AppendixAppendix References

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

James R. Beattie${}^{\href https://orcid.org/0000-0001-9199-7771\,1}$, Mark R.
Krumholz${}^{\href https://orcid.org/0000-0003-3893-854X\,1,2}$, Raphael
Skalidis${}^{\href https://orcid.org/0000-0003-2337-0277\,3,4}$, Christoph
Federrath${}^{\href https://orcid.org/0000-0002-0706-2306\,1,2}$, Amit
Seta${}^{\href https://orcid.org/0000-0001-9708-0286\,1}$ Roland M.
Crocker${}^{\href https://orcid.org/0000-0002-2036-2426\,1}$, Philip
Mocz${}^{\href https://orcid.org/0000-0001-6631-2566\,5}$, and Neco
Kriel${}^{\href https://orcid.org/0000-0002-3558-3926\,1}$
1Research School of Astronomy and Astrophysics, Australian National
University, Canberra, ACT 2611, Australia
2Australian Research Council Centre of Excellence in All Sky Astrophysics
(ASTRO3D), Canberra, ACT 2611, Australia
3Institute of Astrophysics, Foundation for Research and Technology-Hellas,
Vasilika Vouton, GR-70013 Heraklion, Greece
4Department of Physics & ITCP, University of Crete, GR-70013, Heraklion,
Greece
5Lawrence Livermore National Laboratory, 7000 East Ave, Livermore, CA 94550,
USA
E-mail<EMAIL_ADDRESS>
(Accepted XXX. Received YYY; in original form ZZZ)

###### 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, $\delta\mathrm{{\mn@boldsymbol{\mathit{B}}}}$ and large-scale,
$\operatorname{\mathrm{{\mn@boldsymbol{\mathit{B}}}}_{0}}$ fields,
$\operatorname{\left\langle(\delta\mathrm{{\mn@boldsymbol{\mathit{B}}}}\cdot\operatorname{\mathrm{{\mn@boldsymbol{\mathit{B}}}}_{0}})^{2}\right\rangle^{1/2}_{\mathcal{V}}}$.
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-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.

###### keywords:

MHD – turbulence – ISM: kinematics and dynamics – ISM: magnetic fields –
dynamo

††pubyear: 2022††pagerange: Energy balance and Alfvén Mach numbers in
compressible magnetohydrodynamic turbulence with a large-scale magnetic
field–C

##  1 Introduction

Magnetohydrodynamic (MHD) turbulence is pervasive across the Universe, and for
this reason the study of MHD turbulence is a necessary prerequisite for
understanding a broad range of astrophysical processes. For example, each of
the planets in our Solar System probably assembled as the protoplanetary disc
underwent hydrodynamical and magnetohydrodynamical (MHD) instabilities,
driving turbulence and establishing the initial conditions for planet
formation (Lyra & Umurhan, 2019, and references therein). The Sun maintains a
magnetised and turbulent heliosphere, with decades of scale-free velocity and
magnetic fluctuations that play an important role in the generation of solar
winds, plasma heating, and particle acceleration (Bruno & Carbone, 2013, and
references therein). Just like the planets, the Sun was born in a turbulent
plasma environment.

In the context of star formation, turbulent density fluctuations in the cool
molecular gas clouds of galaxies seed the over-densities that fragment, become
gravitationally unstable, and collapse to form stars (Krumholz & McKee, 2005;
Padoan & Nordlund, 2011; Hennebelle et al., 2011; Federrath & Klessen, 2012;
Federrath, 2015; Hopkins, 2013; Burkhart, 2018; Mocz & Burkhart, 2018). The
turbulent motions themselves steepen or flatten the initial mass function
(IMF) of stars (Padoan et al., 1997; Hennebelle & Chabrier, 2009; Hopkins,
2012; Federrath et al., 2017b; Nam et al., 2021) and potentially underlie the
universality that we observe for the IMF by setting the density correlation
scale for star-forming regions (Jaupart & Chabrier, 2021), or more generally,
from the universality of the supersonic turbulence energy cascade in the
interstellar medium (ISM) (Padoan et al., 1997; Federrath, 2013). On scales
above the neutral-ion decoupling scale, magnetic fields are approximately
flux-frozen into the gas, and fluctuate, tangle and become turbulent with the
gas velocities, hence magnetic fields also play an important role in all these
processes (Hennebelle & Inutsuka, 2019; Krumholz & Federrath, 2019).

In the ISM magnetic fields and turbulence coexist in a partnership. Extremely
weak, primordial magnetic fields were potentially formed through a battery
process (e.g., Biermann, 1950), or a phase transition in the early Universe
(Subramanian, 2016, 2019), and, once generated, they are hard to destroy due
to the lack of magnetic monopoles (Parker, 1970; Beck & Wielebinski, 2013;
Acharya et al., 2022). Instead, turbulent motions of gas exponentially amplify
the weak seed fields, growing them through the turbulent dynamo and
magnetising the plasma (see McKee et al. 2020 for a recent review).
Turbulence, through the dynamo process, likely continues to maintain the
magnetic fields found in the present-day Universe, ensuring they are roughly
in energy equipartition with the turbulent motions, i.e., the saturated state
of the turbulent dynamo (Federrath et al., 2014; Schober et al., 2015;
Federrath, 2016; Xu & Lazarian, 2016; McKee et al., 2020; Seta et al., 2020;
Seta & Federrath, 2020; Achikanath Chirakkara et al., 2021; Seta & Federrath,
2021a).

Given the importance of MHD turbulence in the ISM, it is not surprising that
the ISM community has built numerous tools for measuring turbulent properties
(for a recent review, see Burkhart 2021). These include, but are certainly not
limited to: techniques that relate starlight polarisation dispersion to plane-
of-sky magnetic fields strengths, such as the methods described in Skalidis &
Tassis (2020), Davis (1951) and Chandrasekhar & Fermi (1953) (for recent
reviews, extensions and modifications, see Lazarian et al. 2020, 2022);
inference of magnetic field strengths and plasma energetics from local
velocity centroids or intensity fluctuations (Lazarian et al., 2018) and
density gradients of dust continuum maps (Soler et al., 2013); ascertaining
the ratio of compressive to solenoidal modes of turbulent driving sources from
the deprojected column density (Federrath et al., 2009; Brunt et al., 2010a,
b; Brunt & Federrath, 2014; Körtgen & Soler, 2020; Menon et al., 2021; Sharda
et al., 2021); and data-driven statistical techniques that can capture
abstract features of sub- or super-Alfvénic turbulence using wavelet
scattering transforms (Allys et al., 2019; Saydjari et al., 2021) or deep
convolutional neural networks (Peek & Burkhart, 2019). All of these
diagnostics rely upon a thorough, physical understanding of the underlying
phenomenology of MHD turbulence.

However, the parameter space of MHD turbulence is large, and there need not be
a universal phenomenology that captures the richness of the topic (see the
eloquent review from Schekochihin 2020 about phenomenologies and two-point
statistical models for incompressible MHD turbulence, which continue to be
subject to debate; Iroshnikov 1964; Kraichnan 1965; Sridhar & Goldreich 1994;
Goldreich & Sridhar 1995; Boldyrev 2006). In this study we aim to explore
energy balance in a particular part of the parameter space relevant to the
ISM: isothermal, highly-compressible MHD turbulence, driven with a mixture of
compressible and solenoidal modes, and that is threaded by a large-scale
magnetic field, $\operatorname{\mathrm{{\mn@boldsymbol{\mathit{B}}}}_{0}}$,
flux-frozen on the system scale. Such a description is potentially applicable
to any of the approximately isothermal phases of the ISM (Wolfire et al.,
1995; Omukai et al., 2005). In this context, we can identify several distinct
energy reservoirs, but in this study our main aim is to understand the
correlation between the large-scale and turbulent magnetic field;
mathematically, this term takes the form
$\operatorname{\delta\mathrm{{\mn@boldsymbol{\mathit{B}}}}\cdot\operatorname{\mathrm{{\mn@boldsymbol{\mathit{B}}}}_{0}}}$,
where $\delta\mathrm{{\mn@boldsymbol{\mathit{B}}}}$ and
$\operatorname{\mathrm{{\mn@boldsymbol{\mathit{B}}}}_{0}}$ are the fluctuating
and large-scale fields, respectively. We henceforth refer to this as the
“magnetic coupling term”, or simply the “coupling term”. This term has been
neglected previously in the literature (e.g., Zweibel & McKee, 1995) because
when averaging over a volume $\operatorname{\mathcal{V}}$ that contains a few
turbulent correlation scales,
$\left\langle\operatorname{\delta\mathrm{{\mn@boldsymbol{\mathit{B}}}}\cdot\operatorname{\mathrm{{\mn@boldsymbol{\mathit{B}}}}_{0}}}\right\rangle_{\operatorname{\mathcal{V}}}=0$.
However, we show that when one instead considers the $2^{\rm nd}$ moments of
the energy equation (the fluctuations of energy), which maintains the
positivity for all of the contributions to the energy, including
$\operatorname{\delta\mathrm{{\mn@boldsymbol{\mathit{B}}}}\cdot\operatorname{\mathrm{{\mn@boldsymbol{\mathit{B}}}}_{0}}}$,
the coupling term plays a leading order role in the energy balance when
$\operatorname{\mathrm{{\mn@boldsymbol{\mathit{B}}}}_{0}}$ is strong,
corresponding to sub-to-trans-Alfvénic turbulence.

Skalidis & Tassis (2020) and Skalidis et al. (2021b, hereafter S+2021)
recently showed that the coupling term is important for measuring the plane-
of-sky magnetic field using polarisation dispersion techniques for
interstellar gas, especially in highly-magnetised regions of the ISM (Li et
al., 2013; Federrath et al., 2016; Hu et al., 2019; Heyer et al., 2020; Hwang
et al., 2021; Hoang et al., 2021; Skalidis et al., 2021a). In this paper, we
show that by constructing a set of analytical models for the coupling term and
turbulent magnetic fluctuations, based on kinetic and magnetic energy balance,
one can derive strong constraints on the magnetic fluctuations and Alfvén Mach
numbers $\operatorname{\mathcal{M}_{\text{A}}}$ in the plasma. We also study
the impact of a large-scale magnetic field on the turbulence by analysing the
turbulent, total and mean-field $\operatorname{\mathcal{M}_{\text{A}}}$, and
the relationships between them. Beyond significantly suppressing the turbulent
component of the magnetic field as the large-scale field grows in energy in a
power-law fashion, $\delta B\propto B_{0}^{-1}$, we show that having a strong
large-scale field is a necessary prerequisite for sub-Alfvénic turbulence,
i.e., a plasma can only be in the sub-Alfvénic regime when the large-scale,
ordered field contains almost all of the magnetic energy, making the magnetic
fluctuations highly super-Alfvénic and hence dynamically sub-dominant.

This study is organised as follows: in Section 2 we outline the compressible
MHD turbulence simulations that we will use. In Section 3 we review the basics
of energy balance between magnetic and kinetic energy in MHD turbulence. We
focus upon the coupling term, justify why it ought to be considered in the
energy balance equation, and in Section 4 we provide analytical models for
this term in both the super- and sub-Alfvénic regimes. In Section 5 we turn
our attention to the fluid energetics in the context of the Alfvén Mach
number, highlighting the difference between the turbulent, mean-field, and
total Alfvén Mach numbers and deriving relationships between them. Next, in
Section 6 we discuss the role of the turbulent correlation scale for measuring
magnetic field statistics in simulations and observations. Finally, in Section
7 we summarise the key results of this study.

##  2 Numerical simulations

To test our energy balance models, we use a modified version of the flash code
(Fryxell et al., 2000; Dubey et al., 2008), utilising a second-order
conservative MUSCL-Hancock 5-wave approximate Riemann scheme (Bouchut et al.,
2010; Waagan et al., 2011; Federrath et al., 2021) to solve the 3D, ideal,
isothermal, compressible MHD equations with a stochastic acceleration field
acting to drive non-helical turbulence,

$\displaystyle\frac{\partial\rho}{\partial
t}+\nabla\cdot(\rho\mathrm{{\mn@boldsymbol{\mathit{v}}}})$ $\displaystyle=0,$
(1)
$\displaystyle\rho\frac{\partial\mathrm{{\mn@boldsymbol{\mathit{v}}}}}{\partial
t}-\nabla\cdot\left[\frac{1}{4\pi}\mathrm{{\mn@boldsymbol{\mathit{B}}}}\otimes\mathrm{{\mn@boldsymbol{\mathit{B}}}}-\rho\mathrm{{\mn@boldsymbol{\mathit{v}}}}\otimes\mathrm{{\mn@boldsymbol{\mathit{v}}}}-\left(c_{s}^{2}\rho+\frac{B^{2}}{8\pi}\right)\mathbb{I}\right]$
$\displaystyle=\rho\mathrm{{\mn@boldsymbol{\mathit{f}}}},$ (2)
$\displaystyle\frac{\partial\mathrm{{\mn@boldsymbol{\mathit{B}}}}}{\partial
t}-\nabla\times(\mathrm{{\mn@boldsymbol{\mathit{v}}}}\times\mathrm{{\mn@boldsymbol{\mathit{B}}}})$
$\displaystyle=0,$ (3)
$\displaystyle\nabla\cdot\mathrm{{\mn@boldsymbol{\mathit{B}}}}$
$\displaystyle=0,$ (4)

