Negative Energy Modes Research Articles

On the behaviour of a fluid-loaded cylindrical shell with mean flow

The unsteady behaviour of an infinitely long fluid-loaded elastic plate which is driven by a single-frequency point-force excitation in the presence of mean flow is known to exhibit a number of unexpected features, including absolute instability when the normalized flow speed, U, lies above some critical speed U0, and certain unusual propagation effects for U<U0. In the latter respect Crighton & Oswell (1991) have demonstrated most significantly that for a particular frequency range there exists an anomalous neutral (negative energy) mode which has group velocity pointing towards the driver, in violation of the usual radiation condition of outgoing waves at infinity. They show that the rate of working of the driver can be negative, due to the presence of other negative-energy waves, and can also become infinite at a critical frequency corresponding to a real modal coalescence. In this paper we attempt to extend these results by including, as is usually the case in a practical situation, plate curvature in the transverse direction, by considering a fluid-loaded cylinder with axial mean flow. In the limit of infinite normalized cylinder radius, a, Crighton & Oswell's results are regained, but for finite a very significant modifications are found. In particular, we demonstrate that the additional stiffness introduced by the curvature typically moves the absolute-instability boundary to a much higher flow speed than for the flat-plate case. Below this boundary we show that Crighton & Oswell's anomalous neutral mode can only occur for a>a1(U), but in practical situations it turns out that a1(U) is exceedingly large, and indeed seems much larger than radii of curvature achievable in engineering practice. Other negative-energy waves are seen to exist down to a smaller, but still very large, critical radius a2(U), while the existence of a real modal coalescence point, leading to a divergence in the driver admittance, occurs down to a slightly smaller critical radius a3(U). The transition through these various flow regimes as U and a vary is fully described by numerical investigation of the dispersion relation and by asymptotic analysis in the (realistic) limit of small U. The inclusion of plate dissipation is also considered, and, in common with Abrahams & Wickham (1994) for the flat plate, we show how the flow then becomes absolutely unstable at all flow speeds provided that a>a2(U).

Two‐Stream Instability of Counterrotating Galaxies

The present study of the two-stream instability in stellar disks with counterrotating components of stars and/or gas is stimulated by recently discovered counterrotating spiral and S0 galaxies. Strong linear two-stream instability of tightly wrapped spiral waves is found for one- and two-armed waves with the pattern angular speed of the unstable waves always intermediate between the angular speed of the corotating matter (+Ω) and that of the counterrotating matter (-Ω). The instability arises from the interaction of positive and negative energy modes in the co- and counterrotating components. The unstable waves are in general convective—they move in radius and radial wavenumber space—with the result that amplification of the advected wave is more important than the local growth rate. For a galaxy of corotating stars and counterrotating stars of mass fraction ξ*<½, or of counterrotating gas of mass fraction ξg<½, the largest amplification is usually for the one-armed leading waves (with respect to the corotating stars). For the case of both counterrotating stars and gas, the largest amplifications are for ξ*+ξg ≈ ½, also for one-armed leading waves. The two-armed trailing waves usually have smaller amplifications. The growth rates and amplifications all decrease as the velocity spreads of the stars and/or gas increase. It is suggested that the spiral waves can provide an effective viscosity for the gas causing its accretion.

Parametric up-conversion of electromagnetic waves by a gyrating relativistic electron beam

A gyrating relativistic electron beam supports negative energy modes near the harmonics of the electron cyclotron frequency. An electromagnetic wave passing through such a beam parametrically up-converts into high frequency electromagnetic radiation, producing a beam cyclotron mode. In the case when pump frequency ω0 is away from the harmonics of cyclotron frequency, ω≠lωc/γ0 and ω≫ωc/γ0, the beam response to the pump and sideband waves is obtained by the fluid theory whereas the response at the beam cyclotron mode is kinetic. In the case when ω0∼lωc/γ0 the beam response to all three interacting waves is taken to be fully kinetic. It is seen that the normalized growth rate decreases as ω0−ωc/γ0 increases for Raman as well as Compton regimes of operation.

Stability of the B = 2 hedgehog in the Skyrme model

We study the unstable modes of the baryon number two hedgehog of the Skyrme model on a three dimensional spatial lattice. An expansion of the Skyrme Lagrangian around the hedgehog configuration provides the equations of motion for the fluctuation fields solvable numerically via a relaxation method. We find the negative energy modes and, by evolving the excited hedgehog in time, a breakup into two separated solitonic configurations is obtained. Different paths of descent for the receding Skyrmions are presented and the possibility of determining the metric structure of the collective-coordinate manifold is discussed.

