Longitudinal waves in a perpendicular collisionless plasma shock: I. Cold ions

1970 ◽  
Vol 4 (4) ◽  
pp. 739-751 ◽  
Author(s):  
S. Peter Gary ◽  
J. J. Sanderson
Keyword(s):  
Magnetic Field ◽  
Growth Rate ◽  
Collisionless Plasma ◽  
Maximum Growth ◽  
Maximum Growth Rate ◽  
Wave Number ◽  
Zero Magnetic Field ◽  
Linear Dispersion ◽  
Electrostatic Waves ◽  
Cold Ions

This paper considers electrostatic waves in a Vlasov plasma of unmagnetized ions and magnetized, Maxwellian electrons. The linear dispersion relation is derived for waves in a perpendicular shock such that the most important sources of instability are the E × B and ∇B electron drifts. For the case of cold ions, propagation perpendicular to the applied magnetic field, and the E × B drift alone, a numerical analysis of frequency vs. wave-number is presented. The effects of the ∇B drift are also considered, and it is shown that the maximum growth rate can be larger than the maximum growth rate for the zero magnetic field ion acoustic instabifity under comparable conditions.

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.

Longitudinal waves in a perpendicular collisionless plasma shock Part 4. Gradient B

1972 ◽  
Vol 7 (3) ◽  
pp. 417-425 ◽  
Author(s):  
S. Peter Gary
Keyword(s):  
Dispersion Relation ◽  
Growth Rates ◽  
Collisionless Plasma ◽  
Maximum Growth ◽  
Linear Dispersion ◽  
Electrostatic Waves ◽  
Longitudinal Waves ◽  
Linear Dispersion Relation ◽  
Plasma Shock ◽  
Vlasov Plasma

This paper considers electrostatic waves in a Vlasov plasma of unmagnetized ions and magnetized electrons undergoing E x B and gradient B drifts. The linear dispersion relation is solved numerically for Te & Ti. The results show that, as in the Te <Ti case, increasing electron β decreases the maximum growth rates.

scholarly journals Fourth-order nonlinear evolution equations for surface gravity waves in the presence of a thin thermocline

1997 ◽  
Vol 39 (2) ◽  
pp. 214-229 ◽  
Author(s):  
Sudebi Bhattacharyya ◽  
K. P. Das
Keyword(s):  
Growth Rate ◽  
Fourth Order ◽  
Evolution Equations ◽  
Nonlinear Evolution ◽  
Maximum Growth ◽  
Single Equation ◽  
Maximum Growth Rate ◽  
Nonlinear Evolution Equations ◽  
Wave Number ◽  
Coupled Equations

AbstractTwo coupled nonlinear evolution equations correct to fourth order in wave steepness are derived for a three-dimensional wave packet in the presence of a thin thermocline. These two coupled equations are reduced to a single equation on the assumption that the space variation of the amplitudes takes place along a line making an arbitrary fixed angle with the direction of propagation of the wave. This single equation is used to study the stability of a uniform wave train. Expressions for maximum growth rate of instability and wave number at marginal stability are obtained. Some of the results are shown graphically. It is found that a thin thermocline has a stabilizing influence and the maximum growth rate of instability decreases with the increase of thermocline depth.

Three-dimensional instabilities of oscillatory equatorial zonal shear flows

2009 ◽  
Vol 623 ◽  
pp. 59-74 ◽  
Author(s):  
ANDREI NATAROV ◽  
KELVIN J. RICHARDS
Keyword(s):  
Growth Rate ◽  
Three Dimensional ◽  
Maximum Growth ◽  
Maximum Growth Rate ◽  
Oscillatory Shear ◽  
Resonant Excitation ◽  
Wave Number ◽  
Zonal Flows ◽  
Vertical Scale ◽  
Zonal Wave Number

