Nonlinear Guiding Center Research Articles

Numerical Test of Analytical Theories for Perpendicular Diffusion in Small Kubo Number Turbulence

Abstract In the literature, one can find various analytical theories for perpendicular diffusion of energetic particles interacting with magnetic turbulence. Besides quasi-linear theory, there are different versions of the nonlinear guiding center (NLGC) theory and the unified nonlinear transport (UNLT) theory. For turbulence with high Kubo numbers, such as two-dimensional turbulence or noisy reduced magnetohydrodynamic turbulence, the aforementioned nonlinear theories provide similar results. For slab and small Kubo number turbulence, however, this is not the case. In the current paper, we compare different linear and nonlinear theories with each other and test-particle simulations for a noisy slab model corresponding to small Kubo number turbulence. We show that UNLT theory agrees very well with all performed test-particle simulations. In the limit of long parallel mean free paths, the perpendicular mean free path approaches asymptotically the quasi-linear limit as predicted by the UNLT theory. For short parallel mean free paths we find a Rechester & Rosenbluth type of scaling as predicted by UNLT theory as well. The original NLGC theory disagrees with all performed simulations regardless what the parallel mean free path is. The random ballistic interpretation of the NLGC theory agrees much better with the simulations, but compared to UNLT theory the agreement is inferior. We conclude that for this type of small Kubo number turbulence, only the latter theory allows for an accurate description of perpendicular diffusion.

PERPENDICULAR DIFFUSION OF ENERGETIC PARTICLES IN NOISY REDUCED MAGNETOHYDRODYNAMIC TURBULENCE

Recently a model for noisy reduced magnetohydrodynamic turbulence was proposed. The latter model was already used to study the random walk of magnetic field lines. In the current article we use the same model to investigate the diffusion of energetic particles across the mean magnetic field. To compute the perpendicular diffusion coefficient two analytical theories are used, namely the Non- Linear Guiding Center (NLGC) theory and the Unified Non-Linear Transport (UNLT) theory. It is shown that the two theories provide different results for the perpendicular diffusion coefficient. We also perform test-particle simulations for the aforementioned turbulence model. We show that only the UNLT theory describes perpendicular transport accurately confirming that the latter theory is a powerful tool in diffusion theory.

THE MODIFICATION OF THE NONLINEAR GUIDING CENTER THEORY

We modify the NonLinear Guiding Center (NLGC) theory (Matthaeus et al. 2003) for perpendicular diffusion by replacing the spectral amplitude of the two-component model magnetic turbulence with the 2D component one (following Shalchi 2006), and replacing the constant $a^2$, indicating the degree particles following magnetic field line, with a variable $a^{\prime 2}$ as a function of the magnetic turbulence. We combine the modified model with the NonLinear PArallel (NLPA) diffusion theory (Qin 2007) to solve perpendicular and parallel diffusion coefficients simultaneously. It is shown that the new model agrees better with simulations. Furthermore, we fit the numerical results of the new model with polynomials, so that parallel and perpendicular diffusion coefficients can be calculated directly without iteration of integrations, and many numerical calculations can be reduced.

A Study of the Coefficient a2 in the Extended Nonlinear Guiding Center Theory

The diffusion mechanism of charged particles is one of the very im- portant problems in astrophysics. Recently, Matthaeus et al. (2003) developed a nonlinear guiding center theory (NLGC) of the perpendicular diffusion of en- ergetic charged particles with the parallel diffusion coefficient as an input. This theory agrees with the results of numerical simulations very well. In addition, Qin (2007) developed a nonlinear parallel diffusion theory (NLPA) following the idea of NLGC. Combining the NLGC with the NLPA, a new theory, NLGC-E was developed to solve the parallel and perpendicular diffusion coefficients simultaneously. In the NLGC theory, there is a coefficient a2 which is selected to be 1/3 for the best agreement with the results of numerical simulations. In this work we tried the different values of a2 for the NLGC-E theory to determine the optimal value which fits best the results of numerical simulations.

