Relativistic particle acceleration by resonant Alfven waves which are driven internally in a radio source from fully developed fluid turbulence is considered. It is found that self-similar behavior as described by Lacombe, f(p) varies as p to the -s power but with s approximately 4.5, arises self-consistently when this turbulent wave driving coexists with synchrotron losses. The coupling of the wave and particle distributions provides feedback which drives an arbitrary initial distribution to the form-stable, self-similar form. The model predicts that turbulent plasma in a radio source should evolve toward a synchrotron spectral index, alpha approximately in the range from 0.5 to 1.0 in one particle lifetime, and that the average spectrum of most sources should also be in this range. The theory may also be applicable to other turbulent sites, such as comsic-ray reacceleration in the interstellar medium.
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Eilek et al. (1984) studied this question.