where V^ = dx + dy is the Laplacian operator transverse to the magnetic field, which is assumed to lie along the z-axis, and S_1 is the inverse Laplace and Fourier transform operator. Assuming that fo evolves on much longer time scales than t±, T// and the gyro-period 2n/Q (i.e. |fot/fo| << v±, v//, Q) and on space scales much larger than the mean free paths vt// and vzL, then Eq. 2.27.38 can be approximated by the equation:

which is the pitch-angle evolution equation for fo incorporating the effects of cross-field diffusion (the V2 fo term). Multiplying Eq. 2.27.39 by 2np2 and integrating Eq. 2.27.39 over ^ from ^ = -1 to ^ = 1, using the diffusion approximation, and neglecting the source, or initial value term in Eq. 2.27.39 results in the usual diffusion Eq. 2.27.5 in the form:

f + 1 ("*■// If) + V±(V±N ) = 0, (2.27.40)

where k± and k// are given by Eq. 2.27.9. In the derivation of Eq. 2.27.40 it is also necessary to take the first moment of Eq. 2.27.39 (i.e. multiply Eq. 2.27.39 by 2np2v/ and integrate over ^ from ^ = -1 to ^ = 1, and use the diffusion approximation to find the diffusive streaming parallel to the field). It is clear that accurate approximate solutions of Eq. 2.27.39 can be obtained by expanding the distribution function in terms of Legendre's polynomials, and taking moments of Eq. 2.27.39 (e.g., Gombosi et al. 1993; Lu et al., 2001).

Summarizing the results discussed above, Webb et al. (2001) note that from the explicit solution for Fo in Eq. 2.27.27, the complete solution forfr, p, t) for the case of Dirac-delta initial data in position, pitch-angle, and gyro-phase, can be constructed by Laplace-Fourier inversion. By the first determining Fo from Eq. 2.27.27, and using the result to determine fo from Eq. 2.27.20, and then obtain f from Eq. 2.27.14, followed by Laplace and Fourier inversion - to determine f. A multiple scattering analysis (e.g. Webb et al., 2000) and eigenfunction/moment equation methods should reveal further aspects of the solution. There are several outstanding issues raised by the above analysis. For example, in a non-uniform background magnetic field there is a non-zero contribution to the divergence of the particle current owed to curvature and gradient drifts associated with the antisymmetric diffusion coefficient ka . It is of interest to determine whether the effects of these drifts can be included in a pitch-angle evolution equation analogous to Eq. 2.27.39 in this case. It is also of interest to investigate higher order transport effects in the model, e.g. the incorporation of CR inertial effects in telegraph type equations for CR transport including cross-field diffusion, that generalize the telegraph equation obtained by Gombosi et al. (1993). Other aspects of CR transport theory that are raised by the analysis, concern the form of the pitch-angle evolution equation obtained by Skilling (1975) for particle transport in the solar wind, or its relativistic generalization (e.g. Webb, 1985) when cross field transport is included, and the role of cross-field transport effects on CR viscosity, and non-inertial acceleration effects.

2.28. Influence of magnetic clouds on the CR propagation

The propagation of energetic charged particles through interplanetary space is normally described by a transport equation which considers the effects of propagation parallel to the field, pitch-angle scattering at magnetic field irregularities, and focusing in the diverging interplanetary magnetic field (Roelof, 1969) or, in addition to the above effects, also convection with the solar wind and adiabatic deceleration (Ruffolo, 1995). Focusing is always considered for simple geometries, in general the Archimedean spiral field, although variations in the large scale magnetic field structure, in particular propagating magnetic flux ropes (ejects following coronal mass ejections, CMEs, also called magnetic clouds; for a review see e.g. Burlaga, M1995), modify the local focusing length and therefore also particle propagation.

In their detail investigation Kallenrode (2001a) takes into account that magnetic clouds modify the structure of the interplanetary magnetic field on spatial scales of tenth of AU. Their influence on the transport of energetic charged particles is studied with a numerical model that treats the magnetic cloud as an outward propagating modification of the focusing length. As a rule of thumb the influence of the magnetic cloud on particle intensity and anisotropy profiles increases with decreasing particle mean free path and decreasing particle speed. Special attention is paid to energetic particles running into a magnetic cloud released at an earlier time: here the cloud acts as a barrier storing the bulk of the particles in its downstream medium.

