Particle Larmor Radius Research Articles

Particle Scattering in the Presence of Moderate to Strong Adiabatic Focusing

AbstractStarting from the Vlasov‐Boltzmann equation, we derive corrections to the quasi linear Fokker‐Planck pitch angle scattering coefficient caused by a non‐uniformity of the background magnetic field. Integration of the perturbed particle distribution function is performed along a modified unperturbed trajectory that takes into account the effects of focusing and mirroring on pitch angle and gyrophase. We calculate the changes to the quasi‐linear scattering rate owing to a variations in the field strength and the gyrophase drift that affects the resonance between particles and magnetic fluctuations. It is shown that the scattering coefficient has an oscillating behavior in pitch angle when the focusing lengthscale is comparable to the correlation length of the turbulence and the particle's Larmor radius. The applicability of analytic results are demonstrated with a simple model of charged particles trapped in a magnetic field of a planet. It is suggested that the pitch angle diffusion coefficient could become negative for the case of very strong focusing.

Generalized kinetic equation for tokamak plasma equilibrium distribution function

A generalized kinetic equation for the equilibrium distribution function in a finite beta, arbitrary tokamak plasma is derived. The equation is correct to second order in ρ/L (ρ is the particle Larmor radius and L is the system size). Resolving the finite Larmor radius length scales with no restriction on the ratio of poloidal to total equilibrium magnetic field, Bϑ/B, it generalizes the drift kinetic theory of Hazeltine [Phys. Plasmas 15, 77 (1973)] to the limit of Bϑ/B∼1 (e.g., to ensure validity for spherical tokamaks). Two cases are considered. The first provides the equilibrium distribution function, consistent with the generalized gyrokinetic formalism of Dudkovskaia et al. [Plasma Phys. Controlled Fusion 65, 045010 (2023)], derived specifically to capture neoclassical equilibrium currents in the gyrokinetic stability analyses in strong gradient regions. The second assumes short length scales in the direction perpendicular to the magnetic field, which can occur as a result of small coherent magnetic structures in the plasma, such as neoclassical tearing mode magnetic islands close to threshold. This then extends the drift island equations of Dudkovskaia et al. [Nucl. Fusion 63, 016020 (2023)] for the plasma response to magnetic islands to a spherical tokamak plasma configuration. Resolving ρ∼ρϑ (or Bϑ∼B), where ρϑ is the particle poloidal Larmor radius, is also expected to influence calculations of the magnetic island propagation frequency and the associated contributions to the island onset conditions.

Energetic Particle Perpendicular Diffusion: Simulations and Theory in Noisy Reduced Magnetohydrodynamic Turbulence

The transport of energetic charged particles (e.g., cosmic rays) in turbulent magnetic fields is usually characterized in terms of the diffusion parallel and perpendicular to a large-scale (or mean) magnetic field. The nonlinear guiding center theory has been a prominent perpendicular diffusion theory. A recent version of this theory, based on the random ballistic spreading of magnetic field lines and a backtracking correction (RBD/BC), has shown good agreement with test particle simulations for a two-component magnetic turbulence model. The aim of the present study is to test the generality of the improved theory by applying it to the noisy reduced magnetohydrodynamic (NRMHD) turbulence model, determining perpendicular diffusion coefficients that are compared with those from the field line random walk (FLRW) and unified nonlinear (UNLT) theories and our test particle simulations. The synthetic NRMHD turbulence model creates special conditions for energetic particle transport, with no magnetic fluctuations at higher parallel wavenumbers so there is no resonant parallel scattering if the particle Larmor radius R L is even slightly smaller than the minimum resonant scale. This leads to nonmonotonic variation in the parallel mean free path λ ∥ with R L. Among the theories considered, only RBD/BC matches simulations within a factor of 2 over the range of parameters considered. This accuracy is obtained even though the theory depends on λ ∥ and has no explicit dependence on R L. In addition, the UNLT theory often provides accurate results, and even the FLRW limit provides a very simple and reasonable approximation in many cases.

