Diffusion of Relativistic Charged Particles and Field Lines in Isotropic Turbulence. II. Analytical Models

The theory for the perpendicular diffusion of cosmic rays is not well understood which hampers our understanding of cosmic ray modulation. In this paper, a spherically symmetric cosmic ray modulation model is used to evaluate three different theories for perpendicular cosmic ray diffusion in the ecliptic plane, subject to the limitation of negligible cosmic ray transport in the polar direction. In models for the perpendicular diffusion component, it is assumed that perpendicular diffusion due to large‐scale field line wandering dominates resonant perpendicular diffusion. To test these models, observations of anomalous and galactic helium obtained during a time of relatively small latitudinal gradients (1996) are used as a guideline. The spatial dependence of the parallel and perpendicular cosmic ray diffusion coefficients is determined completely theoretically using a promising hydromagnetic model for the transport of combined slab plus two‐dimensional turbulence in the solar wind [Zank et al., 1996]. The main result of the paper is that the nonperturbative model [Zank et al., 1998] yields the best results. Its success is determined by the following: (1) a perpendicular correlation length derived from two‐dimensional solar wind turbulence that is much larger than the parallel correlation length associated with slab turbulence; (2) a modest radial dependence of the radial diffusion coefficient in the outer heliosphere; and (3) a three interval rigidity dependence of the radial diffusion coefficient in the outer heliosphere with the smallest rigidity dependence in the middle interval and a strong rigidity squared dependence in the other two intervals. The nonperturbative model gives a tentative theoretical basis for the cosmic ray modulation simulations by Moraal et al. [1999], who found empirically that a similar spatial and three stage rigidity dependence for the radial diffusion coefficient is important for reproducing observed anomalous and galactic cosmic ray spectra.

The problem of perpendicular diffusion by a particle in a turbulent plasma is a problem of enduring interest and one that has yet to be fully solved. Analytic models do not agree with either observations or numerical simulations. Recently, a nonlinear guiding center theory (NLGC) was developed byMatthaeus et al.[2003]which, for the first time, appears to be consistent with numerical simulations in both the high‐energy and low‐energy particle regimes, provided that the transverse magnetic field is complex. Flux surfaces with high transverse complexity are characterized by the rapid separation of nearby magnetic field lines, and we show that the combination of slab and two‐dimensional turbulence (a “two‐component” model) is necessary to produce transverse complexity and that slab turbulence alone, for example, is insufficient. The nonlinear theory is expressed through the solution of an integral equation for the perpendicular diffusion coefficient κxx, which we solve approximately. Our approximate solution is in excellent agreement with the exact solution of the integral equation. The physical content of the NLGC theory is revealed clearly by the approximate solution and it is shown how κxxscales with parameters such as the energy density in magnetic fluctuations, mean field strength, particle gyroradius, MHD turbulence correlation length scales, parallel diffusion coefficient, etc. Unlike the integral equation formulation, which is not readily amenable to inclusion in models and numerical codes that require the perpendicular diffusion coefficient explicitly, the approximate model derived here is easily incorporated into, for example, heliospheric cosmic ray modulation models. Finally, the perpendicular diffusion coefficient is used to evaluate (1) the particle acceleration timescale for diffusive shock acceleration at perpendicular shocks and (2) the diffusion coefficient for cosmic ray modulation throughout the heliosphere.

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.

We investigate cosmic ray scattering in the direction perpendicular to a mean magnetic field. Unlike in previous articles we employ a general form of the turbulence wave spectrum with arbitrary behavior in the energy range. By employing an improved version of the nonlinear guiding center theory we compute analytically the perpendicular mean free path. As shown, the energy range spectral index, has a strong influence on the perpendicular diffusion coefficient. If this parameter is larger than one we find for some cases a perpendicular diffusion coefficient that is independent of the parallel mean free path and particle energy. Two applications are considered, namely transport of Galactic protons in the solar system and diffusive particle acceleration at highly perpendicular interplanetary shock waves.

