2. Anomalous transport
In a regular magnetic field topology, the transport of charged particles across the field is due to the collisions of the particles and their finite Larmor radii. However, a perturbation of the magnetic field results in a wandering of the field lines and a potentially much faster transport of the particles. This transport is widely attributed to the effect of the large-scale random field component, which would produce, following the quasilinear theory, a diffusion of the field lines across the direction of the the average field (Jokipii & Parker 1969 , Kadomtsev & Pogutse 1979 ). In fusion applications, it is generally assumed that particle collisions lead to diffusive transport along each individual field line. In astrophysical applications, where collisions can be neglected, it is usual to assume that small scale fluctuations in the magnetic field play this role, so that here too, particles diffuse along field lines (Chuvilgin & Ptuskin 1993 ). As long as the particles remain correlated to a given patch of field lines, the combination of these two diffusions results in sub-diffusion of the particles i.e., , with , as described by Getmantsev (1963 ). (In the general case, sub-diffusion refers to all such processes when .) However, the exponential divergence of neighbouring field lines and the resulting stretching of a field line patch lead to decorrelation of a particle from its field line, and thus to the large-scale diffusion of the particles, as pointed out by Rechester and Rosenbluth (1978 ), with a diffusion coefficient given by the expression:
where , , where is the exponentiation length of the field lines. Here is the quasilinear diffusion coefficient of the field lines, which has the dimensions of a length, is a characteristic wave number of the magnetic turbulence, and are the quasilinear diffusion coefficients of the particles along and across the direction of the local magnetic field, respectively. The transport of particles across a turbulent magnetic field is thus sub-diffusive for short timescales, but crosses over to normal diffusion in the limit of the long timescales. In the following, we investigate the anomalous transport regime, which applies if the natural timescale of a particular problem - such as escape from the galaxy, acceleration at a shock, or loss of energy by synchrotron radiation, is shorter than the decorrelation time.
For diffusive transport, P typically contains a factor , and is the Green's function of the diffusion equation.
For sub-diffusive transport with , Rax & White (1992 ) have determined the propagator by combining two diffusive propagators using a Wiener integration method. They considered the transport in cylindrical symmetry across the z direction (i.e., a 2-dimensional problem, with playing the role of time), and obtained:
where r is the radius in cylindrical coordinates, and the Heaviside function. Duffy et al. (1995 ) considered the transport perpendicular to a plane shock front. In a cartesian coordinate system with the shock in the y -z plane, and assuming no gradients in the y direction are present, they obtained the propagator
with and . For this 1-dimensional problem of sub-diffusion, Chuvilgin & Ptuskin (1993 ) derived an equation describing the evolution of the particle density (their Eq. B.12) and solved it to find the above propagator. Essentially the same equation was also found by Balescu (1995 ). It can be written:
with . This is a non-Markovian diffusion equation: the integral term is characteristic of the long-time memory of the dynamics.
In a realistic situation, the topology of the magnetic field may be more complicated than just a pure stochastic sea with its associated diffusion of the field lines. In fusion plasmas, for example, there can exist ordered structures, 'stability islands', in which particles can be trapped for long periods of time, (referred to as 'sticking'), leading to sub-diffusive large-scale transport of the field lines themselves (e.g., White et al. 1993 ). On the other hand, the field lines may also wander faster than implied by diffusion. In this case, the field lines are said to perform 'flights', during which they maintain almost the same direction for a relatively long distance. Ultimately, the large-scale transport of field lines is the result of competition between 'sticking' and 'flights', and can yield transport regimes of many different kinds, ranging from slow sub-diffusion (almost perfect sticking) to fast supra-diffusion (domination of flights). In terms of , the physically relevant range is , with corresponding to transport completely dominated by a single straight-line flight.
To find the particle propagator, it is necessary to combine the propagator for field lines with that for particle motion along the field, as described above for the case of . Here too, the assumption of diffusive transport can be generalised. Particles may, in fact, undergo no scattering at all, in which case they propagate ballistically along the field lines, as assumed by Achterberg & Ball (1994 ). On the other hand, they may be trapped between magnetic mirrors on a segment of a field line. Once again, the transport can be characterised by an index which lies between 0 and 2. However, as we show below, the combination of two propagators does not extend the overall range of which is permitted. In fact, the effect of superimposing transport with the two indices and is described by a single value .
© European Southern Observatory (ESO) 1997
Online publication: April 8, 1998