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We rescale the generalized Ohm’s law and consider the limits that imply the electric field which moves with an ideal (inviscid and perfectly conducting) plasma be proportional to the time rate of change of the current density. Therefore, we show that those limits are satisfied by a sufficiently low electron number density,
$n_{\text{e}}$
. We also show that the electron–ion collision frequency,
$\unicode[STIX]{x1D708}_{\text{ei}}$
, is much smaller than the ion cyclotron frequency,
$\unicode[STIX]{x1D714}_{\text{ci}}$
. The combination of that condition with the Lawson criterion for a typical deuterium–tritium fusion in ITER reveals a lower bound for the geometric mean of the confinement time,
$\unicode[STIX]{x1D70F}_{\text{C}}$
, and collision interval,
$\sqrt{\unicode[STIX]{x1D70F}_{\text{C}}/\unicode[STIX]{x1D708}_{\text{ei}}}\gg 10^{-5}~\text{s}$
. For that reaction, we estimate that
$n_{\text{e}}\sim 10^{19}~\text{m}^{-3}$
, and contrast typical parameters of our fully ionized gas with those of warm, hot and thermonuclear plasmas. When, in addition,
$n_{\text{e}}$
varies slowly in time and weakly in space, we generalize Alfvén’s theorem, by showing that the frozen-in condition holds true for an effective magnetic field, which depends on a finite electron skin depth. We perform a (divergenceless) helical perturbation on an axisymmetric equilibrium, to derive a dispersion relation in the cylindrical tokamak limit, and, subsequently, apply our analytical formulation to the peaked model, which assumes a logarithmic derivative profile for the poloidal component of the equilibrium magnetic field. In that formulation, the definition of the safety factor in terms of the effective field yields a shift in the magnetic surfaces. We find that the instability peak may triple that predicted on neglect of a finite electron mass. We also find that inertial effects may centuple the radius of the stable cylindrical column.
The results of a basic electron heat transport experiment using multiple localized heat sources in close proximity and embedded in a large magnetized plasma are presented. The set-up consists of three biased probe-mounted crystal cathodes, arranged in a triangular spatial pattern, that inject low energy electrons along a strong magnetic field into a pre-existing, cold afterglow plasma, forming electron temperature filaments. When the three sources are activated and placed within a few collisionless electron skin depths of each other, a non-azimuthally symmetric wave pattern emerges due to interference of the drift-Alfvén modes that form on each filament’s temperature gradient. Enhanced cross-field transport from chaotic (
$\boldsymbol{E}\times \boldsymbol{B}$
, where
$\boldsymbol{E}$
is the electric field and
$\boldsymbol{B}$
the magnetic field) mixing rapidly relaxes the gradients in the inner triangular region of the filaments and leads to growth of a global nonlinear drift-Alfvén mode that is driven by the thermal gradient in the outer region of the triangle. Azimuthal flow shear arising from the emissive cathode sources modifies the linear eigenmode stability and convective pattern. A steady-current model with emissive sheath boundary predicts the plasma potential and shear flow contribution from the sources.
The linear theory stability of different collisionless plasma sheath structures, including the classic sheath, inverse sheath and space-charge limited (SCL) sheath, is investigated as a typical eigenvalue problem. The three background plasma sheaths formed between a Maxwellian plasma source and a dielectric wall with a fully self-consistent secondary electron emission condition are solved by recent developed 1D3V (one-dimensional space and three-dimensional velocities), steady-state, collisionless kinetic sheath model, within a wide range of Maxwellian plasma electron temperature
$T_{e}$
. Then, the eigenvalue equations of sheath plasma fluctuations through the three sheaths are numerically solved, and the corresponding damping and growth rates
$\unicode[STIX]{x1D6FE}$
are found: (i) under the classic sheath structure (i.e.
$T_{e}<T_{ec}$
(the first threshold)), there are three damping solutions (i.e.