FURTHER EXPLORATION ON THE EFFECT OFNEGATIVE-ENERGY MODES IN THENONLINEAR INSTABILITIES

The nonlinear variations of the eigenvectors are discussed on the basis of Ref. [1, 2]. It is shown that relative to a steady wave the nonlinear eigen frequency allows for both forward and backward transports. Their eigenvectors deflect in the Fourier space as compared to the linear case. The positive (P-) and negative (N-) energy modes can be defined in the nonlinear case according to the relative strength of the forward and backward eigenvectors of the perturbations. Based on this definition the effect of N-mode on the nonlinear instabilities is further studied. The results are in consistency with the conclusion in Ref. [1,2]. The bista-bility of wavepacket is associated with the transition of a resonance N-mode to a p-mode, in particular its eigenvector is discontinuous in the parameter space; The Hop f bifurcation of a steady wavepacket is associated with the resonance of a N-mode to a p-mode. As a result re-connections occur between their eigenvectors of forward and backward perturbations respec-tively. With the increase of nonlinearity, the p-mode first and the N-mode following next change their energy types. Consequently there exists a parameter regime where both modes become N-type. This regime coincides with that of the Hopf bifurcation. This fact strongly supports the conjecture that the excitation of N-modes induces the nonlinear (Hopf) bifurcation.

Nonlinear instability and chaos in plasma wave–wave interactions. II. Numerical methods and results

In Part I of this work [Phys. Plasmas 2, 1926 (1995)], the behavior of linearly stable, integrable systems of waves in a simple plasma model was described using a Hamiltonian formulation. Explosive instability arose from nonlinear coupling between positive and negative energy modes, with well-defined threshold amplitudes depending on the physical parameters. In this concluding paper, the nonintegrable case is treated numerically. The time evolution is modeled with an explicit symplectic integrator derived using Lie algebraic methods. For amplitudes large enough to support two-wave decay interactions, strongly chaotic motion destroys the separatrix bounding the stable region in phase space. Diffusive growth then leads to explosive instability, effectively reducing the threshold amplitude. For initial amplitudes too small to drive decay instability, slow growth via Arnold diffusion might still lead to instability; however, this was not observed in numerical experiments. The diffusion rate is probably underestimated in this simple model.

Transition of negative-energy modes to positive-energy modes and bistability of wavepackets

Quantitative descriptions for the positive- and negative-energy modes in nonlinear partial differential systems are proposed. The bistability of steady wave packets in driven nonlinear drift-waves is found to be associated with the discontinuous transition of a resonance negative-energy mode to a positive-energy mode.

Nonlinear instability and chaos in plasma wave–wave interactions. I. Introduction

Conventional linear stability analyses may fail for fluid systems with an indefinite free-energy functional. When such a system is linearly stable, it is said to possess negative energy modes. Instability may then occur either via dissipation of the negative energy modes, or nonlinearly via resonant wave–wave coupling, leading to explosive growth. In the dissipationless case, it is conjectured that intrinsic chaotic behavior may allow initially nonresonant systems to reach resonance by diffusion in phase space. In this and a companion paper (submitted to Phys. Plasmas), this phenomenon is demonstrated for a simple equilibrium involving cold counterstreaming ions. The system is described in the fluid approximation by a Hamiltonian functional and associated noncanonical Poisson bracket. By Fourier decomposition and appropriate coordinate transformations, the Hamiltonian for the perturbed energy is expressed in action-angle form. The normal modes correspond to Doppler-shifted ion-acoustic waves of positive and negative energy. Nonlinear coupling leads to decay instability via two-wave interactions, and to either decay or explosive instability via three-wave interactions. These instabilities are described for various integrable systems of waves interacting via single nonlinear terms. This discussion provides the foundation for the treatment of nonintegrable systems in the companion paper.

Contribution to the nonlinear theory of plasma wave excitation inside the electron foreshock region

The broadband electrostatic waves inside the electron foreshock region well away from its boundaries were explained as a beam mode for which the beam velocity is lower than the thermal velocity of the background plasma. We discuss here the nonlinear saturation of this instability. The waves propagating along the beam are stabilized by a plateau formation on the electron distribution function. The waves that propagate at an angle to the beam are stabilized in a different way. In the course of the development of the instability the beam velocity spreads and an initially negative energy mode, which is initially unstable, experiences increased damping by the beam particles and finally is stabilized due to this effect. We give an estimation of the wave energy in both cases, and we present a numerical modelling to substantiate our analytical estimates. The saturated wave energy level is found to be comparable in both cases and agrees with the levels observed in the foreshock.

