Analytic characterization of sub-Alfv\'enic turbulence energetics

Magnetohydrodynamic (MHD) turbulence is a cross-field process relevant to many systems. A prerequisite for understanding these systems is to constrain the role of MHD turbulence, and in particular the energy exchange between kinetic and magnetic forms. The energetics of strongly magnetized and compressible turbulence has so far resisted attempts to understand them. Numerical simulations reveal that kinetic energy can be orders of magnitude larger than fluctuating magnetic energy. We solve this lack-of-balance puzzle by calculating the energetics of compressible and sub-Alfv\'enic turbulence based on the dynamics of coherent cylindrical fluid parcels. Using the MHD Lagrangian, we prove analytically that the bulk of the magnetic energy transferred to kinetic is the energy stored in the coupling between the ordered and fluctuating magnetic field. The analytical relations are in striking agreement with numerical data, up to second order terms.