$\unicode[STIX]{x1D6FE}_{1}$
,
$\unicode[STIX]{x1D6FE}_{2}$
and
$\unicode[STIX]{x1D6FE}_{3}$
,
$0>\unicode[STIX]{x1D6FE}_{1}>\unicode[STIX]{x1D6FE}_{2}>\unicode[STIX]{x1D6FE}_{3}$
) for most cases, but there is only one growth-rate solution
$\unicode[STIX]{x1D6FE}$
when
$T_{e}\rightarrow T_{ec}$
due to the inhomogeneity of sheath being very weak; (ii) under the inverse sheath structure, which arises when
$T_{e}>T_{ec}$
, there are no background ions in the sheath so that the fluctuations are stable; (iii) under the SCL sheath conditions (i.e.
$T_{e}\geqslant T_{e\text{SCL}}$
, the second threshold), the obvious ion streaming through the sheath region again emerges and the three damping solutions are again found.
We show analytically that for
$\unicode[STIX]{x1D704}$
-profiles similar to the one of the Wendelstein 7-X stellarator, where
$\unicode[STIX]{x1D704}$
is the rotational transform of the equilibrium magnetic field, a highly conducting toroidal plasma is unstable to kinetically mediated pressure-driven long-wavelength reconnecting modes, of the infernal type. The modes are destabilized either by the electron temperature gradient or by a small amount of current, depending on how far from unity the average value of
$\unicode[STIX]{x1D704}$
is, which is assumed to be slowly varying. We argue that, for W7-X, a broad mode with toroidal and poloidal mode numbers
$(n,m)=(1,1)$
can be destabilized due to the strong geometric side-band coupling of the resonant kinetic electron response at locations where
$\unicode[STIX]{x1D704}$
is rational for harmonics that belong to the mode family of the
$(n,m)=(1,1)$
mode itself. In many regimes, the growth rate is insensitive to the plasma density, thus it is likely to persist in high performance W7-X discharges. For a peaked electron temperature, with a maximum of
$T_{e}=5~\text{keV}$
, larger than the ion temperature,
$T_{i}=2.5~\text{keV}$
, and a density
$n_{0}=10^{19}~\text{m}^{-3}$
, instability is found in regimes which show plasma sawtooth activity, with growth rates of the order of tens of kiloHertz. Frequencies are either electron diamagnetic or of the ideal magnetohydrodynamic type, but sub-Alfvénic. The kinetic infernal mode is thus a good candidate for the explanation of sawtooth oscillations in present-day stellarators and poses a new challenge to the problem of stellarator reactor optimization.
These lecture notes were presented by Allan N. Kaufman in his graduate plasma theory course and a follow-on special topics course (Physics 242A, B, C and Physics 250 at the University of California Berkeley). The notes follow the order of the lectures. The equations and derivations are as Kaufman presented, but the text is a reconstruction of Kaufman’s discussion and commentary. The notes were transcribed by Bruce I. Cohen in 1971 and 1972, and word processed, edited and illustrations added by Cohen in 2017 and 2018. The series of lectures is divided into four major parts: (i) collisionless Vlasov plasmas (linear theory of waves and instabilities with and without an applied magnetic field, Vlasov–Poisson and Vlasov–Maxwell systems, Wentzel–Kramers–Brillouin–Jeffreys (WKBJ) eikonal theory of wave propagation); (ii) nonlinear Vlasov plasmas and miscellaneous topics (the plasma dispersion function, singular solutions of the Vlasov–Poisson system, pulse-response solutions for initial-value problems, Gardner’s stability theorem, gyroresonant effects, nonlinear waves, particle trapping in waves, quasilinear theory, nonlinear three-wave interactions); (iii) plasma collisional and discreteness phenomena (test-particle theory of dynamic friction and wave emission, classical resistivity, extension of test-particle theory to many-particle phenomena and the derivation of the Boltzmann and Lenard–Balescu equations, the Fokker–Planck collision operator, a general scattering theory, nonlinear Landau damping, radiation transport and Dupree’s theory of clumps); (iv) non-uniform plasmas (adiabatic invariance, guiding-centre drifts, hydromagnetic theory, introduction to drift-wave stability theory).