RADIAL DEPENDENCE OF PEAK PROTON AND IRON ION FLUXES IN SOLAR ENERGETIC PARTICLE EVENTS: APPLICATION OF THE PATH CODE

The radial dependence of particle peak fluxes in large solar energetic particle (SEP) events is important in determining the potential impact of space weather hazards on space missions. Using the Particle Acceleration and Transport in the Heliosphere code, we model the acceleration and transport of protons and iron ions at evolving coronal mass ejection shocks propagating throughout the inner heliosphere from about 0.1 to 2.5 AU. An example shock with a compression ratio of 3.9 and a speed of 1000 km s{sup -1} close to the Sun is modeled using a two-dimensional MHD ZEUS code. The compression ratio and shock speed weakened to 2.3 and 630 km s{sup -1}, correspondingly, at 2 AU. Shocks with 15 Degree-Sign , 45 Degree-Sign , and 75 Degree-Sign angles between the upstream magnetic field and the shock normal were studied. The shock angle was kept constant throughout the simulation. Both gradual and impulsive events are studied. Diffusive shock acceleration is assumed at the shock and we use a total diffusion coefficient that includes a parallel diffusion coefficient which takes into account the upstream wave amplification, and a perpendicular diffusion coefficient which is based on the NonLinear Guiding Center theory. The transport of particles escapingmore » from the shock is modeled using a Monte Carlo approach. Time-intensity profiles for protons and iron ions are obtained. We analyzed the radial dependence of peak fluxes (J) for both protons and iron ions from 0.5 to 2 AU. We find that the functional dependence is softer than R {sup -3} and is about R {sup -2.9} to R {sup -1.8} in the energy range of 0.3-5 MeV nuc{sup -1}. Quasi-perpendicular shock showed a steeper radial dependence than a quasi-parallel shock. Mixed events show a softer radial dependence at energies above 500 keV nuc{sup -1} for iron ions and above 1 MeV for protons. The values of J(R) depend on seed particle composition, particle energy, shock obliquity, and the interplanetary turbulence level. Consequently, we advocate using SEP event-specific computer modeling rather than empirical formulae for future forecasting of the radiation environment throughout the heliosphere during large SEP events.« less

RANDOM BALLISTIC INTERPRETATION OF NONLINEAR GUIDING CENTER THEORY

Nonlinear guiding center (NLGC) theory has been used to explain the asymptotic perpendicular diffusion coefficient κ⊥ of energetic charged particles in a turbulent magnetic field, which can be applied to better understand cosmic ray transport. Here we re-derive NLGC, replacing the assumption of diffusive decorrelation with random ballistic decorrelation (RBD), which yields an explicit formula for κ⊥. We note that scattering processes can cause a reversal of the guiding center motion along the field line, i.e., backtracking, leading to partial cancellation of contributions to κ⊥, especially for low-wavenumber components of the magnetic turbulence. We therefore include a heuristic backtracking correction (BC) that can be used in combination with RBD. In comparison with computer simulation results for various cases, NLGC with RBD and BC provides a substantially improved characterization of the perpendicular diffusion coefficient for a fluctuation amplitude less than or equal to the large-scale magnetic field.

THE EFFECT OF INTERMITTENT GYRO-SCALE SLAB TURBULENCE ON PARALLEL AND PERPENDICULAR COSMIC-RAY TRANSPORT

Earlier work based on nonlinear guiding center (NLGC) theory suggested that perpendicular cosmic-ray transport is diffusive when cosmic rays encounter random three-dimensional magnetohydrodynamic turbulence dominated by uniform two-dimensional (2D) turbulence with a minor uniform slab turbulence component. In this approach large-scale perpendicular cosmic-ray transport is due to cosmic rays microscopically diffusing along the meandering magnetic field dominated by 2D turbulence because of gyroresonant interactions with slab turbulence. However, turbulence in the solar wind is intermittent and it has been suggested that intermittent turbulence might be responsible for the observation of 'dropout' events in solar energetic particle fluxes on small scales. In a previous paper le Roux et al. suggested, using NLGC theory as a basis, that if gyro-scale slab turbulence is intermittent, large-scale perpendicular cosmic-ray transport in weak uniform 2D turbulence will be superdiffusive or subdiffusive depending on the statistical characteristics of the intermittent slab turbulence. In this paper we expand and refine our previous work further by investigating how both parallel and perpendicular transport are affected by intermittent slab turbulence for weak as well as strong uniform 2D turbulence. The main new finding is that both parallel and perpendicular transport are the net effect of an interplay between diffusivemore » and nondiffusive (superdiffusive or subdiffusive) transport effects as a consequence of this intermittency.« less

Dynamics of charged particles and perpendicular diffusion in turbulent magnetic field

A test particle code is employed to explore the dynamics of charged particles and perpendicular diffusion in turbulent magnetic field, where a three-dimensional (3D) isotropic turbulence model is used in this paper. The obtained perpendicular diffusion at different particle energies is compared with that of the nonlinear guiding center (NLGC) theory. It is found that the NLGC theory is consistent with test particle simulations when the particle energies are small. However, the difference between the NLGC theory and test particle simulations tends to increase when the particle energy is sufficiently large, and the threshold is related to the turbulence bend-over length. In the NLGC theory, the gyrocenter of a charged particle is assumed to follow the magnetic field line. Therefore, when the particle has sufficiently large energy, its gyroradius will be larger than the turbulence bend-over length. Then the particle can cross the magnetic field lines, and the difference between the test particle simulations and NLGC theory occurs.