where $\otimes$ is the tensor product and $\mathbb{I}$, the identity matrix.
We solve the equations on a periodic domain of dimension
$L^{3}\equiv\operatorname{\mathcal{V}}_{L}$, discretised with between
$16^{3}-1152^{3}$ cells, where $\mathrm{{\mn@boldsymbol{\mathit{v}}}}$ is the
fluid velocity, $\rho$ is the gas density,
$\mathrm{{\mn@boldsymbol{\mathit{B}}}}=B_{0}\mathrm{{\mn@boldsymbol{\mathit{\hat{e}}}}}_{\parallel}+\delta\mathrm{{\mn@boldsymbol{\mathit{B}}}}(t)$
is the magnetic field, with mean-field
$B_{0}\mathrm{{\mn@boldsymbol{\mathit{\hat{e}}}}}_{\parallel}$111Note that we
refer to a mean-field coordinate system, as adopted in Hartlep et al. (2000),
where $\operatorname{\mathrm{{\mn@boldsymbol{\mathit{B}}}}_{0}}$ always points
along
$\mathrm{{\mn@boldsymbol{\mathit{\hat{z}}}}}=\mathrm{{\mn@boldsymbol{\mathit{\hat{e}}}}}_{\parallel}$
and hence
$\mathrm{{\mn@boldsymbol{\mathit{\hat{x}}}}}=\mathrm{{\mn@boldsymbol{\mathit{\hat{e}}}}}_{\perp,1}$
and
$\mathrm{{\mn@boldsymbol{\mathit{\hat{y}}}}}=\mathrm{{\mn@boldsymbol{\mathit{\hat{e}}}}}_{\perp,2}$.
The plasma is statistically symmetric in the
$(\mathrm{{\mn@boldsymbol{\mathit{\hat{e}}}}}_{\perp,1},\mathrm{{\mn@boldsymbol{\mathit{\hat{e}}}}}_{\perp,2})$
plane, so we will regularly state quantities for
$\mathrm{{\mn@boldsymbol{\mathit{\hat{e}}}}}_{\perp}$. and turbulent field,
$\delta\mathrm{{\mn@boldsymbol{\mathit{B}}}}(t)$, where
$\left\langle\delta\mathrm{{\mn@boldsymbol{\mathit{B}}}}(t)\right\rangle_{\operatorname{\mathcal{V}}_{L}}=0$,
$c_{s}$ is the sound speed and $\mathrm{{\mn@boldsymbol{\mathit{f}}}}$, the
stochastic turbulent acceleration source term that drives the turbulence,
which, in the ISM could be from, for example, supernova shocks, internal
instabilities in the gas, gravity, galactic-scale shocks and shear, or ambient
pressure from the galactic environment (Brunt et al., 2009; Elmegreen, 2009;
Federrath, 2015; Krumholz & Burkhart, 2016; Padoan et al., 2016; Grisdale et
al., 2017; Jin et al., 2017; Körtgen et al., 2017; Federrath et al., 2017a;
Colling et al., 2018; Schruba et al., 2019; Lu et al., 2020). Here, and
throughout this paper, we use the notation
$\left\langle\ldots\right\rangle_{\operatorname{\mathcal{V}}}$ to indicate the
mean value of some quantity within a specified volume
$\operatorname{\mathcal{V}}$ (which can be the entire simulation volume
$\operatorname{\mathcal{V}}_{L}$, but need not be). We discuss the resolution
of our simulations, and demonstrate that the quantities of interest for us are
converged in them, in Appendix A.

The forcing term $\mathrm{{\mn@boldsymbol{\mathit{f}}}}$ follows an Ornstein-
Uhlenbeck process with finite correlation time, $\tau=\ell_{0}/\left\langle
v^{2}\right\rangle_{\operatorname{\mathcal{V}}_{L}}^{1/2}=L/(2c_{s}\mathcal{M})$,
where $\mathcal{M}$ is the sonic Mach number, such that $\ell_{0}=L/2$ is the
driving, or energy injection scale, and
$\mathrm{{\mn@boldsymbol{\mathit{f}}}}$ is constructed so that we are able to
set $0.5\lesssim\mathcal{M}\lesssim 10$, encapsulating the $\mathcal{M}$
values of supersonic molecular gas clouds in the interstellar medium (e.g.,
Schneider et al., 2013; Federrath et al., 2016; Orkisz et al., 2017; Beattie
et al., 2019) as well as the subsonic, diffuse, warm medium (e.g., Kritsuk et
al., 2017). We force with equal energy in both compressive
$(\nabla\times\mathrm{{\mn@boldsymbol{\mathit{f}}}}=0)$ and solenoidal
$(\nabla\cdot\mathrm{{\mn@boldsymbol{\mathit{f}}}}=0)$ modes. The energy
injection is isotropic, centred on
$|\mathrm{{\mn@boldsymbol{\mathit{k}}}}L/2\pi|=2$ and falling off to zero with
a parabolic spectrum within
$1\leq|\mathrm{{\mn@boldsymbol{\mathit{k}}}}L/2\pi|\leq 3$ (see Federrath et
al. 2008; Federrath et al. 2009; Federrath et al. 2010; Federrath et al. 2022
for turbulence driving details). $\operatorname{\mathcal{M}_{\text{A0}}}$ is
set by fixing $B_{0}$ and using the definition of the mean-field Alfvén
velocity and $\mathcal{M}$,
$\operatorname{\mathcal{M}_{\text{A0}}}=2c_{s}\sqrt{\pi\rho_{0}}\mathcal{M}/B_{0}$.
We vary this value for each of the simulations between
$10^{-2}\lesssim\operatorname{\mathcal{M}_{\text{A0}}}\lesssim 10^{3}$,
resulting in a total of 280 simulations across different grid resolutions,
with 56 unique simulations, which we list in LABEL:tb:simtab. The initial
velocity field is set to
$\mathrm{{\mn@boldsymbol{\mathit{v}}}}(x,y,z,t=0)=\mathrm{{\mn@boldsymbol{\mathit{0}}}}$,
with units $c_{s}=1$, the density field $\rho(x,y,z,t=0)=\rho_{0}$, with units
$\rho_{0}=1$, and
$\delta\mathrm{{\mn@boldsymbol{\mathit{B}}}}(t=0)=\mathrm{{\mn@boldsymbol{\mathit{0}}}}$,
with units $c_{s}\rho_{0}^{1/2}=1$.

We run the simulations for 10 correlation times of
$\mathrm{{\mn@boldsymbol{\mathit{f}}}}$, and report statistics from time-
averages over the last 5 correlation times to ensure that the sub-Alfvénic
mean-field simulations are statistically stationary (Beattie et al., 2021a).
After 5 correlation times large vortical structures develop in the sub-
Alfvénic mean-field simulations, extending along the strong large-scale field
and out to the driving scale perpendicular to the field, which we show, as an
example, in Figure 1 for the M2MA01 simulation.222We use a naming convention
for our simulations whereby the value following the M gives the target sonic
Mach number $\mathcal{M}$ (with decimal points omitted) and the value
following MA gives the target Alfvén Mach number $\mathcal{M}_{\rm A0}$ – thus
run M2MA01 is one where we set the mean magnetic field and tune the forcing to
produce $\mathcal{M}=2$ and $\mathcal{M}_{\rm A0}=0.1$. For more details about
the current simulations, we refer the readers to Beattie & Federrath (2020)
for the anisotropy in $\rho/\rho_{0}$, Beattie et al. (2020) for a detailed
analysis of $\delta\mathrm{{\mn@boldsymbol{\mathit{B}}}}$, Beattie et al.
(2021b) for an anisotropic model of the $\rho/\rho_{0}$ variance, and Beattie
et al. (2021a) for the density intermittency and the $\ln(\rho/\rho_{0})$-PDF.

Figure 1: Typical velocity (red) streamline structure in sub-Alfvénic mean-
field turbulence, with $\mathcal{M}=2$ and
$\operatorname{\mathcal{M}_{\text{A0}}}=0.1$. The direction of the mean-field,
$\operatorname{\mathrm{{\mn@boldsymbol{\mathit{B}}}}_{0}}$, is shown in the
bottom right corner of the box. Slices of the density are shown in grey scale
at the box edges. Ordered vortex structures occupy the full extent of the box
along $\hat{e}_{\parallel}$, and out to the driving scale in
$\hat{e}_{\perp}$. Figure 2: The joint PDF for $\delta B_{\parallel}$ and
$\delta B_{\perp}$ for the M2MA001 simulation, showing a long, asymmetric tail
into the negative values for $\delta B_{\parallel}$. As demonstrated in
S+2021, the $\delta B_{\perp}$ fluctuations are symmetric about $\delta
B_{\perp}=0$, and are analogous to a harmonic oscillator (in magnetic
amplitude space) that is restored by the magnetic tension $\propto\delta
B_{\perp}^{2}$ with a quadratic potential. On the other hand, the $\delta
B_{\parallel}$ amplitude fluctuates anharmonically, $\propto\delta
B_{\parallel}$, with a linear potential. In sub-to-trans-Alfvénic compressible
MHD the anharmonic, parallel magnetic field fluctuations contain most of the
magnetic energy.

##  3 Energy balance

###  3.1 Energy balance basics & averaging

Recent studies have shown that one can use energy balance arguments that
include the large-scale magnetic field,
$\operatorname{\mathrm{{\mn@boldsymbol{\mathit{B}}}}_{0}}$, to derive scaling
laws between the Alfvénic and kinetic fluid quantities (Federrath 2016;
Beattie et al. 2020; Skalidis & Tassis 2020, S+2021). The dimensionless
magnetic energy density, by which we mean the magnetic energy density
normalised to the mean thermal pressure333A natural normalisation for an
isothermal plasma because $\rho_{0}$ and $c_{s}$ are both constant, and
problem dependant. $\rho_{0}c_{s}^{2}$, is

$\displaystyle\operatorname{e_{\rm mag}}=\frac{B^{2}}{8\pi
c_{s}^{2}\rho_{0}}=\frac{1}{8\pi
c_{s}^{2}\rho_{0}}\Big{(}B_{0}^{2}+\underbrace{2\delta\mathrm{{\mn@boldsymbol{\mathit{B}}}}\cdot\mathrm{{\mn@boldsymbol{\mathit{B}}}}_{0}}_{\mathclap{\begin{subarray}{c}\text{coupling}\\\
\text{term}\end{subarray}}}{}+\delta B^{2}\Big{)},$ (5)

where $B_{0}^{2}$ is the large-scale field contribution to the total energy,
$\delta B^{2}$ is the turbulent field contribution and
$2\delta\mathrm{{\mn@boldsymbol{\mathit{B}}}}\cdot\mathrm{{\mn@boldsymbol{\mathit{B}}}}_{0}$
is the coupling term between the two field components. In the linear
perturbation theory limit of the MHD equations, $\delta B^{2}$ includes
contributions from shear Alfvén, fast and slow magnetosonic compressive
eigenmodes (e.g., Landau & Lifshitz, 1959). Because
$\delta\mathrm{{\mn@boldsymbol{\mathit{B}}}}\cdot\mathrm{{\mn@boldsymbol{\mathit{B}}}}_{0}=\delta
B_{\parallel}B_{0}$, the coupling term only contains the component of magnetic
field fluctuations that are parallel to the large-scale field. In linear
theory, both fast and slow magnetosonic compressible modes are able to perturb
the field variables parallel to
$\operatorname{\mathrm{{\mn@boldsymbol{\mathit{B}}}}_{0}}$, so under the lens
of linear theory, the coupling term is the fluctuation contribution from the
compressible modes in the turbulence scaled by $B_{0}$ (Bhattacharjee et al.,
1998). Furthermore, for sub-Alfvénic turbulence Beattie et al. (2021a) showed
that converging, shocked flows along magnetic field lines excite strong
$\delta B_{\parallel}$ fluctuations, which travel roughly at the theoretical
fast Alfvén mode speed. Therefore, it is likely, assuming that $\delta
B_{\parallel}/B_{0}\ll 1$ (this is indeed the case for
$\operatorname{\mathcal{M}_{\text{A0}}}<1$ plasmas; see left panel of Figure 5
in Beattie et al. 2022) where a linear theory may be valid for the magnetic
field, the coupling term contains significant energy contributions from fast
magnetosonic modes excited by shocked gas that converges and forms dense
filaments perpendicular to magnetic field lines.

The excitation of a dominating $\delta B_{\parallel}$ is something
characteristic of sub-Alfvénic compressible turbulence. We demonstrate this by
plotting the time-averaged joint $\delta B_{\parallel}-\delta B_{\perp}$ PDF
in Figure 2, for highly-sub-Alfvénic turbulence
$(\operatorname{\mathcal{M}_{\text{A0}}}=0.01)$. It is evident that the
distributions of $\delta B_{\parallel}$ and $\delta B_{\perp}$ are not the
same444We show the super-Alfvénic version of this plot, which admits to
isotropic fluctuations, in Figure 16.. The reason is straightforward: $\delta
B_{\perp}$ (fluctuations from shear Alfvén waves) is subject to a quadratic
restoring force via the magnetic tension555When $B_{0}\gg\delta B$,
$(\mathrm{{\mn@boldsymbol{\mathit{B}}}}\cdot\nabla)\mathrm{{\mn@boldsymbol{\mathit{B}}}}\approx-\kappa
B_{0}^{2}\mathrm{{\mn@boldsymbol{\mathit{\hat{e}}}}}_{\perp}$, where $\kappa$
is the field line curvature. Hence
$(\mathrm{{\mn@boldsymbol{\mathit{B}}}}\cdot\nabla)\mathrm{{\mn@boldsymbol{\mathit{B}}}}$
acts to strongly dampen shear Alfvén waves. This approximation for
$(\mathrm{{\mn@boldsymbol{\mathit{B}}}}\cdot\nabla)\mathrm{{\mn@boldsymbol{\mathit{B}}}}$
is most appropriate for regions of the plasma where $\nabla_{\parallel}\cdot
v_{\parallel}\approx 0$, because compressions can excite $\delta
B_{\parallel}$, creating parallel gradients in the magnetic field that also
act to increase the tension (Beattie et al., 2021a) (Yuen & Lazarian, 2020;
Beattie et al., 2021a), which results in a symmetry about $\delta
B_{\perp}=0$. However, $\delta B_{\parallel}$ has a linear restoring force and
is forced out of the minimum energy state $\delta B_{\parallel}=-B_{0}$ to
conform to $\left\langle\delta
B_{\parallel}\right\rangle_{\operatorname{\mathcal{V}}}=0$666Note that in the
language of solid state physics, we may consider $\delta B_{\parallel}$ to be
a topologically frustrated field, because the minimum energy state is $\delta
B_{\parallel}=-B_{0}$, but conservation of total magnetic flux requires
$\left\langle\delta
B_{\parallel}\right\rangle_{\operatorname{\mathcal{V}}}=0$. Hence, populations
of parallel magnetic fluctuations can be imagined to compete to get to $\delta
B_{\parallel}=-B_{0}$, but for every $\delta B_{\parallel}$ that comes close
to $-B_{0}$ there must be either another fluctuation that comes close to
$+B_{0}$ or a population of fluctuations that in total add to $+B_{0}$,
ensuring globally that $\left\langle\delta
B_{\parallel}\right\rangle_{\operatorname{\mathcal{V}}}=0$. We do not take
this analogy any further in this study but it may stimulate future works on
magnetic field fluctuation PDFs. (S+2021). This gives rise to a skewed
distribution in $\delta B_{\parallel}$, with a long extended tail of negative
$\delta B_{\parallel}$ values. We will show below that $\delta B_{\parallel}$
contains almost all of the turbulent magnetic energy in the compressible
plasma. Now we turn our attention to what feeds the magnetic field
fluctuations.

