Particle Larmor Radius Research Articles

Effect of rotation on plasma stability in the gas-dynamic trap

The effects of the centrifugal force and finite Larmor radius on plasma stability in the gas-dynamic trap are considered. Estimates show that the stability is governed by the strong effect of the finite Larmor radius of fast particles. If this stabilizing mechanism does not operate, then the instability ceases to be exponential when the sheared plasma rotation becomes sufficiently intense. In this case, the potential perturbation amplitude increases according to a power law, and the instability threshold remains unchanged. These effects do not influence the stability of the first azimuthal perturbation mode in a plasma with a free boundary. But when the plasma is in good electrical contact with the conducting wall of the device and when the plasma density profile is not too peaked at the axis, the first mode is stabilized by the finite Larmor radius effect because of the radial variation of the perturbed electric field.

Magnetic reconnection at the termination shock in a striped pulsar wind

Context. Most of the rotational luminosity of a pulsar is carried away by a relativistic magnetised wind in which the matter energy flux is negligible compared to the Poynting flux. However, observations of the Crab nebula for instance clearly indicate that most of the Poynting flux is eventually converted into ultra-relativistic particles. The mechanism responsible for transformation of the electro-magnetic energy into the particle energy remains poorly understood. Near the equatorial plane of an obliquely rotating pulsar magnetosphere, the magnetic field reverses polarity with the pulsar period, forming a wind with oppositely directed field lines. This structure is called a striped wind; dissipation of alternating fields in the striped wind is the object of our study. Aims. The aim of this paper is to study the conditions required for magnetic energy release at the termination shock of the striped pulsar wind. Magnetic reconnection is considered via analytical methods and 1D relativistic PIC simulations. Methods. An analytical condition on the upstream parameters for partial and full magnetic reconnection is derived from the conservation laws of energy, momentum and particle number density across the relativistic shock. Furthermore, by using a 1D relativistic PIC code, we study in detail the reconnection process at the termination shock for different upstream Lorentz factors and magnetisations. Results. We found a very simple criterion for dissipation of alternating fields at the termination shock, depending on the upstream parameters of the flow, namely, the magnetisation σ , the Larmor radius r B and the wavelength l of the striped wind. The model depends also on a free parameter $\xi>1$, which is the ratio of the current sheet width to the particle Larmor radius. It is found that for $\sigma \gg l / \xi \, r_{\rm B}$, all the Poynting flux is converted into particle energy whereas for $\sigma \ll (l / \xi \, r_{\rm B} )^{2/3}$, no dissipation occurs. In the latter case, the shock can be accurately described by the ideal MHD shock conditions. Finally, 1D relativistic PIC simulations confirm this prediction and enable us to fix the free parameter ξ in the analytical model. Conclusions. Alternating magnetic fields annihilate easily at relativistic highly magnetised shocks. In plerions, our condition for full magnetic dissipation is satisfied at the termination shock so that the Poynting flux may be converted into ultra-relativistic particles not in the pulsar wind but just at the termination shock. The constraints are more severe for the intra-binary shocks in double pulsar systems. Available models explaining observations require low magnetisation in the downstream flow. The condition that the magnetic field dissipates at the intra-binary shock implies an upper limit on the pair multiplicity in the pulsar wind κ . We found $\kappa \lesssim {\rm few} \times 10^4$ for PSR 1259-63 and PSR 1957+20. In the double pulsar PSR 0737-3039, the radio emission from the pulsar B is modulated with the period of the pulsar A, which implies that the striped structure is not erased completely; this gives a lower limit for $\kappa \gtrsim 310$.

Anomalous, non-Gaussian transport of charged particles in anisotropic magnetic turbulence

