Large-amplitude wavetrains and solitary waves in vortices

1990 ◽  
Vol 216 ◽  
pp. 459-504 ◽  
Author(s):  
S. Leibovich ◽  
A. Kribus
Keyword(s):  
Solitary Waves ◽  
Large Amplitude ◽  
Vortex Breakdown ◽  
Continuation Method ◽  
Soliton Solutions ◽  
Numerical Continuation ◽  
Weakly Nonlinear ◽  
Strongly Nonlinear ◽  
Velocity Changes ◽  
Produce Flow

Large-amplitude axisymmetric waves on columnar vortices, thought to be related to flow structures observed in vortex breakdown, are found as static bifurcations of the Bragg–Hawthorne equation. Solutions of this equation satisfy the steady, axisymmetric, Euler equations. Non-trivial solution branches bifurcate as the swirl ratio (the ratio of azimuthal to axial velocity) changes, and are followed into strongly nonlinear regimes using a numerical continuation method. Four types of solutions are found: multiple columnar solutions, corresponding to Benjamin's ‘conjugate flows’, with subcritical–supercritical pairing of wave characteristics; solitary waves, extending previously known weakly nonlinear solutions to amplitudes large enough to produce flow reversals similar to the breakdown transition; periodic wavetrains; and solitary waves superimposed on the conjugate flow that emerge from the periodic wavetrain as the wavelength or amplitude becomes sufficiently large. Weakly nonlinear soliton solutions are found to be accurate even when the perturbations they cause are fairly strong.

scholarly journals Interactions of large amplitude solitary waves in viscous fluid conduits

2014 ◽  
Vol 750 ◽  
pp. 372-384 ◽  
Author(s):  
Nicholas K. Lowman ◽  
M. A. Hoefer ◽  
G. A. El
Keyword(s):  
Solitary Wave ◽  
Viscous Fluid ◽  
Solitary Waves ◽  
Water Waves ◽  
Large Amplitude ◽  
Wave Interactions ◽  
Weakly Nonlinear ◽  
Strongly Nonlinear ◽  
Nonlinear Solitary Waves

AbstractThe free interface separating an exterior, viscous fluid from an intrusive conduit of buoyant, less viscous fluid is known to support strongly nonlinear solitary waves due to a balance between viscosity-induced dispersion and buoyancy-induced nonlinearity. The overtaking, pairwise interaction of weakly nonlinear solitary waves has been classified theoretically for the Korteweg–de Vries equation and experimentally in the context of shallow water waves, but a theoretical and experimental classification of strongly nonlinear solitary wave interactions is lacking. The interactions of large amplitude solitary waves in viscous fluid conduits, a model physical system for the study of one-dimensional, truly dissipationless, dispersive nonlinear waves, are classified. Using a combined numerical and experimental approach, three classes of nonlinear interaction behaviour are identified: purely bimodal, purely unimodal, and a mixed type. The magnitude of the dispersive radiation due to solitary wave interactions is quantified numerically and observed to be beyond the sensitivity of our experiments, suggesting that conduit solitary waves behave as ‘physical solitons’. Experimental data are shown to be in excellent agreement with numerical simulations of the reduced model. Experimental movies are available with the online version of the paper.

scholarly journals Anomalous width variation of rarefactive ion acoustic solitary waves in the context of auroral plasmas

2004 ◽  
Vol 11 (2) ◽  
pp. 219-228 ◽  
Author(s):  
S. S. Ghosh ◽  
G. S. Lakhina
Keyword(s):  
Solitary Waves ◽  
Nonlinear Theory ◽  
Acoustic Waves ◽  
Large Amplitude ◽  
Magnetized Plasma ◽  
Amplitude Variation ◽  
Fully Nonlinear ◽  
Weakly Nonlinear ◽  
Ion Acoustic Solitary Waves ◽  
Ion Acoustic

Abstract. The presence of dynamic, large amplitude solitary waves in the auroral regions of space is well known. Since their velocities are of the order of the ion acoustic speed, they may well be considered as being generated from the nonlinear evolution of ion acoustic waves. However, they do not show the expected width-amplitude correlation for K-dV solitons. Recent POLAR observations have actually revealed that the low altitude rarefactive ion acoustic solitary waves are associated with an increase in the width with increasing amplitude. This indicates that a weakly nonlinear theory is not appropriate to describe the solitary structures in the auroral regions. In the present work, a fully nonlinear analysis based on Sagdeev pseudopotential technique has been adopted for both parallel and oblique propagation of rarefactive solitary waves in a two electron temperature multi-ion plasma. The large amplitude solutions have consistently shown an increase in the width with increasing amplitude. The width-amplitude variation profile of obliquely propagating rarefactive solitary waves in a magnetized plasma have been compared with the recent POLAR observations. The width-amplitude variation pattern is found to fit well with the analytical results. It indicates that a fully nonlinear theory of ion acoustic solitary waves may well explain the observed anomalous width variations of large amplitude structures in the auroral region.