Boundary layers in space and astrophysical plasmas are the location of complex dynamics where different mechanisms coexist and compete, eventually leading to plasma mixing. In this work, we present fully kinetic particle-in-cell simulations of different boundary layers characterized by the following main ingredients: a velocity shear, a density gradient and a magnetic gradient localized at the same position. In particular, the presence of a density gradient drives the development of the lower-hybrid drift instability (LHDI), which competes with the Kelvin–Helmholtz instability (KHI) in the development of the boundary layer. Depending on the density gradient, the LHDI can even dominate the dynamics of the layer. Because these two instabilities grow on different spatial and temporal scales, when the LHDI develops faster than the KHI an inverse cascade is generated, at least in two dimensions. This inverse cascade, starting at the LHDI kinetic scales, generates structures at scale lengths at which the KHI would typically develop. When that is the case, those structures can suppress the KHI itself because they significantly affect the underlying velocity shear gradient. We conclude that, depending on the density gradient, the velocity jump and the width of the boundary layer, the LHDI in its nonlinear phase can become the primary instability for plasma mixing. These numerical simulations show that the LHDI is likely to be a dominant process at the magnetopause of Mercury. These results are expected to be of direct impact to the interpretation of the forthcoming BepiColombo observations.
Beam-driven instabilities are considered in a pulsar plasma assuming that both the background plasma and the beam are relativistic Jüttner distributions. In the rest frame of the background, the only waves that can satisfy the resonance condition are in a tiny range of slightly subluminal phase speeds. The growth rate for the kinetic (or maser) version of the weak-beam instability is much smaller than has been estimated for a relativistically streaming Gaussian distribution, and the reasons for this are discussed. The growth rate for the reactive version of the weak-beam instability is treated in a conventional way. We compare the results with exact calculations, and find that the approximate solutions are not consistent with the exact results. We conclude that, for plausible parameters, there is no reactive version of the instability. The growth rate in the pulsar frame is smaller than that in the rest frame of the background plasma by a factor
$2\unicode[STIX]{x1D6FE}_{\text{s}}$
, where
$\unicode[STIX]{x1D6FE}_{\text{s}}=10^{2}{-}10^{3}$
is the Lorentz factor of the bulk motion of the background plasma, placing a further constraint on effective wave growth. Based on these results, we argue that beam-driven wave growth probably plays no role in pulsar radio emission.
The stability of conducting Taylor–Couette flows under the presence of toroidal magnetic background fields is considered. For strong enough magnetic amplitudes such magnetohydrodynamic flows are unstable against non-axisymmetric perturbations which may also transport angular momentum. In accordance with the often used diffusion approximation, one expects the angular momentum transport to be vanishing for rigid rotation. In the sense of a non-diffusive
$\unicode[STIX]{x1D6EC}$
effect, however, even for rigidly rotating
$z$
-pinches, an axisymmetric angular momentum flux appears which is directed outward (inward) for large (small) magnetic Mach numbers. The internal rotation in a magnetized rotating tank can thus never be uniform. Those particular rotation laws are used to estimate the value of the instability-induced eddy viscosity for which the non-diffusive
$\unicode[STIX]{x1D6EC}$
effect and the diffusive shear-induced transport compensate each other. The results provide the Shakura & Sunyaev viscosity ansatz leading to numerical values linearly growing with the applied magnetic field.
Ion-temperature-gradient-driven (ITG) turbulence is compared for two quasi-symmetric (QS) stellarator configurations to determine the relationship between linear growth rates and nonlinear heat fluxes. We focus on the quasi-helically symmetric (QHS) stellarator HSX and the quasi-axisymmetric (QAS) stellarator NCSX. In normalized units, HSX exhibits higher growth rates than NCSX, while heat fluxes in gyro-Bohm units are lower in HSX. These results hold for simulations made with both adiabatic and kinetic electrons. The results show that HSX has a larger number of subdominant modes than NCSX and that eigenmodes are more spatially extended in HSX. We conclude that the consideration of nonlinear physics is necessary to accurately assess the heat flux due to ITG turbulence when comparing QS stellarator equilibria.