A GENERALIZED NONLINEAR GUIDING CENTER THEORY FOR THE COLLISIONLESS ANOMALOUS PERPENDICULAR DIFFUSION OF COSMIC RAYS

The original nonlinear guiding center (NLGC) theory by Matthaeus et al. was a breakthrough in establishing a theory that promised to reproduce for the first time the anomalous perpendicular diffusion results from test particle trajectory calculations in prescribed static magnetic field turbulence dominated by a two-dimensional component. The assumptions used in this approach guaranteed anomalous diffusion to be a classical process (the variance � Δx 2 �∝ (Δt) α with α = 1). However, Shalchi & Kourakis showed that similar calculations can be even better reproduced within the context of a generalized compound diffusion model for anomalous perpendicular diffusion whereby anomalous diffusion is nonclassical (0 <α< 1 (subdiffusion) or 1 <α< 2 (superdiffusion)). In this paper, it is shown how NLGC theory can be generalized to model compound diffusion conditions consistent with the generalized compound diffusion model in terms of a nonuniform plasma medium. Such a medium is assumed to generate a distribution of cosmic-ray pitch-angle scattering times for cosmic rays interacting resonantly with a minor small-scale slab turbulence component. This suggests that the anomalous perpendicular diffusion from the test particle simulations might be explained best within the framework of anomalous diffusion in a nonuniform plasma medium. It is argued that, during intermediate times when the magnetic field appears to be static in the quiet solar wind near Earth, generalized NLGC theory predicts possibly subdiffusive anomalous perpendicular transport for intermediate-energy (E � 400–500 MeV) cosmic rays because they experience nonuniform scattering conditions. These conditions are suggested to be a product of stochastic wave growth of small-scale slab turbulence resulting intermittently in large patches of intense slab turbulence where scattering times are reduced and cosmic rays are trapped. Classical anomalous diffusion is expected to be restored at high cosmic-ray energies (E � 400–500 MeV) in accordance with original NLGC theory because they encounter more uniform scattering conditions due to weaker stochastic wave growth. Magnetic turbulence observations near Earth during 2003 were used to show that the solar wind can be characterized as a nonuniform medium sustaining a power-law distribution of cosmic-ray pitch-angle scattering times. However, application of generalized NLGC theory to the 2003 observations produced unexpectedly a negative exponent for the variance of the anomalous diffusion ( α< 0). This indicates that particle bunching occurs as cosmic rays are trapped in plasma regions associated with strong particle scattering. Such effects can plausibly be explained by the intermittent presence of strong compressive turbulence downstream of shocks, stream, and interaction regions.

Perpendicular Transport of Energetic Charged Particles in Nonaxisymmetric Two‐Component Magnetic Turbulence

We examine energetic charged particle diffusion perpendicular to a mean magnetic field B0 due to turbulent fluctuations in a plasma, relaxing the common assumption of axisymmetry around B0 and varying the ratio of two fluctuation components, a slab component with parallel wavenumbers and a two-dimensional (2D) component with perpendicular wavenumbers. We perform computer simulations mostly for 80% 2D and 20% slab energy and a fluctuation amplitude on the order of B0. The nonlinear guiding center (NLGC) theory provides a reasonable description of asymptotic perpendicular diffusion as a function of the nonaxisymmetry and particle energy. These values areroughlyproportionaltotheparticlespeedtimesthefieldlinediffusioncoefficient,withaprefactorthatismuchlower than in the classical field line random walk model of particle diffusion. NLGC predicts a prefactor in closer agreement with simulations. Next we consider extreme fluctuation anisotropy and the approach to reduced dimensionality. For 99% slab fluctuation energy, field line trajectories are diffusive, but the particle motion is subdiffusive. For 99% 2D fluctuation energy, both field lines and particle motions are initially subdiffusive and then diffusive, but NLGC gives unreliable results. The time dependence of the running particle diffusion coefficient shows that in all cases asymptotic diffusionisprecededbyfreestreamingandsubdiffusion,but thelatterdiffersfromstandardcompoundsubdiffusion.We can model the time profiles in terms of a decaying negative correlation of the perpendicular velocity due to the possibility of backtracking along magnetic field lines.

