Non-symmetric two-stream instability

1977 ◽  
Vol 17 (1) ◽  
pp. 23-39 ◽  
Author(s):  
S. Cuperman ◽  
L. Gomberoff ◽  
I. Roth ◽  
W. Bernstein
Keyword(s):  
Numerical Solutions ◽  
Cyclotron Frequency ◽  
Particle Density ◽  
Maximum Growth ◽  
Systematic Investigation ◽  
Marginal Stability ◽  
Theoretical Problem ◽  
Electrostatic Waves ◽  
Dirac Delta ◽  
Stream Instability

The recent Plum Brook—NASA experiments on counterstreaming plasma instabilities in which electric field emissions at frequencies [(n + ½) ±α]ωe (n = 1, 2, 3,…, 9, α < 0·5) have been observed, raised the theoretical problem of the non-symmetric two-stream instability. This is the case in which the counterstreaming beams have different velocities in the directions parallel and perpendicular to the static magnetic fields as well as different particle density.We investigate theoretically this problem. The instability of quasi-electrostatic waves at shifted half odd integer multiples of the cyclotron frequency, due to two non-symmetric counterstreaming electron beams, is considered. The beam velocities parallel and perpendicular to the static magnetic field are represented by different double Dirac delta functions; no parameter limitation is imposed. A systematic investigation of (i) the coupling between plasma modes and cyclotron modes and (ii) the coupling between cyclotron modes (beam 1) and cyclotron modes (beam 2) resulting in shifted half odd integer multiples of the cyclotron frequency is carried out.The results include approximate but simple analytical expressions for maximum growth rates and for marginal stability as well as exact numerical solutions for the detailed unstable spectra (1 ≤ m ≤ 20) in both ωτ- and kr-space. The relative weakening or suppression of certain modes is also predicted.

Three component non-symmetric counter-streaming instabilities: longitudinal electron plasma modes

1978 ◽  
Vol 19 (1) ◽  
pp. 1-14 ◽  
Author(s):  
S. Cuperman ◽  
L. Gomberoff ◽  
I. Roth
Keyword(s):  
Numerical Solutions ◽  
Electron Beams ◽  
Particle Density ◽  
Complex Structure ◽  
Wave Spectrum ◽  
Finite Size ◽  
Maximum Growth ◽  
Maximum Growth Rate ◽  
Symmetric Case ◽  
Longitudinal Modes

The counter-streaming instabilities arising in three-component electron plasmas are investigated analytically and numerically in the general non-symmetric case, i.e. when Here n1 n2 and na represent the electron particle density in the first and second beams and in the background (ambient) stationary plasma, respectively; U1 and U2 represent the streaming velocities of the two counter-streaming electron beams. No magnetic or temperature effects are considered; consequently the three components interact only through the electric collective fields and only longitudinal modes are present. The positive ions here represent a stationary neutralizing background. Combined analytical and numerical solutions of the dispersion equation indicate that the basic properties of the unstable plasma modes may change significantly, depending on the values of the dimensionless parameters e, a and g. Thus, the standing wave spectrum (Re ω = 0) which occurs in the symmetric case (ε = a = 1) without background (gr = 0) may be replaced by a mixed travellingstanding wave spectrum having a rather complex structure; the maximum growth rate could be also strongly affected. The transformation of the instability from ‘absolute’ into mixed ‘convective and absolute’ may have significant physical implications, especially for finite size plasma systems or finite length unstable interaction regions. The results are relevant for laboratory and (especially) astrophysical situations in which counter-streaming electron beams having unequal streaming velocities (and particle densities) penetrate plasma regions with significant relative particle concentration.

Low-frequency drift-induced instabilities in a magnetized two-ion plasma

1986 ◽  
Vol 35 (3) ◽  
pp. 393-412 ◽  
Author(s):  
R. Bharuthram ◽  
M. A. Hellberg
Keyword(s):  
Dispersion Relation ◽  
Numerical Solutions ◽  
Cyclotron Frequency ◽  
Low Frequency ◽  
Frequency Drift ◽  
Transition Diagram ◽  
Electrostatic Waves ◽  
Ion Plasma ◽  
Parameter Values ◽  
A Current

Numerical solutions of a dispersion relation for low-frequency electrostatic waves in a current-carrying, cold, weakly collisional, magnetized two-ion plasma are used to discuss the two-stream and resistive natures of the ion-ion hybrid instability. An instability with analogous behaviour is found to be associated with the light ion cyclotron frequency. Analytical results explain the behaviour. A numerically derived transition diagram summarizes the parameter values for which transitions between different modes take place.