Dipole and stellarator geometries are capable of confining plasmas of arbitrary neutrality, ranging from pure electron plasmas through to quasineutral. The diocotron mode is known to be important in non-neutral plasmas and has been widely studied. However, drift mode dynamics, dominating quasineutral plasmas, has received very little by way of attention in the non-neutral context. Here, we show that non-neutral plasmas can be unstable respect to both density-gradient- and temperature-gradient-driven instabilities. A local shearless slab limit is considered for simplicity. A key feature of non-neutral plasmas is the development of strong electric fields, in this local limit of straight field line geometry, the effect of the corresponding
$\boldsymbol{E}\times \boldsymbol{B}$
drift is limited to the Doppler shift of the complex frequency
$\unicode[STIX]{x1D714}\rightarrow \unicode[STIX]{x1D714}-\unicode[STIX]{x1D714}_{E}$
. However, the breaking of the quasineutrality condition still leads to interesting dynamics in non-neutral plasmas. In this paper we address the behaviour of a number of gyrokinetic modes in electron–ion and electron–positron plasmas with arbitrary degree of neutrality.
Analytic treatment is presented of the electrostatic instability of an initially planar electron hole in a plasma of effectively infinite particle magnetization. It is shown that there is an unstable mode consisting of a rigid shift of the hole in the trapping direction. Its low frequency is determined by the real part of the force balance between the Maxwell stress arising from the transverse wavenumber
$k$
and the kinematic jetting from the hole’s acceleration. The very low growth rate arises from a delicate balance in the imaginary part of the force between the passing-particle jetting, which is destabilizing, and the resonant response of the trapped particles, which is stabilizing. Nearly universal scalings of the complex frequency and
$k$
with hole depth are derived. Particle in cell simulations show that the slow-growing instabilities previously investigated as coupled hole–wave phenomena occur at the predicted frequency, but with growth rates 2 to 4 times greater than the analytic prediction. This higher rate may be caused by a reduced resonant stabilization because of numerical phase-space diffusion in the simulations.
Wave dispersion in a pulsar plasma is discussed emphasizing the relevance of different inertial frames, notably the plasma rest frame
${\mathcal{K}}$
and the pulsar frame
${\mathcal{K}}^{\prime }$
in which the plasma is streaming with speed
$\unicode[STIX]{x1D6FD}_{\text{s}}$
. The effect of a Lorentz transformation on both subluminal,
$|z|<1$
, and superluminal,
$|z|>1$
, waves is discussed. It is argued that the preferred choice for a relativistically streaming distribution should be a Lorentz-transformed Jüttner distribution; such a distribution is compared with other choices including a relativistically streaming Gaussian distribution. A Lorentz transformation of the dielectric tensor is written down, and used to derive an explicit relation between the relativistic plasma dispersion functions in
${\mathcal{K}}$
and
${\mathcal{K}}^{\prime }$
. It is shown that the dispersion equation can be written in an invariant form, implying a one-to-one correspondence between wave modes in any two inertial frames. Although there are only three modes in the plasma rest frame, it is possible for backward-propagating or negative-frequency solutions in
${\mathcal{K}}$
to transform into additional forward-propagating, positive-frequency solutions in
${\mathcal{K}}^{\prime }$
that may be regarded as additional modes.
