where kr = -Q/vp is the resonance wave number, P(k) is the wave power spectrum, Q is the particle gyro-frequency, and other notations are standard (Gary and Feldman, 1978; Kennel and Engelmann, 1966; Lee, 1971).

Four panels of Fig. 2.30.3 show D versus p for various values of SB. The turbulence is compression, and the wave phases are random. When SB is small, D is doubly peaked, as it vanishes at p = 0, -1, and 1. However, as the turbulence amplitude is increased, the diffusion at p = 0 becomes drastically enhanced. At SB ~ 0.3, D is of the same order with respect to p. This is also apparent in Fig. 2.30.4, in which D is plotted against SB. When 0 < p < 1, numerically computed D matches well with Dql (thick broken line), whilst they start to deviate around SB ~ 0.1.

From the numerical results discussed above, Hada et al. (2003a) are tempted to model the pitch-angle diffusion process by a simple equation, df (l) = D* fl _ f l)- f {-?), (2 30 2) dt di di t(i)

where fu) is the distribution function, D* (m) is the modified pitch-angle diffusion coefficient including the resonance broadening effect (and thus D* (0) ï 0), and t (m) is the time scale for the mirror reflection, which may be determined by statistics of compression magnetic field (one should note, however, that the mirror reflection is not always adiabatic as assumed in Eq. 2.30.2). If there is a finite coherence in the MHD turbulence, as evidenced by recent spacecraft data analysis (Hada et al., 2003b), it strongly influences t(u). which in turn modifies the pitch-angle diffusion. io"-

10"] -102 -103 -ioJ-10 s -10° -Q 10'7 -10 s -10 s-

11 iti|-1—I I 11111|-1—r i i 1111 [-1—i i i 11 ri|-1—i——1

Fig. 2.30.4. D versus SB. According to Hada et al. (2003a).

11 iti|-1—I I 11111|-1—r i i 1111 [-1—i i i 11 ri|-1—i——1

Fig. 2.30.4. D versus SB. According to Hada et al. (2003a).

2.31. Particle diffusion across the magnetic field and the anomalous transport of magnetic field lines

2.31.1. On the anomalous transport of magnetic field lines in the quasilinear regime

The transport of CR particles across the regular component of the magnetic field in space is for a large part induced by the transport of the magnetic field lines themselves (so called compound diffusion, see in Jokipii, 1966; Schlickeiser, 1994). At the shock fronts of supernovae like SN1987A the observed acceleration time of GeV electrons suggests a transport also dominated by the wandering of the magnetic field lines, as the inferred diffusion coefficient of the electrons by far exceeds the Bohm value of this coefficient (Ball and Kirk, 1992; Ragot, 2001a,b). Understanding the behavior of magnetic field lines in a turbulence composed of random fluctuations SB superimposed on a regular magnetic field Bo is thus of prime importance to model the propagation of charged particles in space plasma. The case of small magnetic field perturbation is treated by the quasi-linear theory (Jokipii and Parker, 1968) for weak magnetic turbulence. This theory, which neglects the perpendicular displacement of the field lines in the derivation of their spreading (first order derivation in Sb = SB/Bo ), predicts a diffusion of the field lines beyond the parallel correlation length, Lc //, defined as the characteristic scale of the two-point correlation function. There is a strong belief amongst astrophysicists and physicists in general that, as long as the quasi-linear approximation holds, i.e., as long as the perpendicular displacement can be neglected, the quasi-linear theory does predict a diffusion of the magnetic field lines or, more accurately, their linear spreading across the direction of Bo with the distance Az along Bo . However, this diffusive result is conditioned by the existence of a finite correlation length, Lc //, small enough to consider the transport of the field lines on much longer scales. In the papers by Jokipii and Parker (1968), Jokipii and Coleman (1968), this correlation length was estimated as the inverse of the upper wave-number in the low, flat part of the turbulence spectrum. A power spectrum flat below k = L-1 produces indeed a correlation function of the magnetic field perturbation with an exponential cutoff of characteristic scale Lc . Ragot (2001c) note that yet a flattening of the spectrum at sufficiently high frequency is not guaranteed. For instance, in the solar wind the early observations apparently indicating a flattening at 10-5 Hz, which would have given a quasi-linear correlation just short enough, have not been confirmed by more recent measurements which show power-law spectra down to lower frequencies (Goldstein et al., 1995). In general the presence of such extended, projected spectra, relatively smooth but not flat, is expected for an anisotropic turbulence (e.g., Ragot, 1999a), and as the damping rates of many plasma waves depend on the propagation angle of the waves, anisotropic turbulence is likely to be a quite common feature of plasmas. Clearly, in those cases of extended projected spectra a study of the transport of field lines is still needed even in the quasi-linear regime of magnetic field perturbation, as the spreading of the field lines on any relevant scale will be determined by a part of the spectrum that is not flat, hence neglected in the original quasi-linear theory.

