The Fokker-Planck Coefficient for Pitch-Angle Scattering of Cosmic Rays

Abstract

most cases has been evaluated incorrectly in the past. For example, there is no formal justification for the approximations due to Jokipii (1966, 1971), which permit this integral to be expressed simply in terms of a one-dimensional power spectrum of magnetic field fluctuations, evaluated at a resonant wavenumber. By using both analytic and numerical techniques, we correctly evaluate <(AP2) >/At for a general form of the power spectrum, which should provide a good fit to observed power spectra. In the limit of low rigidity, where the ratio of the correlation length for field fluctuations to the particle's gyro-radius is large,,the correct result for <(Ap) 2 >/At differs from that due to Jokipii (1966) by a factor of p (for p 0).At all but very small values of p, this correction will introduce only a small numerical change in predicted values of <(AP) 2 >/At. However, in certain schemes for averaging <(Ap) 2 >/At over p, and for, forming the diffusion coefficient for propagation parallel to the mean field (Jokipii, 1966(Jokipii, , 1971Earl, 1973), this extra factor of p introduces strong divergences as P-0.These divergences no doubt result from the inadequacies near p=0 of the quasi-linear and adiabatic approximations. We also show that quasi-linear/adiabatic theory does not in general predict that <(Ap) 2 >/At =0 at p =0 (e.g. Jokipii, 1966(e.g. Jokipii, , 1971 or that <(Ap) 2 >/At diverges continuously to infinity as p-*0 (Hasselmann and Wibberenz, 1968). Rather, this theory predicts for physically realistic power spectra that < (Ap) 2 >/At contains a delta-function, 6 (p). Finally, we discuss how the above corrections affect our ability to determine an accurate diffusion coefficient for propagation parallel to the mean

Nonresonant Interaction of Charged Energetic Particles With Low-Frequency Noncompressive Turbulence: Numerical Simulation

“…More recently, that picture has been put into question by our finding that for electrons, PA scattering is actually fastest across μ = 0 (where μ is a gyration-averaged PA cosine μ or μ av ; Ragot 1999Ragot , 2005see Figures 3-5 of Ragot 2006a) and does not obey a diffusive law. This is consistent with the earlier finding that the Fokker-Planck PA scattering coefficient, calculated for a finite gyroradius, contains a Dirac δ function in μ (Fisk et al 1974;Klimas & Sandri 1973), further confirmed by Goldstein et al (1975) and related to mirroring. The extent of the nondiffusive time variations of μ, however, is much broader than ever anticipated.…”

Turbulence, Magnetic Reconnection in Turbulent Fluids and Energetic Particle Acceleration

“…In the recent paper of Yan and Lazarian (2008, henceforth YL08), a nonlinear theory (NLT) based on the Völk (1975) suggestion was developed. The new formalism was applied to two major processes of CR acceleration in turbulence, namely, to gyroresonance and to the transient time damping (TTD) Fisk et al (1974); Goldstein et al (1975).…”

Stationarity of magnetohydrodynamic fluctuations in the solar wind

“…Introduction in many areas of 5nlar wind research it is common to assume, explicitly or implicitly, that the medium being studied is either tin y stationary or spatially homogeneous (or both). For example, the thoories of pitch angle scattering of charged particles (Jokipii, 19603;Ftall anO Sturrock, 1967;Hasselmann and Wibberenz, 1968;Klimas and Sandri, 1971;,zones, Kaiser and Birmingham, 1973;Fisk et al, 1974] and field line random walk (dokipii, 1966;J'okipii andParker, 1968, 1969;jokipii, 1971) both utilize the concept of an ensemble average of the interplanetary mag netic fields. Such averages are meaningful only if a particular realization of the magnetic field time series does, in fact, represent a realization of a stationar y process.…”