Electromagnetic cyclotron-loss-cone instability associated with weakly relativistic electrons

1982 ◽  
Vol 28 (3) ◽  
pp. 503-525 ◽  
Author(s):  
H. K. Wong ◽  
C. S. Wu ◽  
F. J. Ke ◽  
R. S. Schneider ◽  
L. F. Ziebell
Keyword(s):  
Cyclotron Frequency ◽  
Maximum Growth ◽  
Basic Mechanism ◽  
Electron Cyclotron ◽  
Relativistic Electrons ◽  
Electron Component ◽  
Loss Cone ◽  
Electron Cyclotron Frequency ◽  
Cone Type ◽  
Consistent Manner

The amplification of fast extraordinary mode waves with frequencies very close to the electron cyclotron frequency is investigated for a plasma which consists of a weakly relativistic electron component with a loss-cone type distribution and a cold background electron component. The basic mechanism of the amplification is attributed to a relativistic cyclotron resonance between the wave and the energetic electrons. The method employed in the present analysis enables us to solve the dispersion relation in a self-consistent manner for arbitrary ratio of the densities of the energetic and background electrons. It is found that the maximum growth rates occur at certain values of ω2pe/Ω2e and the angular dependence of the growth rate is sensitive to the ratios ω2pe/Ω2e and ne/nb. Here ωpe and Ωe are the electron plasma frequency and the electron cyclotron frequency, respectively, and ne and nb denote the number densities of the energetic and background electrons, respectively.

Review of basic quantum physics

Quantum 20/20 ◽  
2019 ◽  
pp. 1-20
Author(s):  
Ian R. Kenyon
Keyword(s):  
Quantum Physics ◽  
Particle Density ◽  
Electromagnetic Coupling ◽  
Black Body ◽  
Black Body Radiation ◽  
Gauge Transformations ◽  
Dirac Delta ◽  
Wave Particle Duality ◽  
Delta Functions ◽  
Lorentz Invariants

Basic experimental evidence is sketched: the black body radiation spectrum, the photoeffect, Compton scattering and electron diffraction; the Bohr model of the atom. Quantum mechanics is reviewed using the Copenhagen interpretation: eigenstates, observables, hermitian operators and expectation values are explained. Wave-particle duality, Schrödinger’s equation, and expressions for particle density and current are described. The uncertainty principle, the collapse of the wavefunction, Schrödinger’s cat and the no-cloning theorem are discussed. Dirac delta functions and the usage of wavepackets are explained. An introduction to state vectors in Hilbert space and the bra-ket notation is given. Abstracts of special relativity and Lorentz invariants follow. Minimal electromagnetic coupling and the gauge transformations are explained.

scholarly journals Numerical complete solution for random genetic drift by energetic variational approach

2019 ◽  
Vol 53 (2) ◽  
pp. 615-634 ◽  
Author(s):  
Chenghua Duan ◽  
Chun Liu ◽  
Cheng Wang ◽  
Xingye Yue
Keyword(s):  
Genetic Drift ◽  
Variational Approach ◽  
Numerical Solutions ◽  
Complete Solution ◽  
Random Genetic Drift ◽  
Least Action Principle ◽  
Dirac Delta ◽  
Splitting Technique ◽  
Least Action ◽  
Convection Dominated

In this paper, we focus on numerical solutions for random genetic drift problem, which is governed by a degenerated convection-dominated parabolic equation. Due to the fixation phenomenon of genes, Dirac delta singularities will develop at boundary points as time evolves. Based on an energetic variational approach (EnVarA), a balance between the maximal dissipation principle (MDP) and least action principle (LAP), we obtain the trajectory equation. In turn, a numerical scheme is proposed using a convex splitting technique, with the unique solvability (on a convex set) and the energy decay property (in time) justified at a theoretical level. Numerical examples are presented for cases of pure drift and drift with semi-selection. The remarkable advantage of this method is its ability to catch the Dirac delta singularity close to machine precision over any equidistant grid.