A Comparison with Theory of Computation and Estimation of Pitch-Angle Diffusion Coefficients from Simulations in Noisy Reduced Magnetohydrodynamic Turbulence

The transport of energetic charged particles in turbulent magnetic fields is a topic of interest in various astrophysical and laboratory plasma contexts. In order to estimate the mean free path λ∥ of a particle in the direction parallel to the mean magnetic field, one can use theoretical expressions that include the pitch-angle diffusion coefficient Dμμ. In this work we evaluate theories for Dμμ in the context of the noisy reduced magnetohydrodynamic (NRMHD) model where turbulent fluctuations are absent at large parallel wavenumbers. For most turbulence models, the standard quasilinear theory predicts zero pitch-angle diffusion only for particles with a 90◦ pitch angle, but for NRMHD a range of pitch angles is affected, leading to infinite λ∥. We examine two theories that include resonance broadening which yield finite λ∥ and compare them with test-particle computer simulations in which the parallel mean free path can be readily obtained. We find that both theories are quite accurate in some regions of the parameter space considered, but neither is particularly good when the particle Larmor radius RL becomes much smaller than the quasilinear resonance limit.

Pitch angle scattering of fast particles by low frequency magnetic fluctuations

The adiabatic invariance of the magnetic moment during particle motion is of fundamental importance to the dynamics of magnetized plasma. The related rate of pitch angle scattering is investigated here for fast particles that thermally stream through static magnetic perturbations. For a uniform magnetic field with a localized perturbation, it is found that the curvature parameter κ2=min(Rc/ρL) does not predict the level of pitch angle scattering. Instead, based on numerical integration of particle orbits in prescribed magnetic fields, we derive predictions for the particle scattering rates, which can be characterized by the relative perturbation amplitude A≃δB/B, and the particle Larmor radius normalized by the field-aligned wavelength of the perturbations, ρL/λ||. Particles with ρL/λ||≃1 are subject to strong pitch angle scattering, while the level of scattering vanishes for both the limits of ρL/λ||≪1 and ρL/λ||≫1. The results are summarized in terms of a scattering operator, suitable for including the described scattering in basic kinetic models.

Gravitational instability in radiative molecular clouds including cosmic ray diffusion and ion Larmor radius corrections

ABSTRACT The effects of cosmic ray (CR) diffusion and finite Larmor radius (FLR) corrections have been studied on the linear gravitational instability of thermally conducting plasmas typically in the H ii regions of molecular clouds. The hydrodynamic fluid–fluid approach is considered for interacting CRs with gravitating, magnetized, and thermally conducting gas in molecular clouds. The magnetohydrodynamic fluid model is formulated considering CR pressure gradients, CR diffusion, and radiative and FLR effects in terms of particle Larmor radius. The dispersion relation of the gravitational instability is analytically derived using the normal mode analysis, and the effects of CRs and FLR corrections have been discussed in longitudinal and transverse modes. It is observed that in the absence of CRs, the FLR effects (magnetic viscosity) reduce the growth rate for wavenumber smaller than a critical value, and above it gets increased. However, the growth rate is strongly suppressed in the presence of combined CRs and FLR effects. The individual behaviour of FLR effects is observed to destabilize the growth rate of the gravitational instability in the presence of CR effects. The CR pressure decreases the growth rates of the gravitational and thermal instabilities, whereas parallel CR diffusion enhances the growth rate of the gravitational instability. The Jeans length of the gravitating gas cloud gets increased due to an increase in the CR-to-gas pressure ratio. It is found that the gravitational collapse of the system is supported by high-energy (above knee) CR particles with the Larmor radii comparable to the cloud size. The present results have been applied to understand the role of CRs and FLR corrections on the gravitational collapse in the H ii regions of molecular clouds.

A Statistical Study of Lower Hybrid Waves in the Earth's Magnetosphere by Van Allen Probes

AbstractThe lower hybrid (LH) waves are electrostatic emissions near the LH resonant frequency. They propagate perpendicularly with a small wavelength comparable to Larmor radius of thermal particles and can be capable of heating both ions and electrons. In this study, we statistically study the global distribution of LH waves in the inner magnetosphere by using Van Allen Probes observation from 2012 to 2018. We find that (a) LH waves are commonly observed in the inner magnetosphere. Most LH waves are confined near the magnetic equator with typical amplitudes of 0.02–0.2 mV/m and occurrence rates up to 10%. (b) LH waves extend to inner L regions with increasing wave amplitudes as AE* increases. (c) Weak LH waves occur at the nightside inside the plasmapause. Moderate and strong LH waves occur at the nightside and noon inside the plasmapause. As AE* increases, they extend to all magnetic local times inside the plasmapause and dawnside outside the plasmapause.

