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Fig. 2.29.2. Theoretical prediction of temporal profiles for the selected NMs using calculated parameters Ao, AA and mean r for each NM. On the ordinate axis are shown expected intensity relative to HO in maximum. According to Fedorov et al. (2002).

Fig. 2.29.3. The NM record and theoretical prediction for South Pole (SP) station, which detected both the high-energy stream and the diffusive tail. The northern Thule (TH) station detected only the diffusive tail. Left panel - observations; right panel - predicted temporal curves for different values of r (in GV) and Ao . According to Fedorov et al. (2002).

Fig. 2.29.3. The NM record and theoretical prediction for South Pole (SP) station, which detected both the high-energy stream and the diffusive tail. The northern Thule (TH) station detected only the diffusive tail. Left panel - observations; right panel - predicted temporal curves for different values of r (in GV) and Ao . According to Fedorov et al. (2002).

From Fig. 2.29.3 (left panel) it can be seen that South Pole records indicate a more complicated structure being probably a mixture of the anisotropic stream of fast particles and a diffusive tail of lower energy particles. In the initial phase SP looked at the source with very narrow width AX = 0.1 and registered the anisotropic stream of particle rigidity up to 10 GV. The double peak structure is probably caused by very fast irregular changes in latitude as well as longitude of the apparent source direction (IMF direction). Thus the SP asymptotic direction could jump from one force-line to another and, therefore, it could repeatedly see the source. The model (right panel in Fig. 2.29.3) predicts the two types of time profile as a function of R . In contrast to the South Pole the Thule station in the north (both with zero vertical cutoff rigidities and with the same receipt parameters) shows only the second, diffusive-like tail (see in Fig. 2.29.3, left panel). Probably the North-South anisotropy of lower energy particles which is not described in the present model and/or the latitudinal component of IMF direction can cause such difference. Therefore it is not possible to consider SP in the simple model with one characteristic (mean rigidity R of a NM station), especially in the initial phase of the event. Thus the real picture of registered profile on Fig. 2.29.3 (left panel) is some mixing of these theoretical curves (presented in the right panel of Fig. 2.29.3).

2.30. Pitch-angle diffusion of energetic particles by large amplitude MHD waves

2.30.1. The matter of the problem

Hada et al. (2003a) consider some fundamental properties of pitch-angle diffusion of charged particles by MHD waves by performing test particle simulations. Even at a moderate normalized turbulence level (turbulence magnetic field energy density normalized to the background field energy density ~ 0.1), both the mirroring and the resonance broadening effects become important, and the diffusion starts to deviate substantially from the standard quasi-linear diffusion model. Generally speaking, the transport of CR charged particles by MHD turbulence is one of the key issues in space and astro-plasma physics. Pitch-angle diffusion is fundamental to other transport processes such as the energy and the parallel diffusion (Jokipii, 1966; Terasawa, 1991; Michalek and Ostrovski, 1996; Tsurutani et al., 2002). For the discussion of the various transport processes of CR in space plasma the quasi-linear theory is frequently used, in which two assumptions are fundamental. First, the turbulence amplitude is sufficiently small, so that truncation at the second power of the turbulence is guaranteed. Second, the wave phases are random (random phase approximation), so that any effect of modemode coherence is destroyed by phase mixing. However, the MHD turbulence in space does not necessarily satisfy these assumptions: in particular, the waves excited near collisionless shocks have the wave magnetic field amplitude comparable to or even larger than the background field. In addition, their waveforms show consequences of strong nonlinear evolution (e.g., the shocklets found in the Earth's foreshock region - according to Hoppe et al., 1981), suggesting the presence of the phase coherence (Hada et al., 2003b). It is a main cause why Hada et al. (2003a) investigate the pitch-angle diffusion of energetic particles by MHD waves, which are not necessarily small amplitude, and their phases not necessarily random, by numerically integrating in time the equations of motion of charged particles under influence of given MHD turbulence.

2.30.2. The model used

Hada et al. (2003a) employ the so called slab model for the MHD turbulence, although this is probably an over-simplification for the turbulence in reality (e.g., in the solar wind according to Matthaeus et al., 1990). Within this model the fluctuation electromagnetic field is given as a superposition of parallel propagating, circularly polarized finite amplitude Alfvén waves, with different wave numbers and different polarizations. Since the typical particle velocity far exceeds the Alfvén wave's speed, it was assumed that the waves to be non-propagating: within this system, particle energy is conserved. For both groups of waves with different polarizations it was assumed that the wave spectrum is given by a power law (with an index y), and their phases be related by the iteration formula defined in Eq. 4 of Kuramitsu and Hada (2000).

2.30.3. Main results of simulation

Fig. 2.30.1 shows the time evolution of distribution of particle pitch-angle cosine, p, defined as an inner product of the unit vectors parallel to the particle velocity and the local magnetic field.

10 -0.5 00 0.5 1.0 -1.0 -0.5 0.0 0.5 1.0 -1.0 -0.5 0.0 0.5 1.0 M(0) H(0) n(0)

Fig. 2.30.1. Time evolution of p for SB = 0.01. According to Hada et al. (2003a).

10 -0.5 00 0.5 1.0 -1.0 -0.5 0.0 0.5 1.0 -1.0 -0.5 0.0 0.5 1.0 M(0) H(0) n(0)

Fig. 2.30.1. Time evolution of p for SB = 0.01. According to Hada et al. (2003a).

For each panel in Fig 2.30.1 the horizontal axis represents the initial distribution, p(0), and the vertical axis denotes the distribution at some later times, p(t ). Each dot represents a single test particle. Important parameters used here are: y = 1.5, c^ =0 (random phase), and the variance of the normalized perpendicular magnetic field fluctuations, SB = 0.01. At t = 1 the distribution of p has not evolved much, and so the dots are almost aligned along the diagonal line. Later at t = 16 pitch-angle diffusion is more evident, but is still absent around p ~ 0 and \p\ ~ 1. The former is owed to the lack of waves which resonate with near 90° pitch-angle, and the latter is simply owing to geometry. At an even later time at t = 256, substantially longer than the pitch-angle diffusion time scale, it is clear that the majority of particles stay within the hemisphere they belonged to initially. Three panels from the left in Fig. 2.30.2 show the same plots as before except that the turbulence level is increased to SB = 0.1, keeping other parameters unchanged.

Fig. 2.30.2. Time evolution of p for SB = 0.1. Non-compressional turbulence is used for the run shown in the right bottom panel. According to Hada et al. (2003a).

Fig. 2.30.2. Time evolution of p for SB = 0.1. Non-compressional turbulence is used for the run shown in the right bottom panel. According to Hada et al. (2003a).

From the comparison of the two runs it is clear that not only the diffusion occurs on a faster time scale but also that many particles traverse the 90° pitch-angle. This is mainly owed to the mirroring and the resonance broadening, both of which are the consequences of finite amplitude waves. These two effects can be separate by making the turbulence non-compression, b'(x) = SB b(x)/|b(x)|, where b(x) is the given compression turbulence (the power spectrum and the phase distribution of b(x) and b'(x) are not exactly the same). The distribution of x as diffused by such a non-compression turbulence is shown in the right bottom panel of Fig. 2.30.2. Although the number of particles crossing the 90° pitch-angle is less compared with the compression case, it is shown that the resonance broadening alone can mix the particles across ¡x = 0. Fig. 2.30.3 as well as Fig. 2.30.4 summarizes the numerically evaluated pitch-angle diffusion coefficient D, compared with the value Dql obtained from the quasi-linear theory,

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