suggests that the energy cascades almost entirely perpendicular to the local magnetic field. As a result, the cyclotron resonance is unimportant in dissipating the turbulent energy. Motivated by the GS cascade, we calculate the linear collisionless dissipation of Alfven waves with frequencies much less than the proton cyclotron frequency, but with perpendicular wavelengths of order the Larmor radius of thermal protons. In plasmas appropriate to hot accretion flows (proton temperature ≫ electron temperature) the dissipated Alfven wave energy primarily heats the protons. For a plasma with β < ∼ 5, however, where β is the ratio of the gas pressure to the magnetic pressure, the MHD assumptions utilized in the GS analysis break down before most of the energy in Alfven waves is dissipated; how the cascade then proceeds is unclear. Hot accretion flows, such as advection dominated accretion flows (ADAFs), are expected to contain significant levels of MHD turbulence. This work suggests that, for β> ∼ 5, the Alfvenic component of such turbulence primarily heats the protons. Significant proton heating is required for the viability of ADAF models. We contrast our results on particle heating in ADAFs with recent work by Bisnovatyi-Kogan & Lovelace (1997). Subject headings: accretion – hydromagnetics – plasmas – turbulence 1
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