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We investigate several key questions of plasma heating in open-field regions of the corona that connect to the solar wind. We present results for a model of Alfven-wave-driven turbulence for three typical open magnetic field structures: a polar coronal hole, an open flux tube neighboring an equatorial streamer, and an open flux tube near a strong-field active region. We compare time-steady, one-dimensional turbulent heating models (Cranmer et al., 2007) against fully time-dependent three-dimensional reduced-magnetohydrodynamics modeling of BRAID (van Ballegooijen et al., 2011). We find that the time-steady results agree well with time-averaged results from BRAID. The time-dependence allows us to investigate the variability of the magnetic fluctuations and of the heating in the corona. The high-frequency tail of the power spectrum of fluctuations forms a power law whose exponent varies with height, and we discuss the possible physical explanation for this behavior. The variability in the heating rate is bursty and nanoflare-like in nature, and we analyze the amount of energy lost via dissipative heating in transient events throughout the simulation. The average energy in these events is 10^21.91 erg, within the "picoflare" range, and many events reach classical "nanoflare" energies. We also estimated the multithermal distribution of temperatures that would result from the heating-rate variability, and found good agreement with observed widths of coronal differential emission measure (DEM) distributions. The results of the modeling presented in this paper provide compelling evidence that turbulent heating in the solar atmosphere by Alfven waves accelerates the solar wind in open flux tubes.