Fig. 2.21.1. Logarithm of the non-resonant mean free path Anr for AkF = kc. The contribution from each 'decade' of the spectrum is given in dashed line starting, in the top of the figure, with the wavenumber interval 10-6kc to io-5kc. According to Ragot (1999a).

Fig. 2.21.1. Logarithm of the non-resonant mean free path Anr for AkF = kc. The contribution from each 'decade' of the spectrum is given in dashed line starting, in the top of the figure, with the wavenumber interval 10-6kc to io-5kc. According to Ragot (1999a).

2.21.3. Resulting mean free path and comparison with gyro-resonant model

By the comparison of the found mean-free-path Anr with the one resulting from gyro-resonance with slab Alfven waves (see Fig. 2.21.2), Ragot (1999a) shows that the non-resonant scattering process becomes much more efficient than the gyro-

IS (rigidity/MV)

Fig. 2.21.2. Mean free path X, for Akp = kc/100 and Aka <kc. Continuous line: X ; thick dashed line: effective X. Above a few 100 MV, X results from gyro-resonance at ¡< va/v with Alfvén waves. Below 1 MV, for electrons, it is determined by TTD at ¡> va/v. From Ragot (1999a).

Further comparison with the mean-free path derived from the smaller pitch-angles indicates that the slowest scattering process does not occur around ^ = 0 any longer for rigidities less than ~ 1 MV. This again results from the upper steepening of the spectrum. Indeed, when the particles' rigidity is really low, gyro-resonance also becomes impossible around |u| = 1. As the transit-time damping interaction, which is owed to the compressive component of the magnetic field - along Bo - is very inefficient at these large it produces a relatively large mean-free-path.

Note that this mean-free-path is constant below ~ 0.1 MV. Ragot (1999a) notes also that the developed theory can reproduce the main features of the parallel mean free path as a function of the particles rigidity - for electrons, down to 1 keV, and for protons, down to 1 or 10 MV. The flatness of the curve between 1 MV and 103 MV seems to require a steepening of the fast mode component of the turbulence spectrum above about 10-2kc. This is, in the presence of low-energy CR, plausible, given that the fast magneto-sonic waves in the range [i0-2 kc, kc ] give the main contribution to the transit-time damping acceleration process.

2.21.4. Contribution from slab and oblique Alfvén waves to the non-resonant pitch-angle scattering

In Ragot (1999c) was shown that the slab Alfvén component of the spectrum, for reasons of symmetry, does not contribute to the non-resonant scattering and also oblique Alfvén waves do not produce any significant scattering by the non-resonant scattering process. In a turbulence of slab Alfvén waves the fluctuating fields consist of transversal left- and right-hand polarized waves propagating parallel and anti-parallel to the homogeneous magnetic field Bo . The polarization of the waves is circular. It follows that the integral (over k) describing the variations of ^ has an oscillatory integrand in exp(± i(y/k + p)), where y/k = kx - mkt + ak and 9 is the gyro-

phase of the particle. As a consequence, averaging over the particle gyro-period -the shortest timescale of the problem, see Ragot (1999a,b) - will just reduce the ^ variation to a negligible contribution. The case of oblique Alfvén waves is close to the one of the oblique fast magneto-sonic waves presented by Ragot (1999a,b). When k is not along the main magnetic field Bo, the Alfvén waves are linearly polarized waves with, if the inertia of the electrons is neglected, an electric field SE normal to Bo, in the plane of k and Bo . The magnetic field SB is normal to SE and

Bo . The different configuration of the magnetic field perturbation results here in an equation for the pitch-angle cosine ^ of a form similar to Eq. 2.21.7, but where cos^k cos(p-pk ) substitutes for cos^k sin(p-pk ), pp being the angle between k and the plane (x, z), with a z -axis along Bo . The averaging of this equation over the particle gyro-period will not permit to extract any constant term of significant amplitude, on the expected timescale of pitch-angle variation. Indeed, an expansion of cos^k sin(p-pk) in Bessel functions only displays oscillatory terms in cos( -pk) or sin( - <pk) with n a strictly positive integer. This, according to Ragot (1999c), shows that the contribution from oblique AlfVen waves to the non-resonant pitch-angle scattering is also negligible.

2.21.5. Parallel mean free path: comparison of the theoretical predictions with the measurements

If the Alfven waves, as argued in the Section 2.21.4, do not produce any significant contribution to the pitch-angle scattering by the non-resonant effect, it means that the result obtained by Ragot (1999a,b) might already provide with a reasonable description of the CR scattering in the solar wind. Ragot (1999c) tried to compare her theoretical predictions with the measurements. The sensitivity of the mean free path to the characteristics of the fast magneto-sonic waves spectrum (in particular, spectral index and cutoff wave number) and the fact that the data obtained from different solar events are often presented together, without reference to the distinct events, makes this comparison difficult (see Fig. 2.21.3).

