Plasma Dispersion Relation Research Articles

Dispersion relations in hot magnetized plasmas

In the framework of hot magnetized collisionless plasmas, dispersion relations have been extensively studied in the past [2,12,13,24,33,34,38]. This subject is still topical in plasma physics [19,27,32,36,42]. The aim of this article is to provide a rigorous derivation of the characteristic variety, based on some asymptotic analysis of the relativistic Vlasov–Maxwell system. Special emphasis is made on the modeling of Tokamaks, with spatial variations of the magnetic field and of the equilibrium distribution function. In order to take into account the inhomogeneities, the problem is formulated in terms of geometrical optics [29,31]. This allows to unify, justify and extend the preceding results. New aspects are indeed included. For instance, the dielectric tensor is defined for real frequencies through singular integrals involving the Hilbert transform.

Hot Plasma Effects on the Cyclotron‐Resonant Pitch‐Angle Scattering Rates of Radiation Belt Electrons Due to EMIC Waves

AbstractTo investigate the hot plasma effects on the cyclotron‐resonant interactions between electromagnetic ion cyclotron (EMIC) waves and radiation belt electrons in a realistic magnetospheric environment, calculations of the wave‐induced bounce‐averaged pitch angle diffusion coefficients are performed using both the cold and hot plasma dispersion relations. The results demonstrate that the hot plasma effects have a pronounced influence on the electron pitch angle scattering rates due to all three EMIC emission bands (H+, He+, and O+) when the hot plasma dispersion relation deviates significantly from the cold plasma approximation. For a given wave spectrum, the modification of the dispersion relation by hot anisotropic protons can strongly increase the minimum resonant energy for electrons interacting with O+ band EMIC waves, while the minimum resonant energies for H+ and He+ bands are not greatly affected. For H+ band EMIC waves, inclusion of hot protons tends to weaken the pitch angle scattering efficiency of >5 MeV electrons. The most crucial differences introduced by the hot plasma effects occur for >3 MeV electron scattering rates by He+ band EMIC waves. Mainly due to the changes of resonant frequency and wave group velocity when the hot protons are included, the difference in scattering rates can be up to an order of magnitude, showing a strong dependence on both electron energy and equatorial pitch angle. Our study confirms the importance of including hot plasma effects in modeling the scattering of ultra‐relativistic radiation belt electrons by EMIC waves.

One‐Dimensional Full Wave Simulation of Equatorial Magnetosonic Wave Propagation in an Inhomogeneous Magnetosphere

AbstractThe effect of the plasmapause on equatorially radially propagating fast magnetosonic (MS) waves in the Earth's dipole magnetic field is studied by using finite difference time domain method. We run 1‐D simulation for three different density profiles: (1) no plasmapause, (2) with a plasmapause, and (3) with a plasmapause accompanied with fine‐scale density irregularity. We find that (1) without plasmapause the radially inward propagating MS wave can reach ionosphere and continuously propagate to lower altitude if no damping mechanism is considered. The wave properties follow the cold plasma dispersion relation locally along its trajectory. (2) For simulation with a plasmapause with a scale length of 0.006 RE compared to wavelength, only a small fraction of the MS wave power is reflected by the plasmapause. WKB approximation is generally valid for such plasmapause. (3) The multiple fine‐scale density irregularities near the outer edge of plasmapause can effectively block the MS wave propagation, resulting in a terminating boundary for MS waves near the plasmapause.

An improved approximation for the analytical treatment of the local linear gyro-kinetic plasma dispersion relation in toroidal geometry

The analytical treatment of plasma kinetic linear instabilities in toroidal geometry is commonly tackled employing a power series expansion of the resonant part of the dispersion relation. This expansion is valid under the assumption that the modulus of the mode frequency is smaller than the magnitude of the frequencies characterising the system (the drift, bounce and transit frequencies for example). We will refer to this approximation as high frequency approximation (HFA). In this paper the linear plasma dispersion relation is derived in the framework of the gyro-kinetic model, for the electrostatic case, in the local limit, in the absence of collisions, for a non rotating plasma, considering adiabatic electrons, in toroidal circular geometry, neglecting the parallel dynamics effect. A systematic analysis of the meaning and limitations of the HFA is performed. As already known, the HFA is not valid for tokamak relevant parameters. A new way to approximate the resonant part of the dispersion relation, called here Improved high frequency approximation (IHFA), is therefore proposed. A quantitative analysis of the ion temperature gradient (ITG) instability is presented. The IHFA is shown to be applicable to the treatment of the ITG instability in tokamaks.