Instability of strongly nonlinear waves in vortex flows

1994 ◽  
Vol 269 ◽  
pp. 247-264 ◽  
Author(s):  
A. Kribus ◽  
S. Leibovich
Keyword(s):  
Linear Stability ◽  
Nonlinear Waves ◽  
Vortex Breakdown ◽  
Axisymmetric Flow ◽  
Three Dimensional ◽  
Vortex Flows ◽  
Weakly Nonlinear ◽  
Strongly Nonlinear ◽  
Perturbation Problem ◽  
Bending Modes

Weakly nonlinear descriptions of axisymmetric cnoidal and solitary waves in vortices recently have been shown to have strongly nonlinear counterparts. The linear stability of these strongly nonlinear waves to three-dimensional perturbations is studied, motivated by the problem of vortex breakdown in open flows. The basic axisymmetric flow varies both radially and axially, and the linear stability problem is therefore nonseparable. To regularize the generalization of a critical layer, viscosity is introduced in the perturbation problem. In the absence of the waves, the vortex flows are linearly stable. As the amplitude of a wave constituting the basic flow increases owing to variation in the level of swirl, stability is first lost to non-axisymmetric ‘bending’ modes. This instability occurs when the wave amplitude exceeds a critical value, provided that the Reynolds number is larger enough. The critical wave amplitudes for instability typically are large, but not large enough to create regions of closed streamlines. Examination of the most-amplified eigenvectors shows that the perturbations tend to be concentrated downstream of the maximum streamline displacement in the wave, in a position consistent with the observed three-dimensional perturbations in the interior of a bubble type of vortex breakdown.

Nonlinear electron-acoustic waves in a multi-species plasma

1980 ◽  
Vol 24 (1) ◽  
pp. 169-180 ◽  
Author(s):  
B. Buti
Keyword(s):  
Solitary Waves ◽  
Acoustic Waves ◽  
Large Amplitude ◽  
Space Plasmas ◽  
Strongly Nonlinear ◽  
Electron Acoustic Waves ◽  
Electron Acoustic ◽  
Ion Species

Propagation of electron-acoustic waves in a strongly nonlinear magnetoplasma with two ion species is investigated. The presence of the second ion component affects the dynamics of these solitary waves in a variety of ways. Besides solitons, supersonic holes (density depressions) are produced by sufficiently large- amplitude perturbations. Heavier and hotter ions are more favourable to the holes. Applications of the present investigations to space plasmas are pointed out.

scholarly journals Long-term evolution of strongly nonlinear internal solitary waves in a rotating channel

2009 ◽  
Vol 16 (5) ◽  
pp. 587-598 ◽  
Author(s):  
J. C. Sánchez-Garrido ◽  
V. Vlasenko
Keyword(s):  
Solitary Waves ◽  
Nonlinear Effects ◽  
Kelvin Waves ◽  
Internal Solitary Waves ◽  
Kelvin Wave ◽  
Weakly Nonlinear ◽  
Strongly Nonlinear ◽  
Petviashvili Equation ◽  
Rotating Channel

Abstract. The evolution of internal solitary waves (ISWs) propagating in a rotating channel is studied numerically in the framework of a fully-nonlinear, nonhydrostatic numerical model. The aim of modelling efforts was the investigation of strongly-nonlinear effects, which are beyond the applicability of weakly nonlinear theories. Results reveal that small-amplitude waves and sufficiently strong ISWs evolve differently under the action of rotation. At the first stage of evolution an initially two-dimensional ISW transforms according to the scenario described by the rotation modified Kadomtsev-Petviashvili equation, namely, it starts to evolve into a Kelvin wave (with exponential decay of the wave amplitude across the channel) with front curved backwards. This transition is accompanied by a permanent radiation of secondary Poincaré waves attached to the leading wave. However, in a strongly-nonlinear limit not all the energy is transmitted to secondary radiated waves. Part of it returns to the leading wave as a result of nonlinear interactions with secondary Kelvin waves generated in the course of time. This leads to the formation of a slowly attenuating quasi-stationary system of leading Kelvin waves, capable of propagating for several hundreds hours as a localized wave packet.