Longitudinal waves in a perpendicular collisionless plasma shock: II. Vlasov ions

1970 ◽  
Vol 4 (4) ◽  
pp. 753-760 ◽  
Author(s):  
S. Peter Gary
Keyword(s):  
Collisionless Plasma ◽  
Maximum Growth ◽  
Zero Magnetic Field ◽  
Linear Dispersion ◽  
Electrostatic Waves ◽  
Longitudinal Waves ◽  
Acoustic Instability ◽  
Ion Acoustic ◽  
Plasma Shock ◽  
Vlasov Plasma

This paper presents an analysis of the linear dispersion relation for electrostatic waves in a Vlasov plasma of unmagnetized, Maxwellian ions and magnetized, Maxwellian electrons. The electrons undergo E × B and ∇B drifts, and the electron β is small. For propagation in the perpendicular direction, maximum growth rates can be substantially larger than those of the zero magnetic field ion acoustic instability. For propagation outside a few degrees from the perpendicular the dispersion characteristics are essentially those of the ion acoustic instability.

scholarly journals Circular dispersive shock waves in colloidal media

2016 ◽  
Vol 25 (04) ◽  
pp. 1650044 ◽  
Author(s):  
A. Azmi ◽  
T. R. Marchant
Keyword(s):  
Solitary Waves ◽  
Numerical Solutions ◽  
Particle Density ◽  
Modulational Instability ◽  
Particle Interaction ◽  
Packing Fraction ◽  
Effect Of Temperature ◽  
Hard Sphere Model ◽  
Large Radius ◽  
Temperature Dependent

A dispersive shock wave (DSW), with a circular geometry, is studied in a colloidal medium. The colloidal particle interaction is based on the repulsive theoretical hard sphere model, where a series in the particle density, or packing fraction is used for the compressibility. Experimental results show that the particle interactions are temperature dependent and can be either repulsive or attractive, so the second term in the compressibility series is modified to allow for temperature dependent effects, using a power-law relationship. The governing equation is a focusing nonlinear Schrödinger-type equation with an implicit nonlinearity. The initial jump in electric field amplitude is resolved via a DSW, which forms before the onset of modulational instability. A semi-analytical solution for the amplitude of the solitary waves in a DSW of large radius, is derived based on a combination of conservation laws and geometrical considerations. The effect of temperature and background packing fraction on the evolution of the DSW and the amplitude of the solitary waves is discussed and the semi-analytical solutions are found to be very accurate, when compared with numerical solutions.

scholarly journals Numerical Solution for a Non-Fickian Diffusion in a Periodic Potential

2013 ◽  
Vol 13 (2) ◽  
pp. 502-525 ◽  
Author(s):  
Adérito Araújo ◽  
Amal K. Das ◽  
Cidália Neves ◽  
Ercília Sousa
Keyword(s):  
Diffusion Equation ◽  
Periodic Potential ◽  
Space Dimension ◽  
Numerical Solutions ◽  
Brownian Particle ◽  
Particle Density ◽  
Mean Square Displacement ◽  
Hyperbolic Type ◽  
Fickian Diffusion ◽  
Spatial Shape

AbstractNumerical solutions of a non-Fickian diffusion equation belonging to a hyperbolic type are presented in one space dimension. The Brownian particle modelled by this diffusion equation is subjected to a symmetric periodic potential whose spatial shape can be varied by a single parameter. We consider a numerical method which consists of applying Laplace transform in time; we then obtain an elliptic diffusion equation which is discretized using a finite difference method. We analyze some aspects of the convergence of the method. Numerical results for particle density, flux and mean-square-displacement (covering both inertial and diffusive regimes) are presented.

Electromagnetic ion-cyclotron instability vs. electrostatic ion-cyclotron instability in mixed (warm-cold) magnetospheric-like plasmas

1977 ◽  
Vol 18 (3) ◽  
pp. 391-413 ◽  
Author(s):  
S. Cuperman ◽  
L. Gomberoff
Keyword(s):  
Electromagnetic Coupling ◽  
Maximum Growth ◽  
Systematic Investigation ◽  
Maximum Growth Rate ◽  
Plasma Parameters ◽  
Loss Cone ◽  
Cyclotron Instability ◽  
Ion Cyclotron ◽  
Validity Range ◽  
Approximate Expressions

This work presents a systematic investigation and comparison of electromagnetic ion-cyclotron (e.m.) and electrostatic ion-cyclotron (e.s.) instabilities in uniform mixed warm and cold plasmas for magnetospheric-like plasma parameters. The following main aspects are included: Analytical: (i) we derive simple approximate expressions for the maximum growth rate, γmax for the quasi-electrostatic instability in the regime ωr » …p, к┴ » к∥ (к∥ ≠ 0) with the protons being described by mixed loss-cone and cold populations and with inclusion of electromagnetic coupling effects due to electrons; (ii) we analyse another regime in which electrostatic instalilities first increase with addition of cold plasma and decrease only after having reached a maximum, namely a regime with ωr » …p, к = к≠; (iii) we summarize the corresponding analytical results for parallel propagating electromagnetic ion-cyclotron unstable waves and discuss their validity range.

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