and M is the Alfvenic Mach number of the shock and r is the gas compression ratio of the shock. When the downstream Alfvenic Mach number
all waves are able to propagate in the downstream region. For cold upstream plasma this means that M > 42.8 but since there are also considered the downstream modes in the cold plasma approximation, it must restrict ourselves too small gas compression ratios and shocks with r1/2 < M < 2. As an illustrative example, Vainio and Schlickeiser (1999a) consider downstream turbulence consisting of (i) Alfven waves with / << Op being dominated by the backward propagating waves, (ii) forward whistler waves with Op << f << 1, and (iii) equal intensities of forward and backward waves near the cyclotron frequencies. The latter assumption is made since it is not really known how waves with high frequencies and wave numbers interact with shocks and since other wave-generation processes may also be important at high frequencies. For upstream waves it was assumed that waves at low wave numbers dominate in intensity over the waves at high wave numbers. It was assumed also that all upstream waves are backward waves. Using these assumptions for turbulence near the shock and the results of Section 2.21 Vainio and Schlickeiser (1999a) conclude the following: (i) upstream and downstream bulk speed of the energetic (v >> va) ions relative to the plasma is close to the local Alfven speed, V > va ; (ii) upstream bulk speed of energetic electrons is decreasing with momentum from V ~ -9va at v ~ 2wmax to V > -va at ultra-relativistic (y > 200) energies; (iii) downstream bulk speed of energetic electrons is V2 > 0 at non-relativistic energies and V2 = -va at ultra-relativistic energies. The study of Vainio and Schlickeiser (1999a) reveals that (i) for ions and ultra-relativistic (E > 100 MeV) electrons k = r(M - 1)/M + Hcar12), (4.23.6)
where the downstream cross-helicity state is close to Hc 2 = -1 and, thus, r ~ 1 (see
Vainio and Schlickeiser, 1999b,c for a more details); (ii) for less energetic electrons, the first-order Fermi process will be less efficient and will, in fact, turn to deceleration at mildly relativistic or non-relativistic energies. Thus Vainio and Schlickeiser (1999a) expect stochastic acceleration in the downstream region to determine the spectrum of these electrons at the shock.
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