Weak Alfv\'enic turbulence in relativistic plasmas II: Current sheets and dissipation

Alfv\'{e}n waves as excited in black hole accretion disks and neutron star magnetospheres are the building blocks of turbulence in relativistic, magnetized plasmas. A large reservoir of magnetic energy is available in these systems, such that the plasma can be heated significantly even in the weak turbulence regime. We perform high-resolution three-dimensional simulations of counter-propagating Alfv\'{e}n waves, showing that an $E_{B_{\perp}}(k_{\perp}) \propto k_{\perp}^{-2}$ energy spectrum develops as a result of the weak turbulence cascade in relativistic magnetohydrodynamics and its infinitely magnetized (force-free) limit. The plasma turbulence ubiquitously generates current sheets, which act as locations where magnetic energy dissipates. We show that current sheets form as a natural result of nonlinear interactions between counter-propagating Alfv\'{e}n waves. These current sheets form due to the compression of elongated eddies, driven by the shear induced by growing higher order modes, and undergo a thinning process until they break-up into small-scale turbulent structures. We explore the formation of {current sheets} both in overlapping waves and in localized wave packet collisions. The relativistic interaction of localized Alfv\'{e}n waves induces both Alfv\'{e}n waves and fast waves and efficiently mediates the conversion and dissipation of electromagnetic energy in astrophysical systems. Plasma energization through reconnection in current sheets emerging during the interaction of Alfv\'{e}n waves can potentially explain X-ray emission in black hole accretion coronae and neutron star magnetospheres.