Perturbed Distribution Function Research Articles

Instabilities in a self-gravitating granular gas

A kinetic and hydrodynamic descriptions are developed in order to analyze the instabilities in a self-gravitating granular gas. In the kinetic description the Boltzmann equation is coupled with the Poisson equation, while in the hydrodynamic description the Poisson equation is coupled with the balance equations of mass density, hydrodynamic velocity and temperature for an Eulerian fluid. In the background solution for both descriptions the fluid is at rest with constant mass density and gravitational potential while the temperature depends on time through Haff’s law. In the kinetic description the perturbed distribution function and gravitational potential in the Fourier space are related to time dependent small amplitudes. In the hydrodynamic description the perturbed mass density, hydrodynamic velocity and temperature in the Fourier space are functions of time dependent small amplitudes. From the analysis of the system of coupled differential equations for the amplitudes for the two descriptions the time evolution of the density contrast – a parameter that indicate where there are local enhancements in the matter density – is determined. The solutions depend on two parameters, one is the mean free path of the gas particles and another Jeans’ wavelength, which is a function of the gravitational constant, mass density and speed of sound of the gas. It is shown that instabilities due to the inelastic collisions occur when the Jeans and the perturbation wavelengths are larger than the mean free path, while Jeans’ instabilities due to the gravitational field happen when the mean free path and the perturbation wavelength are larger than Jeans’ wavelength.

Phonon limited mobility in 3D Dirac semimetal Cd3As2

We report the numerical calculations of phonon limited electron mobility, μ in Cd3As2 three dimensional Dirac semimetal using Boltzmann transport theory over a temperature range 20<T<300K. By employing the Ritz iteration technique, we have directly solved the 3D linearized Boltzmann transport equation to obtain the first order perturbation distribution function Φ−1(Ek) as a function of electron energy Ek and hence the mobility, μ. we have considered electrons scattering by acoustic phonons via deformation potential scattering and polar optical phonons. The μ due to the latter is dominating the former for T>80K, for electron concentration ne = 1x1018cm−3. Thiscrossover shifts to lower T, with decreasing ne. The calculations are qualitatively agreeing with the experimental results.

Study of up–down poloidal density asymmetry of high- impurities with the new impurity version of XGCa

Addition of multispecies impurity ions to the total-f gyrokinetic particle-in-cell code XGCa is reported, including a cross-verification of neoclassical physics against the NEO code. This new version of the neoclassical gyrokinetic code XGCa is used to benchmark and confirm the previous reduced-equation-based prediction that high- $Z$ impurity particles in the Pfirsch–Schlüter regime can exhibit a significant level of up–down poloidal asymmetry, through the large parallel friction force, and thus influence the radial plasma transport significantly. The study is performed in a plasma with weak toroidal rotation. In comparison, when the impurity particles are in the plateau regime, the up–down poloidal asymmetry becomes weak, with the parallel friction force becoming weaker than the parallel viscous force. It is also found that the linearization of the perturbed distribution function, based on the small poloidal asymmetry assumption, can become invalid. Using the numerical data from XGCa, each term in the parallel fluid force-balance equation have been analysed to find that both the main ions and the electrons respond to the poloidal potential variation adiabatically when the high- $Z$ tungsten possesses large poloidal variation.

Dust-acoustic waves with finite OAM in a self-gravitating dusty plasma with superthermal energetic tails of electrons and ions

The kinetic instability of twisted dust-acoustic (DA) wave with distinct orbital angular momentum states is studied in an unmagnetized self-gravitating superthermal dusty plasma. For this purpose, the linearized Vlasov–Poisson equations are solved together by expressing the perturbed distribution function, electrostatic and gravitational potentials in terms of Laguerre–Gauss functions or solutions. A generalized plasma dielectric constant is derived and the existence conditions for the damping/growth rates are identified. The impact of streaming speed, dust self-gravitation, superthermality and twist parameters are numerically examined on the growth rate instability. It has been found that parametric variation significantly modifies the threshold conditions and associated growth rate instability of twisted DA waves. The relevance of the present results to interstellar dust clouds is illustrated, where massive dust grains follow a Maxwellian distribution in addition to superthermal electrons and ions.