The dimensionless turbulent kinetic energy, normalised by the mean thermal
pressure (similarly to $\operatorname{e_{\rm mag}}$; see Equation 5), is

$\displaystyle\operatorname{e_{\rm kin}}=\frac{1}{2}\left(\frac{\delta
v}{c_{s}}\right)^{2},$ (6)

which acts as an energy reservoir for the magnetic field fluctuations via the
velocity term in the induction equation, Equation 3. Considering our ideal,
isothermal (in our units, the thermal energy is $e_{\rm thermal}=3/2$), MHD
system, the total energy is then

$\displaystyle e_{\rm tot}$ $\displaystyle=\frac{1}{2}\left(\frac{\delta
v}{c_{s}}\right)^{2}+\frac{1}{8\pi
c_{s}^{2}\rho_{0}}\left(B_{0}^{2}+2\delta\mathrm{{\mn@boldsymbol{\mathit{B}}}}\cdot\mathrm{{\mn@boldsymbol{\mathit{B}}}}_{0}+\delta
B^{2}\right)+\frac{3}{2},$ (7)

and for just the ‘total’ turbulent energy,

$\displaystyle e_{\rm turb}$ $\displaystyle=\frac{1}{2}\left(\frac{\delta
v}{c_{s}}\right)^{2}+\frac{1}{8\pi
c_{s}^{2}\rho_{0}}\left(2\delta\mathrm{{\mn@boldsymbol{\mathit{B}}}}\cdot\mathrm{{\mn@boldsymbol{\mathit{B}}}}_{0}+\delta
B^{2}\right),$ (8)

where only the $\delta B^{2}$ and
$\operatorname{\delta\mathrm{{\mn@boldsymbol{\mathit{B}}}}\cdot\operatorname{\mathrm{{\mn@boldsymbol{\mathit{B}}}}_{0}}}$
terms are retained in the magnetic energy, because they contain the turbulent
contribution.

In a fluid with initially weak magnetic fluctuations and $B_{0}=0$,
$\operatorname{e_{\rm kin}}$ (Equation 6) will transfer energy and enhance
$\operatorname{e_{\rm mag}}$ (Equation 5 with $B_{0}$ set to 0) via the small-
scale turbulent dynamo (for a recent review see McKee et al., 2020). A
standard ansatz of dynamo theory is that saturation will be reached between
the turbulent fields, such that,

$\displaystyle\frac{\left\langle\operatorname{e_{\rm
mag}}\right\rangle_{\operatorname{\mathcal{V}}}}{\left\langle\operatorname{e_{\rm
kin}}\right\rangle_{\operatorname{\mathcal{V}}}}=\operatorname{\alpha_{\rm
sat}},$ (9)

where $0\leq\operatorname{\alpha_{\rm sat}}\leq 1$. The value of
$\operatorname{\alpha_{\rm sat}}$ is a function of $\mathcal{M}$, the Alfvén
Mach number $\operatorname{\mathcal{M}_{\text{A}}}$ (a precise definition for
which we defer to Section 5), the nature of the driving mechanism,
$\mathrm{{\mn@boldsymbol{\mathit{f}}}}$, in particular if it is compressive
$\nabla\times\mathrm{{\mn@boldsymbol{\mathit{f}}}}=0$ or solenoidal
$\nabla\cdot\mathrm{{\mn@boldsymbol{\mathit{f}}}}=0$, and the Prandtl and
Reynolds numbers of the fluid (Federrath et al., 2011a; Schober et al., 2012;
Federrath et al., 2014; Schober et al., 2015; Federrath, 2016; Achikanath
Chirakkara et al., 2021; Kriel et al., 2022). The exact physics of the
saturation is still an open problem in dynamo theory, but most likely the
saturation develops due to the effect of strong magnetic fields on both the
amplification (via field line stretching), diffusion of magnetic fields and
instabilities caused by tearing and magnetic reconnection (Schekochihin et
al., 2002; Xu & Lazarian, 2016; Seta & Federrath, 2021a; Galishnikova et al.,
2022); however, the exact value of $\operatorname{\alpha_{\rm sat}}$ and its
dependence on other parameters is not important for our purposes. What is
significant is that, assuming that the energy transfer from
$\operatorname{e_{\rm kin}}$ to $\operatorname{e_{\rm mag}}$ is solely through
the turbulent components of the respective fields, including the turbulent and
large-scale field coupling term for the more general case where $B_{0}\neq 0$,
from Equation 8, Equation 9 becomes

$\displaystyle\frac{1}{8\pi c_{s}^{2}\rho_{0}}\left\langle
2\delta\mathrm{{\mn@boldsymbol{\mathit{B}}}}\cdot\operatorname{\mathrm{{\mn@boldsymbol{\mathit{B}}}}_{0}}+\delta
B^{2}\right\rangle_{\operatorname{\mathcal{V}}}=\frac{\operatorname{\alpha_{\rm
sat}}}{2}\left\langle\left(\frac{\delta
v}{c_{s}}\right)^{2}\right\rangle_{\operatorname{\mathcal{V}}},$ (10)

which naively reduces to

$\displaystyle\frac{1}{8\pi c_{s}^{2}\rho_{0}}\left\langle\delta
B^{2}\right\rangle_{\operatorname{\mathcal{V}}}=\frac{\operatorname{\alpha_{\rm
sat}}}{2}\left\langle\left(\frac{\delta
v}{c_{s}}\right)^{2}\right\rangle_{\operatorname{\mathcal{V}}},$ (11)

if
$\left\langle\delta\mathrm{{\mn@boldsymbol{\mathit{B}}}}\cdot\operatorname{\mathrm{{\mn@boldsymbol{\mathit{B}}}}_{0}}\right\rangle_{\operatorname{\mathcal{V}}}=\left\langle\delta
B_{\parallel}\right\rangle_{\operatorname{\mathcal{V}}}B_{0}=0$ because
$\left\langle\delta B_{\parallel}\right\rangle_{\operatorname{\mathcal{V}}}=0$
when $\operatorname{\mathcal{V}}$ captures a few correlation lengths of the
turbulence, for the regular Reynolds (1895) decomposition of a stochastic
field777S+2021 showed that even this leads to complications because
$\operatorname{\delta\mathrm{{\mn@boldsymbol{\mathit{B}}}}\cdot\operatorname{\mathrm{{\mn@boldsymbol{\mathit{B}}}}_{0}}}$
is analogous to a potential energy, which does not make sense to average
because it is invariant to gauge transforms, not positive definite, nor
symmetric around the minimum energy state for $\delta B_{\parallel}$. See
discussion in §4 of S+2021 for more details.. But this is not necessarily a
sensible result because when the large-scale field is strong the coupling term
is leading order in the turbulent magnetic energy, and all energy reservoirs
should be strictly positive. Because the coupling term is the only term that
is not positive semi-definite in Equation 10 we may want to treat averaging
the equation with more care.

These considerations lead us to consider an alternative ansatz, one that
enforces the positivity of all terms. Our approach is to take the $2^{\rm nd}$
moments of Equation 10, but also taking the square root to ensure that the
units are appropriate for an energy balance,

$\displaystyle\frac{1}{8\pi
c_{s}^{2}\rho_{0}}\left\langle\left(2\delta\mathrm{{\mn@boldsymbol{\mathit{B}}}}\cdot\operatorname{\mathrm{{\mn@boldsymbol{\mathit{B}}}}_{0}}+\delta
B^{2}\right)^{2}\right\rangle_{\operatorname{\mathcal{V}}}^{1/2}=\frac{\operatorname{\alpha_{\rm
sat}}}{2}\left\langle\left(\frac{\delta
v}{c_{s}}\right)^{4}\right\rangle^{1/2}_{\operatorname{\mathcal{V}}}.$ (12)

The physical interpretation of this balance is that instead of balancing the
means of the energy distributions, we balance the root-mean-squared values,
which are a measure of the typical local fluctuations in energy of the plasma.
This method of volume-averaging Equation 10 gives rise to $4^{\rm th}$ order
terms in velocity and magnetic field fluctuations ($2^{\rm nd}$ order in
energies). Note that for finite $\operatorname{\mathcal{M}_{\text{A0}}}$,
$\operatorname{\alpha_{\rm sat}}$ is now different from
$\operatorname{\alpha_{\rm sat}}$ in small-scale dynamo experiments (where
$B_{0}=0$) because it is now sensitive to the large-scale field through the
energy contribution of the coupling term.

To better understand the $4^{\rm th}$ order terms we plot them as a function
of $2^{\rm nd}$ order terms in Figure 3 and show the 1:1 and 1:2 lines with
dashes and dots, in each of the plots, respectively. Using least-squares
fitting we find that $\left\langle\delta
v^{4}\right\rangle^{1/2}_{\operatorname{\mathcal{V}}}=(1.3\pm
0.3)\left\langle\delta v^{2}\right\rangle_{\operatorname{\mathcal{V}}}$, and
$\left\langle\delta
B^{4}\right\rangle^{1/2}_{\operatorname{\mathcal{V}}}=(1.5\pm
0.4)\left\langle\delta B^{2}\right\rangle_{\operatorname{\mathcal{V}}}$,
hence, within $\approx 1\sigma$, the proportionality constants are
approximately unity. Physically, this means that as the mean of the energy
distributions increase, so does the root-mean-squared, or spread of the
distributions. This has been shown before, for example, in Schekochihin et al.
(2004), where they found $\left\langle\delta
B^{4}\right\rangle^{1/2}_{\operatorname{\mathcal{V}}}\approx\sqrt{2}\left\langle\delta
B^{2}\right\rangle_{\operatorname{\mathcal{V}}}$ (see Fig 11, saturated
regime). This is an important point, because it means that the contributions
from the turbulent fields remain approximately the same in both averaging
schemes, Equation 10 and Equation 12, but now we are able to properly include
the energy contribution from the coupling term.

Figure 3: The square root of the $4^{\rm th}$ order turbulent velocity (left)
and magnetic field (right) moments as a function of $2^{\rm nd}$ order
moments, coloured by $\operatorname{\mathcal{M}_{\text{A0}}}$, for all of the
simulations listed in LABEL:tb:simtab. The $2^{\rm nd}$ order moments of the
turbulent fields are approximately equivalent to the scaled $4^{\rm th}$ order
moments, $\left\langle\delta
v^{4}\right\rangle^{1/2}_{\operatorname{\mathcal{V}}}=(1.3\pm
0.3)\left\langle\delta v^{2}\right\rangle_{\operatorname{\mathcal{V}}}$, and
$\left\langle\delta
B^{4}\right\rangle^{1/2}_{\operatorname{\mathcal{V}}}=(1.5\pm
0.4)\left\langle\delta B^{2}\right\rangle_{\operatorname{\mathcal{V}}}$,
rarely deviating by more than a factor of 2 (dotted line) from the 1:1 dashed
line. This means that the standard deviation of the magnetic and kinetic
energy distributions scale with the mean.

###  3.2 Weak and strong B-field limits for rms energy balance

Consider now Equation 12 in the weak
$\operatorname{\mathrm{{\mn@boldsymbol{\mathit{B}}}}_{0}}$ regime, such that
$B_{0}\ll\delta B$, averaged over $\operatorname{\mathcal{V}}$. This means

$\displaystyle\left\langle\left(2\delta\mathrm{{\mn@boldsymbol{\mathit{B}}}}\cdot\operatorname{\mathrm{{\mn@boldsymbol{\mathit{B}}}}_{0}}+\delta
B^{2}\right)^{2}\right\rangle_{\operatorname{\mathcal{V}}}$ $\displaystyle=$
$\displaystyle\left\langle(2\delta\mathrm{{\mn@boldsymbol{\mathit{B}}}}\cdot\operatorname{\mathrm{{\mn@boldsymbol{\mathit{B}}}}_{0}})^{2}\right\rangle_{\operatorname{\mathcal{V}}}+\left\langle
4(\delta\mathrm{{\mn@boldsymbol{\mathit{B}}}}\cdot\operatorname{\mathrm{{\mn@boldsymbol{\mathit{B}}}}_{0}})\delta
B^{2}\right\rangle_{\operatorname{\mathcal{V}}}+\left\langle\delta
B^{4}\right\rangle_{\operatorname{\mathcal{V}}}$ (13) $\displaystyle\sim$
$\displaystyle\left\langle\delta
B^{4}\right\rangle_{\operatorname{\mathcal{V}}},$ (14)

to leading $\delta B^{4}$ order, and therefore

$\displaystyle\frac{\left\langle\delta
B^{4}\right\rangle_{\operatorname{\mathcal{V}}}^{1/2}}{8\pi
c_{s}^{2}\rho_{0}}\approx\frac{\operatorname{\alpha_{\rm
sat}}}{2}\left\langle\left(\frac{\delta
v}{c_{s}}\right)^{4}\right\rangle^{1/2}_{\operatorname{\mathcal{V}}}.$ (15)

Based upon our numerical results this equation can be re-written in terms of
$2^{\rm nd}$ order terms (see Figure 3),

$\displaystyle\frac{\left\langle\delta
B^{2}\right\rangle_{\operatorname{\mathcal{V}}}}{8\pi
c_{s}^{2}\rho_{0}}\approx\frac{\operatorname{\alpha_{\rm
sat}}}{2}\left\langle\left(\frac{\delta
v}{c_{s}}\right)^{2}\right\rangle_{\operatorname{\mathcal{V}}},$ (16)

with the $\left\langle\delta B^{2}\right\rangle_{\operatorname{\mathcal{V}}}$
dominating the balance between the kinetic turbulent energy. Likewise, as
Federrath (2016) framed the relation, the kinetic energy is feeding the
magnetic field through the $\delta B^{2}$ term in this regime.