In this paper, we investigate the linear stability of oscillating zonal flows on the equatorial β-plane in the presence of fully three-dimensional disturbances. To exclude inflection point effects, we focus on the simplest case of a linear meridional shear with time-mean and oscillating components. For purely oscillatory background flows we find that in addition to resonant excitation of ‘additive’ type that occurs in the zonally invariant case, resonant excitation of ‘difference’ type is also possible. For flows with an oscillatory shear superimposed on an unstable time-mean shear it is shown that while the oscillatory shear has a stabilizing influence on disturbances with a small zonal wave number k, at higher k the effect of the oscillating shear diminishes and can even be destabilizing. Overall, a small oscillatory shear tends to reduce the fastest growth rate in the system and pushes the dominant k to higher values. Calculation of dominant zonal and vertical modes shows that the zonally asymmetric modes dominate a large portion of the parameter space, especially at high time-mean background shear and low oscillatory shear. As a result, the dominant vertical mode can have a somewhat larger vertical scale than in the zonally invariant case. At intermediate values of the time-mean shear the growth rate is relatively flat with respect to the zonal mode number, with maximum growth rate occurring in bands of high and low k. We have uncovered a rich assortment of vertical and zonal modes which are likely to play a role in the nonlinear evolution of equatorial flows.

Kelvin–Helmholtz instability under horizontal rotation and magnetic fields

1998 ◽  
Vol 59 (2) ◽  
pp. 193-209
Author(s):  
L. A. DÁVALOS-OROZCO
Keyword(s):  
Magnetic Field ◽  
Growth Rate ◽  
Fluid Layer ◽  
Maximum Growth ◽  
Maximum Growth Rate ◽  
Density Stratification ◽  
Velocity Shear ◽  
Sufficient Condition ◽  
Horizontal Shear ◽  
Helmholtz Instability

The author's previous work on the Rayleigh–Taylor instability is extended to the Kelvin–Helmholtz instability, and the maximum growth rate of a perturbation and an estimate of its upper bound is obtained for an infinite fluid layer under horizontal rotation where the density, horizontal velocity (shear) and magnetic field are continuously stratified in the direction of gravity. Conclusions are drawn about the possibility of stability for some directions of propagation of the perturbation, even in the case of unstably stratified density. It is also shown that the new terms that appear owing to the interaction of the horizontal shear flow, horizontal rotation and stratified magnetic field increase the range of values that contribute to the estimate of the maximum growth rate compared with previous work. Furthermore, a generalization of the sufficient condition for stability under horizontal rotation alone obtained by Johnson is calculated in the presence of density stratification. A new method is also given to obtain a sufficient condition for stability when a magnetic field is present in addition to rotation and density stratification.

scholarly journals Brillouin-shifted third-harmonic backscattering of laser in a magnetized plasma

2015 ◽  
Vol 33 (1) ◽  
pp. 97-102 ◽  
Author(s):  
Alireza Paknezhad
Keyword(s):  
Magnetic Field ◽  
Growth Rate ◽  
Static Magnetic Field ◽  
Maximum Growth ◽  
Maximum Growth Rate ◽  
Magnetized Plasma ◽  
Third Harmonic ◽  
Coupled Equations ◽  
Relativistic Mass ◽  
Weakly Coupled

AbstractThird-harmonic Brillouin backscattering (3HBBS) instability is investigated in the interaction of a picosecond extraordinary laser pulse with a homogeneous transversely magnetized underdense plasma. Nonlinear coupled equations that describe the instability are derived and solved for a weakly coupled regime to find the maximum growth rate. The nonlinearity arises through the combined effect of relativistic mass increase, static magnetic field, and ponderomotive acceleration of plasma electrons. The growth rate is found to decrease as the static magnetic field increases. It also increases by increasing both plasma density and laser intensity. It is also established that the growth rate of 3HBBS instability in a magnetized plasma is lower than that of fundamental Brillouin backscattering instability.

Filamentation of a laser beam in a strongly ionized magnetoplasma

1978 ◽  
Vol 19 (1) ◽  
pp. 55-61 ◽  
Author(s):  
L. A. Pitale
Keyword(s):  
Magnetic Field ◽  
Growth Rate ◽  
Laser Beam ◽  
Spatial Scale ◽  
Large Scale ◽  
Maximum Growth ◽  
Mean Free Path ◽  
Maximum Growth Rate ◽  
Main Beam ◽  
Small Scale

On a time-scale of the order of the energy relaxation time, a high power laser beam, propagating in a strongly ionized magnetoplasma is shown to be unstable for small scale fluctuations. In the domain r0 < [mi/m]½ λm. v2/[ω2c + v2] (r0, λm, v, ωc, and m being respectively the spatial scale of the perturbation, electron mean free path, collision frequency, cyclotron frequency and mass and mi being the ion mass) the main loss of excess electron energy is due to thermal conduction; in the other limit collisional loss dominates. It is shown that for small scale fluctuations the growth rate increases with (i) increasing magnetic field and (ii) increasing r0. For large scale fluctuations the magnetic field does not show any effect; the growth rate, however, diminishes with increasing spatial scale. A maximum growth rate is obtained both for some optimum value of scale length and for intensity of the main beam.