Wave dispersion in a pulsar plasma (a one-dimensional, strongly magnetized, pair plasma streaming highly relativistically with a large spread in Lorentz factors in its rest frame) is discussed, motivated by interest in beam-driven wave turbulence and the pulsar radio emission mechanism. In the rest frame of the pulsar plasma there are three wave modes in the low-frequency, non-gyrotropic approximation. For parallel propagation (wave angle
$\unicode[STIX]{x1D703}=0$
) these are referred to as the X, A and L modes, with the X and A modes having dispersion relation
$|z|=z_{\text{A}}\approx 1-1/2\unicode[STIX]{x1D6FD}_{\text{A}}^{2}$
, where
$z=\unicode[STIX]{x1D714}/k_{\Vert }c$
is the phase speed and
$\unicode[STIX]{x1D6FD}_{\text{A}}c$
is the Alfvén speed. The L mode dispersion relation is determined by a relativistic plasma dispersion function,
$z^{2}W(z)$
, which is negative for
$|z|<z_{0}$
and has a sharp maximum at
$|z|=z_{\text{m}}$
, with
$1-z_{\text{m}}<1-z_{0}\ll 1$
. We give numerical estimates for the maximum of
$z^{2}W(z)$
and for
$z_{\text{m}}$
and
$z_{0}$
for a one-dimensional Jüttner distribution. The L and A modes reconnect, for
$z_{\text{A}}>z_{0}$
, to form the O and Alfvén modes for oblique propagation (
$\unicode[STIX]{x1D703}\neq 0$
). For
$z_{\text{A}}<z_{0}$
the Alfvén and O mode curves reconnect forming a new mode that exists only for
$\tan ^{2}\unicode[STIX]{x1D703}\gtrsim z_{0}^{2}-z_{\text{A}}^{2}$
. The L mode is the nearest counterpart to Langmuir waves in a non-relativistic plasma, but we argue that there are no ‘Langmuir-like’ waves in a pulsar plasma, identifying three features of the L mode (dispersion relation, ratio of electric to total energy and group speed) that are not Langmuir like. A beam-driven instability requires a beam speed equal to the phase speed of the wave. This resonance condition can be satisfied for the O mode, but only for an implausibly energetic beam and only for a tiny range of angles for the O mode around
$\unicode[STIX]{x1D703}\approx 0$
. The resonance is also possible for the Alfvén mode but only near a turnover frequency that has no counterpart for Alfvén waves in a non-relativistic plasma.
The model of the global gyrokinetic particle-in-cell code ORB5 has been extended for the study of pair plasmas. This has been done by including the physics of the Debye shielding, by including the electron polarization density and by retaining the effects of the electron finite Larmor radius. This model is verified against previous numerical results for the cyclone base case tokamak scenario in deuterium plasmas, and for local pair plasma simulations. The linear dynamics of temperature-gradient driven instabilities and geodesic acoustic modes is investigated. Mass dependencies for different Debye lengths are studied.
To explain the possible destabilization of a two-dimensional magnetic equilibrium such as the near-Earth magnetotail, we developed a kinetic model describing the resonant interaction of electromagnetic fluctuations and bouncing electrons trapped in the magnetic bottle. A small-
$\unicode[STIX]{x1D6FD}$
approximation (i.e., the plasma pressure is lower than the magnetic pressure) is used in agreement with a small field line curvature. The linearized gyro-kinetic Vlasov equation is integrated along the unperturbed particle trajectories, including cyclotron and bounce motions. The dispersion relation for drift-Alfvèn waves is obtained through the plasma quasi-neutrality condition and Ampere’s law for the parallel current. It has been found that for a quasi-dipolar configuration (
$\text{L}\sim 8$
corresponds to the set of the Earth’s magnetic field lines, crossing the Earths magnetic equator at 8 Earth radii), unstable electromagnetic modes may develop in the current sheet with a growth rate of the order of a few tenths of a second provided that the typical scale of density gradient slope responsible for the diamagnetic drift effects is over one Earth radius or less. This instability growth rate is large enough to destabilize the current sheet on time scales of 2–4 minutes as observed during substorm onset.
Kinetic treatments of drift tearing modes that match an inner, resonant layer solution to an external magnetohydrodynamic (MHD) solution, characterised by
$\unicode[STIX]{x1D6E5}^{\prime }$
, can fail to match the ideal MHD boundary condition on the parallel electric field,
$E_{\Vert }=0$
. In this paper we demonstrate how consideration of ion sound and ion Landau damping effects achieves this, placing the theory on a firm footing. These effects are found to modify the effective critical
$\unicode[STIX]{x1D6E5}^{\prime }$
for instability of drift tearing modes, in particular for weak electron temperature gradients. The implications for a realistic hot plasma resonant layer model – involving large ion Larmor radius and semi-collisional electron physics (Connor et al., Plasma Phys. Control. Fusion, vol. 54, 2012, 035003) – are determined.
