Energy balance for 6D kinetic ions with adiabatic electrons

This paper explores the energy fluxes in the 6D kinetic Vlasov system. We present a novel method for calculating particle and energy flows within this framework, enabling the direct determination of energy and particle fluxes and the Poynting flux from lower moments of the distribution function, such as kinetic energy density and the momentum transfer tensor. This technique allows for identifying individual contributions of the energy flux while minimizing inherent gyro-oscillations in energy and particle fluxes. The resulting energy and particle flux expressions are compared with simulation results of ion temperature gradient turbulence, using either a local or nonlinear treatment of the temperature gradient. By computing the fluxes and the decay of the initial temperature profile from the simulation, we demonstrate good agreement with the summation of individual contributions, confirming the validity of the new method. The results indicate that energy fluxes are primarily dominated by the E×B heat flux, with additional contributions from the momentum transfer tensor and the Poynting flux. This paper also derives the residual Poynting flux in the electrostatic limit and assesses its contribution to the overall energy flux. From this analysis, we identify an additional contribution that is negligible for low-frequency gyrokinetic modes but may become significant for high-frequency modes, such as ion Bernstein waves.