Energy balance and Alfv\'en Mach numbers in compressible magnetohydrodynamic turbulence with a large-scale magnetic field

Energy equipartition is a powerful theoretical tool for understanding astrophysical plasmas. It is invoked, for example, to measure magnetic fields in the interstellar medium (ISM), as evidence for small-scale turbulent dynamo action, and, in general, to estimate the energy budget of star-forming molecular clouds. In this study we motivate and explore the role of the volume-averaged root-mean-squared (rms) magnetic coupling term between the turbulent, $\delta\mathbf{B}$ and large-scale, $\mathbf{B}_0$ fields, $\left< (\delta\mathbf{B}\cdot\mathbf{B}_0)^{2} \right>^{1/2}_{\mathcal{V}}$. By considering the second moments of the energy balance equations we show that the rms coupling term is in energy equipartition with the volume-averaged turbulent kinetic energy for turbulence with a sub-Alfv\'enic large-scale field. Under the assumption of exact energy equipartition between these terms, we derive relations for the magnetic and coupling term fluctuations, which provide excellent, parameter-free agreement with time-averaged data from 280 numerical simulations of compressible MHD turbulence. Furthermore, we explore the relation between the turbulent, mean-field and total Alfv\'en Mach numbers, and demonstrate that sub-Alfv\'enic turbulence can only be developed through a strong, large-scale magnetic field, which supports an extremely super-Alfv\'enic turbulent magnetic field. This means that the magnetic field fluctuations are significantly subdominant to the velocity fluctuations in the sub-Alfv\'enic large-scale field regime. Throughout our study, we broadly discuss the implications for observations of magnetic fields and understanding the dynamics in the magnetised ISM.