Internal solitary waves in a two-fluid system with a free surface

2016 ◽  
Vol 804 ◽  
pp. 201-223 ◽  
Author(s):  
Tsubasa Kodaira ◽  
Takuji Waseda ◽  
Motoyasu Miyata ◽  
Wooyoung Choi
Keyword(s):  
Free Surface ◽  
Solitary Wave ◽  
Surface Waves ◽  
Solitary Waves ◽  
Silicone Oil ◽  
Internal Solitary Wave ◽  
Internal Solitary Waves ◽  
Weakly Nonlinear ◽  
Two Fluids ◽  
Strongly Nonlinear

Internal solitary waves in a system of two fluids, silicone oil and water, bounded above by a free surface are studied both experimentally and theoretically. By adjusting an extra volume of silicone oil released from a reservoir, a wide range of amplitude waves are generated in a wave tank. Wave profiles as well as wave speeds are measured using multiple wave probes and are then compared with both the weakly nonlinear Korteweg–de Vries (KdV) models and the strongly nonlinear Miyata–Choi–Camassa (MCC) models. As the density difference between the two fluids in the experiment is relatively small (approximately 14 %), but non-negligible, special attention is paid to the effect of the boundary condition at the top surface. The nonlinear models valid for rigid-lid (RL) and free-surface (FS) boundary conditions are considered separately. It is found that the solitary wave of the FS model for a given amplitude is consistently narrower than that of the RL model and it propagates at a slightly lower speed. Due to strong nonlinearity in the internal-wave motion, the weakly nonlinear KdV models fail to describe the measured internal solitary wave profiles of intermediate and large wave amplitudes. The strongly nonlinear MCC-FS model agrees better with the measurements than the MCC-RL model, which indicates that the free-surface boundary condition at the top surface is crucial in describing the internal solitary waves in the experiment correctly. Leaving the top surface free in the experiment allows us to observe small and relatively short wave packets on the top surface, particularly when the amplitude of the internal solitary wave is large. Once excited, the wave packet is located above the front half of the internal solitary wave and propagates with a speed close to that of the internal solitary wave underneath. A simple resonance mechanism between short surface waves and long internal waves without and with nonlinear effects is examined to estimate the characteristic wavelength of modulated short surface waves, which is found to be in good agreement with the observed wavelength when nonlinearity is taken into account. Using ray theory, the evolution of short surface waves in the presence of a background current induced by an internal solitary wave is also investigated to examine the location of the modulated surface wave packet.

Large-amplitude solitary waves in a relativistic nonisothermal plasma with warm ions

2000 ◽  
Vol 78 (4) ◽  
pp. 267-275
Author(s):  
R Roychoudhury ◽  
P Chatterjee
Keyword(s):  
Solitary Waves ◽  
Large Amplitude ◽  
Small Amplitude ◽  
Relativistic Effect ◽  
Soliton Solutions ◽  
Critical Value ◽  
Ion Temperature ◽  
52.35 Tc ◽  
Nonisothermal Plasma ◽  
52.35 Fp

Large amplitude solitary waves in a relativistic plasma with non-isothermal electrons and finite ion temperature are studied using Sagdeev's pseudopotential technique. It is found that there exists a critical value of beta, the ratio of the temperatures of the free and trapped electrons respectively, beyond which soliton solutions cease to exist. This critical value depends on other parameters. Also the relativistic effect and finite ion temperature restrict the region of existence of solitary waves. A small amplitude expansion of the pseudopotential is derived to find different kinds of solitary waves.PACS Nos.: 52.35 Fp, 52.35 Sb, 52.35 Tc

scholarly journals NONLINEAR OBLIQUE INTERACTION OF LARGE AMPLITUDE INTERNAL SOLITARY WAVES

2012 ◽  
Vol 1 (33) ◽  
pp. 19
Author(s):  
Keisuke Nakayama ◽  
Taro Kakinuma ◽  
Hidekazu Tsuji ◽  
Masayuki Oikawa
Keyword(s):  
Solitary Waves ◽  
Water Waves ◽  
Large Amplitude ◽  
Numerical Study ◽  
Incident Angle ◽  
Resonant Interaction ◽  
Internal Solitary Waves ◽  
Weakly Nonlinear ◽  
Wave Angle ◽  
The One