In the strong $\operatorname{\mathrm{{\mn@boldsymbol{\mathit{B}}}}_{0}}$
regime we have $B_{0}\gg\delta B$, and hence, to leading $B_{0}^{2}$ order
Equation 13 becomes

$\displaystyle\left\langle\left(2\delta\mathrm{{\mn@boldsymbol{\mathit{B}}}}\cdot\operatorname{\mathrm{{\mn@boldsymbol{\mathit{B}}}}_{0}}+\delta
B^{2}\right)^{2}\right\rangle_{\operatorname{\mathcal{V}}}\sim
2\left\langle\left(\delta\mathrm{{\mn@boldsymbol{\mathit{B}}}}\cdot\operatorname{\mathrm{{\mn@boldsymbol{\mathit{B}}}}_{0}}\right)^{2}\right\rangle_{\operatorname{\mathcal{V}}},$
(17)

with the
$2\operatorname{\left\langle(\delta\mathrm{{\mn@boldsymbol{\mathit{B}}}}\cdot\operatorname{\mathrm{{\mn@boldsymbol{\mathit{B}}}}_{0}})^{2}\right\rangle^{1/2}_{\mathcal{V}}}$
term dominating the balance. Hence the energy balance must be between

$\displaystyle\frac{\left\langle\left(\delta\mathrm{{\mn@boldsymbol{\mathit{B}}}}\cdot\operatorname{\mathrm{{\mn@boldsymbol{\mathit{B}}}}_{0}}\right)^{2}\right\rangle_{\operatorname{\mathcal{V}}}^{1/2}}{4\pi
c_{s}^{2}\rho_{0}}\approx\frac{\operatorname{\alpha_{\rm
sat}}}{2}\left\langle\left(\frac{\delta
v}{c_{s}}\right)^{4}\right\rangle^{1/2}_{\operatorname{\mathcal{V}}},$ (18)

which we can similarly reduce to $2^{\rm nd}$ order terms,

$\displaystyle\frac{\left\langle(\delta\mathrm{{\mn@boldsymbol{\mathit{B}}}}\cdot\operatorname{\mathrm{{\mn@boldsymbol{\mathit{B}}}}_{0}})^{2}\right\rangle^{1/2}_{\operatorname{\mathcal{V}}}}{4\pi
c_{s}^{2}\rho_{0}}\approx\frac{\operatorname{\alpha_{\rm
sat}}}{2}\left\langle\left(\frac{\delta
v}{c_{s}}\right)^{2}\right\rangle_{\operatorname{\mathcal{V}}}.$ (19)

Note now this is the same relation derived in S+2021, but it comes from
directly considering the rms balanced energy equations, and then invoking the
numerical result that the square root of the $4^{\rm th}$ order velocity and
magnetic terms scale almost perfectly with the $2^{\rm nd}$ order terms. The
$2^{\rm nd}-4^{\rm th}$ moment relation should be accurate to a factor less
than 2, as indicated in Figure 3. Establishing the strong mathematical footing
for this relation is a key result from our study.

We will return to Equation 10 and the two limiting cases, Equation 16 and
Equation 19, throughout this study. Specifically, we will show that by using
this simple energy balance model that includes
$\operatorname{\left\langle(\delta\mathrm{{\mn@boldsymbol{\mathit{B}}}}\cdot\operatorname{\mathrm{{\mn@boldsymbol{\mathit{B}}}}_{0}})^{2}\right\rangle^{1/2}_{\mathcal{V}}}$,
we can learn a great deal about the magnetic and velocity field fluctuations.
First, we start by understanding the nature of the coupling term.

Figure 4: The magnetic coupling term,
$\operatorname{\left\langle(\delta\mathrm{{\mn@boldsymbol{\mathit{B}}}}\cdot\operatorname{\mathrm{{\mn@boldsymbol{\mathit{B}}}}_{0}})^{2}\right\rangle^{1/2}_{\mathcal{V}}}$,
as a function of the $\mathcal{M}$, coloured by
$\operatorname{\mathcal{M}_{\text{A0}}}$, for all of the simulations up to
$\operatorname{\mathcal{M}_{\text{A0}}}=10$. We show the strong-field model,
Equation 20, for the coupling term, indicated with the grey dashed line, which
is valid for the simulations with dark shading, assuming exact energy
equipartition between the turbulent kinetic and the coupling term energy.
Figure 5: The ratio of the magnetic coupling term,
$\operatorname{\left\langle(\delta\mathrm{{\mn@boldsymbol{\mathit{B}}}}\cdot\operatorname{\mathrm{{\mn@boldsymbol{\mathit{B}}}}_{0}})^{2}\right\rangle^{1/2}_{\mathcal{V}}}$,
to the turbulent magnetic energy, $\left\langle\delta
B^{2}\right\rangle_{\operatorname{\mathcal{V}}}$ as a function of
$\operatorname{\mathcal{M}_{\text{A0}}}$, coloured by the $\mathcal{M}$, for
all simulations. We show in red dots the equipartition between the two terms.
The grey dashed line shows the strong-field model, Equation 22, which is valid
for $B_{0}\gg\delta B$, or $\operatorname{\mathcal{M}_{\text{A0}}}\lesssim 2$,
and the grey dot-dashed line for the weak-field model, Equation 25, valid for
$\operatorname{\mathcal{M}_{\text{A0}}}>2$. Figure 6: The magnetic coupling
term,
$\operatorname{\left\langle(\delta\mathrm{{\mn@boldsymbol{\mathit{B}}}}\cdot\operatorname{\mathrm{{\mn@boldsymbol{\mathit{B}}}}_{0}})^{2}\right\rangle^{1/2}_{\mathcal{V}}}$,
compensated by Equation 20, as a function
$\operatorname{\mathcal{M}_{\text{A0}}}$, coloured by $\mathcal{M}$, for all
simulations. This choice of compensation reveals the
$\operatorname{\mathcal{M}_{\text{A0}}}$ dependency in the super-Alfvénic
turbulence regime, which we provide a model for in Section 4.2, shown with the
grey dot-dashed line.

##  4 Models for
$\operatorname{\left\langle(\delta\mathrm{{\mn@boldsymbol{\mathit{B}}}}\cdot\operatorname{\mathrm{{\mn@boldsymbol{\mathit{B}}}}_{0}})^{2}\right\rangle^{1/2}_{\mathcal{V}}}$

###  4.1 Strong mean-field, $B_{0}\gg\delta B$

Assuming that the kinetic energy fluctuations are in energy equipartition with
the coupling term ($\operatorname{\alpha_{\rm sat}}=1$) it immediately follows
from Equation 19 that in the strong-mean-field regime the coupling term is,

$\displaystyle\operatorname{\left\langle(\delta\mathrm{{\mn@boldsymbol{\mathit{B}}}}\cdot\operatorname{\mathrm{{\mn@boldsymbol{\mathit{B}}}}_{0}})^{2}\right\rangle^{1/2}_{\mathcal{V}}}=2c_{s}^{2}\rho_{0}\pi\mathcal{M}^{2}.$
(20)

We plot this predicted relationship, along with the values measured from our
simulations, in Figure 4. The plot is consistent with our expectations:
simulations in the strong-mean-field regime,
$\operatorname{\mathcal{M}_{\text{A0}}}<1$, sit very close to the
equipartition line, while those with
$\operatorname{\mathcal{M}_{\text{A0}}}>1$ sit below it, indicating that the
$\left\langle\delta B^{2}\right\rangle_{\operatorname{\mathcal{V}}}$ term is
playing an increasingly large role in the energy balance as we transition to
the weak mean-field regime.

Even in the strong mean-field regime, we see weak variation with $\mathcal{M}$
in how closely the simulations follow the prediction of Equation 20. For
low-$\mathcal{M}$, the strong-field model works best, but as $\mathcal{M}$
gets larger there is some scatter to lower values of
$\operatorname{\left\langle(\delta\mathrm{{\mn@boldsymbol{\mathit{B}}}}\cdot\operatorname{\mathrm{{\mn@boldsymbol{\mathit{B}}}}_{0}})^{2}\right\rangle^{1/2}_{\mathcal{V}}}$,
even in the sub-Alfvénic simulations. This suggests that for
$\operatorname{\mathcal{M}_{\text{A0}}}\gtrsim 1$ there are some contributions
to the magnetic energy through the $\left\langle\delta
B^{2}\right\rangle_{\operatorname{\mathcal{V}}}$ term, which we neglect in our
model, i.e., the shear Alfvén waves and fast modes that perturb the magnetic
field perpendicular to
$\operatorname{\mathrm{{\mn@boldsymbol{\mathit{B}}}}_{0}}$. Of course, the
turbulence naturally excites such modes but it is plausible that the magnetic
tension significantly suppresses them when $B_{0}/\delta B$ is large.

To further quantify when each of the magnetic terms in Equation 12 contribute
the most to the energy balance we examine the ratio of the two magnetic energy
reservoir terms,
$\operatorname{\left\langle(\delta\mathrm{{\mn@boldsymbol{\mathit{B}}}}\cdot\operatorname{\mathrm{{\mn@boldsymbol{\mathit{B}}}}_{0}})^{2}\right\rangle^{1/2}_{\mathcal{V}}}/\left\langle\delta
B^{2}\right\rangle_{\operatorname{\mathcal{V}}}$. We estimate
$\left\langle\delta B^{2}\right\rangle_{\operatorname{\mathcal{V}}}$ following
the fluctuation models in Federrath (2016) and Beattie et al. (2020),888Note
that anisotropy in the magnetic and velocity fluctuations (decomposing to
perpendicular and parallel field components) was ignored in these studies, as
pointed out by S+2021, but the corrections are of order unity, which we show
in Appendix B, and only become important for more sensitive calculations,
which we discuss later in Section 5. which leads to a predicted relationship

$\displaystyle\left\langle\delta
B^{2}\right\rangle^{1/2}_{\operatorname{\mathcal{V}}}=c_{s}\sqrt{\pi\rho_{0}}\mathcal{M}\operatorname{\mathcal{M}_{\text{A0}}}.$
(21)

The ratio between the coupling term to the energy from the above equation
squared is then,

$\displaystyle\frac{\operatorname{\left\langle(\delta\mathrm{{\mn@boldsymbol{\mathit{B}}}}\cdot\operatorname{\mathrm{{\mn@boldsymbol{\mathit{B}}}}_{0}})^{2}\right\rangle^{1/2}_{\mathcal{V}}}}{\left\langle\delta
B^{2}\right\rangle_{\operatorname{\mathcal{V}}}}=\frac{2}{\mathcal{M}^{2}_{\text{A0}}}=2\frac{e_{\rm
mag,0}}{\operatorname{e_{\rm kin}}},$ (22)

where $\mathcal{M}^{-2}_{\text{A0}}=e_{\rm mag,0}/\operatorname{e_{\rm kin}}$.
This means at $\operatorname{\mathcal{M}_{\text{A0}}}=e_{\rm
mag,0}/\operatorname{e_{\rm kin}}=1$, i.e. when the turbulent and
$\operatorname{\mathrm{{\mn@boldsymbol{\mathit{B}}}}_{0}}$ energy are in
equipartition, we expect
$\operatorname{\left\langle(\delta\mathrm{{\mn@boldsymbol{\mathit{B}}}}\cdot\operatorname{\mathrm{{\mn@boldsymbol{\mathit{B}}}}_{0}})^{2}\right\rangle^{1/2}_{\mathcal{V}}}/\left\langle\delta
B^{2}\right\rangle_{\operatorname{\mathcal{V}}}=2$. We plot the relation
measured in the simulations in Figure 5, showing our predicted scaling in the
strong mean-field regime with the dashed, grey line. Again, the plot shows
excellent agreement between the model and the MHD data between
$0.01\leq\operatorname{\mathcal{M}_{\text{A0}}}\leq 2$, indicating a perfect
balancing act between
$\operatorname{\left\langle(\delta\mathrm{{\mn@boldsymbol{\mathit{B}}}}\cdot\operatorname{\mathrm{{\mn@boldsymbol{\mathit{B}}}}_{0}})^{2}\right\rangle^{1/2}_{\mathcal{V}}}$
and $\left\langle\delta v^{2}\right\rangle_{\operatorname{\mathcal{V}}}$. The
$\operatorname{\mathcal{M}_{\text{A0}}}\approx 2$ transition is where
$\left\langle\delta
B^{2}\right\rangle_{\operatorname{\mathcal{V}}}/B_{0}^{2}\approx 1$, and the
turbulent magnetic field starts to dominate the magnetic energy reservoir. We
will find that this is a reoccurring transition phase for compressible MHD
turbulence.

