Unifying the Parker and the Turbulence Models of Solar Coronal Heating

We present results from a series of three-dimensional simulations investigating heating in coronal loops of various lengths, based on the equations of reduced magnetohydrodynamics, following up on our recent simulations of the Parker model of coronal heating [Ng et al., Astrophys. J. {\bf 747} 109, 2012]. We study the time-averaged energy dissipation rate $\langle W \rangle$ as a function of the length of the loop $L$. We confirm that in the limit of small $L$ ($L \ll V_A \tau_c$, where $V_A$ is the Alfv\'en speed based on the parallel magnetic field and $\tau_c$ is the correlation time of the random photospheric motions), $\langle W \rangle$ agrees well with the scaling derived from the Parker model. In the other limit of $L \gg V_A \tau_c$, we show that $\langle W \rangle$ is given by the photospheric Poynting flux required to launch Alfv\'en waves. In the intermediate range of $L \sim V_A \tau_c$, $\langle W \rangle$ is well described by a scaling based on the Kolmogorov turbulence energy cascade, rather than the Iroshnikov-Kraichnan cascade. We also show that $\langle W \rangle$ can be modeled by combining the Parker heating and the Alfv\'en wave launching power for all range of $L$.