Vol 666 ◽  
pp. 104-145 ◽  
Author(s):  
ROMAIN LAGRANGE ◽  
PATRICE MEUNIER ◽  
FRANÇOIS NADAL ◽  
CHRISTOPHE ELOY
Keyword(s):  
Reynolds Number ◽  
Nonlinear Effects ◽  
Intermittent Flow ◽  
Mean Flow ◽  
Weakly Nonlinear ◽  
Weakly Nonlinear Theory ◽  
Aspect Ratios ◽  
Image Velocimetry ◽  
Precessional Motion ◽  
Weakly Nonlinear Model

In this paper, the instability of a fluid inside a precessing cylinder is addressed theoretically and experimentally. The precessional motion forces Kelvin modes in the cylinder, which can become resonant for given precessional frequencies and cylinder aspect ratios. When the Reynolds number is large enough, these forced resonant Kelvin modes eventually become unstable. A linear stability analysis based on a triadic resonance between a forced Kelvin mode and two additional free Kelvin modes is carried out. This analysis allows us to predict the spatial structure of the instability and its threshold. These predictions are compared to the vorticity field measured by particle image velocimetry with an excellent agreement. When the Reynolds number is further increased, nonlinear effects appear. A weakly nonlinear theory is developed semi-empirically by introducing a geostrophic mode, which is triggered by the nonlinear interaction of a free Kelvin mode with itself in the presence of viscosity. Amplitude equations are obtained coupling the forced Kelvin mode, the two free Kelvin modes and the geostrophic mode. They show that the instability saturates to a fixed point just above threshold. Increasing the Reynolds number leads to a transition from a steady saturated regime to an intermittent flow in good agreement with experiments. Surprisingly, this weakly nonlinear model still gives a correct estimate of the mean flow inside the cylinder even far from the threshold when the flow is turbulent.

Time Simulation of Water Shipping for a Ship Advancing in Large Longitudinal Waves

1993 ◽  
Vol 37 (02) ◽  
pp. 126-137
Author(s):  
Ming-Chung Fang ◽  
Ming-Ling Lee ◽  
Chwang-Kuo Lee
Keyword(s):  
Nonlinear Effects ◽  
Ship Motion ◽  
Irregular Waves ◽  
Wave System ◽  
Time Step ◽  
Large Wave ◽  
Time Simulation ◽  
Head Waves ◽  
Speed Effect

The technique of time-domain numerical simulation for the occurrence of water shipping on board in head waves is presented. The nonlinear effects of the large-amplitude motion are treated. These nonlinear factors include the effect of large wave amplitude, large ship motion, the change of hull configuration below the free surface and the nonlinear resultant wave. Therefore, the variation of the potentials and the hydrodynamic coefficients for a ship at each time step must be carefully treated. While handling the determination of the instantaneous wave surface around the ship hull, the complete incident, diffracted, and radiated wave system is used rather than the incident wave only. The complexity of the ship speed effect on the related terms is also treated at each time step, especially for the radiation problems. An experimental setup is also designed to measure the motion response and the relative motion, and comparisons are made. The results show excellent agreement and the validity of the theory is confirmed. The successful development of the present technique can be extended to analyze the dynamic stability, capsize phenomena, and ship motion in irregular waves

Analysis of Complex Surface Displacements above a Storage Cavern Field in NW-Germany, Observed by InSAR

2021 ◽  
Author(s):  
Markus Even ◽  
Malte Westerhaus ◽  
Verena Simon
Keyword(s):  
Displacement Field ◽  
Pressure Response ◽  
Phase Model ◽  
Basic Method ◽  
Elastic Displacement ◽  
The North ◽  
Geophysical Model ◽  
Surface Displacements ◽  
Pressure Changes ◽  
East West