In Ragot (2001c) is introduced the assumption concerning the existence of a short correlation length and express the spreading of the field lines along the axis x normal to the average magnetic field as a function of the projected power spectrum of turbulence. In the case when this projected spectrum can be described as a power law on an interval of wave-numbers around 1/Az, which is generally assumed in any study of turbulence, it can be then establish a new asymptotic expansion for the variance (Ar2). With this expansion it may be analytically proved that whenever the spectral index of the turbulence does not vanish exactly on an interval of wave-numbers at least two or three decades broad around 1/Az, the transport of the field lines is non-diffusive, or anomalous: (Ar2) increases as (Az)a with a different from 1. This confirms the numerical result obtained by Ragot (1999a,b) for similar power-law spectra. Then can be established simple expressions for the transport exponent a, as well as the transport coefficient Da , defined by (Ar2) = Da (Az )a . These expressions are particularly important for a quantitative comparison with the spreading predicted by the original quasi-linear theory.

Ragot (2001c) consider (as in the paper by Ragot, 1999a) a three-dimensional turbulence in quasi-linear regime with a continuous spectrum; hence unlike Pommois et al. (1999), he always keep the length scale Az much shorter than the inverse of the minimum wave-number, which is an absolute requisite to model a continuous spectrum. Below by drawing by Ragot (2001c) the main lines of the classical quasi-linear derivation.

2.31.2. Quasi-linear theory for magnetic lines diffusion

In the quasi-linear approximation, i.e., if the perpendicular deviation is neglected, the displacement along the axis r of the field line that goes through the point ro = (ro, yo, zo) can be written as

zo where b stands for SB/B0 , and the variance (Ax2\ can be expressed as:

^Ax J= jdz' \dz"(bx(xa,y0,z')bx(xa,y0,z,,) = 2Az J ds|^1-—Rx*(s(2.31.2)

where Az = z - z0 . The brackets ( ) denote an average over a statistical ensemble of systems and Rxx(s) = (bx(x0,y 0,z')bx(x0,y0,z") stands for the two-point correlation function of the magnetic field along x. In the usual quasi-linear theory Rxx is assumed to cut off on the length scale Lc //, known as the parallel correlation length, and the limit Az >> Lc // is taken so that

2Az J xx

It shows that the magnetic field lines diffuse with the diffusion coefficient D on length scales much longer than Lc //. However, it does not prove that Lc // exists and is very much smaller than the size of the system, which happens to be necessary to observe a diffusion in the system. In the following, Ragot (2JJ1c) introduce the assumption concerning the existence of a finite correlation length and derive a general expression for the spreading of magnetic field lines in the quasi-linear regime of turbulence.

2.31.3. Quasi-linear spreading of magnetic field lines

If km and kM denote the lowest and highest wave-numbers in the spectrum, the spreading of the field lines:

Ax2^ = 2km Jdz' Jdz" Jdkb2 (k)cos[[(z'-z")], (2.31.4)

- = Jdk± J dk//bx (k)cos(k±- r + k//z + fa), (2.31.5)

where bx(k)exp(ifa) is the Fourier transform of bx(r) with bx(k)> J. The derivation of Eq. 2.31.4 assumes, as in the quasi-linear theory, that the phases fa decorrelate on the scale km but this assumption of no spectral structuring could of course be given up by introducing a different phase-correlation scale and o -o substituting for the factor km . Integrating now over z' and z'', we obtain in the quasi-linear regime of magnetic field perturbation:

where

Px // (k// )= J d0 J dk±k±b2x (k//, k± ,0) (2.31.7)

is the x-component of the power spectrum projected along Bo, and kmin (k// )= max((0,km - k/2/ j1/2; k c(k// )=(kM- k/2

When the spectrum is smooth enough to be represented as a series of power laws, the right-hand side of Eq. 2.31.6 can be integrated over the parallel wave-numbers to obtain an explicit form of the field lines spreading. For a power-law spectrum

from k1 to it was found for the quasi-linear regime:

Ax2) = 4kmPx//(k1 )kf1 {^ + |k1Az|1+a r(-1 - a)sinf

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