Fig. 2.31.1. Transport exponent a as a function of the spectral index a for ^Az = 10 . From Ragot (2001c).
The spreading of magnetic field lines in the quasi-linear regime of turbulence is only linear for a flat spectrum. For all decreasing power laws (a > 0) the field lines supra-diffuse (a > 1), whereas for inverted power laws the field lines sub-diffuse (a < 1) as long as a > — 2. For a < — 0.5, a is averaged over a broad range of Az. From Fig. 2.31.1 it can be seen that the transport of the magnetic field lines is supra-diffusive (a > 1) for any positive spectral index of turbulence and sub-diffusive (a < 1) for any inverted spectrum, which confirms the results of Ragot (1999a,b). As the spectral index a approaches 1 from below, the term in — (Az)2/(2 — 2a) has a growing weight in Eq. 2.31.11, owing to the factor (2 — 2a)—1. It is dominant for a > 1, therefore its sum with (Az )1+a r(— 1 — a )sin(an/ 2), which is then negative, remains always positive. In the limit of very small ^Az the other term becomes completely negligible and a
converges to 2 as soon as a > 1. For k\Az = 10 as in Fig. 2.31.1, the first term still has a significant weight up to a = 1.5 — 2.0, but for all spectral indexes steeper than 2 the transport exponent a is practically equal to 2.
For \a\ < 0.5 - 0.8, ^Axis accurately determined by the first term. In this range of spectral indexes the transport exponent simply reduces to a = 1 + a. This case is of particular interest since it corresponds to a spectrum that would tend to flatten at low frequency, but not perfectly, as is observed for instance in the solar wind (Goldstein et al., 1995).
Ragot (2001c) note that Px // (k )kf = Px // (k// )k// for any k // in the interval
[ k1, k2] so that the value of Dma does not depend on the lower limit k1 of the interval on which Px // is in k/—a but solely on the level of turbulence in this interval of parallel wave-numbers. The range of validity in a for a = 1 + a, however, does depend on the value of ^Az . If k^Az ^ 0 , it extends from —1 to 1.
The condition established by Ragot (1999a) to observe, in the quasi-linear regime of turbulence, a diffusive spreading of the field lines on at least one decade is confirmed; namely, the spectrum should be flat on at least three decades around
1 Az (2 decades for k;—1 << Az << k—1 plus 1 decade for the variation of Az). This means that even a flattening at 10—5 Hz in the solar wind would not have guaranteed a diffusive spreading of the field lines on a scale of length shorter than the typical distance between strong inhomogeneities, since the sun rotates at a frequency of
(3.2 — 4.6)x 10—7 Hz less than 100 times smaller. This conclusion of no quasi-linear diffusion of magnetic field lines in the inner Heliosphere is not contradicted by the observation of fluctuating field line directions and could account for the lack of mixing of charged particles propagating through the turbulent solar wind (Zurbuchen et al., 2000) recently observed with the SWICS instrument on ACE. A supra-diffusion is indeed characterized by a lower dispersion and ordered fields on many scales, as is considered below in Section 2.31.5.
2.31.5. Comparison with the original quasi-linear prediction
A quantitative comparison of Ragot (2 J J 1c) prediction for the magnetic field line spreading with the prediction of the original quasi-linear theory now is relatively straightforward. For a level of turbulence to be the same at the lower wave-number k1, the ratio of the two predicted variances can be written as
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