Nonlinear Cosmic‐Ray Diffusive Transport in Combined Two‐dimensional and Slab Magnetohydrodynamic Turbulence: A BGK‐Boltzmann Approach

Particle simulations revealed that standard quasi-linear theory cannot adequately describe the parallel and perpendicular diffusion of cosmic rays in three-dimensional magnetohydrodynamic solar wind turbulence, when the turbulence is modeled approximately as a combination of a dominant two-dimensional component and a minor slab component. Recent nonlinear theories based on a Taylor-Green-Kubo formalism, such as the nonlinear guiding center (NLGC) theory, and the weakly nonlinear theory (WNLT) proved to be much more successful. Instead, we follow a BGK-Boltzmann approach to investigate this matter. Going beyond current NLGC theory and WNLT in scope, we derived a complete cosmic-ray transport theory that includes expressions not only for parallel and perpendicular diffusion, but also for drift, convection, adiabatic energy change, and momentum diffusion. The BGK-Boltzmann approach enables one to derive tractable yet complicated expressions for the transport coefficients in both the weak and strong particle scattering limits. We show that the WNLT expressions can be recovered by combining these two limits in such a way that there is weak particle scattering along the magnetic field but strong scattering across the field. It is shown that the complex WNLT expressions can be reduced to simple analytical expressions for parallel and perpendicular diffusion that reproduce approximately the rigidity dependence of particle simulations at low to medium rigidities. These expressions also prove to be consistent with well-known expressions for perpendicular diffusion in the literature. It is discussed how large-scale gradient and curvature drifts get modified by turbulence and how stochastic particle acceleration changes when two-dimensional turbulence is dominant.

Nonlinear Parallel Diffusion of Charged Particles: Extension to the Nonlinear Guiding Center Theory

A nonlinear theory of the parallel diffusion of charged particles with perpendicular scattering and dynamical turbulence is obtained. The combination of the new parallel diffusion theory and the nonlinear guiding center theory shows good agreement with numerical simulations using typical parameters for a solar wind. Furthermore, the combination of the theories has a simpler mathematical form and is more computationally tractable than the weakly nonlinear theory.

Nonlinear perpendicular diffusion in strong turbulent electromagnetic fields

A nonlinear description for perpendicular particle diffusion in strong electromagnetic fluctuations is developed by using the fundamental Newton–Lorentz equation. Although not based on the same approach, the recently presented nonlinear guiding center (NLGC) theory is recovered as a special case. The approach used here is rather based on the argument that a well defined particle gyromotion does not exist in strong fluctuations than on the assumption that the particle gyrocenter follows magnetic field lines which themselves separate diffusively. The assumption of a guiding center motion and the diffusive separation of magnetic field lines is absolutely central to the NLGC theory. It is argued that the NLGC result should provide most accurate results for strong fluctuations. Furthermore, as a direct consequence of the particle equation of motion, it is shown that particle diffusion in one perpendicular direction is governed by the fluctuations in the other normal direction. This results contradicts the NLGC result, where perpendicular diffusion is triggered by fluctuations in the same direction. This is of particular interest for anisotropic perpendicular diffusion in a non-axisymmetric turbulence. Future numerical simulation results for non-axisymmetric magnetic turbulence and their comparison with the approach presented here and the NLGC theory have to provide an answer whether particle diffusion in one perpendicular direction is governed by the fluctuations in the other normal direction or by fluctuations in the same direction.

Nonlinear Guiding Center Theory of Perpendicular Diffusion in Dynamical Turbulence

Recently, a nonlinear theory for perpendicular diffusion of charged particles was presented by Matthaeus and coworkers. This theory is called the Nonlinear Guiding Center (NLGC) theory and provides an integral equation for the perpendicular mean free path. In this paper we consider numerical results of this equation in the case of dynamical turbulence. We consider solutions for two prominent models for dynamical turbulence: the damping model of dynamical turbulence and the random sweeping model. For both models we also use a general power spectrum with an energy, inertial, and dissipation range. In the current paper we also examine the NLGC theory for two different plasma wave dispersion relations, shear Alfven waves and fast magnetosonic waves. We compare all these numerical results with observational results and with solutions of the magnetostatic dissipationless NLGC theory.

Nonlinear guiding center theory of perpendicular diffusion: General properties and comparison with observation

A nonlinear guiding center (NLGC) theory for diffusion of charged particles perpendicular to the mean magnetic field was recently proposed. Here, we draw attention to a number of attractive features of this theory: (1) The theory provides a natural mechanism to connect the perpendicular mean free path with the parallel mean free path. In fact, the parallel mean free path is the only particle property required to determine uniquely the perpendicular mean free path. (2) Under a broad range of conditions, the theory predicts that the perpendicular mean free path will be of order one percent or a few percent of the parallel mean free path, in agreement with numerical simulations of particle transport. (3) For conditions representative of the inner heliosphere, the theory predicts values of the perpendicular mean free path in agreement with observational determinations from Jovian electrons and from modeling Ulysses observations of Galactic protons.