Jeans instability of inhomogeneous dusty plasma with polarization force, ionization and recombination

The self-gravitational Jeans instability has been studied in dusty plasma containing significant background of neutral pressure and recombination of ions and electrons on the dust surface. The full dynamics of charged dust grains, ions and neutral species are employed considering the electrons as Maxwellian. We have derived the general dispersion relation for collisional dusty plasma with ionization, recombination and polarization force. The general dispersion relation describes the effects of considered parameters which are solved in different dusty plasma situations. Further, the dispersion relation is solved numerically. The present work is applicable to understand the structure formation of interstellar molecular clouds in astrophysical plasma.

A Model for Dissipation of Solar Wind Magnetic Turbulence by Kinetic Alfvén Waves at Electron Scales: Comparison with Observations

Abstract In hydrodynamic turbulence, it is well established that the length of the dissipation scale depends on the energy cascade rate, i.e., the larger the energy input rate per unit mass, the more the turbulent fluctuations need to be driven to increasingly smaller scales to dissipate the larger energy flux. Observations of magnetic spectral energy densities indicate that this intuitive picture is not valid in solar wind turbulence. Dissipation seems to set in at the same length scale for different solar wind conditions independently of the energy flux. To investigate this difference in more detail, we present an analytic dissipation model for solar wind turbulence at electron scales, which we compare with observed spectral densities. Our model combines the energy transport from large to small scales and collisionless damping, which removes energy from the magnetic fluctuations in the kinetic regime. We assume wave–particle interactions of kinetic Alfvén waves (KAWs) to be the main damping process. Wave frequencies and damping rates of KAWs are obtained from the hot plasma dispersion relation. Our model assumes a critically balanced turbulence, where larger energy cascade rates excite larger parallel wavenumbers for a certain perpendicular wavenumber. If the dissipation is additionally wave driven such that the dissipation rate is proportional to the parallel wavenumber—as with KAWs—then an increase of the energy cascade rate is counterbalanced by an increased dissipation rate for the same perpendicular wavenumber, leading to a dissipation length independent of the energy cascade rate.

Scattering of relativistic and ultra-relativistic electrons by obliquely propagating Electromagnetic Ion Cyclotron waves

Electromagnetic Ion Cyclotron (EMIC) waves are transverse plasma waves that are generated in the Earth magnetosphere by ring current protons with temperature anisotropy in three different bands: below the H+, He+ and O+ ion gyrofrequencies. EMIC events are enhanced during the main phase of a geomagnetic storm when intensifications in the electric field result in enhanced injections of ions and are usually confined to high-density regions just inside the plasmapause or within drainage plumes. EMIC waves are capable of scattering radiation belt electrons and thus provide an important link between the intensification of the electric field, ion populations, and radiation belt electrons. Bounce-averaged diffusion coefficients computed with the assumption of parallel wave propagation are compared to the results of the code that uses the full cold plasma dispersion relation taking into account oblique propagation of waves and higher-order resonances. We study the sensitivity of the scattering rates to a number of included higher-order resonances, wave spectral distribution parameters, wave normal angle distribution parameters, ambient plasma density, and ion composition. Inaccuracies associated with the neglect of higher-order resonances and oblique propagation of waves are compared to potential errors introduced by uncertainties in the model input parameters.

Using the cold plasma dispersion relation and whistler mode waves to quantify the antenna sheath impedance of the Van Allen Probes EFW instrument

AbstractCold plasma theory and parallel wave propagation are often assumed when approximating the whistler mode magnetic field wave power from electric field observations. The current study is the first to include the wave normal angle from the Electric and Magnetic Field Instrument Suite and Integrated Science package on board the Van Allen Probes in the conversion factor, thus allowing for the accuracy of these assumptions to be quantified. Results indicate that removing the assumption of parallel propagation does not significantly affect calculated plasmaspheric hiss wave powers. Hence, the assumption of parallel propagation is valid. For chorus waves, inclusion of the wave normal angle in the conversion factor leads to significant alterations in the distribution of wave power ratios (observed/ calculated); the percentage of overestimates decreases, the percentage of underestimates increases, and the spread of values is significantly reduced. Calculated plasmaspheric hiss wave powers are, on average, a good estimate of those observed, whereas calculated chorus wave powers are persistently and systematically underestimated. Investigation of wave power ratios (observed/calculated), as a function of frequency and plasma density, reveals a structure consistent with signal attenuation via the formation of a plasma sheath around the Electric Field and Waves spherical double probes instrument. A simple, density‐dependent model is developed in order to quantify this effect of variable impedance between the electric field antenna and the plasma interface. This sheath impedance model is then demonstrated to be successful in significantly improving agreement between calculated and observed power spectra and wave powers.