###  4.2 Weak large-scale field, $B_{0}\lesssim\delta B$

Beyond $\operatorname{\mathcal{M}_{\text{A0}}}\gtrsim 2$ energy balance
arguments only work if the saturation level changes as a function of plasma
parameters, because $\delta\mathrm{{\mn@boldsymbol{\mathit{B}}}}$ is not
constant with $\operatorname{\mathrm{{\mn@boldsymbol{\mathit{B}}}}_{0}}$
(Federrath, 2016; Beattie et al., 2020). We have one free parameter,
$\operatorname{\alpha_{\rm sat}}$, which need not be constant for all
$\operatorname{\mathcal{M}_{\text{A0}}}$ and $\mathcal{M}$ (Federrath, 2016;
Achikanath Chirakkara et al., 2021; Seta & Federrath, 2021a). To extract
$\operatorname{\alpha_{\rm sat}}$, we model the coupling term in the super-
Alfvénic regime by starting with an empirical model that Beattie et al. (2020)
found held universally for $\mathcal{M}$ in the
$\operatorname{\mathcal{M}_{\text{A0}}}\gtrsim 2$ regime, $\left\langle\delta
B^{2}\right\rangle^{1/2}_{\operatorname{\mathcal{V}}}/B_{0}=\mathcal{M}_{A0}^{2/3}$.
This provides an independent estimate of the super-Alfvénic turbulent magnetic
fluctuations,

$\displaystyle\left\langle\delta
B^{2}\right\rangle^{1/2}_{\operatorname{\mathcal{V}}}$
$\displaystyle=\mathcal{M}^{2/3}_{\text{A0}}B_{0}=2c_{s}\sqrt{\pi\rho_{0}}\mathcal{M}\mathcal{M}^{-1/3}_{\text{A0}},$
(23)

and therefore, equating Equation 16 with the square of Equation 23,
$\operatorname{\alpha_{\rm sat}}=\mathcal{M}^{-2/3}_{\text{A0}}$, which
implies that as the large-scale field becomes weaker, the turbulent magnetic
field saturates to smaller and smaller values because there is less total
magnetic energy, consistent with what was qualitatively found in Beattie et
al. (2020). Following the same steps as in Section 4.1, additionally using
$B_{0}/(c_{s}\rho_{0}^{2})=2\sqrt{\pi}\mathcal{M}\mathcal{M}^{-1}_{\text{A0}}$,
the definition of the large-scale field Alfvén Mach number, the coupling term
in the super-Alfvénic regime becomes,

$\displaystyle\operatorname{\left\langle(\delta\mathrm{{\mn@boldsymbol{\mathit{B}}}}\cdot\operatorname{\mathrm{{\mn@boldsymbol{\mathit{B}}}}_{0}})^{2}\right\rangle^{1/2}_{\mathcal{V}}}$
$\displaystyle=2\pi
c_{s}^{2}\rho_{0}\mathcal{M}^{2}\mathcal{M}^{-4/3}_{\text{A0}},$ (24)

and ratio
$\operatorname{\left\langle(\delta\mathrm{{\mn@boldsymbol{\mathit{B}}}}\cdot\operatorname{\mathrm{{\mn@boldsymbol{\mathit{B}}}}_{0}})^{2}\right\rangle^{1/2}_{\mathcal{V}}}/\left\langle\delta
B^{2}\right\rangle_{\operatorname{\mathcal{V}}}$,

$\displaystyle\frac{\operatorname{\left\langle(\delta\mathrm{{\mn@boldsymbol{\mathit{B}}}}\cdot\operatorname{\mathrm{{\mn@boldsymbol{\mathit{B}}}}_{0}})^{2}\right\rangle^{1/2}_{\mathcal{V}}}}{\left\langle\delta
B^{2}\right\rangle_{\operatorname{\mathcal{V}}}}=\frac{1}{2}\mathcal{M}^{-2/3}_{\text{A0}}.$
(25)

We plot Equation 24 and Equation 25, alongside their strong-field
counterparts, in Figure 5 and Figure 6, respectively. Note that Figure 6 shows
the same information as Figure 4, but we have normalised by Equation 20 to
remove the $\mathcal{M}^{2}$ dependency. This allows us to better observe the
dependence on $\operatorname{\mathcal{M}_{\text{A0}}}$ in the $B_{0}\ll\delta
B$ regime. Astonishingly, through this relatively simple analysis both of the
theoretical models describe the data very well, providing excellent agreement
over 3 orders of magnitude in $\operatorname{\mathcal{M}_{\text{A0}}}$, with
no free parameters.

###  4.3 Discussion of Section 4 and caveats

We have established that for sub-to-trans-Alfvénic turbulence, the volume-
averaged turbulent kinetic energy is in exact energy equipartition with the
rms
$\operatorname{\delta\mathrm{{\mn@boldsymbol{\mathit{B}}}}\cdot\operatorname{\mathrm{{\mn@boldsymbol{\mathit{B}}}}_{0}}}$
field. Each of the models in Section 4.1 relies on this assumption
$(\operatorname{\alpha_{\rm sat}}=1)$, and without any further free
parameters, with such a simple model, the agreement to the numerical data in
Figure 4, Figure 5 and Figure 6 is remarkable. The models in the super-
Alfvénic regime critically rely on the empirical result from Beattie et al.
(2020), $\left\langle\delta
B^{2}\right\rangle^{1/2}_{\operatorname{\mathcal{V}}_{L}}/B_{0}=\mathcal{M}^{2/3}_{\text{A0}}$,
but likewise, the models for the coupling and fluctuation terms outlined in
Section 4.2, are in excellent agreement with the data, again, with no free
parameters.

Our results therefore strongly support the Skalidis & Tassis (2020) and S+2021
model for relating the balanced rms magnetic field coupling term to the
volume-averaged turbulent kinetic energy (the energy balance arguments). We
hope that our treatment satisfies other authors’ concerns about the Skalidis &
Tassis (2020) coupling term method. These concerns have taken two forms; one
is that the energy contribution from
$\left\langle\operatorname{\delta\mathrm{{\mn@boldsymbol{\mathit{B}}}}\cdot\operatorname{\mathrm{{\mn@boldsymbol{\mathit{B}}}}_{0}}}\right\rangle_{\operatorname{\mathcal{V}}}\stackrel{{\scriptstyle\rm
must}}{{=}}0$ because
$\left\langle\delta\mathrm{{\mn@boldsymbol{\mathit{B}}}}\right\rangle_{\operatorname{\mathcal{V}}}=0$,
as highlighted in the appendix of Li et al. (2021b). But as has been
extensively discussed in S+2021, and in Section 3 of the current study,
$\left\langle\operatorname{\delta\mathrm{{\mn@boldsymbol{\mathit{B}}}}\cdot\operatorname{\mathrm{{\mn@boldsymbol{\mathit{B}}}}_{0}}}\right\rangle_{\operatorname{\mathcal{V}}}=0$
is not a valid way of understanding the contribution from the coupling term.
The other concern raised in the appendix of Li et al. (2021a), is that energy
balance only involves $2^{\rm nd}$ order quantities; this approach by
definition omits the coupling term contribution, which is first order, in the
energetics. Omission of the coupling term leads to significantly
underestimating the magnetic energy in sub-Alfvénic turbulence999S+2021 showed
that omitting the coupling term in the estimation of the magnetic field
strength in sub-Alfvénic turbulence, or equally applying the Davis (1951) and
Chandrasekhar & Fermi (1953) method, can produce estimates which can be up to
an order of magnitude larger than the actual values.; this is strongly
supported by our numerical results in Figure 5 and 6. In the same figures we
show that $2^{\rm nd}$ order terms become significant only in super-Alfvénic
turbulence. Therefore, one should clearly state the magnetisation level of
turbulence (sub- or super-Alfvénic) before arguing about the relative
contribution of the various terms in the energy balance.

Liu et al. (2021) further argues that self-gravity may modify this energy
balance. We do not include gravity in this study, but it is possible that
gravity may collapse locally bound (by self-gravity) regions in the ISM,
enhancing and creating strong magnetic fields (Sur et al., 2010). This may
make the coupling term even more relevant as the regions collapse, forming
convergent flows parallel to the field lines, and strengthening the magnetic
field and hence the local affect of
$\operatorname{\delta\mathrm{{\mn@boldsymbol{\mathit{B}}}}\cdot\operatorname{\mathrm{{\mn@boldsymbol{\mathit{B}}}}_{0}}}$.
This is speculative, and the exact effects of gravity are unclear; we will
return to this topic in future work. Of course, all of this work is done in
the isothermal context, so our relations are only relevant to individual
phases of the ISM, which are well approximated by an isothermal equation of
state (e.g., Wolfire et al., 1995).

Figure 7: The total Alfvén Mach number, $\mathcal{M}_{A,\text{total}}$
(Equation 29), shown in red, and the turbulent Alfvén Mach number,
$\mathcal{M}_{A,\text{turb}}$ (Equation 28), shown in blue, as a function of
large-scale field Alfvén Mach number, $\operatorname{\mathcal{M}_{\text{A0}}}$
(Equation 27), for all simulations. The grey dashed line is the one-to-one
line between the two Alfvénic Mach numbers and
$\operatorname{\mathcal{M}_{\text{A0}}}$. The dotted line shows the model for
$\mathcal{M}_{A,\text{turb}}$ for the
$\operatorname{\mathcal{M}_{\text{A0}}}\leq 2$ (shown with vertical purple
line) regime. The separation of $\operatorname{\mathcal{M}_{\text{A}}}$ into
field components shows explicitly that the turbulent component of the field is
highly super-Alfvénic in the sub-Alfvénic large-scale field regime (i.e., the
turbulent kinetic energy is much larger than the turbulent magnetic energy,
shown with the horizontal purple line), and the large-scale field dominates
the total magnetic energy,
$\operatorname{\mathcal{M}_{\text{A0}}}\approx\mathcal{M}_{A,\text{total}}$,
which coincides to
$\operatorname{\left\langle(\delta\mathrm{{\mn@boldsymbol{\mathit{B}}}}\cdot\operatorname{\mathrm{{\mn@boldsymbol{\mathit{B}}}}_{0}})^{2}\right\rangle^{1/2}_{\mathcal{V}}}\propto\left\langle\delta
v^{2}\right\rangle_{\operatorname{\mathcal{V}}}$, as discussed in Section 4.1.
The transition into the super-Alfvénic large-scale field regime happens at a
critical point in the
$\operatorname{\mathcal{M}_{\text{A}}}-\operatorname{\mathcal{M}_{\text{A0}}}$
diagram,
$\operatorname{\mathcal{M}_{\text{A0}}}\approx\operatorname{\mathcal{M}_{\text{A,total}}}\approx\operatorname{\mathcal{M}_{\text{A,turb}}}\approx
2$, where there is energy equipartition between the turbulent and large-scale
magnetic field. For $\operatorname{\mathcal{M}_{\text{A0}}}>2$, the turbulent
magnetic energy is greater than the energy in the large-scale field (vertical
purple line), hence
$\mathcal{M}_{A,\text{turb}}\approx\mathcal{M}_{A,\text{total}}$, and the
turbulent kinetic energy is greater than the magnetic energy, corresponding to
the
$\left\langle\delta\mathrm{{\mn@boldsymbol{\mathit{B}}}}^{2}\right\rangle_{\operatorname{\mathcal{V}}}\propto\left\langle\delta
v^{2}\right\rangle_{\operatorname{\mathcal{V}}}$ regime, as discussed in
Section 4.2.

##  5 The three Alfvén Mach Numbers

###  5.1 Definitions and results

The Alfvén Mach number, $\operatorname{\mathcal{M}_{\text{A}}}$, is another
part of the energy balance story, because the quantity itself is directly
related to the energy equilibrium in the plasma,

$\displaystyle\mathcal{M}_{A}^{-2}=\left\langle\frac{B^{2}/(8\pi)}{(1/2)\rho\delta
v^{2}}\right\rangle_{\operatorname{\mathcal{V}}}=\left\langle\frac{\operatorname{e_{\rm
mag}}}{\operatorname{e_{\rm kin}}}\right\rangle_{\operatorname{\mathcal{V}}},$
(26)

which is similar to $\operatorname{\alpha_{\rm sat}}$ in Equation 9, but not
exactly the same, because $\left\langle
X/Y\right\rangle_{\operatorname{\mathcal{V}}}\neq\left\langle
X\right\rangle_{\operatorname{\mathcal{V}}}/\left\langle
Y\right\rangle_{\operatorname{\mathcal{V}}}$ if there are any correlations
between $X$ and $Y$, as is the case for the magnetic and kinetic energy
($\operatorname{\alpha_{\rm sat}}=\mathcal{M}^{-2/3}_{\text{A0}}$, Section
4.2).

Throughout the previous section, we utilised
$\operatorname{\mathcal{M}_{\text{A0}}}$ to construct our models around values
of $\operatorname{\mathrm{{\mn@boldsymbol{\mathit{B}}}}_{0}}$. We could do
this easily because $\operatorname{\mathcal{M}_{\text{A0}}}$ is an input (or
at least controlled, albeit with some small variation due to velocity
fluctuations) in our simulations. However, in many astrophysical turbulence
studies, authors prefer to use $\operatorname{\mathcal{M}_{\text{A}}}$. For
some of these studies, it is not clear if one should interpret this as the
$\operatorname{\mathcal{M}_{\text{A}}}$ with respect to just turbulent
fluctuations, or the total field strength. The difference between these
quantities is rarely appreciated, so we make a point by defining and relating
three different canonical constructions of
$\operatorname{\mathcal{M}_{\text{A}}}$. The three definitions we use101010We
note there are, of course, even more definitions that one could in principle
construct, for example,
$\operatorname{\mathcal{M}_{\text{A}}}=\left\langle\delta
v\right\rangle\sqrt{4\pi\rho_{0}}/\left\langle B\right\rangle$, which we use
to set the Alfvén Mach number in Section 2, or one could even use component-
wise constructions. are

$\displaystyle\operatorname{\mathcal{M}_{\text{A0}}}$
$\displaystyle=\left\langle\frac{\delta
v\sqrt{4\pi\rho}}{B_{0}}\right\rangle_{\operatorname{\mathcal{V}}},$ (27)
$\displaystyle\operatorname{\mathcal{M}_{\text{A,turb}}}$
$\displaystyle=\left\langle\frac{\delta v\sqrt{4\pi\rho}}{\delta
B}\right\rangle_{\operatorname{\mathcal{V}}},$ (28)
$\displaystyle\operatorname{\mathcal{M}_{\text{A,total}}}$
$\displaystyle=\left\langle\frac{\delta
v\sqrt{4\pi\rho}}{B}\right\rangle_{\operatorname{\mathcal{V}}},$ (29)

where the first of the three defines the large-scale field (or mean-field on
the system scale) Alfvén Mach number, which compares the large-scale magnetic
energy with the kinetic energy, the second is the turbulent-field Alfvén Mach
number, which compares the turbulent magnetic energy with the kinetic energy,
and the third, the total field Alfvén Mach number.

