Eq. 2.31.11 shows that the spreading of the field lines is not linear unless a = 0. Moreover, since r(-1 - a)sin(an/2)^n/2 as a ^ 0, the usual quasi-linear diffusion coefficient Pr // (1) is recovered in this limit of a flat spectrum. For a finite upper wave-number k2 one has to subtract
4k3Pr //(k1 Jdk//[1 - cos(k//z)]-2-a from the right-hand side of Eq. 2.31.10,
which can be estimated in a way similar to that as the integral from k1 to However, the first term resulting from the integration of k/-2-a is small compared to the part in k1 as soon as (1/k2 )1+a << 1. As for the other term, it is negligible for ¿2Az >> 1 and a > -1 because of the Riemann-Lebesgue lemma (Bender and
-2-a 1 1 Orszag, M1978), since Jdk//|k//| a exists. Consequently if k2 <<Az << kf k2
and a >-1, the Eq. 2.31.10 and 2.31.11 still apply for a power-law spectrum on a finite interval [ k1, k2 ].
2.31.4. The transport exponent and transport coefficient for magnetic field lines
In accordance with Ragot (2001c) the transport exponent a and transport coefficient Dma, defined by
can be expressed in analytical form on the basis of results in Section 2.31.3. From a =
d (log( Ax2^ j Jd(log Az) there follows for spectral indexes a >-1 or a >-0.5 (depending on how small ^Az is)
A1 = r(-1 - a )sin(an/2), A2 = -V(2 - 2a). (2.31.14)
In the limit of small |a|, namely, |a| < 0.5 for ^Az ~ 10-1 and |a| < 1 for ^Az ^ 0, it gives
whereas in the limit of larger a | (>2 for ^Az = 10-1 and \a\ > 1 for k1Az ^ 0),
where coefficients A and A2 are determined by Eq. 2.31.14. Fig. 2.31.1 shows the transport exponent a as a function of the spectral index a.
Was this article helpful?