Gradient- and flux-driven global gyrokinetic simulations of ITG and TEM turbulence with an improved hybrid kinetic electron model

ORB5 is a global gyrokinetic code being developed by many scientists over the last 20 years. It allows performing so-called gradient-driven simulations, in which initial profiles such as density and temperature are specified and an adaptative source is used to relax the zonal component of the perturbed distribution function so that the total flux-surface-averaged distribution function does not deviate too much from the initial one, . Thus, f0 can be used as a control variate to reduce the sampling noise.Even though the gradient-driven model is useful due to its relatively low numerical cost (there is no need to perform simulations over transport time scales) it does not reflect the experimental reality where fluxes are prescribed by the actual heat sources. Furthermore, comparison between experimental and gradient-driven numerical predictions of the heat fluxes is a challenging task because of the profile stiffness: a small change, within experimental error bars, in the imposed temperature gradients leads to a large variation of the computed heat fluxes.To address the stiffness problem, the ORB5 code has been adapted to be able to run flux-driven simulations. In this model, fluxes are prescribed by a radially localized heat source and sink and the background profiles slowly evolve toward a quasi-steady state. As a first step toward complete flux-driven simulations, a mixed approach is used, in which the run is started in the gradient-driven mode providing estimates for the heat sources/sinks, which are then used for pursuing the simulation in flux-driven mode. This allows us to keep the numerical cost relatively low as a quasi-steady state is more quickly reached.In this work, we present the heat source implemented in ORB5 to perform the flux-driven simulations as well as the numerical technique used to ensure that no other moment is injected by the source up to machine precision. Furthermore, simulation results illustrating the mode switching between gradient- and flux-driven will be shown as well as preliminary results of TEM dominated cases done using an upgraded hybrid electron model.

Projection-operator methods for classical transport in magnetized plasmas. Part 1. Linear response, the Braginskii equations and fluctuating hydrodynamics

An introduction to the use of projection-operator methods for the derivation of classical fluid transport equations for weakly coupled, magnetised, multispecies plasmas is given. In the present work, linear response (small perturbations from an absolute Maxwellian) is addressed. In the Schrödinger representation, projection onto the hydrodynamic subspace leads to the conventional linearized Braginskii fluid equations when one restricts attention to fluxes of first order in the gradients, while the orthogonal projection leads to an alternative derivation of the Braginskii correction equations for the non-hydrodynamic part of the one-particle distribution function. The projection-operator approach provides an appealingly intuitive way of discussing the derivation of transport equations and interpreting the significance of the various parts of the perturbed distribution function; it is also technically more concise. A special case of the Weinhold metric is used to provide a covariant representation of the formalism; this allows a succinct demonstration of the Onsager symmetries for classical transport. The Heisenberg representation is used to derive a generalized Langevin system whose mean recovers the linearized Braginskii equations but that also includes fluctuating forces. Transport coefficients are simply related to the two-time correlation functions of those forces, and physical pictures of the various transport processes are naturally couched in terms of them. A number of appendices review the traditional Chapman–Enskog procedure; record some properties of the linearized Landau collision operator; discuss the covariant representation of the hydrodynamic projection; provide an example of the calculation of some transport effects; describe the decomposition of the stress tensor for magnetised plasma; introduce the linear eigenmodes of the Braginskii equations; and, with the aid of several examples, mention some caveats for the use of projection operators.

Growth rate instabilities with finite OAM in non-Maxwellian plasmas: Saturn's magnetosphere

The electrostatic twisted waves with finite orbital angular momentum (OAM) states are studied in a doubly κ− distributed plasma, containing superthermal hot and cool electrons in addition to positive ions. In the framework of Vlasov-Poisson model, the expression for a generalized plasma response function is derived by considering the Laguerre-Gaussian (LG) solutions for the perturbed distribution function and electrostatic potential. Modified dispersion relations and instability growth rates of the twisted electron-acoustic (TEA) and twisted ion-acoustic (TIA) waves are derived in an unmagnetized double κ−distributed superthermal plasma. For parametric analysis, the effects of spectral indices (κh and κc), twisted parameter (η) and hot-to-cool electron density ratio (f) are numerically examined on the real and imaginary frequencies of TEA and TIA waves. The threshold conditions for TEA and TIA waves are also analyzed to investigate their instability growth rates both analytically and numerically. The relevance of the present results to Saturn's outer magnetosphere is discussed, where two groups of electrons with different finite temperatures are modeled by the double κ−distributions.