To understand the relation between the three quantities, we plot them in
Figure 7 ($\operatorname{\mathcal{M}_{\text{A,total}}}$ in red, and
$\operatorname{\mathcal{M}_{\text{A,turb}}}$ in blue, both as a function of
$\operatorname{\mathcal{M}_{\text{A0}}}$). The dashed, grey line shows the
one-to-one line between $\operatorname{\mathcal{M}_{\text{A0}}}$ and
$\operatorname{\mathcal{M}_{\text{A}}}$. For
$\operatorname{\mathcal{M}_{\text{A0}}}\lesssim 2$,
$\operatorname{\mathcal{M}_{\text{A0}}}\approx\operatorname{\mathcal{M}_{\text{A,total}}}$,
which means the energetics of the fluid are completely dominated by the large-
scale field, and not the turbulence at all.
$\operatorname{\mathcal{M}_{\text{A,turb}}}$ follows a power-law in
$\operatorname{\mathcal{M}_{\text{A0}}}$ which prevents the fluctuating
magnetic field from ever becoming stronger than
$\operatorname{\mathcal{M}_{\text{A,turb}}}\approx 2$. Once the turbulent
field has reached $\operatorname{\mathcal{M}_{\text{A}}}\approx 2$, it then
begins weakening again, but this time
$\operatorname{\mathcal{M}_{\text{A,turb}}}\approx\operatorname{\mathcal{M}_{\text{A,total}}}$,
with
$\operatorname{\mathcal{M}_{\text{A,total}}}\lesssim\operatorname{\mathcal{M}_{\text{A0}}}$,
i.e., transitioning into a turbulent magnetic field dominant regime as the
$\mathrm{{\mn@boldsymbol{\mathit{B}}}}$-field becomes tangled and more energy
dense than the large-scale field. As discussed in Section 4.1, the
$\operatorname{\mathcal{M}_{\text{A0}}}=2$ transition between the sub-and-
super-Alfvénic regimes defines exactly when the $B_{0}$ and $\delta B$ field
are equal energy in energy, and the transition between
$\operatorname{\left\langle(\delta\mathrm{{\mn@boldsymbol{\mathit{B}}}}\cdot\operatorname{\mathrm{{\mn@boldsymbol{\mathit{B}}}}_{0}})^{2}\right\rangle^{1/2}_{\mathcal{V}}}\propto\left\langle\delta
v^{2}\right\rangle_{\operatorname{\mathcal{V}}}$ and $\left\langle\delta
B^{2}\right\rangle_{\operatorname{\mathcal{V}}}\propto\left\langle\delta
v^{2}\right\rangle_{\operatorname{\mathcal{V}}}$, as annotated in the plot.

We are able to derive the relation between
$\operatorname{\mathcal{M}_{\text{A,turb}}}$ and
$\operatorname{\mathcal{M}_{\text{A0}}}$ in the sub-Alfvénic regime by
rearranging the coupling magnetic components,
$\operatorname{\left\langle(\delta\mathrm{{\mn@boldsymbol{\mathit{B}}}}\cdot\operatorname{\mathrm{{\mn@boldsymbol{\mathit{B}}}}_{0}})^{2}\right\rangle^{1/2}_{\mathcal{V}}}=\left\langle\delta
B_{\parallel}^{2}\right\rangle^{1/2}_{\operatorname{\mathcal{V}}}B_{0}$, on
the RHS and turbulent components on the LHS of Equation 19,

$\displaystyle\frac{2\left\langle\delta
B_{\parallel}^{2}\right\rangle^{1/2}_{\operatorname{\mathcal{V}}}}{\delta
v\sqrt{4\pi\rho_{0}}}=\operatorname{\mathcal{M}_{\text{A0}}}.$ (30)

But now we need to use total fluctuating magnetic field, not just the parallel
field, to get the complete $\operatorname{\mathcal{M}_{\text{A,turb}}}$. In
Appendix B we directly measure the different field components and relate them
to the total fields. The most strongly-anisotropic regime, in the highly-sub-
Alfvénic turbulence, corresponds to $(1/3)\left\langle\delta
B_{\parallel}^{2}\right\rangle^{1/2}_{\operatorname{\mathcal{V}}}\leq\left\langle\delta
B_{\perp}^{2}\right\rangle^{1/2}_{\operatorname{\mathcal{V}}}\leq(2/3)\left\langle\delta
B_{\parallel}^{2}\right\rangle^{1/2}_{\operatorname{\mathcal{V}}}$ and
$\left\langle\delta
v_{\parallel}^{2}\right\rangle^{1/2}_{\operatorname{\mathcal{V}}}\approx(2/3)\left\langle\delta
v_{\perp}^{2}\right\rangle^{1/2}_{\operatorname{\mathcal{V}}}$. For the
magnetic fluctuations, we pick the average between these two values,
$\left\langle\delta
B_{\perp}^{2}\right\rangle^{1/2}_{\operatorname{\mathcal{V}}}=(1/2)\left\langle\delta
B_{\parallel}^{2}\right\rangle^{1/2}_{\operatorname{\mathcal{V}}}$, and then
propagate both the magnetic anisotropy through the regular vector magnitude
equations, which gives $\left\langle\delta
B^{2}\right\rangle^{1/2}_{\operatorname{\mathcal{V}}}=\sqrt{3/2}\left\langle\delta
B_{\parallel}^{2}\right\rangle^{1/2}_{\operatorname{\mathcal{V}}}$.
Substituting this back into Equation 30 gives,

$\displaystyle\operatorname{\mathcal{M}_{\text{A,turb}}}=\sqrt{6}\mathcal{M}^{-1}_{\text{A0}},\,\,\operatorname{\mathcal{M}_{\text{A0}}}\leq
2,$ (31)

which we plot with the grey dotted line in Figure 7. This simple model
intersects with the
$\operatorname{\mathcal{M}_{\text{A}}}=\operatorname{\mathcal{M}_{\text{A0}}}$
line at the
$\operatorname{\mathcal{M}_{\text{A,total}}}\approx\operatorname{\mathcal{M}_{\text{A,turb}}}$
transition. At lower $\operatorname{\mathcal{M}_{\text{A0}}}$ there is some
deviation away from the model, which is because of the stronger than average
magnetic fluctuation anisotropy present in the
$\operatorname{\mathcal{M}_{\text{A0}}}\ll 1$ data.

###  5.2 Hypothesis on limiting behaviour

The discussion in the preceding sections leads us to propose a hypothesis
regarding the limiting behaviour of MHD turbulent systems in Figure 7, which
we illustrate schematically in Figure 8.

####  5.2.1 $\operatorname{\mathcal{M}_{\text{A0}}}\rightarrow 0$

As $\operatorname{\mathcal{M}_{\text{A0}}}\rightarrow 0$, the turbulent field
should continue to become weaker and weaker
($\operatorname{\mathcal{M}_{\text{A,turb}}}\gg 1$). The reason for this is
that the magnetic fluctuations (specifically the shear Alfvén waves) are
smoothed out by the increasing magnetic tension,
$(\mathrm{{\mn@boldsymbol{\mathit{B}}}}\cdot\nabla)\mathrm{{\mn@boldsymbol{\mathit{B}}}}$,
and are reduced in degrees of freedom since they are perfectly flux-frozen
into $\operatorname{\mathrm{{\mn@boldsymbol{\mathit{B}}}}_{0}}$. Therefore,
instead of any field-line stretching, coherent magnetic field-lines are
randomly walked in the perpendicular plane to
$\operatorname{\mathrm{{\mn@boldsymbol{\mathit{B}}}}_{0}}$ (field line random
walk, Jokipii & Parker 1968). 2D planar motions cannot instigate dynamo action
(Zel’dovich 1957 theorem), so it is unlikely that $\delta B$ ever grows
irreversibly111111Note that it may grow locally, through reversible processes
such as compression, but these ought to average out over time. again in this
limit, and we find $\left\langle\delta
B^{2}\right\rangle^{1/2}_{\operatorname{\mathcal{V}}_{L}}\propto B_{0}^{-1}$
(Equation 31; consistent with qualitative observations in previous studies,
Haugen et al. 2004), until the magnetic field only has a large-scale component
and $B=B_{0}$.

####  5.2.2 $\operatorname{\mathcal{M}_{\text{A0}}}\rightarrow\infty$

In the $\operatorname{\mathcal{M}_{\text{A0}}}\rightarrow\infty$
$(B_{0}\rightarrow 0)$ limit, we reach the results from the small-scale dynamo
community. Very broadly speaking, in these studies, where $B_{0}=0$, and hence
$\operatorname{\mathcal{M}_{\text{A0}}}\rightarrow\infty$, the equilibrium
magnetic field strength asymptotes to a value that depends on the sonic Mach
number and ratio of compressive to solenoidal modes in the turbulence, and
plasma Reynolds numbers (if they are finite). In this limit, the maximally
efficient turbulent dynamo is for the most sub-sonic, solenoidal flows and the
least efficient for the highest-$\mathcal{M}$, most compressible flows
(Federrath et al., 2014; Schober et al., 2015; Federrath, 2016; Achikanath
Chirakkara et al., 2021). Hence, as $\operatorname{\mathcal{M}_{\text{A0}}}$
becomes larger, we should expect to observe the
$\operatorname{\mathcal{M}_{\text{A}}}(\operatorname{\mathcal{M}_{\text{A0}}})$
curves separate and asymptote to different constant values of
$\operatorname{\mathcal{M}_{\text{A,turb}}}$ as a function of $\mathcal{M}$,
which is what we find in Figure 7. Since the most efficient turbulent
dynamo121212Note this is in absence of magnetic helicity, which may
significantly change the saturation of the dynamo given that there are more
degrees of freedom to store magnetic energy than in non-helical turbulence and
magnetic modes above the outer scale of the turbulence can be energised (e.g.,
§4-5 in Rincon, 2019). leads to saturation of
$\operatorname{\mathcal{M}_{\text{A,total}}}\approx 2$, (see highly sub-sonic,
solenoidal experiments in Achikanath Chirakkara et al. 2021) this defines a
floor that bounds $\operatorname{\mathcal{M}_{\text{A}}}$ from below as
$\operatorname{\mathcal{M}_{\text{A0}}}\rightarrow\infty$.

Figure 8: Schematic for the
$\operatorname{\mathcal{M}_{\text{A,turb}}}-\operatorname{\mathcal{M}_{\text{A0}}}$
plane, showing the small-scale dynamo saturation in the
$\operatorname{\mathcal{M}_{\text{A0}}}\rightarrow\infty$ limit, and the
$\operatorname{\mathcal{M}_{\text{A,turb}}}\propto\mathcal{M}^{-1}_{\text{A0}}$
$(\left\langle\delta
B^{2}\right\rangle^{1/2}_{\operatorname{\mathcal{V}}_{L}}\propto B_{0}^{-1})$
power-law in the $\operatorname{\mathcal{M}_{\text{A0}}}\rightarrow 0$ limit,
derived using energy balance. The separation between
$\operatorname{\mathcal{M}_{\text{A,turb}}}(\operatorname{\mathcal{M}_{\text{A0}}})$
curves, $\Delta\operatorname{\mathcal{M}_{\text{A,turb}}}$, changes with
different plasma parameters that control the small-scale dynamo saturation.

###  5.3 Sub-Alfvénic turbulent fields do not exist

We find that the fluctuating magnetic field becomes extremely weak at
$\operatorname{\mathcal{M}_{\text{A0}}}<2$, and is bounded from below by the
most efficient saturation of the small-scale dynamo in the limit
$\operatorname{\mathcal{M}_{\text{A0}}}\to\infty$. An immediate consequence of
these two limiting behaviours is that there is no room in the
$\operatorname{\mathcal{M}_{\text{A}}}-\operatorname{\mathcal{M}_{\text{A0}}}$
plane for $\operatorname{\mathcal{M}_{\text{A,turb}}}\lesssim 2$ turbulence.
Hence the only sub-Alfvénic turbulence that is possible in this parameter
space is sub-Alfvénic large-scale field (or coherent field) fluid turbulence.
We show that this kind of turbulence is highly super-Alfvénic with respect to
the turbulent velocity fluctuations (shocks and vortices),
$\operatorname{\mathcal{M}_{\text{A,turb}}}\gg 1$, and it is only through the
non-turbulent components of the plasma that the magnetic energy is able to be
sufficiently stronger than the kinetic energy. As a caveat of this analysis,
in this section, we volume-averaged the plasma beyond the correlation scale of
the turbulence, hence this does not rule out that the turbulence can be sub-
Alfvénic on scales much smaller than the correlation scale. In the next
section we explore averaging below the correlation scale, and discuss how sub-
Alfvénic turbulence can emerge by taking statistics below the correlation
scale.

Figure 9: Top: The turbulent correlation scale, $\ell_{\text{cor},v}$
(Equation 32), in units of turbulent driving scale, $\ell_{0}$, as a function
of $\operatorname{\mathcal{M}_{\text{A0}}}$, for all simulations. We find
$\ell_{\text{cor},v}\approx\ell_{0}$, with some systematic deviation at
low-$\operatorname{\mathcal{M}_{\text{A0}}}$, most likely due to the strong,
global anisotropy in those simulations. Bottom: The same, but for the
correlation scale of the magnetic field, $\ell_{\text{cor},B}$. The scatter at
each $\operatorname{\mathcal{M}_{\text{A0}}}$ is determined by $\mathcal{M}$,
which ranges between $\mathcal{M}\approx 0.5-10$ and only weakly changes the
correlation scale of the turbulence. Figure 10: Coupling and mean squared
values of magnetic fluctuation terms in units of large-scale field as a
function of averaging scale normalised to the correlation scale of the
turbulence, $\ell_{\text{cor},v}$, for the 10 randomly sampled regions in the
M2MA001 (top) and M2MA10 (bottom) simulations. The square magnitude of
$\delta\mathrm{{\mn@boldsymbol{\mathit{B}}}}$ on scale
$\ell/\ell_{\text{cor},v}$ is shown in red, and the mean of
$\operatorname{\delta\mathrm{{\mn@boldsymbol{\mathit{B}}}}\cdot\operatorname{\mathrm{{\mn@boldsymbol{\mathit{B}}}}_{0}}}$
on that scale is shown in black. The yellow shaded region indicates where
numerical dissipation effects may influence the field statistics. Due to the
spatial correlation of the magnetic field, on scales
$\ell<\ell_{\text{cor},v}$ (volumes
$\operatorname{\mathcal{V}}\lesssim\ell_{\text{cor},v}^{3})$, turbulent fields
are converted into effective mean-fields. This means
$\left\langle\operatorname{\delta\mathrm{{\mn@boldsymbol{\mathit{B}}}}\cdot\operatorname{\mathrm{{\mn@boldsymbol{\mathit{B}}}}_{0}}}\right\rangle_{\operatorname{\mathcal{V}}}\neq
0$ on scales below $\ell_{\text{cor},v}$, acting as an effective mean-field on
that scale. On scales above $\ell_{\text{cor},v}$ the Reynolds rule of
averaging holds and
$\left\langle\operatorname{\delta\mathrm{{\mn@boldsymbol{\mathit{B}}}}\cdot\operatorname{\mathrm{{\mn@boldsymbol{\mathit{B}}}}_{0}}}\right\rangle_{\operatorname{\mathcal{V}}}=0$
as expected.

