The effect of plasma compressibility on the Kelvin-Helmholtz instability

The Kelvin-Helmholtz (KH) instability of a tangential discontinuity between two compressible plasmas in relative motion is investigated, by solving the dispersion equation for two cases. In the first, neutrals are excluded; in the second, collisions between neutrals and ions are introduced in the form of a drag force in the momentum equation. The velocity of neutrals is assumed to be perpendicular to the interface. In both cases the growth rate of the KH instability is obtained as a function of the density jump between the plasmas. Although it has often been remarked that compressibility should, in general, stabilize a plasma, it is found that this ceases to be true when allowance is made for a significant density jump at the interface. Thus, for a large density jump and a large velocity shear, the instability growth rate in a compressible plasma may considerably exceed the growth rate obtained when incompressibility is assumed. Collisions, it is shown, may either stabilize or destabilize a tangential discontinuity, depending on the change in the product of density and collision frequency (pv), as one moves with the neutrals across the interface; when pv decreases, the instability is enhanced (and vice versa).

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Magnetic field generation in a jet-sheath plasma via the kinetic Kelvin-Helmholtz instability

Annales Geophysicae ◽

10.5194/angeo-31-1535-2013 ◽

2013 ◽

Vol 31 (9) ◽

pp. 1535-1541 ◽

Cited By ~ 14

Author(s):

K.-I. Nishikawa ◽

P. Hardee ◽

B. Zhang ◽

I. Duţan ◽

M. Medvedev ◽

...

Keyword(s):

Magnetic Field ◽

Growth Rate ◽

Magnetic Fields ◽

Electric Field ◽

Flow Direction ◽

Transverse Magnetic ◽

Velocity Shear ◽

Helmholtz Instability ◽

Magnetic Field Generation ◽

Relativistic Jet

Abstract. We have investigated the generation of magnetic fields associated with velocity shear between an unmagnetized relativistic jet and an unmagnetized sheath plasma. We have examined the strong magnetic fields generated by kinetic shear (Kelvin–Helmholtz) instabilities. Compared to the previous studies using counter-streaming performed by Alves et al. (2012), the structure of the kinetic Kelvin–Helmholtz instability (KKHI) of our jet-sheath configuration is slightly different, even for the global evolution of the strong transverse magnetic field. In our simulations the major components of growing modes are the electric field Ez, perpendicular to the flow boundary, and the magnetic field By, transverse to the flow direction. After the By component is excited, an induced electric field Ex, parallel to the flow direction, becomes significant. However, other field components remain small. We find that the structure and growth rate of KKHI with mass ratios mi/me = 1836 and mi/me = 20 are similar. In our simulations saturation in the nonlinear stage is not as clear as in counter-streaming cases. The growth rate for a mildly-relativistic jet case (γj = 1.5) is larger than for a relativistic jet case (γj = 15).

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Kelvin–Helmholtz instability under horizontal rotation and magnetic fields

Journal of Plasma Physics ◽

10.1017/s0022377897006314 ◽

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.

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The role of a density jump in the Kelvin—Helmholtz instability of compressible plasma

Journal of Plasma Physics ◽

10.1017/s0022377800017888 ◽

1994 ◽

Vol 52 (2) ◽

pp. 223-244 ◽

Cited By ~ 15

Author(s):

A. G. González ◽

J. Gratton

Keyword(s):