The transport of energetic particles in a mean magnetic field and in the presence of anisotropic magnetic turbulence is studied numerically, for parameter values relevant to astrophysical plasmas. A numerical realization of magnetic turbulence is set up, in which the degree of anisotropy is varied by changing the correlation lengths lx, ly, and lz. The ratio ρ∕λ of the particle Larmor radius ρ over the turbulence correlation length λ is also varied. It is found that for lx,ly⪢lz, and for ρ∕λ≲10−2 transport can be non-Gaussian, with superdiffusion along the average magnetic field and subdiffusion perpendicular to it. In addition, the spatial distribution of particles is clearly non-Gaussian. Such regimes are characterized by a Levy statistics, with diverging second-order moments. Decreasing the ratio lx∕lz, nearly Gaussian (normal) diffusion is obtained, showing that the transport regime depends on the turbulence anisotropy. Changing the particle Larmor radius, normal diffusion is found for 10−2≲ρ∕λ≲1 because of increased pitch angle diffusion. New anomalous superdiffusive regimes appear when ρ∕λ≳1 showing that the interaction between particles and turbulence decreases in these cases. A new regime, called generalized double diffusion, is proposed for the cases when particles are able to trace back field lines. A summary of the physical conditions which lead to non-Gaussian transport is given.

On the Efficiency of Fermi Acceleration at Relativistic Shocks

It is shown that Fermi acceleration at an ultra-relativistic shock wave cannot operate on a particle for more than 1 1/2 Fermi cycle (i.e., u -> d -> u -> d) if the particle Larmor radius is much smaller than the coherence length of the magnetic field on both sides of the shock, as is usually assumed. This conclusion is shown to be in excellent agreement with recent numerical simulations. We thus argue that efficient Fermi acceleration at ultra-relativistic shock waves requires significant non-linear processing of the far upstream magnetic field with strong amplification of the small scale magnetic power. The streaming or transverse Weibel instabilities are likely to play a key role in this respect.

Anomalous particle diffusion and Lévy random walk of magnetic field lines in three-dimensional solar wind turbulence

Plasma transport in the presence of turbulence depends on a variety of parameters such as the fluctuation level, δB/B0, the ratio between the particle Larmor radius and the turbulence correlation length, and the turbulence anisotropy. In this paper, we present the results of numerical simulations of plasma and magnetic field line transport in the case of anisotropic magnetic turbulence, for parameter values close to those of the solar wind. We assume a uniform background magnetic field B0 = B0 ez and a Fourier representation for magnetic fluctuations, which includes wavectors oblique with respect to B0. The energy density spectrum is a power law, and in k space it is described by the correlation lengths lx, ly, lz, which quantify the anisotropy of turbulence. For magnetic field lines, transport perpendicular to the background field depends on the Kubo number R = (δB/B0) (lz/lx). For small Kubo numbers, R ≪ 1, anomalous, non-Gaussian transport regimes (both sub- and superdiffusive) are found, which can be described as a Lévy random walk. Increasing the Kubo number, i.e. the fluctuation level, δB/B0, or the ratio lz/lx, we find first a quasilinear regime and then a percolative regime, both corresponding to Gaussian diffusion. For particles, we find that transport parallel and perpendicular to the background magnetic field depends heavily on the turbulence anisotropy and on the particle Larmor radius. For turbulence levels typical of the solar wind, δB/B0≃ 0.5–1, when the ratio between the particle Larmor radius and the turbulence correlation lengths is small, anomalous regimes are found in the case lz/lx ⩽ 1, with a Lévy random walk (superdiffusion) along the magnetic field and subdiffusion in the perpendicular directions. Conversely, for lz/lx > 1 normal Gaussian diffusion is found. A possible expression for generalized double diffusion is discussed.

Stopping power for arbitrary angle between test particle velocity and magnetic field

Using the longitudinal dielectric function derived previously for charged test particles in helical movement around magnetic field lines, the numerical convergence of the series involved is found and the double numerical integrations on wave vector components are performed yielding the stopping power for arbitrary angle between the test particle velocity and magnetic field. Calculations are performed for particle Larmor radius larger and shorter than Debye length, i.e., for protons in a cold magnetized plasma and for thermonuclear α particles in a dense, hot, and strongly magnetized plasma. A strong decrease is found for the energy loss as the angle varies from 0 to π∕2. The range of thermonuclear α particles as a function of the velocity angle with respect to the magnetic field is also given.

Interplanetary Transport Mechanisms of Solar Energetic Particles

Numerical simulations of charged particle trajectories in a Parker field with model turbulence are used to study the transport mechanisms of solar energetic particles (SEPs): magnetic focusing, pitch-angle diffusive transport due to magnetic fluctuations, and adiabatic cooling. The results of the simulations are compared with analytical formulae for the transport effects. We find that adiabatic cooling is significant for SEPs traveling from the Sun to 1 AU. The simulation results agree much better with the anisotropic adiabatic cooling effect theory than with the isotropic one. Also, we show that the simulation results agree well with the theory of magnetic focusing with pitch-angle scattering by magnetic fluctuations. In addition, in order for quasilinear theory to be valid, the average particle's Larmor radius must be much smaller than the correlation scale of the two-dimensional component of turbulence.

