Transverse instabilities of deep-water solitary waves

2006 ◽  
Vol 462 (2071) ◽  
pp. 2039-2061 ◽  
Author(s):  
Bernard Deconinck ◽  
Dmitry E Pelinovsky ◽  
John D Carter
Keyword(s):  
Growth Rate ◽  
Deep Water ◽  
Wave Train ◽  
Two Dimensional ◽  
Nls Equation ◽  
One Dimensional ◽  
Hill’S Method ◽  
Real Growth ◽  
Dimensional Soliton ◽  
Complex Growth Rate

The dynamics of a one-dimensional slowly modulated, nearly monochromatic localized wave train in deep water is described by a one-dimensional soliton solution of a two-dimensional nonlinear Schrödinger (NLS) equation. In this paper, the instability of such a wave train with respect to transverse perturbations is examined numerically in the context of the NLS equation, using Hill's method. A variety of instabilities are obtained and discussed. Among these, we show that the solitary wave is susceptible to an oscillatory instability (complex growth rate) due to perturbations with arbitrarily short wavelength. Further, there is a cut-off on the instability with real growth rates. We show analytically that the nature of this cut-off is different from what is claimed in previous works.

scholarly journals Transverse instability of gravity–capillary solitary waves on deep water in the presence of constant vorticity

2019 ◽  
Vol 871 ◽  
pp. 1028-1043
Author(s):  
M. Abid ◽  
C. Kharif ◽  
H.-C. Hsu ◽  
Y.-Y. Chen
Keyword(s):  
Growth Rate ◽  
Solitary Wave ◽  
Solitary Waves ◽  
Deep Water ◽  
Capillary Waves ◽  
Two Dimensional ◽  
Transverse Instability ◽  
Constant Vorticity ◽  
Dimensional Gravity ◽  
Wave Properties

The bifurcation of two-dimensional gravity–capillary waves into solitary waves when the phase velocity and group velocity are nearly equal is investigated in the presence of constant vorticity. We found that gravity–capillary solitary waves with decaying oscillatory tails exist in deep water in the presence of vorticity. Furthermore we found that the presence of vorticity influences strongly (i) the solitary wave properties and (ii) the growth rate of unstable transverse perturbations. The growth rate and bandwidth instability are given numerically and analytically as a function of the vorticity.

Tsunami generation

1973 ◽  
Vol 59 (4) ◽  
pp. 817-828 ◽  
Author(s):  
R. D. Braddock ◽  
P. Van Den Driessche ◽  
G. W. Peady
Keyword(s):  
Stationary Phase ◽  
Wave Front ◽  
Integral Transform ◽  
Wave Train ◽  
Asymptotic Approximations ◽  
Two Dimensional ◽  
Tsunami Generation ◽  
One Dimensional ◽  
Series Representations ◽  
Surface Disturbances

The problems of tsunami generation are treated by standard integral-transform and modified stationary-phase methods to yield asymptotic approximations to the surface disturbances. The effects of asymmetry are considered in a one-dimensional ocean. Series representations are used to produce sets of normal-mode oscillations in a two-dimensional ocean, and the magnitudes of the wave front and wave train are discussed in relation to the asymmetry of the generating region.

Weak quadratic interactions of two-dimensional waves

1971 ◽  
Vol 50 (1) ◽  
pp. 107-132 ◽  
Author(s):  
Young Yuel Kim ◽  
Thomas J. Hanratty
Keyword(s):  
Shallow Water ◽  
Deep Water ◽  
Wave Train ◽  
Water Layer ◽  
Rate Of Change ◽  
Two Dimensional ◽  
Low Frequencies ◽  
Resonant Interactions ◽  
Water Layers ◽  
Phase Angles