The Interplay of Magnetically Dominated Turbulence and Magnetic Reconnection in Producing Nonthermal Particles

Abstract Magnetized turbulence and magnetic reconnection are often invoked to explain the nonthermal emission observed from a wide variety of astrophysical sources. By means of fully kinetic 2D and 3D particle-in-cell simulations, we investigate the interplay between turbulence and reconnection in generating nonthermal particles in magnetically dominated (or, equivalently, “relativistic”) pair plasmas. A generic by-product of the turbulence evolution is the generation of a nonthermal particle spectrum with a power-law energy range. The power-law slope p is harder for larger magnetizations and stronger turbulence fluctuations, and it can be as hard as p ≲ 2. The Larmor radius of particles at the high-energy cutoff is comparable to the size l of the largest turbulent eddies. Plasmoid-mediated reconnection, which self-consistently occurs in the turbulent plasma, controls the physics of particle injection. Then, particles are further accelerated by stochastic scattering off turbulent fluctuations. The work done by parallel electric fields—naturally expected in reconnection layers—is responsible for most of the initial energy increase and is proportional to the magnetization σ of the system, while the subsequent energy gain, which dominates the overall energization of high-energy particles, is powered by the perpendicular electric fields of turbulent fluctuations. The two-stage acceleration process leaves an imprint in the particle pitch-angle distribution: low-energy particles are aligned with the field, while the highest-energy particles move preferentially orthogonal to it. The energy diffusion coefficient of stochastic acceleration scales as D γ ∼ 0.1σ(c/l)γ 2, where γ is the particle Lorentz factor. This results in fast acceleration timescales t acc ∼ (3/σ)l/c. Our findings have important implications for understanding the generation of nonthermal particles in high-energy astrophysical sources.

Nonlinear gyrokinetic Coulomb collision operator

A gyrokinetic Coulomb collision operator is derived, which is particularly useful to describe the plasma dynamics at the periphery region of magnetic confinement fusion devices. The derived operator is able to describe collisions occurring in distribution functions arbitrarily far from equilibrium with variations on spatial scales at and below the particle Larmor radius. A multipole expansion of the Rosenbluth potentials is used in order to derive the dependence of the full Coulomb collision operator on the particle gyroangle. The full Coulomb collision operator is then expressed in gyrocentre phase-space coordinates, and a closed formula for its gyroaverage in terms of the moments of the gyrocentre distribution function in a form ready to be numerically implemented is provided. Furthermore, the collision operator is projected onto a Hermite–Laguerre velocity space polynomial basis and expansions in the small electron-to-ion mass ratio are provided.

Perpendicular relativistic shocks in magnetized pair plasma

Perpendicular relativistic ($\gamma_0=10$) shocks in magnetized pair plasmas are investigated using two dimensional Particle-in-Cell simulations. A systematic survey, from unmagnetized to strongly magnetized shocks, is presented accurately capturing the transition from Weibel-mediated to magnetic-reflection-shaped shocks. This transition is found to occur for upstream flow magnetizations $10^{-3}<\sigma<10^{-2}$ at which a strong perpendicular net current is observed in the precursor, driving the so-called current-filamentation instability. The global structure of the shock and shock formation time are discussed. The MHD shock jump conditions are found in good agreement with the numerical results, except for $10^{-4} < \sigma < 10^{-2}$ where a deviation up to 10\% is observed. The particle precursor length converges toward the Larmor radius of particles injected in the upstream magnetic field at intermediate magnetizations. For $\sigma>10^{-2}$, it leaves place to a purely electromagnetic precursor following from the strong emission of electromagnetic waves at the shock front. Particle acceleration is found to be efficient in weakly magnetized perpendicular shocks in agreement with previous works, and is fully suppressed for $\sigma > 10^{-2}$. Diffusive Shock Acceleration is observed only in weakly magnetized shocks, while a dominant contribution of Shock Drift Acceleration is evidenced at intermediate magnetizations. The spatial diffusion coefficients are extracted from the simulations allowing for a deeper insight into the self-consistent particle kinematics and scale with the square of the particle energy in weakly magnetized shocks. These results have implications for particle acceleration in the internal shocks of AGN jets and in the termination shocks of Pulsar Wind Nebulae.

