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Alfv\'{e}n wave collisions are the primary building blocks of thenon-relativistic turbulence that permeates the heliosphere and low-to-moderateenergy astrophysical systems. However, many astrophysical systems such asgamma-ray bursts, pulsar and magnetar magnetospheres, and active galacticnuclei have relativistic flows or energy densities. To better understand thesehigh energy systems, we derive reduced relativistic MHD equations and employthem to examine weak Alfv\'{e}nic turbulence, dominated by three-waveinteractions, in reduced relativistic magnetohydrodynamics, including theforce-free, infinitely magnetized limit. We compare both numerical andanalytical solutions to demonstrate that many of the findings fromnon-relativistic weak turbulence are retained in the relativistic system. But,an important distinction in the relativistic limit is the inapplicability of aformally incompressible limit, i.e, there exists finite coupling to thecompressible fast mode regardless of the strength of the magnetic field. Sincefast modes can propagate across field lines, this mechanism provides a routefor energy to escape strongly magnetized systems, e.g., magnetarmagnetospheres. However, we find that the fast-Alfv\'{e}n coupling isdiminished in the limit of oblique propagation.

Weak Alfv\'{e}nic turbulence in relativistic plasmas. Part 1. Dynamical equations and basic dynamics of interacting resonant triads