This paper reports on weak quadratic interactions which can occur with two-dimensional waves on shallow water layers and in the capillary-gravity range on deep water layers. It supplies experimental support of theoretical predictions for resonant interactions, but, perhaps of more significance, it explores in detail interactions which occur under conditions near resonance.Waves of approximately sinusoidal form are introduced on the surface of water in a long rectangular tank. For deep water a rapid distortion in the sinusoidal wave and sometimes additional crests are observed because of energy exchange among the first, second and third harmonics at frequencies where both surface tension and gravity are important (7·5–13 c/s). An even greater exchange of energy can be observed on shallow water layers at low frequencies. For example, a wave train with seven secondary crests can be observed when the wave maker is operated at 3·04 c/s in a water layer of 0·65 cm.Measured amplitudes and phase angles of the Fourier components of the wave train are described by a system of equations using only quadratic interactions among participating harmonics. The exchange of energy among Fourier components under certain conditions is explained in terms of the rate of change of relative phase angles of the different harmonics.

scholarly journals Stability analysis of three-dimensional breather solitons in a Bose–Einstein condensate

2005 ◽  
Vol 461 (2063) ◽  
pp. 3561-3574 ◽  
Author(s):  
M Matuszewski ◽  
E Infeld ◽  
G Rowlands ◽  
M Trippenbach
Keyword(s):  
Three Dimensional ◽  
Einstein Condensate ◽  
Dimensional Case ◽  
Two Dimensional ◽  
One Dimensional ◽  
Stability Properties ◽  
Bose Einstein Condensate ◽  
The Stability ◽  
Bose Einstein ◽  
Dimensional Soliton

We investigated the stability properties of breather soliton trains in a three-dimensional Bose–Einstein condensate (BEC) with Feshbach-resonance management of the scattering length. This is done so as to generate both attractive and repulsive interaction. The condensate is confined only by a one-dimensional optical lattice and we consider strong, moderate and weak confinement. By strong confinement we mean a situation in which a quasi two-dimensional soliton is created. Moderate confinement admits a fully three-dimensional soliton. Weak confinement allows individual solitons to interact. Stability properties are investigated by several theoretical methods such as a variational analysis, treatment of motion in effective potential wells, and collapse dynamics. Armed with all the information forthcoming from these methods, we then undertake a numerical calculation. Our theoretical predictions are fully confirmed, perhaps to a higher degree than expected. We compare regions of stability in parameter space obtained from a fully three-dimensional analysis with those from a quasi two-dimensional treatment, when the dynamics in one direction are frozen. We find that in the three-dimensional case the stability region splits into two parts. However, as we tighten the confinement, one of the islands of stability moves toward higher frequencies and the lower frequency region becomes more and more like that for the quasi two-dimensional case. We demonstrate these solutions in direct numerical simulations and, importantly, suggest a way of creating robust three-dimensional solitons in experiments in a BEC in a one-dimensional lattice.

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 Plasma sheet thinning due to loss of near-Earth magnetotail plasma

2021 ◽  
Vol 87 (1) ◽  
Author(s):  
Rudolf Tretler ◽  
Tomo Tatsuno ◽  
Keisuke Hosokawa
Keyword(s):  
Rarefaction Wave ◽  
Plasma Sheet ◽  
Wave Train ◽  
Two Dimensions ◽  
Two Dimensional ◽  
One Dimensional ◽  
Current Disruption ◽  
Auroral Breakup ◽  
Thinning Process ◽  
D Model

A one-dimensional (1-D) model for thinning of the Earth's plasma sheet (Chao et al., Planet. Space Sci., vol. 25, 1977, p. 703) according to the current disruption (CD) model of auroral breakup is extended to two dimensions. A rarefaction wave, which is a signature component of the CD model, is generated with an initial disturbance. In the 1-D gas model, the rarefaction wave propagates tailward at sound velocity and is assumed to cause thinning. Extending to a two-dimensional (2-D) gas model of a simplified plasma sheet configuration, the rarefaction wave is weakened, and the thinning ceases to propagate. Extending further to a 2-D plasma model by adding magnetic field into the lobes, the rarefaction wave is quickly lost in the plasma sheet recompression, but the plasma sheet thinning is still present. It propagates at a slower velocity than a 1-D model suggests, behind a wave train of pulses of increased pressure generated by the thinning process itself. This shows that the dynamics of plasma sheet thinning may be dominated by sheet–lobe interactions that are absent from the 1-D model and may not support the behaviour assumed by the CD model.