Estimates of arrival directions of giant air showers

The arrival directions of giant air showers generated in the atmosphere by the primary cosmic ray particles with energies above ∼5 · 10 19 eV may be connected with possible extragalactic sources, because the Larmor radius of such particles is too large. Besides it was suggested that the primary particles with enormous energies may be neutrons. In this case it is possible to avoid the energy loss in interactions with the microwave background radiation and arrival directions will strongly point to sources. Thus it is of primary importance to decrease possible errors in the estimates of arrival directions of giant air showers. To estimate the arrival direction of a giant air shower one has to have any model of its space–time structure. The simplest model of the shower time front is a model of the flat front when all particles are located in this front plane. It was shown that possible errors in estimates of the zenith and azimuth angles which characterize the arrival directions may be as large as 5 ◦ or even more. The χ 2 method gives very large values of χ 2 . That means that this model is inconsistent with the data. A much more realistic model of a shower time front was suggested by Linsley. Calculations displayed that both the shower disk thickness and the average time delay depend on the energy of the primary particles. So the calculated time front of the shower for both electrons and muons may fit the data and thus provides better accuracy. The standard mathematical procedure to interpret data is the χ 2 method. This method leads to reasonable estimates of the zenith and azimuth angles with uncertainties of 2÷3 ◦ . In some cases the minimax procedure may be utilized to interpret data. It was shown that the possible error in estimates of the zenith and azimuth angles may be decreased up to 0.5 ◦ . At last the fuzzy uncertain variables and the possibility theory are suggested here to be used for interpreting the data. Calculations were carried out in terms of the quark-gluon string model for primary protons and an observation level of 1020 g/cm. The Landau–Pomeranchuk–Migdal effect and interactions of neutral pions with nuclei in the atmosphere at high energies are taken into account. The Monte Carlo method was used for primary protons while cascades from numerous charged pions were considered with the help of cascade equations. Though experimental statistics is very low no evidence is found to prefer any directions. Thus the isotropic distribution of the arrival directions of giant air showers with

Particle canonical variables and guiding center Hamiltonian up to second order in the Larmor radius

A generating function, expressed as a power series in the particle Larmor radius, is used to relate an arbitrary set of magnetic field line coordinates to particle canonical variables. A systematic procedure is described for successively choosing the generating function at each order in the Larmor radius so that the transformed particle Hamiltonian is independent of the Larmor phase angle. The particle guiding center Hamiltonian up to second order in the Larmor radius is thereby derived. The analysis includes finite electric fields in which the particle “electric drifts” can be of the order of the particle velocity. The transformations which relate an arbitrary set of toroidal magnetic flux coordinates to particle canonical variables are also discussed.

Fast particle finite orbit width and Larmor radius effects on low-n toroidicity induced Alfvén eigenmode excitation

The effects of finite drift orbit width (FOW) and Larmor radius (FLR) of fast particles on the stability of low-n toroidicity-induced Alfvén eigenmodes (TAE) are studied. The formulation is based on the solution of the low frequency gyrokinetic equation (ω≪ωc, where ωc is particle cyclotron frequency). A quadratic form has been derived in terms of invariant variables; energy E, magnetic moment μ, and toroidal angular momentum Pφ. The growth rate of the TAE is computed perturbatively using numerical averaging over the fast particle drift orbit. This new computational capability improves the NOVA-K code [G. Y. Fu, C. Z. Cheng, and K. L. Wong, Phys. Fluids B 5, 4040 (1994)] which included FOW effects in the growth rate calculation based on small radial orbit width approximation. The new NOVA-K version has been benchmarked for different regimes of particle TAE excitation. It is shown that both FOW and FLR effects are typically stabilizing; the TAE growth rate can be reduced by as much as a factor of 2 for tokamak fusion test reactor supershots [D. J. Grove and D. M. Meade, Nucl. Fusion 25, 1167 (1985)]. However, FOW may be destabilizing for the global modes, which are localized at the plasma edge.