##  6 The averaging scale

We have shown it is important to average Equation 9 in such a way that all of
the terms are positive-definite, and that
$\operatorname{\left\langle(\delta\mathrm{{\mn@boldsymbol{\mathit{B}}}}\cdot\operatorname{\mathrm{{\mn@boldsymbol{\mathit{B}}}}_{0}})^{2}\right\rangle^{1/2}_{\mathcal{V}}}$
balances the kinetic turbulent energy perfectly in the sub-Alfvénic large-
scale field turbulence regime. However, these methods critically rely on an
averaging scale $\operatorname{\mathcal{V}}$. In this section we highlight the
importance of $\operatorname{\mathcal{V}}$ and show that even if one adopts
the traditional ansatz that does not enforce positivity, Equation 10, the
coupling term can make non-zero contributions to the turbulent energy.

The fundamental reason for this is that the volume averaging scale for
$\left\langle\ldots\right\rangle_{\operatorname{\mathcal{V}}}$ is important.
In simulations, we regularly report volume averaged statistics over a few
turbulent correlation scales, $\ell_{\text{cor},v}$, i.e.,
$\operatorname{\mathcal{V}}=\bigcup_{i=1}^{N}(\ell_{\text{cor},v}^{3})_{i}$,
where $N$ is a few, or directly at the full size of a simulation box,
$\operatorname{\mathcal{V}}=\operatorname{\mathcal{V}}_{L}$. However, for many
observations of the ISM, the region sampled is far smaller than the turbulent
correlation scale. For example, magnetic fields in star-forming clouds are
observed to be correlated on scales up to $\sim 100\,\rm{pc}$ (Li et al.,
2014), comparable to the scale height and the outer scale of turbulence in
galactic discs (e.g. Karlsson et al., 2013; Falceta-Gonçalves et al., 2014;
Krumholz & Ting, 2018). Dust polarisation observations using Herschel (e.g.,
the Herschel Gould Belt Survey – André et al. 2010) generally sample131313In
observations we can define a sampling scale as the maximum spatial separation
of observational data in the plane of the sky. much smaller fields of view (of
order $10\,\rm{pc}$); over size scales typically probed by such observations,
there is no sign of a flattening in the velocity dispersion-size relation
(Federrath et al., 2021; Yun et al., 2021; Zhou et al., 2021), clear evidence
that the region being studied is much smaller than the correlation scale.

To explore the implications of this, we directly compute the correlation
scales of the turbulence $\ell_{\text{cor},v}$ and magnetic field
$\ell_{\text{cor},B}$ in our simulations, and plot them as a function of
$\operatorname{\mathcal{M}_{\text{A0}}}$, coloured by $\mathcal{M}$ in Figure
9. We compute both of them in the textbook manner, directly from the energy
spectra, $\mathscr{P}_{v}(k)$, and $\mathscr{P}_{B}(k)$ as

$\displaystyle\frac{\ell_{\text{cor}}}{\ell_{0}}=\frac{L}{\ell_{0}}\frac{\displaystyle\int_{0}^{\infty}\operatorname{d}\\!{k}\,k^{-1}\mathscr{P}(k)}{\displaystyle\int_{0}^{\infty}\operatorname{d}\\!{k}\,\mathscr{P}(k)},$
(32)

where $\mathscr{P}(k)$ is replaced by $\mathscr{P}_{v}(k)$ for the turbulence
correlation scales, and $\mathscr{P}_{B}(k)$ for the magnetic field
correlation scales.

We focus first on the top panel of Figure 9, the turbulence correlation scale.
The super-Alfvénic large-scale field experiments have
$\ell_{\text{cor},v}\approx\ell_{0}$, with a small dip at
$\operatorname{\mathcal{M}_{\text{A0}}}=2$ as the turbulence transitions
between $\operatorname{\mathrm{{\mn@boldsymbol{\mathit{B}}}}_{0}}$ and
$\delta\mathrm{{\mn@boldsymbol{\mathit{B}}}}$ dominated (as discussed in
Section 3). The sub-Alfvénic turbulence has correlations scales above the
driving scale, most likely due to the system-scale vortices that develop in
the flow (Beattie et al., 2020; Beattie et al., 2021a). If we interpret this
experiment at face value, this means that if individual clouds are sub-
Alfvénic, we should expect correlated turbulent velocities beyond the extent
of the entire sub-Alfvénic region, even if the driving scale is not itself
larger than the individual clouds. For the ISM in general, which is probably
trans-Alfvénic-to-super-Alfvénic
($\operatorname{\mathcal{M}_{\text{A0}}}\approx 2$) and trans-sonic on average
(Gaensler et al., 2011; Krumholz et al., 2020; Liu et al., 2021; Seta &
Federrath, 2021b), we should expect turbulent motions to be correlated out to
the driving scale of the largest turbulent motions. This, of course, is a
natural repercussion of one of the central tenets of turbulence: the energy
cascade from large (galactic, in this context) to small (molecular cloud and
smaller, Armstrong et al., 1995) scales.

Similar to the turbulence correlation scales, the magnetic correlation scales
of the sub-Alfvénic simulations are on larger scales than the driving scale,
indicating, as we showed in Section 5, that the fluctuating magnetic field is
negligible in the sub-Alfvénic regime, and is strongly suppressed by the
large-scale field. The super-Alfvénic simulations show a decaying power-law
$\ell_{\text{cor},B}\propto\ell_{0}\mathcal{M}^{-1/2}_{\text{A0}}$, which
demonstrates that as the large-scale field weakens, the causally connected
regions in the magnetic field move to smaller and smaller scales
(qualitatively consistent with previous expectations, e.g., Lazarian &
Beresnyak, 2006). This is likely due to the strong turbulent motions tangling
the magnetic field (e.g., §4.1.1. in Sampson et al., 2022), increasing the net
curvature (Yuen & Lazarian, 2020) and facilitating a smaller scale
field141414In the sense that the ratio of the magnetic energy in the low $k$
modes to the magnetic energy in the high $k$ modes is shrinking.. This means,
to place the correlation scale of the magnetic field on comparable scales of a
$10\,\rm{pc}$ observation (e.g., Federrath et al., 2016; Panopoulou et al.,
2016; Beattie et al., 2019; Hu et al., 2019), for $\ell_{0}=100\,\rm{pc}$ we
require that $\operatorname{\mathcal{M}_{\text{A0}}}\gtrsim 100$, which on
average is unrealistically high for the
$\operatorname{\mathcal{M}_{\text{A0}}}$ in the disc of Milky Way analogues
(Wibking & Krumholz, 2021; Hopkins et al., 2021). Likewise, for average ISM
parameters that we use from the above discussion ($\ell_{0}=100\,\rm{pc}$,
$\operatorname{\mathcal{M}_{\text{A0}}}=2$),
$\ell_{\text{cor},B}\propto\ell_{0}\mathcal{M}^{-1/2}_{\text{A0}}$ gives
$\ell_{\text{cor},B}\approx 70\,\rm{pc}$, which determines the largest scale
in which the magnetic field can be casually connected via magnetic field
fluctuations, when driven at $100\,\rm{pc}$.

The significance of this for magnetic energy balance, in both simulations and
observations, is that we are often dealing with volumes
$\operatorname{\mathcal{V}}\ll\operatorname{\mathcal{V}}_{L}$, and while
$\left\langle\operatorname{\delta\mathrm{{\mn@boldsymbol{\mathit{B}}}}\cdot\operatorname{\mathrm{{\mn@boldsymbol{\mathit{B}}}}_{0}}}\right\rangle_{\operatorname{\mathcal{V}}_{L}}=0$,
in general
$\left\langle\operatorname{\delta\mathrm{{\mn@boldsymbol{\mathit{B}}}}\cdot\operatorname{\mathrm{{\mn@boldsymbol{\mathit{B}}}}_{0}}}\right\rangle_{\operatorname{\mathcal{V}}}\neq
0$ when $\operatorname{\mathcal{V}}<\operatorname{\mathcal{V}}_{L}$. More
generally, for random fields $\mathrm{{\mn@boldsymbol{\mathit{X}}}}$ and
$\mathrm{{\mn@boldsymbol{\mathit{Y}}}}$, averaged on volumes
$\operatorname{\mathcal{V}}<\operatorname{\mathcal{V}}_{L}$,
$\langle\left\langle\mathrm{{\mn@boldsymbol{\mathit{X}}}}\right\rangle_{\operatorname{\mathcal{V}}}\mathrm{{\mn@boldsymbol{\mathit{Y}}}}\rangle_{\operatorname{\mathcal{V}}}\neq\left\langle\mathrm{{\mn@boldsymbol{\mathit{X}}}}\right\rangle_{\operatorname{\mathcal{V}}}\left\langle\mathrm{{\mn@boldsymbol{\mathit{Y}}}}\right\rangle_{\operatorname{\mathcal{V}}}$
and without loss of generality,
$\left\langle\delta\mathrm{{\mn@boldsymbol{\mathit{X}}}}\right\rangle_{\operatorname{\mathcal{V}}}\neq
0$, i.e., the Reynolds rule of averaging is no longer valid (Germano, 1992;
Hollins et al., 2018). We show an explicit example of this for sub-Alfvénic
M2MA001 (top) and super-Alfvénic M2MA10 (bottom) simulations, mimicking the
trans-sonic average ISM in Figure 10. We provide a full description of the
methodology for performing the experiment in Appendix C, but to summarise
here, we use real-space spherical top-hat filters initialised at random
coordinates in each the simulation, each with diameters
$\ell/\ell_{\text{cor},v}$, to compute the mean-squared (red) and mean (black)
of the filtered $\delta\mathrm{{\mn@boldsymbol{\mathit{B}}}}$ and
$\operatorname{\delta\mathrm{{\mn@boldsymbol{\mathit{B}}}}\cdot\operatorname{\mathrm{{\mn@boldsymbol{\mathit{B}}}}_{0}}}$
fields, in units of the large-scale field, respectively; $\ell_{\text{cor},v}$
is computed independently, directly from the velocity power spectra, Equation
32. We indicate in yellow the scales for which numerical dissipation may
influence the rms statistics, which can be up to $\sim 30\,$grid cells for our
(and other grid) simulation solvers (see §2 in Kitsionas et al., 2009;
Federrath et al., 2011b, for further details).

Figure 11: The same as the bottom panel of Figure 10, but for the scale-
dependent Alfvén Mach numbers of the large-scale field, Equation 27, and the
turbulent field, Equation 28.

For both simulations, we find that on scales $\ell<\ell_{\text{cor},v}$,
$\left\langle\operatorname{\delta\mathrm{{\mn@boldsymbol{\mathit{B}}}}\cdot\operatorname{\mathrm{{\mn@boldsymbol{\mathit{B}}}}_{0}}}(\ell/\ell_{\text{cor},v})\right\rangle_{\operatorname{\mathcal{V}}}\neq
0$, coupled with a small
$\left\langle\delta\mathrm{{\mn@boldsymbol{\mathit{B}}}}^{2}(\ell/\ell_{\text{cor},v})\right\rangle_{\operatorname{\mathcal{V}}}$
compared to the system-scale
$\left\langle\delta\mathrm{{\mn@boldsymbol{\mathit{B}}}}^{2}\right\rangle_{\operatorname{\mathcal{V}}}$.
This is a natural repercussion of finite spatial correlation in the plasma,
where local regions in the turbulence, $\ell^{3}<\ell_{\text{cor},v}^{3}$, can
have fields that appear ordered, even though they are part of the global
fluctuating field, as evident from Figure 10. When averaging over these
filtered regions the spatially correlated fluctuating field acts as an
effective large-scale field on that scale. The main difference between the two
simulations is the size of the fluctuating field, which is orders of magnitude
smaller on all scales in the sub-Alfvénic simulation, as expected from our
previous discussions in Section 5. Our analysis also illustrates the
difficulty of distinguishing between large and small-scale magnetic fields
when one is making observations well below $\ell_{\text{cor},v}$, and provides
a very clear reason why
$\operatorname{\delta\mathrm{{\mn@boldsymbol{\mathit{B}}}}\cdot\operatorname{\mathrm{{\mn@boldsymbol{\mathit{B}}}}_{0}}}$
may be an important quantity for magnetic field observations made over a
finite field of view. This finding has strong implications for the
interpretation of observations. As we discussed in Section 1, some ISM
observations suggest that clouds are in a sub-Alfvénic state (Li et al., 2013;
Federrath et al., 2016; Hu et al., 2019; Heyer et al., 2020; Hwang et al.,
2021; Hoang et al., 2021; Skalidis et al., 2021a). Based on our analysis in
Section 5, this means that a very strong, large-scale field must be present.
Naively, the small-scale dynamo, which is generally invoked to explain the
magnetic field strengths in the ISM, should not be able to maintain such a
system (the most efficient dynamo saturates at
$\operatorname{\mathcal{M}_{\text{A}}}\approx 2$, Federrath, 2016), with all
of the magnetic energy being stored in the large-scale field. An
$\alpha$-$\Omega$ dynamo that can grow a large-scale, coherent magnetic field
through the Parker loops $(\alpha$; Parker 1979) or differential, possibly
galactic, rotation $(\Omega$; see §2.6 in Beck & Wielebinski 2013) may be
required to grow such a field at the kpc scale, that is coupled to the ISM of
the galaxy, piercing individual clouds and making them highly magnetised.