Scattering of galactic cosmic rays by a magnetic cloud injected into interplanetary space during active solar processes

The scattering of charged particles as they pass through areas of the interplanetary magnetic field with large-scale inhomogeneities (magnetic clouds) is studied using the Monte Carlo method and the numerical simulation of trajectories. Charged particles (protons) with energies ranging from 1 to 100 GeV in magnetic clouds with sizes of 0.01–0.1 a.u. and magnetic flux densities of 5 to 50 nT are modeled. It is established that an important factor in determining the nature of galactic cosmic ray scattering is the relationship between the Larmor radii of particles, the size of a magnetic cloud, and the degree of magnetic field inhomogeneity.

Hot plasma dielectric response to radio-frequency fields in inhomogeneous magnetic field

Hot plasma dielectric response models, which are now used in most linear full wave codes, are formulated in Fourier space assuming that particle's Larmor radius is much smaller than the scale of spatial nonuniformity of magnetic field. Such approximation assumes that the spatial scale of plasma dielectric response to the RF field is limited to a few Larmor radii, which is accurate for a limited range of wave frequencies ω. The scale of plasma dielectric response along the magnetic field line could be comparable to the scale of the magnetic field nonuniformity when ω is close to the particle's cyclotron frequency ωc or when ω is much smaller than ωc, which requires the use of a more accurate model. In the present approach, the hot plasma dielectric response is formulated in configuration space without limiting approximations by numerically calculating the plasma conductivity kernel based on the solution of the linearized Vlasov equation in nonuniform magnetic field. Results of the conductivity kernel calculation in hot collisionless plasma are presented for 1-D mirror and 2-D tokamak magnetic field configurations for ω∼ωc. Self-consistent simulation of RF fields using the calculated conductivity kernel of 1-D mirror magnetic field is made. A new parallel full wave RF code, based on the presented approach of accurate self-consistent modeling of plasma dielectric response in configuration space, is under development.

Cosmic ray propagation in sub-Alfvénic magnetohydrodynamic turbulence

This work has the main objective to provide a detailed investigation of cosmic ray propagation in magnetohydrodynamic turbulent fields generated by forcing the fluid velocity field at large scales. It provides a derivation of the particle mean free path dependences in terms of the turbulence level described by the Alfv\'enic Mach number and in terms of the particle rigidity. We use an upgrade version of the magnetohydrodynamic code {\tt RAMSES} which includes a forcing module and a kinetic module and solve the Lorentz equation for each particle. The simulations are performed using a 3 dimension periodical box in the test-particle and magnetostatic limits. The forcing module is implemented using an Ornstein-Uhlenbeck process. An ensemble average over a large number of particle trajectories is applied to reconstruct the particle mean free paths. We derive the cosmic ray mean free paths in terms of the Alfv\'enic Mach numbers and particle reduced rigidities in different turbulence forcing geometries. The reduced particle rigidity is $\rho=r_L/L$ where $r_L$ is the particle Larmor radius and $L$ is the simulation box length related to the turbulence coherence or injection scale $L_{inj}$ by $L \sim 5 L_{inj}$. We have investigated with a special attention compressible and solenoidal forcing geometries. We find that compressible forcing solutions are compatible with the quasi-linear theory or more advanced non-linear theories which predict a rigidity dependence as $\rho^{1/2}$ or $\rho^{1/3}$. Solenoidal forcing solutions at least at low or moderate Alfv\'enic numbers are not compatible with the above theoretical expectations and require more refined arguments to be interpreted. It appears especially for Alfv\'enic Mach numbers close to one that the wandering of field lines controls perpendicular mean free path solutions whatever the forcing geometry.

