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Fig. 2.28.4. Comparison between model (dashed) and observations, see text. According to Kallenrode (2001a).

In Fig. 2.28.4 the passage of the magnetic cloud is marked by a filled rectangle; interplanetary field lines draped around the cloud are represented by the adjacent open rectangles. These latter field lines are the ones which can be approximated in this model while the field lines inside the cloud are not considered. For modeling, this event is a challenge in so far as it shows a rather strong increase in intensity towards the shock combined with a drop in intensity short before the arrival of the magnetic cloud. This is impossible to model in a simple transport model because particles cannot be removed fast enough to get a significant decrease in intensity. if the magnetic cloud is considered, however, not only intensities upstream of the cloud can be fitted but also the fast decrease of intensity associated with the arrival of the field lines draped around the cloud and the reduced intensity in the cloud's downstream medium can be described properly. The description fails, by definition, right inside the cloud since the model only gives intensities along the field lines draped around the cloud but not inside the cloud; the satellite, on the other hand, cuts right through the cloud. Kallenrode (2001a) came to following conclusions:

(1) If the cloud follows the particle source, the upstream intensity is increased by a few percent for 10 MeV protons under average scattering conditions (X = 0.1 AU);

(2) This increase increases with decreasing energy and increasing scattering;

(3) The downstream intensities are reduced by about an order of magnitude;

(4) if the cloud is ahead of the particle source, it is an effective barrier for particle propagation;

(5) The model allows the fitting of observations, although by definition intensities and anisotropies inside the cloud are not described correctly.

All these properties can be understood from the modified focusing: viewed from the outside, the bottleneck configuration shows a converging field and thus reflects part of the particles. Consequently, the cloud is a barrier that separates the upstream and downstream medium and allows for markedly different intensities in both of them. intensities are higher on that side of the cloud where the source is located (upstream in case of a traveling shock, downstream in case of a magnetic cloud from an earlier event). At the bottleneck, intensities are reduced because only the relatively small number of particles just in transit can propagate through. In addition, anisotropies are relatively high because only particles with small pitch-angle can propagate into the bottleneck. Changes in intensity and anisotropy related to the presence of the cloud increase with decreasing energy and mean free path because in these cases particles stay longer in the vicinity of the cloud and thus can perform multiple interactions. For weak scattering and high energies, on the other hand, once a particle has passed the cloud it has only a small return probability. The enhancement of the barrier function of the cloud with increasing scattering also had been proposed by Lario et al. (1999). A relatively unexpected effect was the strong barrier action of a cloud ahead of the particle source. Since SOHO observations show a large number of CMEs during solar maximum (about

2 per day, see in St.Cyr et al., 2000), magnetic clouds in interplanetary space ahead of a particle source might be a relatively common feature. Fits of a transport equation on particle events neglecting the influence of a magnetic cloud might be faulty. This might explain part of the discrepancy between particle mean free paths determined from fits and particle mean free paths determined from the analysis of magnetic field fluctuations (Wanner and Wibberenz, 1993). In addition, the barrier properties of the magnetic cloud as demonstrated in Fig. 2.28.3 can be used to simulate rogue events where converging shocks lead to unusual high particle intensities as described for the August 1972 event by Levy et al. (1976). First examples are described in Kallenrode and Cliver (2001).

2.29. Non-diffusive CR particle pulse transport

2.29.1. The matter of the problem

In Fedorov et al. (2002) there are developed a theory of the transport of an anisotropic pulse of CR charged particles injected into moved space plasma with frozen in magnetic field (with applications to the anisotropic ground level solar CR events). For these events the kinetic regime is considered when the mean free path is comparable with the distance from particle source to detector. The problem is that in many cases the ground-level neutron monitors network detects complicated temporal solar CR intensity profiles, when the profile starts with narrow peak of 'direct particles' with a following diffusion tail of many times scattered particles (Fisk and Axford, 1969; Lupton and Stone, 1973; see reviews in Dorman and Miroshnichenko, M1968; Dorman, M1978; Dorman and Venkatesan, 1993, Miroshnichenko, M2001). In these cases a strong anisotropic pitch-angle distribution of particles in the interplanetary magnetic field is observed which implies a need to consider non-diffusive particle transport (Earl, 1994; Fedorov and Shakhov, 1994; Fedorov et al., 2002), because the mean free path determined by the collision integral is comparable with the distance from the source to the detector. Some of the ground level events are characterized by an impulse peak, which has been observed by the ground-based neutron monitors at Kerguelen and Apatity during the solar proton event on 7-8 December 1982, and at Deep River and Apatity during the event of 16 February 1984 (Borovkov et al., 1987; Smart et al., 1987; Smart and Shea, 1990; Perez-Pereza et al., 1992). Unlike these events, during the event on 22 October 1989 neutron monitors at the South Pole and Calgary have registered a short intensive peak after which a basic enhancement followed (Bieber et al., 1990; Fl├╝ckiger and Kobel, 1993). A similar event, which was observed on 24 May 1990, is also characterized by a strong anisotropy of particle angular distribution as well as by very complicated temporal structure (Morishita et al., 1995; Torsti et al., 1996; Debrunner et al., 1992, 1997). Fedorov et al. (2002) attempt to give simplified model of these events on the basis of the kinetic theory approach of Fedorov and Shakhov (1994), Fedorov and Stehlik (1997), Fedorov et al. (1995). Their solution includes both the angle distribution of injected particles and the angular response function of NM as well as a finite time of the particle injection in the source.

2.29.2. Kinetic equation

According to Fedorov et al. (2002), in the theoretical consideration the regular IMF is taken to be homogeneous. Nevertheless, the formation of an initial angle distribution of particles into narrow stream along the regular IMF (Lumme et al.. 1986) owed to the magnetic focusing of force-lines near the Sun is included, see below. Thus, evolution of the particle distribution function f (y,r,M) follows from the kinetic equation written in the drift approximation, in which the particle scattering on stable magnetic inhomogeneities is supposed to be isotropic (Fedorov et al., 1995):

1 1 v dTf + Mdyf + f -- J fdv = ~sS(y WW), (2.29.1)

2 -1 v where y and t are the coordinate along regular IMF and the time, respectively (in the dimensionless units y = zvs/v, T=vst; vs is the collision frequency of particles with the magnetic clouds; z is the coordinate along the IMF), p = cos 6, 6 is the particle pitch-angle. The right-hand side of Eq. 2.29.1 describes an instantaneous injection of CR particles with an initial angular distribution

where p e (-1, 1). The value of the constant aM, which depends on a maximal value direction of juo and a width Au, can be found from the normalization

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