Pitch angle scattering near energy resonances in the geomagnetic tail

In sharp field reversals where the particle Larmor radius is larger than the magnetic field line curvature radius, particles may not conserve their magnetic moment (first adiabatic invariant). We examine the pitch angle scattering which affects the particles under such conditions. We show that this scattering alternatively leads to a clear three‐branch pattern of magnetic moment variations or to unstructured variations. The three‐branch pattern is characterized by systematic enhancements of magnetic moment at relatively small (up to ∼30°) pitch angles, negligible change at large pitch angles and, in between, either enhancement or damping depending on phase. We show that this three‐branch pattern emerges near the energy resonances reported by Burkhart and Chen [1991], where particles escape after transient oscillations inside the reversal. Away from resonance, the three‐branch pattern gradually expands and affects an increasing volume of the velocity space. Such a structuring of pitch angle scattering is similar to that obtained in weak field reversals (i.e., for Larmor radii comparable to the field line curvature radius), even though the particle orbits in sharp and weak reversals are significantly different. This fact allows us to extend the centrifugal impulse model developed for weak reversals to sharp ones. In this model, nonadiabatic behavior is viewed as the result of perturbation of the particle gyromotion by an impulsive centrifugal force. We demonstrate that this model reproduces essential features of the particle dynamics in sharp reversals and, in particular, the energy resonance phenomenon. We show that in a like manner to weak reversals, three‐branch dynamics in sharp reversals has significant implications for particle injection into the loss cone and gyrophase bunching near the magnetotail midplane.

Ion and electron distribution functions in the distant magnetotail: modeling of Geotail observations

Ion and electron distribution functions measured by the Geotail spacecraft in the distant magnetotail are investigated considering single‐particle dynamics in sharp field reversals. The observed H+ distribution functions exhibit two components, namely, a low‐energy core and a high‐energy population propagating tailward. In agreement with previous studies, we show that ion dynamics in a two‐dimensional (2‐D) reconnection field geometry naturally leads to such distributions and that the neutral line location can be analytically inferred from the velocity space structure. As for electron distribution functions, Geotail observations reveal characteristic “flattop” profiles with enhanced flux in the parallel direction at low energies and in the perpendicular direction at high energies. We interpret these distributions as the result of the electron chaotic behavior in the reconnection region. This chaotic regime, which occurs when the particle Larmor radius is comparable to the magnetic field line curvature radius, can lead to prominent pitch angle scattering. Near the neutral line, we show that at a given energy, this regime yields flux depletion in the parallel direction and enhancement in the perpendicular one. The flattop profile arises from this behavior affecting only the (relatively) high‐energy part of the electron distribution. Conversely, the threshold energy above which electrons exhibit such a behavior contains information on the field reversal characteristics. The electron distribution functions obtained numerically are in agreement with the observations, which support this interpretation framework and allow us to estimate the thickness of the neutral sheet.

Fluid model of collisionless plasma with finite Larmor radius effects

A closed set of moment equations for the plasma density, velocity and pressure is proposed by imposing linear closure for the heat flux and viscosity. The formulation is valid for arbitrary finite Larmor radius parameter k⊥ρ (k⊥ is the perpendicular wave vector, ρ is the particle Larmor radius) and retains main convective nonlinearities. Reduced moment equations relevant to the drift-wave modes are derived in the low frequency limit (ω≪ωB, ωB is the particle cyclotron frequency).

Numerical modeling of the 22‐year variation of the cosmic ray intensity and anisotropy

We have solved the two‐dimensional time‐dependent diffusion‐convection equation numerically to obtain the distribution and anisotropy of cosmic rays in the heliosphere. We have assumed that the parallel and perpendicular mean free paths are proportional to the particle Larmor radius, and we have treated each proportionality constant (a, b) as a parameter. We have found that the set (a, b) = (4, 2) gives the steady state solution compatible with observations on the intensity and the solar diurnal anisotropy of cosmic rays in 0.5‐ to 10‐GeV range as obtained at the earth. This set of (a, b) corresponds to the ratio of the diffusion coefficients D∥/D⊥ = 10. In our solution the intensity for the (pre‐1980) interplanetary magnetic field (IMF) state where the solar magnetic dipole and the angular velocity vector are parallel is higher than for the (post‐1980) state where they are antiparallel, while the phase of the diurnal anisotropy is about 15 hours for the parallel state and about 18 hours for the antiparallel state. We have also reproduced the observed small radial gradient for each IMF state. We discuss the nature of the solution in order to understand the effect of the density gradient drift motion on the cosmic ray distribution.

