Weak Alfv\'{e}nic turbulence in relativistic plasmas I: asymptotic solutions.

Alfv\'{e}n wave collisions are the primary building blocks of the non-relativistic turbulence that permeates the heliosphere and low-to-moderate energy astrophysical systems. However, many astrophysical systems such as gamma-ray bursts, pulsar and magnetar magnetospheres, and active galactic nuclei have relativistic flows or energy densities. To better understand these high energy systems, we derive reduced relativistic MHD equations and employ them to examine asymptotically weak Alfv\'{e}nic turbulence through third order in reduced relativistic magnetohydrodynamics, including the force-free, infinitely magnetized limit. We compare both numerical and analytical asymptotic solutions to demonstrate that many of the findings from non-relativistic weak turbulence are retained in the relativistic system. But, an important distinction in the relativistic limit is finite coupling to the compressible fast mode regardless of the strength of the magnetic field, i.e., the modes remain coupled even in the force-free limit. Since fast modes can propagate across field lines, this mechanism provides a route for energy to escape strongly magnetized systems, e.g., magnetar magnetospheres. However, we find that the fast-Alfv\'{e}n coupling is diminished in the limit of oblique propagation.