General Procedure ◽

Velocity Shear ◽

Geometrical Parameters ◽

Uniform Density ◽

Density Jump ◽

Helmholtz Instability ◽

Laboratory Plasmas ◽

Velocity Jump ◽

General Dispersion

The hydromagnetic Kelvin–Helmholtz instability is relevant in many complex situations in astrophysical and laboratory plasmas. Many cases of interest are very complicated, since they involve the combined role of velocity shear, of density and magnetic field stratification, and of various geometries in compressible plasmas. In the present work we continue investigating the influence of various physical and geometrical parameters of the plasma on the Kelvin–Helmholtz modes. We use the general dispersion relation for the ideal compressible MHD modes localized near a velocity discontinuity between two uniform plasmas. We study analytically the existence and properties of the modes and their stability, for a velocity jump combined with a density jump, and for any relative orientation of B, u and k (B is continuous). Stability is analysed by means of a general procedure that allows discussion of any configuration and all kinds of perturbations. The boundaries between modes of different kinds are discussed. In contrast to the case of uniform density, for a density jump there are no monotonically unstable modes, only overstabilities. The unstable modes belong to two types. Those with the largest growth rates tend to monotonically unstable modes in the limit of uniform density, and are related to the torsional Alfvén mode. The other overstable modes have no analogue among the purely incompressible modes, and occur in a range of U that is stable in the incompressible limit. We derive bounds for the growth rate of the instability. The present results may serve as a guide to interpret results in more complicated and realistic situations as those occurring in laboratory and natural plasmas.

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Suppression of Baroclinic Instabilities in Buoyancy-Driven Flow over Sloping Bathymetry

Journal of Physical Oceanography ◽

10.1175/jpo-d-15-0240.1 ◽

2017 ◽

Vol 47 (1) ◽

pp. 49-68 ◽

Cited By ~ 22

Author(s):

Robert D. Hetland

Keyword(s):

Growth Rate ◽

Linear Instability ◽

Linear Stability Theory ◽

Buoyancy Frequency ◽

Bottom Slope ◽

Coriolis Parameter ◽

Plume Front ◽

Instability Theory ◽

Instability Growth ◽

Flow Configurations

AbstractBaroclinic instabilities are ubiquitous in many types of geostrophic flow; however, they are seldom observed in river plumes despite strong lateral density gradients within the plume front. Supported by results from a realistic numerical simulation of the Mississippi–Atchafalaya River plume, idealized numerical simulations of buoyancy-driven flow are used to investigate baroclinic instabilities in buoyancy-driven flow over a sloping bottom. The parameter space is defined by the slope Burger number S = Nf−1α, where N is the buoyancy frequency, f is the Coriolis parameter, and α is the bottom slope, and the Richardson number Ri = N2f2M−4, where M2 = |∇Hb| is the magnitude of the lateral buoyancy gradients. Instabilities only form in a subset of the simulations, with the criterion that SH ≡ SRi−1/2 = Uf−1W−1 = M2f−2α 0.2, where U is a horizontal velocity scale and SH is a new parameter named the horizontal slope Burger number. Suppression of instability formation for certain flow conditions contrasts linear stability theory, which predicts that all flow configurations will be subject to instabilities. The instability growth rate estimated in the nonlinear 3D model is proportional to ωImaxS−1/2, where ωImax is the dimensional growth rate predicted by linear instability theory, indicating that bottom slope inhibits instability growth beyond that predicted by linear theory. The constraint SH 0.2 implies a relationship between the inertial radius Li = Uf−1 and the plume width W. Instabilities may not form when 5Li > W; that is, the plume is too narrow for the eddies to fit.

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The Study of Thermal Conditions on Weibel Instability

Advances in High Energy Physics ◽

10.1155/2015/428013 ◽

2015 ◽

Vol 2015 ◽

pp. 1-6

Author(s):

M. Mahdavi ◽

H. Khanzadeh

Keyword(s):

Growth Rate ◽

Thermal Conditions ◽

Ion Scattering ◽

Weibel Instability ◽

Anisotropic Temperature ◽

Instability Growth ◽

Limiting Condition ◽

Limiting Case ◽

Background Ions ◽

Electromagnetic Instability

Weibel electromagnetic instability has been studied analytically in relativistic plasma with high parallel temperature, where|α=(mc2/T∥)(1+p^⊥2/m2c2)1/2|≪1and while the collision effects of electron-ion scattering have also been considered. According to these conditions, an analytical expression is derived for the growth rate of the Weibel instability for a limiting case of|ζ=α/2(ω′/ck)|≪1, whereω′is the sum of the wave frequency of instability and the collision frequency of electrons with background ions. The results show that in the limiting conditionα≪1there is an unusual situation of the Weibel instability so thatT∥≫T⊥, while in the classic Weibel instabilityT∥≪T⊥. The obtained results show that the growth rate of the Weibel instability will be decreased due to an increase in the number of collisions and a decrease in the anisotropic temperature by the increasing of plasma density, while the increase of the parameterγ^⊥=(1+p^⊥2/m2c2)1/2leads to the increase of the Weibel instability growth rate.

