Dispersion Relation For Linear Waves Research Articles

Ion acoustic solitary waves and double layers with nonextensive electrons and thermal positrons

By using Sagdeev’s pseudopotential technique, the problem of arbitrary amplitude ion acoustic solitary waves (IASWs) is discussed for a plasma comprising nonextensive electrons and thermal positrons. The standard normal-mode analysis is used to study the dispersion relation for linear waves. It is found that the present plasma model supports IASWs having positive as well as negative potential well. The influence of nonextensive electrons on the present plasma model is investigated for the existence of solitary waves. The investigation shows that the solitary structure ceases to exist when the parameter q crosses a certain limit. It is also found that both the small amplitude compressive and rarefactive double layer solution can exist in such a plasma system in some parametric region. It is shown that solitary structures and double layers are affected by nonextensivity, as well as by relevant plasma parameters.

Downward transfer of momentum by wind-driven waves

A model for the downward transfer of wind momentum is derived for growing waves. It is shown that waves, which grow due to an uneven pressure distribution on the water surface or a wave-coherent surface shear stress have horizontal velocities out of phase with the surface elevation. Further, if the waves grow in the x-direction, while the motion is perhaps time-periodic at any fixed point, the Reynolds stresses associated with the organized motion are positive. This is in agreement with several field and laboratory measurements which were previously unexplained, and the new theory successfully links measured wave growth rates and measured sub-surface Reynolds stresses. Wave coherent air pressure (and/or surface shear stress) is shown to change the speed of wave propagation as well as inducing growth or decay. From air pressure variations that are in phase with the surface elevation, the influence on the waves is simply a phase speed increase. For pressure variations out of phase with surface elevation, both growth (or decay) and phase speed changes occur. The theory is initially developed for long waves, after which the velocity potential and dispersion relation for linear waves in arbitrary depth are given. The model enables a sounder model for the transfer to storm surges or currents of momentum from breaking waves in that it does not rely entirely on ad-hoc turbulent diffusion. Future models of atmosphere-ocean exchanges should also acknowledge that momentum is transferred partly by the organized wave motion, while other species, like heat and gasses, may rely totally on turbulent diffusion. The fact that growing wind waves do in fact not generally obey the dispersion relation for free waves may need to be considered in future wind wave development models.

Linear wave dispersion relation of a magnetized relativistic rectilinear electron beam-dielectric system

A general dispersion relation is derived for a relativistic rectilinear electron beam of arbitrary momentum distributions interacting with a dielectric in a guide magnetic field, on the basis of Maxwell equations and the relativistic Vlasov equation. The instability occurs when the beam velocity exceeds the wave phase velocity in the medium. The linear wave dispersion relation, growth rate, spatial growth rate are studied analytically for delta and Lorentzian distributions of beam momentums in detail. The results are of importance for a new kind of high-power microwave generation or amplification devices based on anomalous Doppler effect.

Interface dynamics of a two-component Bose-Einstein condensate driven by an external force

The dynamics of an interface in a two-component Bose-Einstein condensate driven by a spatially uniform time-dependent force is studied. Starting from the Gross-Pitaevskii Lagrangian, the dispersion relation for linear waves and instabilities at the interface is derived by means of a variational approach. A number of diverse dynamical effects for different types of driving force is demonstrated, which includes the Rayleigh-Taylor instability for a constant force, the Richtmyer-Meshkov instability for a pulse force, dynamic stabilization of the Rayleigh-Taylor instability and onset of the parametric instability for an oscillating force. Gaussian Markovian and non-Markovian stochastic forces are also considered. It is found that the Markovian stochastic force does not produce any average effect on the dynamics of the interface, while the non-Markovian force leads to exponential perturbation growth.

Wave turbulence on the surface of a ferrofluid in a horizontal magnetic field

We report observations of wave turbulence on the surface of a ferrofluid submitted to a magnetic field parallel to the fluid surface. The magnetic wave turbulence shows several differences compared to the normal field case reported recently. The inertial zone of the magnetic wave turbulence regime is notably found to be strongly increased with respect to the normal field case and to be well described by our theoretical predictions. The dispersion relation of linear waves is also measured and differs from the normal field case due to the absence of the Rosensweig instability.

Electron acoustic solitary waves and double layers with superthermal hot electrons

The problem of arbitrary amplitude electron acoustic solitary waves (EASWs) are discussed using Sagdeev’s pseudopotential technique for a plasma comprising cold electrons, superthermal hot electrons, and stationary ions. The standard normal-mode analysis is used to study the dispersion relation for linear waves. It is found that the present plasma model supports EASWs having negative potential. The influence of superthermal hot electrons on the present plasma model is investigated for the existence of solitary waves. The investigation shows that the solitary structure ceases to exist when the parameter κ crosses a certain limit. It is also found that the small amplitude double layer solution can exist in such a plasma system in some parametric regions. It is shown that solitary structures and double layers are affected by superthermality, as well as by relevant plasma parameters.