PDRK: A General Kinetic Dispersion Relation Solver for Magnetized Plasma**supported by the National Magnetic Confinement Fusion Science Program of China (Nos. 2015GB110003, 2011GB105001, 2013GB111000), National Natural Science Foundation of China (No. 91130031), the Recruitment Program of Global Youth Experts

A general, fast, and effective approach is developed for numerical calculation of kinetic plasma linear dispersion relations. The plasma dispersion function is approximated by J-pole expansion. Subsequently, the dispersion relation is transformed to a standard matrix eigenvalue problem of an equivalent linear system. Numerical solutions for the least damped or fastest growing modes using an 8-pole expansion are generally accurate; more strongly damped modes are less accurate, but are less likely to be of physical interest. In contrast to conventional approaches, such as Newton's iterative method, this approach can give either all the solutions in the system or a few solutions around the initial guess. It is also free from convergence problems. The approach is demonstrated for electrostatic dispersion equations with one-dimensional and two-dimensional wavevectors, and for electromagnetic kinetic magnetized plasma dispersion relation for bi-Maxwellian distribution with relative parallel velocity flows between species.

Dispersion relations in cold magnetized plasmas

Starting from kinetic models of cold magnetized collisionless plasmas, we provide a complete description of the characteristic variety sustaining electromagnetic wave propagation. As in [ 4 , 13 , 17 ], the analysis is based on some asymptotic calculus exploiting the presence at the level of dimensionless relativistic Vlasov-Maxwell equations of a large parameter: the electron gyrofrequency. The method is inspired from geometric optics [ 29 , 33 ]. It allows to unify all the preceding results [ 8 , 12 , 38 , 31 , 37 , 40 ], while incorporating new aspects. Specifically, the non trivial effects [ 5 , 9 , 10 , 24 ] of the spatial variations of the background density, temperature and magnetic field are exhibited. In this way, a comprehensive overview of the dispersion relations becomes available, with important possible applications in plasma physics [ 7 , 28 , 30 ].

Gyro-viscosity and linear dispersion relations in pair-ion magnetized plasmas

A fluid theory has been developed by taking account of gyro-viscosity to study wave propagation characteristics in a homogeneous pair-ion magnetized plasma with a cylindrical symmetry. The exact dispersion relations derived by the Hankel-Fourier transformation are shown comparable with those observed in the experiment by Oohara and co-workers. The gyro-viscosity is responsible for the change in propagation characteristics of the ion cyclotron wave from forward to backward by suppressing the effect of the thermal pressure which normally causes the forward nature of dispersion. Although the experiment has been already explained by a kinetic theory by the present authors, the kinetic derivations are so involved because of exact particle orbits in phase space, finite Lamor radius effects, and higher order ion cyclotron resonances. The present fluid theory provides a simple and transparent structure to the dispersion relations since the gyro-viscosity is renormalized into the ion cyclotron frequency which itself indicates the backward nature of dispersion. The usual disadvantage of a fluid theory, which treats only fundamental modes of eigen-waves excited in a system and is not able to describe higher harmonics that a kinetic theory does, is compensated by simple derivations and clear picture based on the renormalization of the gyro-viscosity.

Regime transition of ion Bernstein instability driven by ion shell velocity distributions

AbstractLinear kinetic dispersion theory is used to investigate a regime transition of the ion Bernstein instability driven by a proton velocity distribution with positive slopes with respect to the perpendicular velocity, ∂fp(v∥∼0,v⊥)/∂v⊥>0. The unstable waves arising from this instability are ion Bernstein waves with proton cyclotron harmonic dispersion. However, in the inner magnetosphere, particularly inside of the plasmapause where plasmas are dominated by a cold background, the instability leads to ion Bernstein waves which approximately follow the cold plasma dispersion relation for fast magnetosonic waves and are, therefore, fast magnetosonic‐like. Subsequently, the relevant waves have been termed fast magnetosonic waves and many studies have assumed the cold plasma dispersion relation to describe them. On the other hand, how the dispersion properties of ion Bernstein waves become fast magnetosonic‐like has not yet been well understood. To examine this regime transition of the instability, we perform linear dispersion analyses using a two‐component proton model of fp(v) = fM(v) + fs(v), where fM is a Maxwellian velocity distribution and fs is an isotropic shell velocity distribution. The results show that the unstable waves are essentially ion Bernstein waves; however, when the shell proton concentration becomes sufficiently small (less than 10), the unstable waves approach the cold plasma dispersion relation for fast magnetosonic waves and become fast magnetosonic‐like. Although a reduced proton‐to‐electron mass ratio of 100 has been used for convenience, which reduces the number of unstable modes involved by lowering the lower hybrid frequency, this does not change the overall regime transition picture revealed in this study.