The Simulation of Mixing Layers Driven by Compound Buoyancy and Shear

1996 ◽  
Vol 118 (2) ◽  
pp. 370-376 ◽  
Author(s):  
D. M. Snider ◽  
M. J. Andrews
Keyword(s):  
Growth Rate ◽  
Flow Field ◽  
Numerical Solution ◽  
Mixing Layer ◽  
Measured Data ◽  
Mixing Layers ◽  
Two Dimensional ◽  
One Dimensional ◽  
Self Similar ◽  
Accurate Numerical Solution

Fully developed compound shear and buoyancy driven mixing layers are predicted using a k-ε turbulence model. Such mixing layers present an exchange of equilibrium in mixing flows. The k-ε buoyancy constant Cε3 = 0.91, defined in this study for buoyancy unstable mixing layers, is based on an approximate self-similar analysis and an accurate numerical solution. One-dimensional transient and two-dimensional steady calculations are presented for buoyancy driven mixing in a uniform flow field. Two-dimensional steady calculations are presented for compound shear and buoyancy driven mixing. The computed results for buoyancy alone and compound shear and buoyancy mixing compare well with measured data. Adding shear to an unstable buoyancy mixing layer does not increase the mixing growth rate compared with that from buoyancy alone. The nonmechanistic k-ε model which balances energy generation and dissipation using constants from canonical shear and buoyancy studies predicts the suppression of the compound mixing width. Experimental observations suggest that a reduction in growth rate results from unequal stream velocities that skew and stretch the normally vertical buoyancy plumes producing a reduced mixing envelope width.

scholarly journals On Determining the Onset and Strength of Breaking for Deep Water Waves. Part I: Unforced Irrotational Wave Groups

2002 ◽  
Vol 32 (9) ◽  
pp. 2541-2558 ◽  
Author(s):  
Jin-Bao Song ◽  
Michael L. Banner
Keyword(s):  
Growth Rate ◽  
Nonlinear Wave ◽  
Deep Water ◽  
Water Waves ◽  
Wave Group ◽  
Two Dimensional ◽  
Wave Groups ◽  
Average Growth ◽  
Deep Water Waves

Abstract Finding a robust threshold variable that determines the onset of breaking for deep water waves has been an elusive problem for many decades. Recent numerical studies of the unforced evolution of two-dimensional nonlinear wave trains have highlighted the complex evolution to recurrence or breaking, together with the fundamental role played by nonlinear intrawave group dynamics. In Part I of this paper the scope of two-dimensional nonlinear wave group calculations is extended by using a wave-group-following approach applied to a wider class of initial wave group geometries, with the primary goal of identifying the differences between evolution to recurrence and to breaking onset. Part II examines the additional influences of wind forcing and background shear on these evolution processes. The present investigation focuses on the long-term evolution of the maximum of the local energy density along wave groups. It contributes a more complete picture, both long-term and short-term, of the approach to breaking and identifies a dimensionless local average growth rate parameter that is associated with the mean convergence of wave-coherent energy at the wave group maximum. This diagnostic growth rate appears to have a common threshold for all routes to breaking in deep water that have been examined and provides an earlier and more decisive indicator for the onset of breaking than previously proposed breaking thresholds. The authors suggest that this growth rate may also provide an indicative measure of the strength of wave breaking events.

A one-dimensional self-gravitating stellar gas

1966 ◽  
Vol 25 ◽  
pp. 46-48 ◽  
Author(s):  
M. Lecar
Keyword(s):  
Gravitational Field ◽  
Phase Space ◽  
Motion Picture ◽  
Space Distribution ◽  
Two Dimensional ◽  
One Dimensional ◽  
Phase Space Distribution ◽  
Dimensional Phase Space ◽  
The Mean

“Dynamical mixing”, i.e. relaxation of a stellar phase space distribution through interaction with the mean gravitational field, is numerically investigated for a one-dimensional self-gravitating stellar gas. Qualitative results are presented in the form of a motion picture of the flow of phase points (representing homogeneous slabs of stars) in two-dimensional phase space.

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