Global diffusion of cosmic rays in random magnetic fields

The propagation of charged particles, including cosmic rays, in a partially ordered magnetic field is characterized by a diffusion tensor whose components depend on the particle's Larmor radius $R_L$ and the degree of order in the magnetic field. Most studies of the particle diffusion presuppose a scale separation between the mean and random magnetic fields (e.g., there being a pronounced minimum in the magnetic power spectrum at intermediate scales). Scale separation is often a good approximation in laboratory plasmas, but not in most astrophysical environments such as the interstellar medium (ISM). Modern simulations of the ISM have numerical resolution of order 1 pc, so the Larmor radius of the cosmic rays that dominate in energy density is at least $10^{6}$ times smaller than the resolved scales. Large-scale simulations of cosmic ray propagation in the ISM thus rely on oversimplified forms of the diffusion tensor. We take the first steps towards a more realistic description of cosmic ray diffusion for such simulations, obtaining direct estimates of the diffusion tensor from test particle simulations in random magnetic fields (with the Larmor radius scale being fully resolved), for a range of particle energies corresponding to $10^{-2}\lesssim R_L/l_c \lesssim 10^{3}$, where $l_c$ is the magnetic correlation length. We obtain explicit expressions for the cosmic ray diffusion tensor for $R_L/l_c \ll 1$, that might be used in a sub-grid model of cosmic ray diffusion. The diffusion coefficients obtained are closely connected with existing transport theories that include the random walk of magnetic lines.

Testing the magnetotail configuration based on observations of low‐altitude isotropic boundaries during quiet times

AbstractWe investigate the configuration of the geomagnetic field on the nightside magnetosphere during a quiet time interval based on National Oceanic and Atmospheric Administration Polar Orbiting Environment Satellites Medium Energy Proton and Electron Detector (NOAA/POES MEPED) measurements in combination with numerical simulations of the global terrestrial magnetosphere using the Space Weather Modeling Framework. Measurements from the NOAA/POES MEPED low‐altitude data sets provide the locations of isotropic boundaries; those are used to extract information regarding the field structure in the source regions in the magnetosphere. In order to evaluate adiabaticity and mapping accuracy, which is mainly controlled by the ratio between the radius of curvature and the particle's Larmor radius, we tested the threshold condition for strong pitch angle scattering based on the MHD magnetic field solution. The magnetic field configuration is represented by the model with high accuracy, as suggested by the high correlation coefficients and very low normalized root‐mean‐square errors between the observed and the modeled magnetic field. The scattering criterion, based on the values of ratio at the crossings of magnetic field lines, associated with isotropic boundaries, with the minimum B surface, predicts a critical value of kCR∼33. This means that, in the absence of other scattering mechanisms, the strong pitch angle scattering takes place whenever the Larmor radius is ∼33 times smaller than the radius of curvature of the magnetic field, as predicted by the Space Weather Modeling Framework.

ENERGETIC PARTICLE ANISOTROPIES AT THE HELIOSPHERIC BOUNDARY. II. TRANSIENT FEATURES AND RIGIDITY DEPENDENCE

In the preceding paper, we showed that large second-order anisotropies of heliospheric ions measured by the Voyager 1 space probe during the August 2012 boundary crossing event could be explained by a magnetic shear across the heliopause preventing particles streaming along the magnetic field from escaping the inner heliosheath. According to Stone et al., the penetration distance of heliospheric ions into the outer heliosheath had a strong dependence on the particle's Larmor radius. By comparing hydrogen, helium, and oxygen ions with the same energy per nucleon, these authors argued that this effect must be attributed to larger cyclotron radii of heavier species rather than differences in velocity. We propose that gradient drift in a nonuniform magnetic field was the cause of both the large second-order anisotropies and the spatial differentiation based on the ion's rigidity. A latitudinal gradient of magnetic field strength of about 10% per AU between 2012.7 and 2012.9 could have provided drift motion sufficient to match both LECP and CRS Voyager 1 observations. We explain the transient intensity dropout observed prior to the heliocliff using flux tube structures embedded in the heliosheath and magnetically connected to interstellar space. Finally, this paper reports a new indirect measurement of the plasma radial velocity at the heliopause on the basis of the time difference between two cosmic-ray telescopes measuring the same intensity dropout.

