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Astron. Astrophys. 327, 432-440 (1997)

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1. Introduction

Many of the properties of cosmic rays can be understood on the basis of a propagation model in which they execute isotropic diffusion between injection at sources distributed in the galactic disk and escape at certain boundaries whilst possibly being carried out of the galaxy in a wind (Lerche & Schlickeiser 1982 Berezinskii et al. 1990 , Webber et al. 1992 , Bloemen et al. 1993 ). On the other hand, it has been known for many years that these particles are 'magnetised' in the sense that their Larmor radius is much smaller than the mean free path between 'collisions' implied by the diffusion coefficient. This means that particles diffuse primarily along the magnetic field, and only very slowly across it. Since the galactic magnetic field lies predominantly in the galactic plane (e.g., Zweibel & Heiles 1997 ), it is surprising that isotropic diffusion describes cosmic ray propagation so well. One might instead expect cosmic rays to be transported anomalously slowly in the direction normal to the disk (Getmantsev 1963 ). The success of the simple diffusion model of propagation (or even of the more primitive but related 'leaky-box' model) appears to show that this is not the case, at least for the bulk of cosmic rays. This is presumably due to fluctuations in the magnetic field, possibly connected with instabilities, which cause a given field line to wander out of the galactic plane, and give rise to an effective diffusion across the direction of the mean magnetic field (Jokipii & Parker 1969 , Jokipii 1973 ).

Nevertheless, it may not always be the case that the transport process can be described by an effective diffusion coefficient. Since its first mention by Getmantsev (1963 ), anomalous transport, i.e., transport in which the mean square deviation of a particle from its position at a given time [FORMULA] does not depend linearly on the elapsed time, has been discussed in two astrophysical contexts. Firstly, Chuvilgin & Ptuskin (1993 ) derived a kinetic equation for cosmic ray transport for the case in which [FORMULA], but pointed out that this type of transport can be expected only when considering timescales short compared to the time taken for a particle to effectively decorrelate from a given magnetic field line. Many effects such as particle drifts, temporal changes in the magnetic field or just the chaotic structure of the field itself can cause decorrelation, but although it is very difficult to estimate this time in an astrophysical situation, the success of the diffusion model suggests that cosmic rays do indeed decorrelate from the field faster than they escape from the galaxy (which takes about [FORMULA] years).

The second astrophysical context is that of particle acceleration at a shock front. In this case, the natural timescale is the acceleration time, which is thought to be a strong function of energy, and to vary over several orders of magnitude for those cosmic rays accelerated at supernova shocks (Dendy et al 1995 , Duffy et al. 1995 ). Here too, it is difficult to make a realistic estimate of the decorrelation time of a particle from the magnetic field. However, predictions can be made of the spectrum and spatial distribution of those particles undergoing anomalous transport, which might enable them to be distinguished from diffusing particles (Kirk et al. 1996a ).

A problem which is similar in many respects arises in the confinement of plasma in fusion devices, and in this context there has been considerable interest in recent years in anomalous, non-diffusive transport models (Rechester & Rosenbluth 1978 , Kadomtsev & Pogutse 1979 , Isichenko 1991a , 1991b , Rax & White 1992 , Wang et al. 1995 ). In particular, use of the formalism developed for continuous time random walks (CTRW - see Montroll & Weiss 1965 ) has been advanced (Balescu 1995 ). This approach is widely used in the problem of anomalous transport in random media (e.g., Bouchaud & Georges 1990 , Shlesinger et al. 1993 ). In this paper, we discuss these techniques and show how they relate to the work already done on anomalous transport in astrophysics. We point out that they offer a more general approach to the problem, enabling one to relax the rather restrictive assumptions concerning the statistical properties of the magnetic field employed hitherto. As an example, we examine a new astrophysical application - that of the synchrotron emission of a population of relativistic electrons which lose energy whilst being transported away from the site of their acceleration. Both the spatial extent of the radiating electrons and the observed spectrum depend on the nature of the transport process. These calculations are especially relevant to the interpretation of high resolution radio observations in several astrophysical situations, including spiral galaxies seen 'edge-on' (e.g., Hummel et al. 1991 ) as well the diffuse emission from clusters of galaxies (Schlickeiser et al. 1987 , Kirk et al. 1996b ).

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© European Southern Observatory (ESO) 1997

Online publication: April 8, 1998