Evolution of perturbations of a charged interface between immiscible inviscid fluids in the interelectrode gap

Perturbations of the interface between two immiscible ideal fluids of finite thickness (the lower and upper fluids are the conductor and the dielectric, respectively) located in the gap between two electrodes are considered. In the cases of the “shallow” and “deep” upper fluid the dispersion relations of linear waves and their longwave expansions are found. The methods of determining the space-time evolution of an initial surface perturbation are developed on the basis of the linear approximation. In the cases of the “shallow” and “deep” upper fluid examples of the development of an initial perturbation of the “step” type are given. The development of an initial perturbation of the “step” type are also considered in the near-critical electric fields and in the case of degeneration of cubic dispersion.

Generation of waves by a neutron beam in a two-component system formed by charged particles of nonzero spin

Dispersion relations for linear waves in amedium formed by electrons and ions and traversed by a beam of neutrons whose velocity has a nonzero constant component are derived by methods of quantum hydrodynamics. It is shown that, along with stable eigenmodes, vibrational excitations growing with time can be generated in the system owing to the interaction of spin components.

PML absorbing boundary conditions for the linearized and nonlinear Euler equations in the case of oblique mean flow

AbstractFor the case of uniform mean flow in an arbitrary direction, perfectly matched layer (PML) absorbing boundary conditions are presented for both the linearized and nonlinear Euler equations. Although linear perfectly matched side layers with an oblique mean flow have been studied in previous works, we propose in the present paper a construction of corner layer equations that are dynamically stable. Stability issues are investigated by examining the dispersion relations of linear waves supported by the corner layer equations. For increased efficiency, a pseudo mean flow is included in the derivation of the PML equations for the nonlinear case. Numerical examples are given to support the validity of the proposed equations. Specifically, the linear PML formulation is tested for the case of acoustic, vorticity, and entropy waves traveling with an oblique mean flow. The nonlinear formulation is tested with an isentropic vortex moving diagonally with a constant velocity. Copyright © 2008 John Wiley & Sons, Ltd.

One-parameter localized traveling waves in nonlinear Schrödinger lattices

We address the existence of traveling single-humped localized solutions in the spatially discrete nonlinear Schrödinger (NLS) equation. A mathematical technique is developed for analysis of persistence of these solutions from a certain limit in which the dispersion relation of linear waves contains a triple zero. The technique is based on using the Implicit Function Theorem for solution of an appropriate differential advance–delay equation in exponentially weighted spaces. The resulting Melnikov calculation relies on a number of assumptions on the spectrum of the linearization around the pulse, which are checked numerically. We apply the technique to the so-called Salerno model and the translationally invariant discrete NLS equation with a cubic nonlinearity. We show that the traveling solutions terminate in the Salerno model whereas they generally persist in the translationally invariant NLS lattice as a one-parameter family of solutions. These results are found to be in a close correspondence with numerical approximations of traveling solutions with zero radiation tails. Analysis of persistence also predicts the spectral stability of the one-parameter family of traveling solutions under time evolution of the discrete NLS equation.

Propagation of linear waves in relativistic anisotropic magnetohydrodynamics.

Gedalin [Phys. Rev. E 47, 4354 (1993)] derived a dispersion relation for linear waves in relativistic anisotropic Magnetohydrodynamics (MHD). This dispersion relation is used to point out the regions where the relativistic anisotropic MHD leads to new results that cannot be obtained using usual collisional relativistic MHD. This is highlighted by plotting a Fresnal ray surface. Conditions for the onset of firehose and mirror instabilities are also indicated. Such a study can be applied to astrophysical features such as pulsar winds, propagation of cosmic rays, etc.

Multidimensional and surface solitons in a nonlinear elastic medium

Nonlinear shear waves in two-dimensional systems (in particular, surface waves) are investigated with allowance for the spatial dispersion of the elastic medium. It is shown that the dispersion plays an important role in the structural and modulational stability of the nonlinear waves and to a large degree determines the directions of localization of phonons in a nonlinear localized wave and, in particular, the possibility of existence of elastic surface solitons. By means of an asymptotic procedure, solutions are found for small-amplitude two-dimensional elastic shear solitons of the one-parameter stationary-profile type and for envelope solitons and also for surface solitons localized near an ideal surface of an elastic half space. Localized excitations of this kind can exist only in a medium with a “focusing” (soft) nonlinearity and positive dispersion ∂2ω/∂k2>0, where ω(k) is the dispersion relation for linear waves. A procedure is proposed for finding solutions for surface envelope solitons localized near a surface covered with a layer of another substance. A comparison is made between the structures of the surface shear solitons at an ideal surface and at a surface with a film coating.

