Cyclotron resonance in coronal holes - 2. A two-proton description

In a cold plasma, the ion-cyclotron mode does not extend above the proton cyclotron frequency. As a consequence, for waves propagating outward from the Sun, only protons that are moving slower than the mean proton wind speed can resonate with this mode. Thus only roughly half of the proton distribution function can be in resonance at any instant of time. The proton distribution function is then expected to depart significantly from a bi-Maxwellian, which is usually assumed to provide closure to a set of fluid equations. Here we consider the effects of the ion-cyclotron resonance on protons in a coronal hole. We calculate the trajectories of individual protons in the electric, magnetic, and gravitational fields, and we include the resonant heating and acceleration for an average particle that is diffusing in phase space. For closure we consider two protons, which are proxies for the resonant and nonresonant halves of the distribution. Elementary arguments show that the two protons tend to approach nearly the same radial velocity. When the waves are dispersive, this means that the resonant wavenumber k(res) increases. For a power spectrum that is a power law in wavenumber, and if the dissipation is determined only by the resonant particles, then the resonant effects become very weak as k(res) becomes large and there is little heating or acceleration of the coronal plasma. On the other hand, if the dissipation is determined by a turbulent cascade, k(res) mainly controls the relative importance of resonant acceleration and resonant heating. Such models can yield good agreement with what is known about the behavior of protons in coronal holes.