2.18. Anomalous diffusion: modes of CR diffusion propagation
2.18.1. Three modes of particle propagation: classical diffusion, superdiffusion and sub-diffusion
According to Otsuka and Hada (2003), anomalous diffusion is observed in many branches of science, e.g., anomalous diffusion in rotating flows (Solomon et al., 1993), particle motion in nonlinear dynamical systems (Klafter et al., 1993), chaotic phase diffusion in Josephson junctions (Geisel et al., 1985), field line diffusion in solar wind magnetic turbulence (Pommois et al., 2001), and transport in turbulent plasmas (Balescu, 1995). The term 'anomalous' is used to emphasize deviation from classical (normal) diffusion, in which the mean squared displacement of particles increases proportional with time. Namely, if we define the diffusion coefficient k as where Ar is the particle displacement within the time scale t, and the bracket denotes an ensemble average, then p = 0 for the classical diffusion. This is a consequence of the well-known central limit theorem, which states that in the long time limit the p.d.f. of Ar approaches a normal (Gaussian) distribution with its variance « t. On the other hand, when a particle can travel long distances ballistically, the so-called Levy flights or Levy walks can arise, and the resultant diffusion process of the ensemble of particles becomes super-diffusive (P > 0). When a particle can be trapped within a certain bounded region for a long time, then the sub-diffusion (P < 0) emerges.
2.18.2. Simulation of particle propagation in a two-dimensional static magnetic field turbulence
Otsuka and Hada (2003) compute numerically orbits of CR particles in a two-dimensional static magnetic field turbulence, and show that anomalous diffusion can appear in general. The result may have an important implication for plasma astrophysics, since, up to now, various diffusion processes (including the cross-field diffusion) are almost always discussed within the framework of the quasi-linear theory, which in principle is a combination of the classical diffusion equation for particles and an evolution equation for the energy of the turbulence, which in turn determines the diffusion coefficient. The spatial diffusion problem, in particular, is important for the shock acceleration of CR charged particles (see Chapter 4).
Although the cross-field diffusion in reality is a three-dimensional problem, Otsuka and Hada (2003) limit their discussion to the case where all the physical variables depend only on two spatial coordinates (x and y), and the magnetic field
lines are perpendicular to the x-y plane. By taking such geometry, the effective cross-field diffusion resulting from parallel motion along the twisted field lines (field-line random walk (Jokipii, 1966)) will be excluded. In general, in a model with only two spatial dimensions, particles are tied to the magnetic field lines since the canonical momentum associated with the ignorable coordinate becomes an invariant of the motion (Jokipii et al., 1993). Since the energetic particle velocities considered are much larger than the MHD velocities it may be assumed that the field turbulence is time stationary (fossil turbulence). The particle energy is then conserved, and the position r = (x, y) and the velocity V = (vx ,Vy ) obey the equations of motion
where z is a unit vector in the z direction, b is the fluctuation part of the normalized magnetic field, and time is normalized to the reciprocal of the average particle gyro-frequency. The turbulence field is given by b(x, y ) = A(k )cos(mx + ny + (f>(m, n )), (2.18.3)
In Eq. 2.18.3 and 2.18.4 there are k Y for kmin - k - kmax kmin for ksys - k - kmin-
and L is the system size. Boundary conditions are periodic, and phases <(m,n) are random. The magnetic field correlation length Lc was defined as
There were chosen y = 1.5, L = 512, and (bj = 0.01, then Lc ~ 61. By giving different velocities to the particles, it was running with different rL/Lc , where rL is the Larmor radius.
The results are summarized in Fig. 2.18.1. The upper panels show the expected diffusion coefficient k defined in Eq. 2.18.1 versus t in logarithmic scales for three different regimes of rL/Lc , and the lower panels are the y component of the guiding center rg = r + Vxz versus time t, plotted for some particles. The orbits look quite different for the three runs. When rL/Lc = 10 the orbits look more or less similar to a Brownian motion, whilst for rL/Lc = 1 and rL/Lc = 0.1 they are composed of segments with different characters - sometimes almost ballistic, sometimes trapped at a certain locations, and sometimes like a Brownian motion. This diversity of the types of the orbits is a reflection of the presence of multi-scales in the magnetic field turbulence. Namely, if a particle is guided by a large scale inhomogeneity of the magnetic field for a longer time period than the 'observation' time scale t, then its orbit will appear to be almost ballistic, while a particle trapped by a small scale inhomogeneity will appear as trapped if it makes many rotations around the inhomogeneity within t.
