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Fig. 4.16.1. Illustration of the acceleration times (a), and resulting energy spectrum (b). According to Jokipii et al. (2003).

It is apparent from Fig. 4.16.1(a) that the average acceleration rate decreases rapidly for a wavenumber much less than 1 (diffusion too slow), and asymptotically approaches a constant which is about unity for larger wavenumbers (when the diffusion becomes more important). A net acceleration occurs in spite of the balancing of compression and expansion.

Fig. 4.16.1. Illustration of the acceleration times (a), and resulting energy spectrum (b). According to Jokipii et al. (2003).

To determine the energy spectrum for a simple confinement model Jokipii et al. (2003) consider next the solution to Eq. 4.16.27 for the case in which the system is not strictly periodic but that there are diffusive loss boundaries at x = +15 and for n = 2. The velocity is of the form u(x, t)_ asin(?¡x -1), (4.16.30)

where a = 0.6, which is essentially the same as the periodic system used above, but is a propagating wave. The particles are injected continuously and uniformly in x at a momentum po = 1, so the source term in Eq. 4.16.27:

Figure 4.16.1(b) illustrates the energy spectrum obtained for this model system.

From Fig. 4.16.1 it can be seen that the significant acceleration by non-shock compressions, even in the presence of comparable expansions, is possible as long as the diffusion scale is comparable to the scale of the fluid variations, or larger. This diffusive compression acceleration has some similarities with 2nd-order Fermi acceleration and with shock acceleration, but is different from both. It appears to produce naturally a power law-like spectrum, similar to that in shock acceleration, over a broad range of parameters. According to Jokipii (2001), Jokipii et al. (2003), this acceleration can occur in a number of circumstances. For example, in the inner Heliosphere near 1 AU, for = 1 MeV galactic CR where k ~ 1020 cm2/sec, and where compressive velocities should be of the order of Alfvén velocity va ~ 50 km/sec, we have £ > 1 for scales Lc < 1 AU. Observations suggest that there may be significant compressive fluctuations over these scales. in the interstellar medium, where the diffusion coefficient is typically > 1026 cm2/sec, and typical fluid velocities are = 100 km/sec or so, scales of several parsecs to tens of parsecs can correspond to £ > 1. Jokipii et al. (2003) conclude that compressive variations may contribute to the acceleration of energetic particles in many places in the Universe.

4.16.6. Acceleration at fluid compressions and comparison with shock acceleration

The study of particle acceleration at a shock discontinuity raises the question of acceleration at continuous fluid compressions that have not yet developed into shocks. This has previously been examined for a magnetic field parallel to the shock normal (Krulls and Achterberg, 1994). Particle acceleration was examined in a general, steady-state context in a preliminary report by Klappong et al. (2001), whilst in Jokipii (2001) and Giacalone et al. (2002) the problem for the situation of co-rotating interaction regions in the interplanetary space has examined and they were able to explain observed time-intensity profiles. Malakit et al. (2003) examines steady-state particle acceleration at continuous fluid compressions of varying width in comparison with that at a discontinuous shock for various shock-field angles. The configurations are shown in Fig. 4.16.2; in the compression case magnetic field lines have a hyperbolic shape and width (the semi-conjugate axis) b along the field.

In comparison with shocks the narrow compressions exhibit quantitatively similar particle acceleration, leading smoothly to the shock results as the width is reduced. However, compressions do not naturally yield a power law particle spectrum; rather, the resulting spectrum is sensitive to the velocity dependence of the mean free path of scattering. The study of acceleration at compression leads to better understanding of shock acceleration, especially regarding the effect of magnetic mirroring on the distribution function and hardening of the particle spectrum.

The transport and acceleration of energetic charged particles near a fluid compression in Malakit et al. (2003) is studied by numerically solving a time-dependent pitch angle transport equation for a general, static magnetic field. The numerical methods are based on those of Ruffolo (1999) and Nutaro et al. (2001). For a shock the transport equation is greatly simplified, but care is required when treating particles crossing the shock. The particle orbits are considered as they cross the shock, using a transfer matrix to assign the distribution function to the appropriate ^ and z cells after the shock encounter. In a stringent test of the accuracy of the pitch-angle treatment, the simulations have been able to explain observed 'loss-cone' precursors to Forbush decreases (Leerungnavarat et al., 2003). Although the key results of the work of Malakit et al. (2003) are derived from a more fundamental treatment of pitch angle transport and diffusion-convection treatment. The diffusion-convection transport equation for the plane parallel configuration is an ordinary differential equation, which can be solved analytically for a shock, and can readily be solved numerically for a compression. In the paper Malakit et al. (2003) approximate diffusion-convection results are shown specifically to highlight the role of magnetic mirroring, which is neglected by diffusion-convection included in the full pitch angle treatment.

shock compression region

Fig. 4.16.2. Sample mean magnetic field configurations for a shock (left) and a compression region (right). According to Malakit et al. (2003).

In the results it was found that particle spectra from shocks, as predicted by pitch angle treatment, are not exactly power laws as predicted by diffusion-convection (Krymsky, 1977). The spectra are hardened at low energy, especially for the quasi-perpendicular (Q-Perp) case (upstream shock-field angle &i= 75.96°). However, the particle spectrum in the case of a quasi-parallel (Q-Par) shock ( = 0.57°), predicted by pitch angle treatment, is still a power-law (see Fig. 4.16.3).

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