HEATING RATE SCALING OF TURBULENCE IN THE PROTON KINETIC REGIME

Three-dimensional numerical hybrid simulations with particle protons and quasi-neutralizing, fluid electrons are conducted for a freely decaying turbulence. The main results are obtained from a series of runs as a function of the initial total rms fluctuation amplitude. In the turbulent phase and at a corresponding nonlinear time dependent on the amplitude, the scaling of the proton perpendicular heating rate is examined as a function of the spectral value of the electron bulk perpendicular speed integrated in wavenumbers about the inverse thermal proton gyroradius. The perpendicular direction is relative to the background magnetic field. The obtained spectral value is normalized to the proton thermal speed and ranges from 0.06 to 0.16. The scaling of the perpendicular heating rate with this spectral value is fitted with a power law, which has an index of -3.3 +/- 0.2. The fit is consistent with the scaling of the total heating rate as a function of total rms amplitude, which has an index of -3.06 +/- 0.12. The power-law index is near the turbulent hydrodynamic-like prediction for the energy cascade rate as a function of amplitude. The heating rate, then, obeys a power law with amplitude or spectral value regardless of whether that quantity is evaluated at large scales or at the proton gyroradius scales.