Energy balance and Alfvén Mach numbers in compressible magnetohydrodynamic turbulence with a large-scale magnetic field

Abstract

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, δB and largescale, B 0 fields, (δBV . 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-Alfvénic 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 Alfvén Mach numbers, and demonstrate that sub-Alfvénic turbulence can only be developed through a strong, large-scale magnetic field, which supports an extremely super-Alfvénic turbulent magnetic field. This means that the magnetic field fluctuations are significantly subdominant to the velocity fluctuations in the sub-Alfvénic 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.

Ion Alfvén velocity fluctuations and implications for the diffusion of streaming cosmic rays

“…As M A0 increases, and the strength of the large-scale field decreases the turbulent component of the magnetic field is able to grow, and we can observe some structure in the B-PDFs. Beattie et al , 2022, however small asymmetries between volumes with B > B 0 and B < B 0 develop and negatively skew the PDF. These features become extremely pronounced at M A0 = 10.0 (similar non-Gaussian features are very pronounced for M A0 = ∞; , where δB 2 B 2 0 , and hence, when the nonlinear terms in the fluid equations are the strongest (Oughton et al, 1994).…”

“…Due to the small-scale dynamo action, magnetic field fluctuations that are roughly at equipartition with the turbulent fluctuations are ubiquitous in both incompressible and compressible MHD turbulence across the Universe (Beck and Wielebinski, 2013;Subramanian, 2016Subramanian, , 2019. Based on a balance between the magnetic and kinetic fluctuations, in an isothermal plasma, the amplitudes of magnetic field fluctuations scale with Mf (M A0 ), where f (M A0 ) is a linear function for turbulence with a sub-Alfvénic large-scale field and a power law with exponent −1/3 in the super-Alfvénic regime (Federrath, 2016;Beattie et al, , 2022.…”

“…Magnetic fluctuations parallel to B 0 seem to play a very important role in compressible sub-Alfvénic large-scale field turbulence (Federrath, 2016;Skalidis and Tassis, 2020;Skalidis et al, 2021;Beattie et al, 2021bBeattie et al, , 2022, which, as we have discussed in Section 1, is relevant to cold molecular gas in the ISM Federrath et al, 2016;Heyer et al, 2020;Hwang et al, 2021;Hoang et al, 2021;Skalidis et al, 2021). The parallel fluctuations contain roughly an order of magnitude more energy than their perpendicular counterparts when the plasma is very sub-Alfvénic (and the same when the turbulence is super-Alfvénic) and hence become important for kinetic and magnetic energy practically speaking, the s-PDF is approximately normal.…”

“…balance in these plasma environments (Beattie et al, 2022). They also play a vital role in the formation of large-scale, non-turbulent structures in the plasma, e.g., for a sub-Alfvénic large-scale field Yang et al (2019) found that ≈77% of the total energy was in non-propagating structures (those that do not follow a theoretical wave dispersion relation).…”

“…As well as naturally inducing large-scale (at the system scale) anisotropy on the plasma, the large-scale field actively influences the very nature of the magnetized turbulence. For example, the rms turbulent field follows a power law with the large-scale field strength δB 2 1/2 ∝ B −1 0 , and hence the turbulence with respect to the fluctuating magnetic field component is extremely super-Alfvénic when the large-scale field is strong (Beattie et al, 2022). This is a natural repercussion of B 0 reducing the nonlinear terms (responsible for the turbulence) in the MHD fluid equations, and amplifying the linear terms (Oughton et al, 1994).…”