Magnetic Energy Transfer and Distribution between Protons and Electrons for Alfv\'enic Waves at Kinetic Scales in Wavenumber Space

Turbulent dissipation is considered a main source of heating and acceleration in cosmological plasmas. The alternating current Joule-like term, $\langle\delta j \cdot \delta E\rangle$, is used to measure the energy transfer between electromagnetic (EM) fields and particles. Because the electric field depends on the reference frame, in which frame to calculate $\langle\delta j\cdot \delta E\rangle$ is an important issue. We compute the scale-dependent energy transfer rate spectrum in wavevector space, and investigate the electric-field fluctuations in two reference frames: $\delta E$ in the mean bulk flow frame and $\delta E'$ in the local bulk flow frame (non-inertial reference frame). Considering Alfv\'enic waves, we find that $\langle\delta j\cdot\delta E^\prime\rangle$, which neglects the contribution of work done by the ion inertial force, is not consistent with the magnetic field energy damping rate ($2\gamma \delta B^2$) according to linear Maxwell-Vlasov theory, while $\langle\delta j\cdot \delta E\rangle$ is exactly the same as $2\gamma \delta B^2$ in wavenumber space $(k_\parallel, k_\perp)$, where $\gamma$ is the linear damping rate. Under typical conditions of solar wind at 1 au, we find in our theoretical calculation that the field energy is mainly converted into proton kinetic energy leaving the residual minor portion for electrons. Although the electrons gain energy in the direction perpendicular to the mean magnetic field, they return a significant fraction of their kinetic energy in the parallel direction. Magnetic-field fluctuations can transfer particle energy between the parallel and perpendicular degrees of freedom. Therefore, $\langle\delta j_\parallel\cdot \delta E_\parallel\rangle$ and $\langle\delta j_\perp\cdot \delta E_\perp\rangle$ cannot solely describe the energy transfer in parallel direction and perpendicular direction, respectively.