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Abstract Supersonic isothermal turbulence is a common process in astrophysical systems. In this work, we explore the energy in such systems. We show that the conserved energy is the sum of the kinetic energy ( K ) and Helmholtz free energy ( F ). We develop analytic predictions for the probability distributions, P ( F ) and P ( K ), as well as their nontrivial joint distribution, P ( F , K ). We verify these predictions with a suite of driven turbulence simulations, finding excellent agreement. The turbulence simulations were performed at Mach numbers ranging from 1 to 8, and three modes of driving: purely solenoidal, purely compressive, and mixed. We find that P ( F ) is discontinuous at F = 0, with the discontinuity increasing with Mach number and compressive driving. P ( K ) resembles a lognormal with a negative skew. The joint distribution, P ( F , K ), shows a bimodal distribution, with gas either existing at high F and high K or at low F and low K .
Rabatin et al. (2024) studied this question.