Fig. 2.18.1. Diffusion coefficients k (upper panels) and typical guiding center trajectories (lower panels) for (a) rL/Lc = 0.1, (b) rL/ic = 1, and (c) rL/Lc = 10. According to Otsuka and Hada (2003).
The diffusion coefficients represent the different characteristics of the orbits. Let us first look at the case c in Fig. 2.18.1, rL/Lc = 10. When (i) t < 103, the value of k is still influenced by Larmor rotations (large amplitude oscillations in the figure), and so there is no sense in discussing statistics in this regime. For longer time scale (ii) t > 103, the value of k became almost constant, suggesting that the diffusion is almost classical and the orbits are essentially Brownian. This is reasonable, since when rL/Lc = 10, a particle traverses many inhomogeneities of the magnetic field during one gyration, and the force acting on the particle, which will be a sum of many fluctuations, will be random and incoherent.
When rL/Lc = 0.1 (case a), the gyration regime (i) is followed by two distinct regimes (ii) and (iii) as t is increased. In (ii) the process is slightly super-diffusive (P > 0 in Eq. 2.18.1) since within this time scale the majority of the particles gradient-H drift around the field inhomogeneities without making a complete rotation. For longer time scales many particles are trapped (as seen in the lower panel), resulting in the sub-diffusion (P < 0 in Eq. 2.18.1). The values of P and the transition time scale which separates regimes (ii) and (iii) depend on the parameters for the turbulence.
The case b, rL¡Lc = 1, illustrates the possibility that even more distinct types of orbits can exist. In the super-diffusive regime (iii) some particles 'percolate' along infinitely long open paths, which result from the assumed periodicity of the simulation system. At longer time scales the percolation orbits start to mix (percolation random walk), and thus the diffusion becomes classical again.
Otsuka and Hada (2003) came to the conclusion that in two-dimensional static magnetic field turbulence different types of cross-field diffusion of energetic particles are observed for different regimes of rL¡Lc, and for a finite observation time scale t. When rL/Lc > 1 the diffusion is classical asymptotically (t ^ w), whilst at super- and sub-diffusion can be realized when rL/Lc < 1 and when LLc ~1-
2.19. Energetic particle mean free path in the Alfvén wave heated space plasma
2.19.1. Space plasma heated by Alfvén waves and how it influences particle propagation and acceleration
Vainio et al. (2003a) present a simple analytical expressions for the power spectrum of cascading Alfvén waves and the resulting CR energetic particle mean free path in the solar wind. The model can reproduce the short coronal mean free path required for efficient acceleration of charged particles in coronal shock waves (see Chapter 4, Section 4.20) as well as a longer interplanetary mean free path required for a rapid propagation of the accelerated particles to a distance 1 AU from the Sun. Recent observations of high and anisotropic ion temperatures in the solar corona (Kohl et al., 1998) give observational support to models employing the cyclotron heating mechanism to heat the plasma on open magnetic field lines. In these models the energy input for heating the plasma comes from Alfvén waves created at the solar surface. The waves propagate until their frequency is comparable to the local ion-cyclotron frequency, and the wave energy is absorbed by the plasma ions via the cyclotron-resonance. Energetic particles interact strongly with the waves responsible for heating the corona
(Vainio et al., 2003b). The same waves that heat the solar corona can help to rapidly accelerate charged particles in coronal shock waves, and thus explain particle acceleration in small SEP events, where self-generated waves (see Chapter 3) can not explain the rapid acceleration. On the other hand, observed parameters of SEP transport in the solar wind give constraints on the wave-heating models, limiting the magnitude and spatial extent of wave heating in the solar wind.
2.19.2. Determining of the Alfven wave power spectrum
Vainio et al. (2003a) considered AlfVen waves propagating in the solar corona and solar wind in the framework of the model developed in Hu et al. (1999). An equation governing the power spectrum P(f, r) of outward propagating AlfVen waves in the solar wind in the steady state, is given according to Tu et al. (1984) by va d
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