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Orbit width scaling of TAE instability growth rate

10.2172/453518 ◽

1995 ◽

Author(s):

H.V. Wong ◽

H.L. Berk ◽

B.N. Breizman

Keyword(s):

Growth Rate ◽

Instability Growth Rate ◽

Instability Growth

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A new laser based technique for instability growth rate evaluation in liquid jets

Experiments in Fluids ◽

10.1007/bf00194013 ◽

1993 ◽

Vol 14 (4) ◽

pp. 233-240 ◽

Cited By ~ 4

Author(s):

G. E. Cossali ◽

A. Coghe

Keyword(s):

Growth Rate ◽

Instability Growth Rate ◽

Liquid Jets ◽

Instability Growth ◽

Rate Evaluation

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Effect of intermediate fluid layer on Kelvin–Helmholtz instability

Canadian Journal of Physics ◽

10.1139/cjp-2017-0606 ◽

2018 ◽

Vol 96 (10) ◽

pp. 1145-1154 ◽

Cited By ~ 2

Author(s):

Ying Zhang ◽

Wenqiang Shang ◽

Mengjun Yao ◽

Boheng Dong ◽

Peisheng Li

Keyword(s):

Growth Rate ◽

Stokes Equations ◽

Fluid Layer ◽

Immiscible Fluids ◽

Navier Stokes ◽

Helmholtz Instability ◽

Flow Configuration ◽

Front Tracking Method ◽

Internal Disturbance ◽

Over Time

Two-dimensional K-H (Kelvin–Helmholtz) instability of the three-component immiscible fluids with an intermediate fluid layer is numerically studied using the front-tracking method (FTM). The instability is governed by the Navier–Stokes equations and the conservation of mass equation for incompressible flow. A finite difference method is used to discretize the governing system. This study focuses on the influence of flow configuration, the thickness of intermediate fluid layer and the distribution of intermediate fluid layer on K-H instability. It is shown that the larger the initial horizontal velocity difference is, the faster the internal disturbance increases, and the characteristic form of K-H instability becomes more obvious for different flow configuration. It is also observed that the thickness of the intermediate fluid layer is negatively correlated to the billow height and the numerical growth rate. In addition, when the intermediate fluid layer is thicker than 0.4 times the disturbance wavelength, the billow height and the numerical growth rate for the K-H instability of the upper and lower interfaces change over time synchronously. The higher the initial height of the lower interface is, the greater the growth rate and billow height of the upper interface are. Besides, the upper and lower interfaces are rolled up synchronously over time when the intermediate fluid layer is symmetrically distributed with y = 0.5 in the fluid system.

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Numerical investigations on the effects of geometrical parameters on free electron laser instability

Journal of Plasma Physics ◽

10.1017/s0022377808007447 ◽

2008 ◽

Vol 74 (6) ◽

pp. 741-747

Author(s):

B. S. SHARMA ◽

N. K. JAIMAN

Keyword(s):

Growth Rate ◽

Electron Beam ◽

Free Electron ◽

Free Electron Laser ◽

Instability Growth Rate ◽

Magnetized Plasma ◽

Geometrical Parameters ◽

Uniform Density ◽

Backward Wave Oscillator ◽

Instability Growth

AbstractIn this paper we numerically investigate the effects of various geometrical parameters of a backward wave oscillator (BWO), filled with a magnetized plasma of uniform density and driven by a mild relativistic solid electron beam, on the instability growth rate (Γ) of a free electron laser (FEL). The FEL instability is numerically calculated and the result is compared with the instability growth rate of an annular electron beam for the same set of parameters. The instability growth for a solid electron beam scales inversely to the seventh power of relativistic gamma factor γ0 and directly proportional to the corrugation amplitude.

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