Fig. 2.21.3. Parallel mean free path versus particle rigidity, in logarithm, for various solar events. The dots represent parallel mean free paths derived above 10 MV from proton observations, and below from electron observations, as published by Bieber et al. (1994). The theoretical curve, in the dashed line, has been obtained with cutoff of the Alfven and fast magneto-sonic waves spectra at 0.4kc and 0.003kc, respectively; 5ba = 0.1, 5bF = 0.13, and va = 10-4c . The extension of the plateau at very low rigidities is directly related to the cut-off wavenumber of the Alfven spectrum. This cut-off value is observed in the solar wind at about kc. A value of 0.4kc to produce the best fit presented here is reasonable, since the precise characteristics of the turbulence spectra might vary during a solar event from those of the 'quiet' solar wind. From Ragot (1999c).

Fig. 2.21.3. Parallel mean free path versus particle rigidity, in logarithm, for various solar events. The dots represent parallel mean free paths derived above 10 MV from proton observations, and below from electron observations, as published by Bieber et al. (1994). The theoretical curve, in the dashed line, has been obtained with cutoff of the Alfven and fast magneto-sonic waves spectra at 0.4kc and 0.003kc, respectively; 5ba = 0.1, 5bF = 0.13, and va = 10-4c . The extension of the plateau at very low rigidities is directly related to the cut-off wavenumber of the Alfven spectrum. This cut-off value is observed in the solar wind at about kc. A value of 0.4kc to produce the best fit presented here is reasonable, since the precise characteristics of the turbulence spectra might vary during a solar event from those of the 'quiet' solar wind. From Ragot (1999c).

From Fig. 2.21.3 it can be seen that the theoretical curve globally fits the data points. The dispersion of the points around the theoretical curve presented on Fig.

2.21.3 should not be interpreted as uncertainty of the measurements, or inappropriateness of the theory to fit all the data. The data shown on Fig. 2.21.3 have been obtained from many different solar events. Their dispersion only indicates that the turbulence spectrum in the solar wind varies from one event to another. Ragot (1999c) has studied how the theoretical prediction is modified by variations of the turbulence spectrum, both fast magneto-sonic and Alfven component. There was found a rather strong sensitivity of the theoretical prediction on the precise shape of the spectra. Even if the main features of the curve in Fig. 2.21.3 are preserved (e.g., separation in three domains where the transit-time damping, non-resonant and gyro-resonant interactions successively determine the parallel mean free path), it is always possible to find a curve which will fit one subset of data points, keeping reasonable turbulence spectra. Ragot (1999c) fits in Fig. 2.21.4 one particular event, namely Nov 22, 1977, which looks very similar to Dec 27, 1977, and Apr 11, 1978 (see Beeck et al., 1987; Valdes-Galicia et al., 1988; Dröge et al., 1993). All the measurements, for this particular event, appear to be in the range where the non-resonant interaction with the fast magneto-sonic waves dominates. It would be necessary, in order to validate the theory and obtain the whole information on the turbulence spectra, to have data for single events over a broader range of rigidities, spanning the intervals where the transit-time damping

(below 1 MV) and gyro-resonant (above parallel mean free path.

103 MV) interactions determine the

The circles represent measurements for electrons, and the disks for protons. The theoretical curve remains valid on the whole range of rigidities for electrons. It only holds above about 10 MV for protons, but all the data for protons are obtained above 20 MV, so the theory is consistent with the observations presented in this figure and Fig. 2.21.3. The measurements for this particular event appear to be in the range where the non-resonant interaction with the fast magneto-sonic waves dominates. The theoretical curve in thick dashed line is calculated with an Alfven spectrum of Kolmogorov type up to kc, and a fast mode wave spectrum damped above 3.2x10-3kc, with a spectral index of 1.35 below. The continuous line plots the mean free path resulting from the non-resonant interaction alone, assuming that the slowest scattering process occurs at small p. From Ragot (1999c).