A conservative scheme for electromagnetic simulation of magnetized plasmas with kinetic electrons

A conservative scheme has been formulated and verified for gyrokinetic particle simulations of electromagnetic waves and instabilities in magnetized plasmas. An electron continuity equation derived from the drift kinetic equation is used to time advance the electron density perturbation by using the perturbed mechanical flow calculated from the parallel vector potential, and the parallel vector potential is solved by using the perturbed canonical flow from the perturbed distribution function. In gyrokinetic particle simulations using this new scheme, the shear Alfvén wave dispersion relation in the shearless slab and continuum damping in the sheared cylinder have been recovered. The new scheme overcomes the stringent requirement in the conventional perturbative simulation method that perpendicular grid size needs to be as small as electron collisionless skin depth even for the long wavelength Alfvén waves. The new scheme also avoids the problem in the conventional method that an unphysically large parallel electric field arises due to the inconsistency between electrostatic potential calculated from the perturbed density and vector potential calculated from the perturbed canonical flow. Finally, the gyrokinetic particle simulations of the Alfvén waves in sheared cylinder have superior numerical properties compared with the fluid simulations, which suffer from numerical difficulties associated with singular mode structures.

Kinetic theory of two-temperature polyatomic plasmas

We investigate the kinetic theory of two-temperature plasmas for reactive polyatomic gas mixtures. The Knudsen number is taken proportional to the square root of the mass ratio between electrons and heavy-species, and thermal non-equilibrium between electrons and heavy species is allowed. The kinetic non-equilibrium framework also requires a weak coupling between electrons and internal energy modes of heavy species. The zeroth-order and first-order fluid equations are derived by using a generalized Chapman–Enskog method. Expressions for transport fluxes are obtained in terms of macroscopic variable gradients and the corresponding transport coefficients are expressed as bracket products of species perturbed distribution functions. The theory derived in this paper provides a consistent fluid model for non-thermal multicomponent plasmas.

A quasilinear operator retaining magnetic drift effects in tokamak geometry

The interaction of radio frequency waves with charged particles in a magnetized plasma is usually described by the quasilinear operator that was originally formulated by Kennel &amp; Engelmann (Phys. Fluids, vol. 9, 1966, pp. 2377–2388). In their formulation the plasma is assumed to be homogenous and embedded in a uniform magnetic field. In tokamak plasmas the Kennel–Engelmann operator does not capture the magnetic drifts of the particles that are inherent to the non-uniform magnetic field. To overcome this deficiency a combined drift and gyrokinetic derivation is employed to derive the quasilinear operator for radio frequency heating and current drive in a tokamak with magnetic drifts retained. The derivation requires retaining the magnetic moment to higher order in both the unperturbed and perturbed kinetic equations. The formal prescription for determining the perturbed distribution function then follows a novel procedure in which two non-resonant terms must be evaluated explicitly. The systematic analysis leads to a diffusion equation that is compact and completely expressed in terms of the drift kinetic variables. The equation is not transit averaged, and satisfies the entropy principle, while retaining the full poloidal angle variation without resorting to Fourier decomposition. As the diffusion equation is in physical variables, it can be implemented in any computational code. In the Kennel–Engelmann formalism, the wave–particle resonant delta function is either for the Landau resonance or the Doppler shifted cyclotron resonance. In the combined gyro and drift kinetic approach, a term related to the magnetic drift modifies the resonance condition.

Quasi-electrostatic twisted waves in Lorentzian dusty plasmas

The quasi electrostatic modes are investigated in non thermal dusty plasma using non-gyrotropic Kappa distribution in the presence of helical electric field. The Laguerre Gaussian (LG) mode function is employed to decompose the perturbed distribution function and helical electric field. The modified dielectric function is obtained for the dust ion acoustic (DIA) and dust acoustic (DA) twisted modes from the solution of Vlasov-Poisson equation. The threshold conditions for the growing modes is also illustrated.

