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In this paper, we report our detailed analysis of the new strong-couplingregime between Alfv\'en and fast modes in Poynting-dominated plasma turbulence,reported in our previous work Takamoto & Lazarian (2016), which is an importanteffect for many relativistic plasma phenomena, and calls for new theories ofPoynting-dominated MHD turbulence. We performed numerical simulations ofrelativistic MHD turbulence in isothermal plasmas, and analyzed the ratio offast to Alfv\'en mode energy. We found that the increase of the fast mode withthe background $\sigma$-parameter can be observed even in isothermal plasma,showing that such a phenomena is universal in trans-Alfv\'enic turbulence inPoynting-dominated plasmas. To study the detailed energy conversion process, wealso performed a series of simulations of decaying turbulence injecting pureAlfv\'en, fast, and slow modes, respectively, and investigated the developmentof the mode conversion from the each mode. We also found that the modeconversion between Alfv\'en and fast modes is nearly insensitive to thebackground temperature. Finally, we report a result of a simulation withinitially fast mode dominated turbulence. It developed into a temporallystrong-coupling regime, which is a strong evidence for the existence of oursuggesting strong-coupling regime of fast and Alfv\'en modes. Our resultsuggests that the strong turbulence in Poynting-dominated plasma is verydifferent from that in the non-relativistic plasma. It will also give animportant guidance to studies of particle acceleration and non-thermal photonemission from Poynting-dominated plasma.

Strong Coupling of Alfv\'en and Fast Modes in Compressible Relativistic Magnetohydrodynamic Turbulence in Magnetically-Dominated Plasmas