Solitary waves are typical nonlinear long waves in the ocean. The two-dimensional interaction of solitary waves has been shown to be essentially different from the one-dimensional case and can be related to generation of large amplitude waves (including ‘freak waves’). Concerning surface-water waves, Miles (1977) theoretically analyzed interaction of three solitary waves, which is called “resonant interaction” because of the relation among parameters of each wave. Weakly-nonlinear numerical study (Funakoshi, 1980) and fully-nonlinear one (Tanaka, 1993) both clarified the formation of large amplitude wave due to the interaction (“stem” wave) at the wall and its dependency of incident angle. For the case of internal waves, analyses using weakly nonlinear model equation (ex. Tsuji and Oikawa, 2006) suggest also qualitatively similar result. Therefore, the aim of this study is to investigate the strongly nonlinear interaction of internal solitary waves; especially whether the resonant behavior is found or not. As a result, it is found that the amplified internal wave amplitude becomes about three times as much as the original amplitude. In contrast, a "stem" was not found to occur when the incident wave angle was more than the critical angle, which has been demonstrated in the previous studies.

A sensitivity study of vortex breakdown onset to upstream boundary conditions

2010 ◽  
Vol 645 ◽  
pp. 81-119 ◽  
Author(s):  
BENJAMIN LECLAIRE ◽  
DENIS SIPP
Keyword(s):  
Boundary Conditions ◽  
Vortex Breakdown ◽  
Axisymmetric Flow ◽  
Sensitivity Study ◽  
Numerical Continuation ◽  
Smooth Transition ◽  
Inviscid Fluid ◽  
Weakly Nonlinear ◽  
Wide Range ◽  
New Type

This paper theoretically investigates the influence of the upstream boundary conditions on the bifurcation structure leading to vortex breakdown. The axisymmetric flow of an inviscid fluid in a pipe of constant cross-section and finite axial length is considered. Solutions bifurcating from the columnar solution at criticality are analysed via a weakly nonlinear expansion and computed in the fully nonlinear regime using numerical continuation, until a centreline recirculation is found at the pipe outlet. Bifurcation diagrams are determined for a parametric family of inflows describing a wide range of axial and azimuthal profiles, the third inlet condition being chosen either as a fixed azimuthal vorticity or as a vanishing radial velocity. Including the traditional picture given by Wang & Rusak (J. Fluid Mech., vol. 340, 1997a, p. 177), six different diagrams are found to be possible. In particular, a scenario of smooth transition to breakdown may exist as the swirl is increased, with no loss of stability and no hysteresis, breakdown appearing for swirl levels larger than the critical swirl in a pipe. This transition involves a new type of flow akin to a pre-breakdown flow. Our results, furthermore, suggest that rigidly rotating Poiseuille flow could correspond to the limit for which breakdown is impossible because it is predicted at infinitely large swirl numbers. We finally find that flows with a large rotational core are particularly sensitive to an accurate modelling of the upstream boundary conditions, weakly confined vortices being much more robust.

A Novel Method to Improve the Multiple-Scales Solution of the Forced Nonlinear Oscillators

2019 ◽  
Vol 16 (04) ◽  
pp. 1843010
Author(s):  
Hai-En Du ◽  
Guo-Kang Er ◽  
Vai Pan Iu
Keyword(s):  
Multiple Scales ◽  
Continuation Method ◽  
Inertial Force ◽  
Multiple Scales Method ◽  
Numerical Continuation ◽  
Strong Nonlinearity ◽  
Restoring Force ◽  
Strongly Nonlinear ◽  
Forced Oscillators ◽  
Novel Method

We propose a novel procedure to improve the solutions obtained by perturbation methods for analyzing the solutions of strongly nonlinear systems in this paper. The proposed procedure is presented and then combined with the multiple-scales method for the optimum solutions of a class of forced oscillators with strong nonlinearity. The solutions obtained from conventional multiple-scales method and the proposed method are examined by the results from numerical continuation method. The results show that the proposed method is effective for the oscillators with nonlinear restoring force as well as nonlinear inertial force even if the nonlinearities are strong. Numerical results and comparison show that the proposed method can improve the solution a lot in comparison to the solution obtained by conventional multiple-scales method.

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