Predicted impacts of proton temperature anisotropy on solar wind turbulence

Particle velocity distributions measured in the weakly collisional solar wind are frequently found to be non-Maxwellian, but how these non-Maxwellian distributions impact the physics of plasma turbulence in the solar wind remains unanswered. Using numerical solutions of the linear dispersion relation for a collisionless plasma with a bi-Maxwellian proton velocity distribution, we present a unified framework for the four proton temperature anisotropy instabilities, identifying the associated stable eigenmodes, highlighting the unstable region of wavevector space and presenting the properties of the growing eigenfunctions. Based on physical intuition gained from this framework, we address how the proton temperature anisotropy impacts the nonlinear dynamics of the Alfvénic fluctuations underlying the dominant cascade of energy from large to small scales and how the fluctuations driven by proton temperature anisotropy instabilities interact nonlinearly with each other and with the fluctuations of the large-scale cascade. We find that the nonlinear dynamics of the large-scale cascade is insensitive to the proton temperature anisotropy and that the instability-driven fluctuations are unlikely to cause significant nonlinear evolution of either the instability-driven fluctuations or the turbulent fluctuations of the large-scale cascade.

Quantum effects on propagation of bulk and surface waves in a thin quantum plasma film

Abstract The propagation of bulk and surface plasma waves in a thin quantum plasma film is investigated, taking into account the quantum effects. The generalized bulk and surface plasma dispersion relation due to quantum effects is derived, using the quantum hydrodynamic dielectric function and applying appropriate additional boundary conditions. The quantum mechanical and film geometric effects on the bulk and surface modes are discussed. It is found that quantum effects become important for a thin film of small thickness.

Applying the cold plasma dispersion relation to whistler mode chorus waves: EMFISIS wave measurements from the Van Allen Probes.

Most theoretical wave models require the power in the wave magnetic field in order to determine the effect of chorus waves on radiation belt electrons. However, researchers typically use the cold plasma dispersion relation to approximate the magnetic wave power when only electric field data are available. In this study, the validity of using the cold plasma dispersion relation in this context is tested using Electric and Magnetic Field Instrument Suite and Integrated Science (EMFISIS) observations of both the electric and magnetic spectral intensities in the chorus wave band (0.1–0.9 fce). Results from this study indicate that the calculated wave intensity is least accurate during periods of enhanced wave activity. For observed wave intensities >10−3 nT2, using the cold plasma dispersion relation results in an underestimate of the wave intensity by a factor of 2 or greater 56% of the time over the full chorus wave band, 60% of the time for lower band chorus, and 59% of the time for upper band chorus. Hence, during active periods, empirical chorus wave models that are reliant on the cold plasma dispersion relation will underestimate chorus wave intensities to a significant degree, thus causing questionable calculation of wave-particle resonance effects on MeV electrons.

Two Stream Instability as a Source of Coronal Heating

Recent observation of oscillating the two stream instability (TSI) in a solar type III radio bursts and spatial damping of Langmuir oscillations has made this instability as an important candidate to understand the coronal heating problem. This instability has been studied by several authors for cold plasma found to be stable for high frequencies (greater than plasma frequency ωp). In this paper, we prove that this instability is unstable for warm plasma for higher frequencies (greater than plasma frequency ωp) and much suitable to study the solar coronal heating problem. We have derived a general dispersion relation for warm plasma and discussed the various methods analyzing the instability conditions. Also, we derived an expression for the growth rate of TSI and analyzed the growth rate for photospheric and coronal plasmas. A very promising result is that the ion temperature is the source of this instability and shifts the growth rate to high frequency region, while the electron temperature does the reverse. TSI shows a high growth rate for a wide frequency range for photosphere plasma, suggesting that the electron precipitation by magnetic reconnection current, acceleration by flares, may be source of TSI in the photosphere. But for corona, these waves are damped to accelerate the ions and further growing of such instability is prohibited due to the high conductivity in coronal plasma. The TSI is a common instability; the theory can be easily modifiable for multi-ion plasmas and will be a useful tool to analyze all the astrophysical problems and industrial devices, too.