Stability of magnetohydrodynamic equilibria with flow in the presence of a resistive wall

The effect of a resistive wall on the stability of a magnetohydrodynamic equilibrium with flow is examined. Gyroscopically stabilized negative energy modes are shown to become unstable in the presence of a resistive wall. An estimate for the growth rate is obtained. The effect of this destabilization on the limits on the safety factor and the plasma beta for a rigidly rotating plasma column are studied.

Negative-energy modes in a magnetically confined plasma in the framework of Maxwell-drift kinetic theory.

The general expression for the second-order perturbation energy of a Maxwell-drift kinetic system derived by Pfirsch and Morrison [Phys. Fluids B 3, 271 (1991)] is evaluated for the case of a magnetically confined plasma for which the equilibrium quantities depend on one Cartesian coordinate y. The conditions for the existence of negative-energy modes with vanishing initial field perturbations are also obtained. If the equilibrium guiding center distribution function ${\mathit{f}}_{\mathit{g}\ensuremath{\nu}}^{(0)}$ of any particle species \ensuremath{\nu} has locally the property ${\mathit{v}}_{\mathrm{\ensuremath{\parallel}}}$(\ensuremath{\partial}${\mathit{f}}_{\mathit{g}\ensuremath{\nu}}^{(0)}$/\ensuremath{\partial}${\mathit{v}}_{\mathrm{\ensuremath{\parallel}}}$)g0, where ${\mathit{v}}_{\mathrm{\ensuremath{\parallel}}}$ is the guiding center velocity parallel to the magnetic field, and if this holds in the minimum-energy reference frame, parallel and oblique negative-energy modes exist with no essential restriction on either the orientation or magnitude of the wave vector. This condition also holds for the equilibria of a homogeneous magnetized plasma and an inhomogeneous force-free plasma with sheared magnetic field. If ${\mathit{v}}_{\mathrm{\ensuremath{\parallel}}}$(\ensuremath{\partial}${\mathit{f}}_{\mathit{g}\ensuremath{\nu}}^{(0)}$/\ensuremath{\partial}${\mathit{v}}_{\mathrm{\ensuremath{\parallel}}}$)0, the oblique negative-energy modes possible in a magnetically confined plasma are nearly perpendicular. The condition for purely perpendicular negative-energy modes reads as (${\mathit{dP}}^{(0)}$/dy)(\ensuremath{\partial}${\mathit{f}}_{\mathit{g}\ensuremath{\nu}}^{(0)}$/\ensuremath{\partial}y)0, where ${\mathit{P}}^{(0)}$ is the plasma pressure.

Inverse method for the investigation of nonlinear instabilities associated with negative-energy modes.

Earlier work on explosively unstable similarity solutions of Hamilton's equations with Hamiltonians homogeneous of degree N and satisfying resonance conditions is applied to study the nonlinear stability of linearly stable equilibria with neighboring positive- and negative-energy waves. A multiple-time-scale expansion near equilibrium yields a Hamiltonian system of the assumed structure. In the inverse method an explosively unstable similarity solution is assumed and one solves for the coefficients of the terms in a Hamiltonian of some given structure. Through some general arguments and many examples one concludes that explosively unstable solutions occur generally for wide ranges of coefficient values. Hence the original equilibrium is nonlinearly unstable for wide ranges of interaction parameters.

Nonlinear instabilities relating to negative-energy modes.

The nonlinear instability of general linearly stable systems allowing linear negative-energy perturbations is investigated with the aid of a multiple-time-scale formalism. It is shown that the basic equations thus obtained imply resonance conditions and possess inherent symmetries which lead to the existence of similarity solutions of these equations. These solutions can be of an explosive type, oscillatory, or static. It is demonstrated that at least some of the oscillatory and static solutions are normally linearly unstable.