<p>The cavern field at Epe has been brined out of a salt deposit belonging to the lower Rhine salt flat, which extends under the surface of the North German lowlands and part of the Netherlands, and is used to store e.g. natural gas, brine and petroleum. Cavern convergence and operational pressure changes cause surface displacements that have been studied for this work with the help of SAR interferometry (InSAR) using distributed and persistent scatterers. Vertical and East-West movements have been determined based on Sentinel-1 data from ascending and descending orbit. Simple geophysical modeling is used to support InSAR processing and helps to interpret the observations. In particular, an approach is presented that allows to relate the deposit pressures with the observed surface displacements. Seasonal movements occurring over a fen situated over the western part of the storage site further complicate the analysis. Findings are validated with ground truth from levelling and groundwater level measurements.</p><p>For porous storage sites the geomechanic response can be described as elastic: displacement is almost proportional to reservoir pressure and displays the same pronounced seasonal behavior. At Epe the visco-elastic response of the salt layer has to be considered. The general appearance of the surface displacement is that of a strongly smoothed and shifted version of the cavern pressure curve. To deal with this situation a temporal model for displacement with pressure changes (pressure response) is derived that relates cavern pressure with observed displacement based on the theory for visco-elastic behavior of a Kelvin-Voigt body.</p><p>In order to deal successfully with the challenging displacement field at Epe several algorithmic improvements were implemented. To obtain a more complete picture of the displacement field DS pre-processing has been combined with StaMPS. Furthermore, StaMPS was modified in order to support unwrapping with a phase model composed of linear trend, pressure response and a seasonal component (caused by ground water level changes). Finally, refining the iterative estimation scheme of StaMPS helped avoiding leakage of the displacement signal to the spatially correlated noise term.</p><p>Determining vertical and east-west displacements from InSAR line-of-sight displacements is fundamental for interpretation and integration with levelling data. In this study, a basic method of orbit combination and another one supported by a simplistic geophysical model were applied in order to obtain 2D-displacements. For the basic method the north-south component was handled as if it were zero, while the geophysical model predicts the LOS effect of NS displacements. It assumes that caverns act as spherical pressure/volume sources embedded in an elastic half space (“Mogi” sources). To incorporate the visco-elastic component, each cavern is encompassed by a spherical salt shell that obeys the Kelvin-Voigt differential equations. The model is used here to describe either the parameters of the linear component of the displacement model or of the pressure response. A novelty of the orbit combinations implemented for this study is that the different components of the phase model are combined separately. This allows for a better understanding of the phenomena that contribute to the displacement field.</p>

A weakly nonlinear framework to study shock–vorticity interaction

2022 ◽  
Vol 933 ◽  
Author(s):  
Pranav Thakare ◽  
Vineeth Nair ◽  
Krishnendu Sinha
Keyword(s):  
Numerical Simulations ◽  
Acoustic Waves ◽  
Significant Contribution ◽  
Interaction Analysis ◽  
Nonlinear Effects ◽  
Normal Shock ◽  
Linear Interaction ◽  
Weakly Nonlinear ◽  
High Incidence ◽  
Vorticity Generation

Linear interaction analysis (LIA) is routinely used to study the shock–turbulence interaction in supersonic and hypersonic flows. It is based on the inviscid interaction of elementary Kovásznay modes with a shock discontinuity. LIA neglects nonlinear effects, and hence it is limited to small-amplitude disturbances. In this work, we extend the LIA framework to study the fundamental interaction of a two-dimensional vorticity wave with a normal shock. The predictions from a weakly nonlinear framework are compared with high-order accurate numerical simulations over a range of wave amplitudes ( $\epsilon$ ), incidence angles ( $\alpha$ ) and shock-upstream Mach numbers ( $M_1$ ). It is found that the nonlinear generation of vorticity at the shock has a significant contribution from the intermodal interaction between vorticity and acoustic waves. Vorticity generation is also strongly influenced by the curvature of the normal shock wave, especially for high incidence angles. Further, the weakly nonlinear analysis is able to predict the correct scaling of the nonlinear effects observed in the numerical simulations. The analysis also predicts a Mach number dependent limit for the validity of LIA in terms of the maximum possible amplitude of the upstream vorticity wave.

scholarly journals Complex Surface Displacements above the Storage Cavern Field at Epe, NW-Germany, Observed by Multi-Temporal SAR-Interferometry

2020 ◽  
Vol 12 (20) ◽  
pp. 3348
Author(s):  
Markus Even ◽  
Malte Westerhaus ◽  
Verena Simon
Keyword(s):  
Ground Truth ◽  
Complex Surface ◽  
Sar Interferometry ◽  
Storage Site ◽  
Geophysical Modeling ◽  
The North ◽  
Persistent Scatterers ◽  
Surface Displacements ◽  
Multi Temporal ◽  
Pressure Changes

The storage cavern field at Epe has been brined out of a salt deposit belonging to the lower Rhine salt flat, which extends under the surface of the North German lowlands and part of the Netherlands. Cavern convergence and operational pressure changes cause surface displacements that have been studied for this work with the help of SAR interferometry (InSAR) using distributed and persistent scatterers. Vertical and East-West movements have been determined based on Sentinel-1 data from ascending and descending orbit. Simple geophysical modeling is used to support InSAR processing and helps to interpret the observations. In particular, an approach is presented that allows to relate the deposit pressures with the observed surface displacements. Seasonal movements occurring over a fen situated over the western part of the storage site further complicate the analysis. Findings are validated with ground truth from levelling and groundwater level measurements.

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