Cosmic Ray transport in curved background magnetic fields is investigated using numerical Monte-Carlo simulation techniques. Special emphasis is laid on the Solar system, where the curvature of the magnetic field can be described in terms of the Parker spiral. Using such geometries, parallel and perpendicular diffusion coefficients have to be re-defined using the arc length of the field lines as the parallel displacement and the distance between field lines as the perpendicular displacement. Furthermore, the turbulent magnetic field is incorporated using a WKB approach for the field strength. Using a test-particle simulation, the diffusion coefficients are then calculated by averaging over a large number of particles starting at the same radial distance from the Sun and over a large number of turbulence realizations, thus enabling one to infer the effects due to the curvature of the magnetic fields and associated drift motions.

On the basis of a recently developed nonlinear guiding center theory for the perpendicular spatial diffusion coefficient κ⊥ used to describe the transport of energetic particles, we construct a model for diffusive particle acceleration at highly perpendicular shocks, i.e., shocks whose upstream magnetic field is almost orthogonal to the shock normal. We use κ⊥ to investigate energetic particle anisotropy and injection energy at shocks of all obliquities, finding that at 1 AU, for example, parallel and perpendicular shocks can inject protons with equal facility. It is only at highly perpendicular shocks that very high injection energies are necessary. Similar results hold for the termination shock. Furthermore, the inclusion of self‐consistent wave excitation at quasiparallel shocks in evaluating the particle acceleration timescale ensures that it is significantly smaller than that for highly perpendicular shocks at low to intermediate energies and comparable at high energies. Thus higher proton energies are achieved at quasiparallel rather than highly perpendicular interplanetary shocks within 1 AU. However, both injection energy and the acceleration timescale at highly perpendicular shocks are sensitive to assumptions about the ratio of the two‐dimensional (2‐D) correlation length scale to the slab correlation length scale λ2D/λ∥. Model proton spectra and intensity profiles accelerated by a highly perpendicular interplanetary shock are compared to an identical but parallel interplanetary shock, revealing important distinctions. Finally, we present observations of highly perpendicular interplanetary shocks that show that the absence of upstream wave activity does not inhibit particle acceleration at a perpendicular shock. The accelerated particle distributions closely resemble those expected of diffusive shock acceleration, and observed at oblique shocks, an example of which is shown.

Observations by the TRACE spacecraft have shown that coronal emission in the extreme ultraviolet is characterized by filamentary structures within coronal loops, with transverse sizes close to the instrumental resolution. Starting from the observed filament widths and using the concepts of braided magnetic fields, an estimate of the turbulence level in the coronal loops can be obtained. Magnetic turbulence in the presence of a background magnetic field can be strongly anisotropic, and such anisotropy influences the separation of magnetic field lines, as well as the magnetic field line diffusion coefficient. Careful computations of the magnetic field line diffusion coefficient Dm and of the rate of exponential separation of magnetic field lines h, also allowing for the possibility of anisotropic magnetic turbulence, enable computation of the effective perpendicular diffusion coefficient for electrons. When compared with observations this yields magnetic turbulence levels on the order of δB/B0 = 0.05–0.7, which are larger than previous estimates. These values of the magnetic fluctuation level support the idea that magnetic turbulence can contribute to coronal heating by means of MHD turbulence dissipation. It is also found that field line transport is not governed by the quasilinear regime, but by a nonlinear regime which includes an intermediate and the percolation regimes.

A numerical study of chaotic field line diffusion in a tokamak with an ergodic magnetic limiter is described. The equilibrium model field is analytically obtained by solving a Grad–Schlüter–Shafranov equation in toroidal polar coordinates, and the limiter field is determined by supposing its action as a sequence of delta-function pulses. A symplectic twist mapping is introduced to analyze the mean square radial deviation of a bunch of field lines in a predominantly chaotic region. The formation of a stochastic layer and field diffusivity at the plasma edge are investigated. Field line transport is initially subdiffusive and becomes superdiffusive after a few iterations. The field lines are lost when they collide with the tokamak inner wall; their decay rate is exponential with Poisson statistics.