In this paper, the dynamics of electrons emitted by a spherical object when the total charge of the system is constant is studied in detail. In particular, the condition for which the total electron charge presents damped oscillations is deduced rigorously by considering a perturbation with respect to the steady-state solution. The results obtained can be of utility in simulating the expansion of a spherical plasma by separating the ion and electron time scales.
In a magnetized, collisionless plasma, the magnetic moment of the constituent particles is an adiabatic invariant. An increase in the magnetic-field strength in such a plasma thus leads to an increase in the thermal pressure perpendicular to the field lines. Above a
$\unicode[STIX]{x1D6FD}$
-dependent threshold (where
$\unicode[STIX]{x1D6FD}$
is the ratio of thermal to magnetic pressure), this pressure anisotropy drives the mirror instability, producing strong distortions in the field lines on ion-Larmor scales. The impact of this instability on magnetic reconnection is investigated using a simple analytical model for the formation of a current sheet (CS) and the associated production of pressure anisotropy. The difficulty in maintaining an isotropic, Maxwellian particle distribution during the formation and subsequent thinning of a CS in a collisionless plasma, coupled with the low threshold for the mirror instability in a high-
$\unicode[STIX]{x1D6FD}$
plasma, imply that the geometry of reconnecting magnetic fields can differ radically from the standard Harris-sheet profile often used in simulations of collisionless reconnection. As a result, depending on the rate of CS formation and the initial CS thickness, tearing modes whose growth rates and wavenumbers are boosted by this difference may disrupt the mirror-infested CS before standard tearing modes can develop. A quantitative theory is developed to illustrate this process, which may find application in the tearing-mediated disruption of kinetic magnetorotational ‘channel’ modes.
Processes of laser energy absorption and electron heating in an expanding plasma in the range of irradiances
$I\unicode[STIX]{x1D706}^{2}=10^{15}{-}10^{16}~\text{W}\,\cdot \,\unicode[STIX]{x03BC}\text{m}^{2}/\text{cm}^{2}$
are studied with the aid of kinetic simulations. The results show a strong reflection due to stimulated Brillouin scattering and a significant collisionless absorption related to stimulated Raman scattering near and below the quarter critical density. Also presented are parametric decay instability and resonant excitation of plasma waves near the critical density. All these processes result in the excitation of high-amplitude electron plasma waves and electron acceleration. The spectrum of scattered radiation is significantly modified by secondary parametric processes, which provide information on the spatial localization of nonlinear absorption and hot electron characteristics. The considered domain of laser and plasma parameters is relevant for the shock ignition scheme of inertial confinement fusion.
This paper describes a model of electron energization and cyclotron-maser emission applicable to astrophysical magnetized collisionless shocks. It is motivated by the work of Begelman, Ergun and Rees [Astrophys. J. 625, 51 (2005)] who argued that the cyclotron-maser instability occurs in localized magnetized collisionless shocks such as those expected in blazar jets. We report on recent research carried out to investigate electron acceleration at collisionless shocks and maser radiation associated with the accelerated electrons. We describe how electrons accelerated by lower-hybrid waves at collisionless shocks generate cyclotron-maser radiation when the accelerated electrons move into regions of stronger magnetic fields. The electrons are accelerated along the magnetic field and magnetically compressed leading to the formation of an electron velocity distribution having a horseshoe shape due to conservation of the electron magnetic moment. Under certain conditions the horseshoe electron velocity distribution function is unstable to the cyclotron-maser instability [Bingham and Cairns, Phys. Plasmas 7, 3089 (2000); Melrose, Rev. Mod. Plasma Phys. 1, 5 (2017)].