This ratio is plotted in Fig. 2.31.2 for various values of ^Az .
What strikes at once is that the supra-diffusion (a > 0) seems to give a much slower spreading of the field lines than would be expected for the diffusion of the original quasi-linear theory, whereas the sub-diffusion apparently gives a much faster transport. Whilst this might not be entirely accurate (one chooses a lower turbulence amplitude for larger a by taking the same value of Px// (k1), it serves the purpose of Ragot (2001c) pretty well here. He emphasizes the following. A supradiffusion does not necessarily mean that the transport is faster, nor does a subdiffusion imply a slower transport. This very much depends on the value of the transport coefficient. Supra-diffusion is characterized by a lesser dispersion of the field lines which tend to behave in a more orderly manner. Whilst the small-scale irregularities still exist and might give the impression that the field lines are 'diffusing' in an erratic and uncorrelated way, the large-scale transport is significantly influenced by the lower part of the spectrum and ordered behavior occurs on all scales, even the largest ones. The propagator derived by Ragot and Kirk (1997) illustrates this property with a peaked shape shifted away from zero. For comparison the propagator of diffusion is the well known Gaussian centered around the origin. In the sub-diffusive case (a < 0) the propagator is more widespread and peaks at the origin. The transport is dominated by the small scales and long ordered 'flights' are extremely rare. A greater dispersion might still result, though, from some field lines being able to wander relatively quickly in some part of the space while others (the majority) are trapped in smaller-scale domains on longer length scales.
Ragot (2001c) have shown analytically that in the quasi-linear regime of turbulence the transport of magnetic field lines is anomalous on the length scale Az whenever the projected spectrum of turbulence is not perfectly flat below the parallel wave-number 10/Az. The field line spreading (ax2) varies as (Az)a with a^ 1, 0 < a < 2. A decreasing spectrum results in a supra-diffusion of the field lines
(a > 1), whereas an inverted spectrum implies a sub-diffusion (a < 1). For a spectrum that takes the form of a power-law on an interval of parallel wave-numbers around (Az )-1, there were established new, simple expressions for the transport exponent and coefficient (Eq. 2.31.13-2.31.15). These expressions generalize the quasi-linear prediction for the spreading of magnetic field lines.
2.32. CR transport in the fractal-like medium
In papers Lagutin et al. (2001b,d, 2005), Erlykin et al. (2003), Lagutin and Uchaikin (2003) a model of phenomenological anomalous diffusion, in which the high energy CR propagation in the galactic medium is simulated as fractal walks, has been developed. The anomalous diffusion results from large free paths ('Levy flights') of particles between magnetic domains-traps of the returned type. These paths are distributed according to power law
being an intrinsic property of fractal structures. Here R is the particle's magnetic rigidity. It is also assumed that the particle can spend a long time in the trap. A long time means that the distribution of the particles staying in traps, q(r, R), has a tail of power-law type q(T, R ) B(R,P)z-^~X (2.32.2)
with ¡3 < 1 at t —> ro (Levy trapping time).
Without energy losses and nuclear interactions, the propagator G(r, t, R; Ro), describing such a process, obeys the equation (Lagutin and Tyumentsev, 2004):
dG = — KR,a,0)D1^3(— a)2G(r,t,R;Ro)+ S(r)S()S(( — Ro). (2.32.3) dt
Here, K(R,a,0) is the anomaly diffusion coefficient, Dl+ denotes the Riemann-
Liouville fractional derivative, and (—A)a/2 is the fractional Laplacian, so called 'Riss' operator (see in Samko et al., M1987).
In the case of punctual impulse source of duration T with inverse power spectrum
S (r, t, R) = SoR—pS(r )©(( — t )©(t), (2.32.4)
where ©(() is the Heviside function, CR concentration is n(,t,r^Kfp* i ^aa^iirK^yt^y, (2.32.5)
where the scaling function Y3(a,^)(r)
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