Finite gyroradius corrections in the theory of perpendicular diffusion 2. Strong velocity diffusion

Charged energetic particles propagating in solar wind magnetic fields span field irregularities down to very short turbulent scales, not described by the original quasi-linear theory for weak magnetic turbulence. This theory only predicts a field line diffusion on the largest scales, well above the correlation length, inverse of the spectral flattening wavenumber. The quasi-linear prediction for the transport and behavior of magnetic field lines is generalized here to all scales and arbitrary three-dimensional turbulence spectra. New analytical expressions are derived for the field line mean square cross-field displacement Δx2, and analytical proof is presented for the anomalous transport of the field lines. We find Δx2 ∝ (Δz)β, where Δz is the elapsed distance along the average field and β, the transport exponent, can take any value between 0 and 2. A decreasing turbulence spectrum results in a field line supradiffusion (β > 1), while an inverted spectrum implies a subdiffusion (β < 1). Simple expressions are derived for the transport exponent and coefficient. A powerful new method is presented to compute magnetic field lines in the quasi-linear regime of turbulence that allows rapid computation of field lines generated from any three-dimensional turbulence spectrum, including some 1015 modes and more. Individual field lines computed with this method show how a spectral steepening results in a smoothing of the field lines and how harder spectra give increasingly more short-scale fluctuations. The field line self-similarity, characteristic of power-law spectra, is demonstrated visually, and the anomalous transport of the field lines is confirmed numerically.

Over the past two decades scientists have significantly improved our understanding of the transport of energetic particles across a mean magnetic field. Due to test-particle simulations, as well as powerful nonlinear analytical tools, our understanding of this type of transport is almost complete. However, previously developed nonlinear analytical theories do not always agree perfectly with simulations. Therefore, a correction factor a 2 was incorporated into such theories with the aim to balance out inaccuracies. In this paper a new analytical theory for perpendicular transport is presented. This theory contains the previously developed unified nonlinear transport theory, the most advanced theory to date, in the limit of small Kubo number turbulence. New results have been obtained for two-dimensional turbulence. In this case, the new theory describes perpendicular diffusion as a process that is sub-diffusive while particles follow magnetic field lines. Diffusion is restored as soon as the turbulence transverse complexity becomes important. For long parallel mean-free paths, one finds that the perpendicular diffusion coefficient is a reduced field line random walk limit. For short parallel mean-free paths, on the other hand, one gets a hybrid diffusion coefficient that is a mixture of collisionless Rechester &amp; Rosenbluth and fluid limits. Overall, the new analytical theory developed in the current paper is in agreement with heuristic arguments. Furthermore, the new theory agrees almost perfectly with previously performed test-particle simulations without the need of the aforementioned correction factor a 2 or any other free parameter.

Analytic forms of the cosmic ray perpendicular diffusion coefficient with implicit contribution of slab modes

The existing solution for the Langevin equation of an anisotropic fluid allowed the evaluation of the position-dependent perpendicular and parallel diffusion coefficients, using molecular dynamics data. However, the time scale of the Langevin dynamics and molecular dynamics are different and an ansatz for the persistence probability relaxation time was needed. Here we show how the solution for the average persistence probability obtained from the backward Smoluchowski-Fokker-Planck equation (SE), associated to the Langevin dynamics, scales with the corresponding molecular dynamics quantity. Our SE perpendicular persistence time is evaluated in terms of simple integrals over the equilibrium local density. When properly scaled by the perpendicular diffusion coefficient, it gives a good match with that obtained from molecular dynamics.

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.

Ulysses observations of Jovian relativistic electrons in the interplanetary space near Jupiter: Determination of perpendicular particle transport coefficients and their energy dependence