Instabilities of Alfvén and magnetosonic waves in dusty cometary plasmas with an ion ring beam

The effects of charged dust grains on the instabilities of electromagnetic waves in the solar wind plasma due to ring-beam cometary ion velocity distributions are considered. The waves are assumed to propagate parallel to the solar wind magnetic field. Collisionless cold plasma theory is used to derive the dispersion relations of linear waves in the solar wind, for a delta-function ring-beam velocity distribution of cometary water-group ions. The resonant and nonresonant instabilities are considered, and it is shown that the dust has its greatest effect on the nonresonant firehose instability at very long wavelength. The dust can cause an unstable dust-free firehose mode to be stable, and vice versa. It is also shown that as the frequency approaches the cometary ion cyclotron frequency, there is an upper limit to the wave number for the unstable firehose mode. For the left-hand polarized wave, the presence of dust increases this upper wave number limit, until the firehose mode merges with the resonant mode. The right-hand polarized firehose mode also has an upper wave number limit, which is almost independent of the presence of dust. It is shown that, although there is a high frequency resonant instability of the normal whistler mode induced by the ring beam, there is no corresponding instability of the dust-induced whistler-like mode at low frequencies.

Consequences of nongyrotropy in magnetohydrodynamics

It is assumed in the magnetohydrodynamic set of equations that a plasma is both isotropic and gyrotropic at any instant of time. However, this cannot be always justified for applications in space plasmas which are essentially collisionless. By introducing two relaxation time scales, one for isotropization and another for gyrotropization, we propose a model which describes evolution of the field and the plasma variables, including all the pressure tensor elements. We discuss briefly the linear wave dispersion relation of the system, and then attempt to apply the model to the magnetic reconnection.

The acceleration of energetic particles by transit-time damping

Recently, the quasilinear transport and acceleration parameters for cosmic ray particles interacting with oblique propagating fast magnetosonic waves, in a small-β plasma, have been calculated anew, using a linear wave dispersion relation. Including the dispersive part in the wave dispersion relation, we improve the quasilinear calculation of the transit-time damping acceleration time. We find that, if the turbulence spectrum extends to sufficiently high wavenumbers, the efficiency of the process of acceleration by the fast magnetosonic branch is significantly enhanced for electrons below 10 3 V A/c MeV, making this process a good candidate for injection.

Linear Waves in Force-Free Fibrils

AbstractThe advanced thin flux tube approximation for force-free thin magnetic flux tubes is used to derive a dispersion relation for linear waves. All wave modes appear to be coupled in a twisted flux tube. In the case of a weakly twisted flux tube, it has been found that torsional Alfvén waves have dispersion and produce pressure and temperature fluctuations. The effect of tube rotation is pointed out. These properties of linear waves have an impact on prominence oscillations.

Wave Numbers of Linear Progressive Waves

An explicit solution for wave number and wave speed is derived. The method of solution is to expand the linear wave dispersion relation in Taylor series in the neighborhoods of deep and shallow water limits. The two series are matched at the inflection point of the dispersion function, where the shoaling wave height is a minimum value. Numerical comparisons indicate that the present solution is computationally efficient and has an error of less than 0.05%. Two applications are made to solve problems, involving nearshore spectral analysis and refraction of ocean waves.

Solution of a linearized kinetic model for an ultrarelativistic gas

A linearized model of the Boltzmann equation for a relativistic gas is shown to be reducible, in the ultrarelativistic limit and for (1 + 1)-dimensional problems, to a system of three uncoupled transport equations, one of which is well known. A general method for solving these equations is recalled, with a few new details, and applied to the solution of two boundary value problems. The first of these describes the propagation of an impulsive change in a half space and is shown to give an explicit example of the recently proved result that no signal can propagate with speed larger than the speed of light, according to the relativistic Boltzmann equation. The second problem deals with steady oscillations in a half space and illustrates the meaning of certain recent results concerning the dispersion relation for linear waves in relativistic gas.

Dispersion surfaces

The dispersion relation of linear waves in a non-relativistic, collisionless and homogeneous magnetized plasma is solved by numerical methods. Both electrostatic and electromagnetic waves with frequencies from below the ion gyrofrequency to above the electron gyrofrequency are studied for all angles of propagation. Modes occurring in a cold plasma as well as waves dependent on thermal effects are included. Dispersion surfaces, that is plots of frequency versus wave vector components, are presented for some plasma models. This presentation shows all interesting waves clearly and reveals how different modes are related. The dispersion surfaces can be used as an aid, for example, when comparing wave observations and theory, and they may also help to avoid confusing nomenclature.

Plasma Instabilities Generated by Streaming Particles

It is well known that particles streaming along the ambient magnetic field in space plasmas may generate waves with frequencies of the order of the local ion gyrofrequency (ion waves), (Kindel and Kennel 1971). In this study we analyze the dispersion relation of these waves numerically and discuss mechanisms for damping and instability. All numerical results in this report are obtained with the computer code WHAMP (Rönnmark 1982a,b), which solves the dispersion relation of linear waves in a homogeneous plasma for a complex frequency as a function of a real wavevector. As a specific example we consider the S3-3 satellite observations of banded electrostatic ion waves, associated with streaming particles (Kintner et. al. 1979, Cattell 1981).