Since Kallenrode (2001a) is concerned with particles with energies in the MeV and tens of MeV range, solar wind effects such as convection and adiabatic deceleration are of minor importance (Ruffolo, 1995), in particular, if there are concerned with a long-lasting injection from a propagating interplanetary shock (Lario et al., 1998; Kallenrode, 2001b). For a first approach on the influence of a magnetic cloud, they started from the model of focused transport (Roelof, 1969):

dt r ds 2s df df df j where /(t,s,p) being the distribution function, t time, s distance along the Archimedian magnetic field spiral, vp particle speed, ^ pitch-cosine, k(s,^i) pitch-angle diffusion coefficient, and g(s )=- B(s )/(dB/ ds) (2.28.2)

the focusing length. The terms in the transport Eq. 2.28.1 from left to right describe the field parallel propagation, focusing in a magnetic field with focusing length g(s) depending on distance, and pitch-angle scattering. The source term is allowed to propagate along the field line, simulating the long lasting injection of energetic particles from a shock as described in Kallenrode and Wibberenz (1997), the transport of energetic particles through the shock front is treated as described in Kallenrode (2001b). The magnetic cloud is assumed to be of spherical cross section with the interplanetary magnetic field draped symmetrically around it (see Fig. 2.28.1).

Meridional Cross-Section

Undisturbed divergent field

Kallenrode (2001a) note that the main change is a compression of the interplanetary magnetic field at the flanks of the cloud. The magnetic cloud is characterized by its diameter dc as a certain fraction of the distance rs of the shock from the Sun, the distance rcs of its leading edge from the shock, also expressed as a certain fraction of rs, and its magnetic compression rB at the flanks. For applications these data can be inferred from the observations; for the numerical study below Kallenrode (2001a) used dc = 0.2 and rcs = 0.1 (Bothmer, 1993). With si = s(rs -{dc + rcs )) and s2 = - rcs )

this configuration than is translated into a sinusoidal variation of the focusing length

for s < si, for si < s < s2, for s > s2,

and a corresponding elongation of the interplanetary magnetic field line. The ± allows for the consideration of the magnetic cloud or a void in the field instead of the cloud. Asymmetric draping of field lines (Vandas et al., 1996) can be considered by assuming a stronger (or weaker) compression of the magnetic field with a more (or less) pronounced elongation of the field line. Kallenrode (2001a) note that this approach allows to describe the particle propagation in a flux tube draped around the magnetic cloud but not the features of energetic particles directly inside the cloud. It also does not consider the cross-field transport of energetic particles from the ambient medium into the magnetic cloud.

Fig. 2.28.2 shows intensity and anisotropy profiles for a solar energetic particle event (lower set of curves; observer at 1 AU, particle speed vp = 1 AU/h corresponding to ~ 10 MeV protons, radial mean free path Xr = 0.1 AU,

S-injection on the Sun) followed by a magnetic cloud with a constant speed of 800 km/s (no shock with particle acceleration considered here). The upper set of curves is for particles accelerated at a shock with constant speed of 800 km/s and constant acceleration efficiency, followed by a magnetic cloud. All other parameters are the same as for the solar event. The shock arrives at the drop in particle intensity around 50 h. The solid line gives the particle event without ejecta, the dotted lines are for a cloud geometry with a magnetic compressions at its flank of 1.3 (lower amplitude) and 2. The latter value is in agreement the values inferred from numerical simulations (Vandas and Romashets, 2001).

Fig. 2.28.2. Solar energetic particle event (lower set of curves) and shock accelerated particles (upper set of curves) followed by a magnetic cloud. The upper panel gives intensities for the two scenarios, the lower ones anisotropies (shifted with respect to each other). According to Kallenrode (2001a).

Fig. 2.28.2. Solar energetic particle event (lower set of curves) and shock accelerated particles (upper set of curves) followed by a magnetic cloud. The upper panel gives intensities for the two scenarios, the lower ones anisotropies (shifted with respect to each other). According to Kallenrode (2001a).

The presence of the magnetic cloud leads to: (1) a slight increase in intensities upstream of the cloud by a few percent, (2) a strong drop in intensities downstream of the cloud by about an order of magnitude, depending on the strength of the magnetic compression, and (3) a sharp drop of intensity at the time of cloud passage (remember, this is at the flanks not inside the cloud) combined with a strong anisotropy indicating a net-streaming of particles from the cloud's upstream medium (where intensities are high) into its downstream medium (where intensities are low). Note that these effects are very similar for a simple solar injection as well as for the continuous particle injection from a propagating interplanetary shock. Quantitatively the influence of the magnetic cloud depends on particle speed and strength of the interplanetary scattering. With increasing scattering the increase in upstream intensities increases while the drop in downstream intensities decreases. The intensity drop at the time of cloud passage is independent of scattering while the anisotropy decreases with increasing scattering. With decreasing particle speed both upstream intensity increases and downstream intensity drops increase and the negative anisotropy inside the cloud becomes more pronounced. Thus faster particles are less influenced by the presence of the magnetic cloud than are slower ones. Fig. 2.28.3 gives the same set of curves as Fig. 2.28.2 except that the ejection has started 24 hours prior to the release of the energetic particles in a different solar event. In this case the ejection is running ahead of the particles and is at a radial distance of about 0.5 AU at the start of the particle injection. Again, solid lines are calculated without ejects, dotted ones with.

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