where Kj is the diffusion tensor, Uj is the flow velocity of the background plasma, and Qs and Qi represent any additional sources and losses. This equation applies if there is enough scattering for the distribution function to remain nearly isotropic, even at discontinuities such as current sheets and shock waves. Particle acceleration is contained in the term 9u;/dxj . Application of Eq. 4.16.27 to a one-dimensional system having a planar shock, where the flow velocity changes discontinuously, yields all of the results of diffusive shock acceleration. If the disturbance is not a discontinuity, but instead is a more-gradual compression having a characteristic length scale Lc, one can show that in the limit in which the ratio of the diffusive skin depth Ld = Kxxjux to the length scale Lc is large, or, equivalently, £ = Kxx/ (uxLc )>>1, the solution for the CR distribution function f goes over to the standard diffusive shock solution. In the opposite limit £ << 1 the CR are closely tied to the convecting fluid, and simply compress adiabatically.
Then consider the case Ld > Lc, but where the flow varies smoothly. Note that the scattering mean free path Asc does not appear explicitly in this inequality. So it is possible to have Asc small compared with the compression length scales Lc (so that the diffusion approximation applies) but where the diffusive skin depth Ld is of the order of Lc or larger. It was found that such non-shock compressions may be efficient accelerators even if there are associated expansions. The physical basis of the acceleration is the interplay between a) the energy change caused by the compression or expansion of the fluid and b) the diffusion into or away from the region of compression or expansion. Rapid diffusion leads to a particle being able to diffuse away from a region of compression or expansion before the compensating expansion or compression can occur. Hence statistically some few particles will be fortunate enough to gain energy in several compression regions. In this process, for large k, the accelerations dominate the particle energy change, even in those cases where the compressions and expansions are equally present in the fluid flow. This is because statistically some particles can reach very high energies, but they cannot be decelerated to energies lower than zero. Note also that this acceleration can take place for any orientation of the magnetic field. Gradient and curvature drifts can in general significantly affect the particle trajectories as they are accelerated.
To illustrate this process, Jokipii et al. (2003) consider the simple, periodic one-dimensional velocity profile ux (x ) = uo (1 + a sin(kx)), (4.16.28)
and Kxx independent of x or p. There are not been able to solve this analytically for general parameters, but it is simple to solve numerically, and the solutions depend only on the dimensionless parameters
X=(Uo lKxx k T = (U«/Kxx ^ V = (xx/Uo )k , (4.16.29)
and the amplitude a in Eq. 4.16.28. The solutions are clearly periodic in x with a period 2n/n. Illustrated in Fig. 4.16.1(a) is the initial rate of acceleration d ln(p)/dT (in units of 1/t ), averaged over x and plotted as a function of normalized wavenumber n, for the case in which the parameter a = 0.6, which corresponds to a ratio of maximum density (or velocity) to minimum density (or velocity) of 4.
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