A conservative scheme of drift kinetic electrons for gyrokinetic simulation of kinetic-MHD processes in toroidal plasmas

A conservative scheme of drift kinetic electrons for gyrokinetic simulations of kinetic-magnetohydrodynamic processes in toroidal plasmas has been formulated and verified. Both vector potential and electron perturbed distribution function are decomposed into adiabatic part with analytic solution and non-adiabatic part solved numerically. The adiabatic parallel electric field is solved directly from the electron adiabatic response, resulting in a high degree of accuracy. The consistency between electrostatic potential and parallel vector potential is enforced by using the electron continuity equation. Since particles are only used to calculate the non-adiabatic response, which is used to calculate the non-adiabatic vector potential through Ohm's law, the conservative scheme minimizes the electron particle noise and mitigates the cancellation problem. Linear dispersion relations of the kinetic Alfvén wave and the collisionless tearing mode in cylindrical geometry have been verified in gyrokinetic toroidal code simulations, which show that the perpendicular grid size can be larger than the electron collisionless skin depth when the mode wavelength is longer than the electron skin depth.

Nonlinear co-existence of beta-induced Alfvén eigenmodes and beta-induced Alfvén-acoustic eigenmodes

The nonlinear co-existence of β–induced Alfvén eigenmode (BAE) and β–induced Alfvén-acoustic eigenmode (BAAE) is found in simulations using the gyrokinetic toroidal code, which provides a new mechanism responsible for BAAE excitation in tokamaks. Here, the normalized pressure β is the ratio between plasma thermal pressure and magnetic pressure. The nonlinear simulation results show that the BAAE branch emerges after the BAE branch is saturated. The mode structure's evolution shows that existence of BAAE will change the original BAE mode structure. The perturbed distribution functions in the velocity phase space show that a new resonance region manifesting the wave-particle resonance in the BAAE branch appears during the nonlinear co-existence stage.

Twisted modes instability of electron–positron shell interacted with moving ion background

AbstractIn this paper, the instability of electrostatic twisted modes carrying orbital angular momentum in the moving electron–positron–ion plasma is investigated. In the kinetic theory approach, the general dispersion relation of twisted modes is derived by using Laguerre–Gaussian perturbed distribution function and electrostatic potential in the paraxial limit. Utilizing the obtained general dispersion relation for a specific case of electron–positron (e–p) shell with temperature anisotropy interacted with moving ion background, the effects of angular mode number, electrons and positrons temperature, and positron concentration on the group velocity and instability growth rate of twisted waves are illustrated, numerically. The results of the present investigation will greatly attribute to the understanding of e–p jet dynamic in astrophysical environments and laboratory experiments where the twisted modes can play a central role as a perturbed term.

Distribution functions for Galactic disc stellar populations in the presence of non-axisymmetric perturbations

AbstractThe present-day response of a Galactic disc stellar population to a non-axisymmetric perturbation of the potential, in the form of a bar or spiral arms, can be treated, away from the main resonances, through perturbation theory within the action-angle coordinates of the unperturbed axisymmetric system. The first order moments of such a perturbed distribution function (DF) in the presence of spiral arms give rise to non-zero radial and vertical mean stellar velocities, called breathing modes. Such an Eulerian linearized treatment however diverges at resonances. The Lagrangian approach to the impact of non-axisymmetries at resonances avoids this problem. It is based on the construction of new orbital tori in the resonant trapping region, which come complete with a new system of angle-action variables. These new tori can be populated by phase-averaging the unperturbed DF over the new tori. This boils down to phase-mixing the DF in terms of the new angles, such that the DF for trapped orbits only depends on the new set of actions. This opens the way to quantitatively fitting the effects of the bar and spirals to Gaia data with an action-based DF.

Combining electromagnetic gyro-kinetic particle-in-cell simulations with collisions

It has been an open question whether for electromagnetic gyro-kinetic particle-in-cell (PIC) simulations pitch–angle collisions and the recently introduced pullback transformation scheme (Mishchenko et al., 2014; Kleiber et al., 2016) are consistent.This question is positively answered by comparing the PIC code EUTERPE with an approach based on an expansion of the perturbed distribution function in eigenfunctions of the pitch-angle collision operator (Legendre polynomials) to solve the electromagnetic drift-kinetic equation with collisions in slab geometry.It is shown how both approaches yield the same results for the frequency and damping rate of a kinetic Alfvén wave and how the perturbed distribution function is substantially changed by the presence of pitch-angle collisions.