2.22. On the cosmic ray cross-field diffusion in the presence of highly perturbed magnetic fields

According to Michalek and Ostrowski (1999), the investigation of CR transport in highly perturbed magnetic fields raises a number of issues which are poorly understood. In particular, an analytic theory enabling derivation of particle diffusion across the magnetic field is still not available. The quantitative analytical derivations of the cross-field diffusion coefficient k1 in turbulent magnetic fields are limited to small perturbation amplitudes, SB << Bo (e.g. Jokipii, 1971; Achterberg and Ball, 1994). A significant result in this respect was achieved by Giacalone and Jokipii (1994), Jones et al. (1998). They provided a proof that the cross-field diffusion requires a three-dimensional nature of the turbulent field. A process of particle cross-field diffusion in high amplitude Alfvenic turbulence is considered in Michalek and Ostrowski (1997, 1998, 1999) using the Monte Carlo particle simulations. They derive the cross-field diffusion coefficient k1 in the presence of different 1-D, 2-D and 3-D turbulent wave field models. Vanishing of k1 in 1-D turbulence models is used as an accuracy check for the numerical computations. They found substantial differences in the cross-field diffusion efficiency at the same perturbation amplitude, depending on the detailed form of the turbulent field considered. Michalek and Ostrowski (1997, 1998, 1999) reproduced the expected increase of k1 with the growing power of waves propagating perpendicular to Bo. Substantially larger values of k1 appear in the presence of long compressive fast-mode waves in comparison with the Alfven waves. This result was interpreted in terms of particle drifts in non-uniform magnetic fields. In some cases an initial regime of sub-diffusive transport appears in the simulations.

2.22.2. Description of Monte Carlo particle simulations

Michalek and Ostrowski (1997, 1998, 1999) considered an infinite region of tenuous plasma with a uniform mean magnetic field along the z-axis. It is perturbed by propagating MHD waves (described below in Section 2.22.3 for different turbulence models). Test particles are injected at random positions into this turbulent magnetized plasma and their trajectories are followed by integrating particle equations of motion in space and momentum. By averaging over a large number of trajectories one derives the required diffusion coefficients for turbulent wave fields. In the simulations 500 relativistic particles were used with the same initial velocity vin = 0.99c in an individual run.

For high amplitude waves there are no analytic models available reproducing the turbulent field structure. Because of that, in Michalek and Ostrowski (1997, 1998, 1999) approximate models representing such fields are considered, with turbulence represented as a superposition of Alfven or fast-mode waves. The wave parameters (wave vectors k, wave amplitudes SB0 and initial phases <) are drawn in a random manner from the flat ( F(k)« k-1 ) or the Kolmogorov ( F(k) « wave spectra. The wave vectors are expressed in units of the 'resonance' wave vector kres = 2n/ rg ((B), p0 ) (2.22.1)

for the injected particle with momentum p = po in the mean magnetic field

The wave vectors are selected from the range 0.08kres < k < 8.0kres. Integration time is expressed in units of Q0 = eB0 /ymc. The magnetic field fluctuation vector related to the wave 'i', SB(i), is given in the form:

SB(i ) = Sb( ) sin<k<< ¡r - co(i V - <<( ¡). (2.22.3)

In Michalek and Ostrowski (1997, 1998, 1999) are considered the following turbulence models:

(i) Linearly polarized plane waves (model A)

In this model are considered superposition of plane Alfven waves propagating with the same intensity along the z-axis, in the positive (forward) and the negative (backward) direction.

(ii) 'Wave-packets' models (two models B1 and B2)

It was proposed a simple extension of the above model A to three dimensions by considering wave packets, involving wave modulation in one direction perpendicular to the propagation direction by using Eq. 2.22.3 for ¿B(i ), where the phase parameter is subject to sinusoidal modulation. Two types of modulation (presented for the x-components in Eq. 2.22.3) are considered: model B1 with the 'smooth' sinusoidal modulation characterized by

and model B2 with the 'sharp-edged' modulation characterized by

The y-components can be obtained from Eq. 2.22.4 and Eq. 2.22.5 by interchanging x and y. Vectors k() and k() are drawn in a random manner from the respective wave-vector range for k(().

(iii) Oblique MHD waves (four models C-AF, C-AK, C-MF, and C-MK)

There was considered a superposition of plane MHD waves propagating obliquely to the average magnetic Bo = Boez . The wave propagation angle with respect to Bo is randomly chosen from a uniform distribution within a cone ('wave-cone') along the mean field. For a given simulation two symmetric cones are considered centered along Bo, with the opening angle 2a, directed parallel and anti-parallel to the mean field direction. The same number of waves is selected from each cone in order to model the symmetric wave field. For the model (iii) four different turbulent fields were considered characterized with parameters a and SB and labeled as follows:

• Alfven waves with the flat wave spectrum (model C-AF),

• Alfven waves with the Kolmogorov spectrum (model C-AK),

• Fast-mode magneto-sonic waves with the flat spectrum (model C-MF),

• Fast-mode magneto-sonic waves with the Kolmogorov spectrum (model C-MK).

2.22.4. Simulations for Alfvenic turbulence models A, B1, B2

Examples of the derived formal (e.i., the derived particle dispersion squared and divided by the integration time multiplied by two) cross-field diffusion coefficients versus the integration time are presented for the considered Alfvenic turbulence models A, B1, and B2 in Fig. 2.22.1.

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