Compressional Alfvén and ion–ion hybrid waves in tokamak plasmas with two ion species

Compressional Alfvén and ion–ion hybrid waves excited by energetic beam ions are studied in plasmas with two ion species. In our experiment, a hydrogen-deuterium (H-D) plasma is used to produce instabilities similar to those likely to be present in the burning deuterium-tritium plasmas of future tokamaks. Modes are suppressed in the deuterium cyclotron frequency range with increasing hydrogen gas puffing. In plasmas with H/D concentrations of 2.57 or higher, short-lived modes with small and predominantly negative toroidal mode numbers are observed at frequencies ω/ωβD0 ≈ 2.25, where ωβD0 = ωβD(R0) is the on-axis deuterium cyclotron frequency. These are the highest mode frequencies yet detected in the ion cyclotron range in a spherical tokamak. Modeling of the transparency regions and plasma resonances using the cold plasma dispersion relation explains the observed features. Mode conversion at ion–ion hybrid resonances and subsequent kinetic damping is believed to be responsible for mode suppression. The high frequency modes are present due to excitation by wave-particle resonances within the transparency region for high hydrogen concentrations. The absence of other wave-particle resonances explains significant features of our experiment. This technique has possible applications in plasma heating, current drive and real-time diagnosis of relative ion concentration in the plasma core.

Surface current balance and thermoelectric whistler wings at airless astrophysical bodies: Cassini at Rhea.

Sharp magnetic perturbations found by the Cassini spacecraft at the edge of the Rhea flux tube are consistent with field-aligned flux tube currents. The current system results from the difference of ion and electron gyroradii and the requirement to balance currents on the sharp Rhea surface. Differential-type hybrid codes that solve for ion velocity and magnetic field have an intrinsic difficulty modeling the plasma absorber's sharp surface. We overcome this problem by instead using integral equations to solve for ion and electron currents and obtain agreement with the magnetic perturbations at Rhea's flux tube edge. An analysis of the plasma dispersion relations and Cassini data reveals that field-guided whistler waves initiated by (1) the electron velocity anisotropy in the flux tube and (2) interaction with surface sheath electrostatic waves on topographic scales may facilitate propagation of the current system to large distances from Rhea. Current systems like those at Rhea should occur generally, for plasma absorbers of any size such as spacecraft or planetary bodies, in a wide range of space plasma environments. Motion through the plasma is not essential since the current system is thermodynamic in origin, excited by heat flow into the object. The requirements are a difference of ion and electron gyroradii and a sharp surface, i.e., without a significant thick atmosphere.Key PointsSurface current balance condition yields a current system at astronomical bodiesCurrent system possible for sharp (airless) objects of any sizeCurrent system is thermoelectric and motion through the plasma nonessential

Kinetic transverse dispersion relation for relativistic magnetized electron-positron plasmas with Maxwell-Jüttner velocity distribution functions

We use a kinetic treatment to study the linear transverse dispersion relation for a magnetized isotropic relativistic electron-positron plasma with finite relativistic temperature. The explicit linear dispersion relation for electromagnetic waves propagating along a constant background magnetic field is presented, including an analytical continuation to the whole complex frequency plane for the case of Maxwell-Jüttner velocity distribution functions. This dispersion relation is studied numerically for various temperatures. For left-handed solutions, the system presents two branches, the electromagnetic ordinary mode and the Alfvén mode. In the low frequency regime, the Alfvén branch has two dispersive zones, the normal zone (where ∂ω/∂k > 0) and an anomalous zone (where ∂ω/∂k < 0). We find that in the anomalous zone of the Alfvén branch, the electromagnetic waves are damped, and there is a maximum wave number for which the Alfvén branch is suppressed. We also study the dependence of the Alfvén velocity and effective plasma frequency with the temperature. We complemented the analytical and numerical approaches with relativistic full particle simulations, which consistently agree with the analytical results.

Mode conversion and absorption of fast waves at high ion cyclotron harmonics in inhomogeneous magnetic fields

The propagation and absorption of high harmonic fast waves is of interest for non-inductive current drives in fusion experiments. The fast wave can be coupled with the ion Bernstein wave that propagates in the high magnetic field side of an ion cyclotron harmonic resonance layer. This coupling and the absorption are analyzed using the hot plasma dispersion relation and a wave equation that was converted from an approximate dispersion relation for the case where λi=k⊥2ρi2/2≳1 (where k⊥ is the perpendicular wave number and ρi is the ion Larmor radius). It is found that both reflection and conversion may occur near the harmonic resonance layer but that they decrease rapidly, giving rise to a sharp increase in the absorption as the parallel wave number increases.