PITCH-ANGLE SCATTERING OF ENERGETIC PARTICLES WITH ADIABATIC FOCUSING

Understanding turbulent transport of charged particles in magnetized plasmas often requires a model for the description of random variations in the particle's pitch angle. The Fokker-Planck coefficient of pitch-angle scattering, which is used to describe scattering parallel to the mean magnetic field, is therefore of central importance. Whereas quasi-linear theory assumes a homogeneous mean magnetic field, such a condition is often not fulfilled, especially for high-energy particles. Here, a new derivation of the quasi-linear approach is given that is based on the unperturbed orbit found for an adiabatically focused mean magnetic field. The results show that, depending on the ratio of the focusing length and the particle's Larmor radius, the Fokker-Planck coefficient is significantly modified but agrees with the classical expression in the limit of a homogeneous mean magnetic field.

Mechanisms of Spontaneous Reconnection: From Magnetospheric to Fusion Plasma

Very often space plasma is treated as collisionless. We check the validity of this paradigm considering various regimes of tearing mode (spontaneous reconnection) including effects of particle collisions and shear of magnetic field. We briefly describe Pitaevskii’s effect of effective modification of collision frequency due to the finite particle Larmor radius in the presence of magnetic field. This effect results in a significant increase of the role of collisionality, especially in a weakly magnetized systems. Another popular paradigm is related with application of MHD description to collisionless or weakly collisional systems. We show, that for current sheets observed in the Earth magnetotail and magnetopause as well as for current sheets formed in Solar corona and in laboratory devices most appropriate is the kinetic semi-collisional tearing regime. Role of “collisions” could play usual Coulomb pair collisions of electrons and ions (e.g. in Solar corona) or effective collisions (scattering) of electrons with the microturbulence wave modes. Transition to real MHD modes requires either very large collisions frequencies and/or very large amplitudes of the magnetic field shear. The largest domain in the parameter space is occupied by the kinetic regimes of tearing mode growth where dissipation is provided either by Landau damping or by real (or effective) collisions.

The scaling of collisionless magnetic reconnection in an electron–positron plasma with non-scalar pressure

AbstractCollisionless magnetic reconnection via tearing instability in non-relativistic electron–positron (pair) plasma with an anisotropic pressure is investigated. The equilibrium magnetic field is considered to be sheared force-free, and a set of linearized collisionless Magnetohydrodynamics equations describes the evolution of reconnection dynamics. A linear analytical analysis, based on scaling, demonstrates that in such a pair plasma, breaking the frozen in flow constraint for field lines can be mainly provided by the non-gyrotropic pressure of electrons and positrons (rather than the particle bulk inertia) when the current sheet width is smaller than the particle Larmor radius (Δx &lt; rL). This condition is satisfied when β &gt; d2 (d = c/ωp is the particle skin-depth with the electron/positron frequency ωp and β = 8πP(0)/B02 ⪡ 1). Meanwhile, on top of the Lorentz force and in the absence of the reconnection facilitating mechanism of the Hall effect, non-scalar pressure force can accelerate bulk plasma into the diffusion region at the scale lengths of the order of dx. Therefore, the respective regime of tearing instability proceeds much faster compared with the case of an isotropic pressure with a new dimensionless growth rate of (γτA) ~ d.

Collisionless current generation in the center of the tokamak plasma by an isotropic source of α-particles

The density of the noninductive current generated due to collisionless motion of α-particles in the tokamak magnetic field is calculated. The analysis is based on fully three-dimensional calculations of charged particle trajectories without simplifying assumptions typical for drift and neoclassical approaches. The current is calculated over the entire cross section of the plasma column, including the magnetic axis. It is shown that the current density is not a function of a magnetic surface and is strongly polarized over the poloidal angle. The current density distribution in the tokamak poloidal cross section is obtained, and the current density as a function of the safety factor, the tokamak aspect ratio, and the ratio of the particle Larmor radius on the axis to the tokamak minor radius is determined. It is shown that, when the source of α-particles is spatially nonuniform, the current density in the center of the tokamak is nonzero due to asymmetry of the phase-space boundary between trapped and passing particles. The current density scaling in the tokamak center differs from the known approximations for the bootstrap current and is sensitive to the spatial distribution of α-particles.