Hamiltonian Structure of the Vlasov-Einstein System and the Problem of Stability for Spherical Relativistic Star Clusters

The Hamiltonian formulation of the Vlasov-Einstein system, which is appropriate for collisionless, self-gravitating systems like clusters of stars that are so dense that gravity must be described by the Einstein equation, is presented. In particular, it is demonstrated explicitly in the context of a 3 + 1 splitting that, for spherically symmetric configurations, the Vlasov-Einstein system can be viewed as a Hamiltonian system, where the dynamics is generated by a noncanonical Poisson bracket, with the Hamiltonian generating the evolution of the distribution function ƒ (a noncanonical variable) being the conserved ADM mass-energy H ADM. This facilitates a geometric understanding of the evolution of ƒ in an infinite-dimensional phase space, providing thereby a natural interpretation of the constraints associated with conservation of phase space. This geometric interpretation also facilitates the derivation of improved criteria for linear stability by focusing on dynamically accessible perturbations δƒ which satisfy all the constraints of phase space conservation. An explicit expression is derived for the energy δ (2) H ADM associated with an arbitrary spherical phase space preserving perturbation of an arbitrary spherical equilibrium, and it is shown that the equilibrium must be linearly stable if δ (2) H ADM is positive semi-definite. Insight into the Hamiltonian reformulation is provided by a description of general finite degree of freedom systems. Intuition derived from simple finite models clarifies several features of the Vlasov-Einstein system; for example, how, negative energy modes preclude necessary and sufficient conditions for stability and why, unlike the Newtonian case, the existence of negative energy perturbations for some static, isotropic equilibrium apparently signals the Onset of a linear instability. An Appendix exhibits the construction of a completely covariant bracket which generates the Vlasov-Einstein system for arbitrary configurations in a form independent of any assumed 3 + 1 splitting.

Explosive free-electron-laser instability in a plasma-filled waveguide.

A relativistic electron beam passing through a plasma excites the microwave with phase velocity ${\mathit{scrv}}_{\mathrm{ph}}$${\mathit{scrv}}_{0}$, where ${\mathit{scrv}}_{0}$ is the velocity of the relativistic electrons. At moderately high amplitude this microwave couples to a fast free-electron-laser mode (with ${\mathit{scrv}}_{\mathrm{ph}}$&gt;c) via a negative-energy beam mode, and this process leads to the explosive growth of all the three waves at the expense of the energy of the beam. The explosion time can be decreased as long as reasonable parameters are chosen.

Nonperturbative strong-field effects in tensor-scalar theories of gravitation.

It is shown that a wide class of tensor-scalar theories can pass the present weak-field gravitational tests and exhibit nonperturbative strong-field deviations away from general relativity in systems involving neutron stars. This is achieved without requiring either large dimensionless parameters, fine tuning, or the presence of negative-energy modes. This gives greater significance to tests of the strong gravitational field regime, notably binary pulsar experiments.

THE EFFECT OF NEGATIVE ENERGY MODE ON NONLINEAR INSTABILITY IN DRIVEN DRIFT WAVES (Ⅱ)——EXCHANGE WITH POSITIVE-ENERGY MODE,‘AVOIDED-CROSSING' AND THE HOPF BIFURCATION

In the nonlinear drift-waves driven by a sinucoidal wave, with variations of the driving frequency and strength, crossing and ‘avoided crossing' may appear between the nonlinear complex eigenvalues of a negative-energy mode and a positive-energy mode. While the ‘avoided crossing' occurs, the positions of these two modes are somehow exchanged. Such nonlinear resonance is related to the Hopf bifurcation in weak dissipations.

THE EFFECT OF NEGATIVE-ENERGY MODE ON NONLINEAR INSTABILITY IN DRIVEN DRIFT WAVES (I)——TRANSITION TO POSITIVE-ENERGY MODE AND THE BISTABILITY

In the nonlinear drift-waves driven by a sinucoidal wave, every mode has a resonance regime correspondingly in the parameter space of the driving frequency and strength. Feeding of the applied wave energy causes the transition of a resonant negative-energy mode to positive-energy one. This transition is relevant to the development of bistability of the system in the corresponding regime.

Reversal of phase velocity of a negative-energy mode and bistability in a driven-damped drift wave system

The eigenvalues of the steady states in a nonlinear drift wave system driven by a sinusoidal wave are investigated. A relevance is found between the appearance of bistability and the fact that the negative linear phase velocity of one resonance mode reverses nonlinearity in the frame following the driving wave. The system energy can be enhanced when this negative-energy mode is changed into a positive one. The wavenumber for the occurrence of the resonance varies with the driving frequency of the external wave. This fact leads to the repeated appearances of groups of bistable states in the parameter space.

Theoretical problems in nonsymmetric gravitational theory.

It has recenty been noted that the nonsymmetric metric model of gravity faces severe observational constraints. We show here that it is also subject to physically unacceptable formal difficulties even as an effective field theory: When expanded about a Riemannian background, the model exhibits curvature-coupled negative-energy (ghost) modes and unacceptable asymptotic behavior.