Kinetic theory of the stability of collisional plasma in curved magnetic field

1969 ◽  
Vol 3 (2) ◽  
pp. 155-160 ◽  
Author(s):  
K. Jungwirth
Keyword(s):  
Magnetic Field ◽  
Collisionless Plasma ◽  
Maximum Growth ◽  
Maximum Growth Rate ◽  
Ion Temperature ◽  
Gravitation Field ◽  
Ion Collisions ◽  
Large Gradient ◽  
Field Lines ◽  
The Stability

The effect of curvature of magnetic field lines on plasma instabilities due to a large gradient of the ion temperature has been studied. It is shown that none of the discussed effects can be obtained so long as the curvature of magnetic field lines is simulated by a fictional gravitation field. In configurations with large enough minimum-B, the usual temperature-gradient instability is stabilized in collisionless plasma. However, a new dissipative instability arises due to ion–ion collisions with the maximum growth rate γmax ˜ Vii/40.

scholarly journals α-Stability Analysis of Hydromagnetic Instabilities in Stars with Toroidal Magnetic Fields

1980 ◽  
Vol 58 ◽  
pp. 667-672
Author(s):  
M. Goossens ◽  
D. Biront
Keyword(s):  
Magnetic Field ◽  
Growth Rate ◽  
Stability Analysis ◽  
Magnetic Fields ◽  
Sufficient Conditions ◽  
Maximum Growth ◽  
Maximum Growth Rate ◽  
Toroidal Magnetic Field ◽  
Displacement Fields ◽  
Necessary And Sufficient

Abstractα-stability analysis is used to investigate the adiabatic stability of a star containing an axisymmetric toroidal magnetic field. Necessary and sufficient conditions for α-stability are derived. Special attention is devoted to the typical hydromagnetic instabilities that can be introduced by a weak toroidal magnetic field in a star that is stably stratified in the absence of any magnetic field. An expression for the maximum growth rate of instability is derived and the basic properties of the displacement fields associated with the instabilities are indicated.

Saturation mechanism and generated viscosity in gravito-turbulent accretion disks

2020 ◽  
Vol 640 ◽  
pp. A53
Author(s):  
L. Löhnert ◽  
S. Krätschmer ◽  
A. G. Peeters
Keyword(s):  
Growth Rate ◽  
Numerical Simulations ◽  
Accretion Disks ◽  
Gravitational Instability ◽  
Turbulent Viscosity ◽  
Radial Distance ◽  
Maximum Growth ◽  
Wave Number ◽  
Radiative Cooling ◽  
Mixing Length

Here, we address the turbulent dynamics of the gravitational instability in accretion disks, retaining both radiative cooling and irradiation. Due to radiative cooling, the disk is unstable for all values of the Toomre parameter, and an accurate estimate of the maximum growth rate is derived analytically. A detailed study of the turbulent spectra shows a rapid decay with an azimuthal wave number stronger than ky−3, whereas the spectrum is more broad in the radial direction and shows a scaling in the range kx−3 to kx−2. The radial component of the radial velocity profile consists of a superposition of shocks of different heights, and is similar to that found in Burgers’ turbulence. Assuming saturation occurs through nonlinear wave steepening leading to shock formation, we developed a mixing-length model in which the typical length scale is related to the average radial distance between shocks. Furthermore, since the numerical simulations show that linear drive is necessary in order to sustain turbulence, we used the growth rate of the most unstable mode to estimate the typical timescale. The mixing-length model that was obtained agrees well with numerical simulations. The model gives an analytic expression for the turbulent viscosity as a function of the Toomre parameter and cooling time. It predicts that relevant values of α = 10−3 can be obtained in disks that have a Toomre parameter as high as Q ≈ 10.

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