Study of Oblique Propagating Whistler Mode Waves in Presence of Parallel DC Electric Field in Magnetosphere of Saturn

In this paper whistler mode waves have been investigated in magnetosphere of Saturn. The derivation for perturbed distribution function, dispersion relation and growth rate have been determined by using the method of characteristic and kinetic approach. Analytical expressions for growth rate and real frequency of whistlers propagating oblique to magnetic field direction are attained. Calculations have been performed at 6 radial distances in plasma sheet region of Saturn’s magnetosphere as per data provided by Cassini. Work has been extended for bi-Maxwellian as well as Loss-cone distribution function. Parametric analysis show that temperature anisotropy, increase in number density, energy density and angle of propagation increases the growth rate of whistler waves along with significant shift in wave number. In case of Loss-cone distribution, increase in growth rate of whistlers is significantly more than for bi-Maxwellian distribution function. Generation of second harmonics can also be seen in the graphs plotted. It is concluded that parallel DC field stabilizes the wave and temperature anisotropy, angle of propagation, number density and energy density of electrons enhances the growth rate. Thus the results are of importance in analyzing observed VLF emissions over wide spectrum of frequency range in Saturnian magnetosphere. The analytical model developed can also be used to study various types of instabilities in planetary magnetospheres.

Kinetic study of electrostatic twisted waves instability in nonthermal dusty plasmas

The kinetic theory of electrostatic twisted waves' instability in a dusty plasma is developed in the presence of orbital angular momentum of the helical (twisted) electric field in plasmas with kappa distributed electrons, ions, and dust particles. The kappa distributed electrons are considered to have a drift velocity. The perturbed distribution function and helical electric field are decomposed by Laguerre-Gaussian mode functions defined in cylindrical geometry. The Vlasov-Poisson equation is obtained and solved analytically to investigate the growth rates of the electrostatic twisted waves in a non-thermal dusty plasma. The growth rates of the dust ion acoustic twisted mode (DIATM) and dust acoustic twisted mode (DATM) are obtained analytically and also pictorial presented numerically. The instability condition for the DIATM and DATM is also discussed with different plasma parameters. The growth rates of DIATM and DATM are larger when the drifted electrons are non-Maxwellian distributed and smaller for the Maxwellian distributed drifted electrons in the presence of the helical electric field.

Electrostatic drift instability in a magnetotail configuration: The role of bouncing electrons

To understand the possible destabilization of two-dimensional current sheets, a kinetic model is proposed to describe the resonant interaction between electrostatic modes and trapped electrons that bounce within the sheet. This work follows the initial investigation by Tur, Louarn, and Yanovsky [Phys. Plasmas 17, 102905 (2010)] and Fruit, Louarn, and Tur [Phys. Plasmas 20, 022113 (2013)] that is revised and extended. Using a quasi-dipolar equilibrium state, the linearized gyro-kinetic Vlasov equation is solved for electrostatic fluctuations with a period of the order of the electron bounce period. Using an appropriated Fourier expansion of the particle motion along the magnetic field, the complete time integration of the non-local perturbed distribution functions is performed. The dispersion relation for electrostatic modes is then obtained through the quasineutrality condition. It is found that for a mildly stretched configuration (L∼8), strongly unstable electrostatic modes may develop in the current sheet with the growth rate of the order of a few seconds provided that the background density gradient responsible for the diamagnetic drift effects is sharp enough: typical length scale over one Earth radius or less. However, when this condition in the density gradient is not met, these electrostatic modes grow too slowly to be accountable for a rapid destabilization of the magnetic structure. This strong but finely tuned instability may offer opportunities to explain features in magnetospheric substorms.

Study of whistler mode waves for ring distribution function in Saturn’s magnetosphere

In present paper whistler mode waves in magnetosphere of Saturn have been investigated. Linear properties of ring velocity distribution function are used to derive dispersion relation for whistler mode waves in ambient magnetic field of Saturn in extended plasma sheet of planet’s magnetosphere. The derivation for perturbed distribution function, dispersion relation and growth rate have been determined by using the method of characteristic and kinetic approach. Analytical expressions for growth rate and real frequency of whistlers propagating parallel to magnetic field direction are attained. The analysis shows that temperature anisotropy, increases in number density and energy density increases the growth rate of whistler waves along with significant shift in wave number. It has been shown that whistler mode waves have grown due to loss of perpendicular kinetic energy of ring electrons. Calculations have been performed at two radial distances in Saturn’s magnetosphere. The results are of importance in analyzing observed VLF emissions over wide spectrum of frequency range in Saturnian magnetosphere. The analytical model developed can also be used to study various types of instabilities in planetary magnetospheres.