Motion of a charged particle in a nearly axisymmetric magnetic field

The motion of a charged particle in a static magnetic field is studied by means of repeated canonical transformations of a Hamiltonian system. Adiabatic invariants are generated based on the assumption that the particle Larmor radius is small compared with the characteristic distance over which the magnetic field varies. Unlike many earlier treatments the transformations presented here preserve the axisymmetry of the dynamics when the magnetic field is axisymmetric. It is assumed that the magnetic field consists of a small nonaxisymmetric part plus the axisymmetric toroidal and poloidal parts. After the introduction of the magnetic moment adiabatic invariant, the motion of the guiding center is studied. The results depend sensitively on the ratio of the poloidal magnetic field to the total magnetic field. In some cases a second adiabatic invariant exists and direct inferences concerning long time particle drifts are possible. In one case where a second adiabatic invariant fails to exist, long term drifts are studied by conventional perturbation expansions. At some points, resonance or lack of resonance phenomena appear and determine the drift effects.

Excitation of Alfvén waves by fast particles in a finite pressure plasma of adiabatic traps

Resonant excitation of Alfvén waves by fast particles in a finite pressure plasma in a non-uniform magnetic field is studied. Plasma compressibility in the wave field is determined both by the curvature of the magnetic lines of force and finite Larmor radius of fast particles. A general expression for the instability growth rate is obtained and analyzed; the applicability of the results obtained in the previous paper has also been studied. The finite pressure stabilization of the trapped particles instability has been found. The bounce-resonance effects are analyzed.

Cyclotron oscillations of a large-Larmor-radius plasma

A method is proposed for investigating plasma oscillations in axially symmetric systems with a longitudinal magnetic field, which can be used for an arbitrary ratio of the Larmor radius of charged particles to the radius of the system. An anisotropic ion cyclotron instability and a modified negative-mass instability are analysed with this method. The analysis was performed for a system with parameters similar to those of the Ogra-3 device. It is shown, in particular, that in the case of a modified negative-mass instability, the threshold density for oscillations of the first azimuthal mode has a lower value than for the zero mode. This result corresponds to experimental data.

Absorption of Plasma Waves

The propagation of waves through a plasma, wherein the density and/or magnetic field strength are slowly varying functions of position is discussed. If the local propagation constant, kx, is a slowly varying function of x, the adiabatic approximation will be valid. However, kx2 may pass through zero as a function of x. Using the WKB linear turning point connection formulas, examination shows that an incoming plasma wave is totally reflected in the region where kx2 ≈ 0. A similar analysis for the case where kx2 is a singular function of x shows that absorption of an incoming wave occurs in the vicinity of the singularity. Such singular behavior in kx2 can occur for propagation along the magnetic field when the wave frequency is equal to the local ion or electron cyclotron frequency. For propagation transverse to the magnetic field, an apparent singularity occurs at a frequency somewhat below the ion cyclotron frequency, and at the two hybrid frequencies of Auer, Hurwitz, and Miller. A detailed examination, including higher order effects in electron mass ÷ ion mass, finite electron and ion temperatures, and ion-ion and ion-electron collisions shows that the absorption will take place at the apparent singularity only if the physical damping processes are strong enough to swamp the reactive effects of the higher order corrections. Otherwise the higher order reactive effects introduce a new propagation mode into the dispersion equation with a root which, in the vicinity of the apparent singularity, is conjugate to the root of the original mode. Partial or total reflection now occurs at the apparent singularity instead of absorption. It is, however, conjectured that some of the original mode energy may be reflected into the new mode. As the new mode recedes from the region of the apparent singularity, its wavelength can become comparable to the particle Larmor radius. Energy in this mode may then be absorbed by phase-mixing processes which are of high order in the quantity (Larmor radius ÷ wavelength). Wave reflection from the apparent singularities will then heat ions in the case of the transverse ion cyclotron mode, and electrons in the case of the upper hybrid frequency.