Our current analysis suggests an alternative possibility: one way of creating
an effective mean-field, which may act like a large-scale field for scales
below it (e.g., for a sub-Alfvénic plasma embedded in a super-Alfvénic
plasma), is by taking filtered statistics of the turbulence, and hence
observing the fluctuating field well below the correlation scale of the
turbulence. Because this process turns fluctuating field into an effective
large-scale field, it facilitates the perfect conditions for moving left in
Figure 7, with sub-dominant magnetic field fluctuations and a strong coherent
field. In Figure 11 we show the same filtered turbulence calculation as in
Figure 10 but now instead with
$\operatorname{\mathcal{M}_{\text{A0}}}(\ell/\ell_{\text{cor},v})$ (black
curve) and
$\operatorname{\mathcal{M}_{\text{A,turb}}}(\ell/\ell_{\text{cor},v})$ (blue
curve). We use the M2MA10 simulation, which is globally super-Alfvénic,
$\operatorname{\mathcal{M}_{\text{A0}}}(L/\ell_{\text{cor},v})\approx 10$, as
indicated to the far right of the black curve. On scales smaller than the
$\ell_{\text{cor},v}$, a majority of the random samples exhibit
$\operatorname{\mathcal{M}_{\text{A,turb}}}(\ell/\ell_{\text{cor},v})<1$, and
likewise for
$\operatorname{\mathcal{M}_{\text{A0}}}(\ell/\ell_{\text{cor},v})$, albeit
over a narrower range in $\ell/\ell_{\text{cor},v}$. In the context of
simulations, this shows that the statistics of small regions in the turbulence
can be effectively sub-Alfvénic, even in a globally super-Alfvénic plasma. In
the context of observations of the cold, molecular ISM, it means that even
though individual clouds may be observed to be sub-Alfvénic, the magnetic
fields in these clouds may still be the result of a small-scale dynamo
process, saturating at super-Alfvénic values, but operating on scales much
larger than the cloud being observed. Thus observing a cloud to be sub-
Alfvénic, $\operatorname{\mathcal{M}_{\text{A}}}\la 2$ does not automatically
mean that the field in that cloud is the product of an $\alpha$-$\Omega$ or
similar large-scale dynamo; one can conclude that such a process is at work
only if one recovers $\operatorname{\mathcal{M}_{\text{A}}}\la 2$ on scales
larger than the turbulent correlation length.

As a final calculation for this study, we compute the energy ratio of the
coupling term to the kinetic energy,
$\operatorname{\left\langle(\delta\mathrm{{\mn@boldsymbol{\mathit{B}}}}\cdot\operatorname{\mathrm{{\mn@boldsymbol{\mathit{B}}}}_{0}})^{2}\right\rangle^{1/2}_{\mathcal{V}}}/(2c_{s}^{2}\rho_{0}\pi\mathcal{M}^{2})$
(the ratio of the left- to right-hand side of Equation 19) in a sub-Alfvénic
plasma as a function of scale, just as we did in the previous paragraphs for
the other rms statistics. This tests if the energy balance we presented in
Section 3 is valid over a range of scales, necessary for making it a useful
relation for applications. We plot
$\operatorname{\left\langle(\delta\mathrm{{\mn@boldsymbol{\mathit{B}}}}\cdot\operatorname{\mathrm{{\mn@boldsymbol{\mathit{B}}}}_{0}})^{2}\right\rangle^{1/2}_{\mathcal{V}}}/(2c_{s}^{2}\rho_{0}\pi\mathcal{M}^{2})$
as a function of $\ell/\ell_{\text{cor},v}$ in Figure 12, for the same
simulation as in the top panel of Figure 10. We find that on the interval
between the system scale $L$ and the scale in which numerical dissipation
effects exist $\ell_{\nu}$ (the largest scale which is shaded yellow) that our
filtered samples of the turbulence mostly fall within the 1:2 to 2:1 interval
(blue, dot-dashed lines). On average, across all samples and
$\ell\in[\ell_{\nu},L]$, we find
$\operatorname{\left\langle(\delta\mathrm{{\mn@boldsymbol{\mathit{B}}}}\cdot\operatorname{\mathrm{{\mn@boldsymbol{\mathit{B}}}}_{0}})^{2}\right\rangle^{1/2}_{\mathcal{V}}}/(2c_{s}^{2}\rho_{0}\pi\mathcal{M}^{2})=1.5\pm
1$, capturing the exact equipartition within 0.5$\sigma$. Below $\ell_{\nu}$
most of the samples become highly-magnetised, due to the kinetic energy being
dominated by numerical dissipation and the large-scale magnetic field
permeating through all of the scales in the plasma.

Figure 12: The same as the top panel of Figure 10, but for the scale-dependent
energy balance, in the strong-field limit Equation 19. We show lines of 1:1
(blue, dashed), 2:1 and 1:2 (blue, dot-dashed), highlighting that between the
largest scales in the simulations and the dissipation scales (indicated with
the yellow shaded region) almost all of the random regions sampled fall within
a factor of 2 in exact equipartition.

##  7 Summary and key findings

Motivated by recent works on measuring and modelling magnetic fields in the
ISM (Beattie & Federrath, 2020; Skalidis & Tassis, 2020; Skalidis et al.,
2021b), we provide a theoretical and numerical exposition of the root-mean-
squared (rms) energy balance between the kinetic and magnetic energy,
highlighting the role of the magnetic coupling term,
$\operatorname{\left\langle(\delta\mathrm{{\mn@boldsymbol{\mathit{B}}}}\cdot\operatorname{\mathrm{{\mn@boldsymbol{\mathit{B}}}}_{0}})^{2}\right\rangle^{1/2}_{\mathcal{V}}}$,
which describes the energy contained in magnetic fluctuations
$\delta\mathrm{{\mn@boldsymbol{\mathit{B}}}}$ coupled to the large-scale
magnetic field $\operatorname{\mathrm{{\mn@boldsymbol{\mathit{B}}}}_{0}}$. We
discuss the significance of this term in the context of the $1^{\rm st}$
(comparing volume-averaged energies) and $2^{\rm nd}$ (comparing magnitudes of
energy fluctuations) moments of the energy balance equations, deriving its
typical value directly from the $2^{\rm nd}$ moment equations, which preserve
the positivity of each energy contribution. From this argument we derive a
number of analytical models with no free parameters, for the coupling term and
fluctuating magnetic field, $\delta\mathrm{{\mn@boldsymbol{\mathit{B}}}}$, and
demonstrate that these yield outstanding agreement with the results of a large
suite of MHD simulations. Our analysis demonstrates that
$\operatorname{\left\langle(\delta\mathrm{{\mn@boldsymbol{\mathit{B}}}}\cdot\operatorname{\mathrm{{\mn@boldsymbol{\mathit{B}}}}_{0}})^{2}\right\rangle^{1/2}_{\mathcal{V}}}$
plays an important role in sub-to-trans-Alfvénic large-scale field turbulence,
regardless of the sonic Mach number $\mathcal{M}$. This term becomes less
important for $\operatorname{\mathcal{M}_{\text{A0}}}>2$, where $\delta B^{2}$
becomes dominant, but the large-scale field still has an effect. In Section 5
we explore three different formulations of the Alfvénic Mach number
$\operatorname{\mathcal{M}_{\text{A}}}$, and the relations between them,
showing that sub-Alfvénic large-scale field turbulence, supports an extremely
super-Alfvénic turbulent field, suggesting that the magnetic field
fluctuations are smaller than velocity fluctuations in this limit. We present
a heuristic for understanding the whole turbulent and large-scale field Alfvén
Mach number parameter plane and discuss the implications for interpreting ISM
observations and sub-Alfvénic turbulence. We list the key results of this
study below:

  * •

We provide theoretical models for the volume-averaged fluctuating and coupling
magnetic fields,
$\left\langle\delta\mathrm{{\mn@boldsymbol{\mathit{B}}}}^{2}\right\rangle$ and
$\operatorname{\left\langle(\delta\mathrm{{\mn@boldsymbol{\mathit{B}}}}\cdot\operatorname{\mathrm{{\mn@boldsymbol{\mathit{B}}}}_{0}})^{2}\right\rangle^{1/2}_{\mathcal{V}}}$,
assuming energy equipartition between
$\operatorname{\left\langle(\delta\mathrm{{\mn@boldsymbol{\mathit{B}}}}\cdot\operatorname{\mathrm{{\mn@boldsymbol{\mathit{B}}}}_{0}})^{2}\right\rangle^{1/2}_{\mathcal{V}}}$
and the volume-averaged velocity fluctuations, $\left\langle\delta
v^{2}\right\rangle_{\operatorname{\mathcal{V}}}$ in the sub-Alfvénic regime
(Equation 20, Equation 22), and
$\left\langle\delta\mathrm{{\mn@boldsymbol{\mathit{B}}}}^{2}\right\rangle_{\operatorname{\mathcal{V}}}$
and $\left\langle\delta v^{2}\right\rangle_{\operatorname{\mathcal{V}}}$ in
the super-Alfvénic regime (Equation 24, Equation 25). These models are free of
parameters, but rely on the numerical observation that $\left\langle\delta
v^{2}\right\rangle\approx\left\langle\delta v^{4}\right\rangle^{1/2}$ and
$\left\langle\delta B^{2}\right\rangle\approx\left\langle\delta
B^{4}\right\rangle^{1/2}$, i.e., that the average energy scales with the
magnitude of energy fluctuations, which we demonstrate in Figure 3. Our models
show excellent agreement with numerical compressible MHD data, over a very
broad range of plasma values, in Figure 4, Figure 5 and Figure 6. We discuss
how this provides strong support for the polarisation dispersion models (DCF-
like methods) derived in Skalidis & Tassis (2020) and Skalidis et al. (2021b).

  * •

We define large-scale field, turbulent and total Alfvén Mach numbers (Section
5) and propose that we can completely define the whole
$\operatorname{\mathcal{M}_{\text{A,turb}}}-\operatorname{\mathcal{M}_{\text{A0}}}$
data plane, shown in Figure 8, based on the small-scale dynamo saturation as
$\operatorname{\mathcal{M}_{\text{A0}}}\rightarrow\infty$, and an analytical
model that we derive using energy balance for
$\operatorname{\mathcal{M}_{\text{A0}}}\rightarrow 0$, Equation 31, which
implies $\left\langle\delta
B^{2}\right\rangle^{1/2}_{\operatorname{\mathcal{V}}_{L}}\propto B_{0}^{-1}$.
Critically, we show that the turbulent magnetic field never becomes sub-
Alfvénic, and it is only through a strong, large-scale magnetic field that the
turbulence can transition into this regime. We show that the turbulence
becomes highly super-Alfvénic in the sub-Alfvénic large-scale field regime,
and discuss the implications for sub-Aflvénic ISM observations in Section 5.3.
We suggest that a contributing factor to sub-Alfvénic ISM observations may be
from measuring a trans-to-super Alfvénic average ISM, which, unlike sub-
Alfvénic turbulence, may be supported by a small-scale dynamo, well below the
correlation scale of the turbulence. We show that this is true for simulations
of globally super-Alfvénic turbulence in Figure 11.

  * •

In Figure 10 we explicitly show that by measuring filtered magnetic field
statistics below the correlation scale of the turbulence (Equation 32), which
is roughly equal to the driving scale with some slight deviations in the sub-
Alfvénic regime, (Figure 9), one turns the fluctuating field into an effective
mean-field. We show this is true for both the sub-Alfvénic and super-Alfvénic
regime. This highlights that for quantities such as
$\left\langle\operatorname{\delta\mathrm{{\mn@boldsymbol{\mathit{B}}}}\cdot\operatorname{\mathrm{{\mn@boldsymbol{\mathit{B}}}}_{0}}}\right\rangle_{\operatorname{\mathcal{V}}}$
or
$\left\langle\delta\mathrm{{\mn@boldsymbol{\mathit{B}}}}^{2}\right\rangle_{\operatorname{\mathcal{V}}}$,
if the volume averaging scale, $\operatorname{\mathcal{V}}$, does not resolve
the correlation scale of the turbulence, then the volume average need not be
zero, as previously discussed in Germano (1992) and Hollins et al. (2018).
Furthermore, in Figure 12, we show our sub-Alfvénic energy balance model
provides good agreement across the resolved scales available to us in the sub-
Alfvénic regime.

## Acknowledgements

We thank the anonymous referee for the useful suggestions that increased the
quality of our study. J. R. B. thanks Christoph Federrath’s and Mark
Krumholz’s research groups for many productive discussions and acknowledges
financial support from the Australian National University, via the Deakin PhD
and Dean’s Higher Degree Research (theoretical physics) Scholarships and the
Australian Government via the Australian Government Research Training Program
Fee-Offset Scholarship. M. R. K. acknowledges support from the Australian
Research Council’s Discovery Projects scheme, award DP190101258. R. S.
acknowledges financial support from the European Research Council (ERC) under
the European Unions Horizon 2020 research and innovation programme under grant
agreement No. 771282. C. F. acknowledges funding provided by the Australian
Research Council (Future Fellowship FT180100495), and the Australia-Germany
Joint Research Cooperation Scheme (UA-DAAD). J. R. B. and C. F. further
acknowledge high-performance computing resources provided by the Leibniz
Rechenzentrum and the Gauss Centre for Supercomputing (grants pr32lo, pn73fi,
and GCS Large-scale project 22542), and the Australian National Computational
Infrastructure (grant ek9) in the framework of the National Computational
Merit Allocation Scheme and the ANU Merit Allocation Scheme.

The simulation software, flash, was in part developed by the Flash Centre for
Computational Science at the Department of Physics and Astronomy of the
University of Rochester. Data analysis and visualisation software used in this
study: C++ (Stroustrup, 2013), numpy (Oliphant, 2006; Harris et al., 2020),
matplotlib (Hunter, 2007), cython (Behnel et al., 2011), visit (Childs et al.,
2012), scipy (Virtanen et al., 2020), scikit-image (van der Walt et al.,
2014).

## Data Availability

The data underlying this article will be shared on reasonable request to the
corresponding author.

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Notes: All simulations listed are run with grid resolutions of $16^{3}$,
$36^{3}$, $72^{3}$, $144^{3}$ and $288^{3}$. All statistics are spatially
averaged over the entire domain,
$\operatorname{\mathcal{V}}=\operatorname{\mathcal{V}}_{L}$, and are computed
for 51 time realisations, across 5 correlation times of the Ornstein-Uhlenbeck
forcing function. From the distributions in time, we report the values for the
$16^{\rm th}$, $50^{\rm th}$, and $84^{\rm th}$ percentiles. This process
minimises the possibility of using statistics that are undergoing temporally
intermittent turbulent events (Beattie et al., 2021a). Column (1): the
simulation ID, used throughout this study. Column (2): the turbulent Mach
number, $\mathcal{M}\equiv\left\langle(\delta
v/c_{s})^{2}\right\rangle^{1/2}_{\operatorname{\mathcal{V}}_{L}}$. Column (3):
the correlation scale of the turbulence, $\ell_{\text{cor},v}$, in units of
the driving scale, $\ell_{0}$, defined directly from the power spectra in
Equation 32. Column (4): the Alfvén Mach number of the mean magnetic field,
$\operatorname{\mathcal{M}_{\text{A0}}}\equiv\left\langle(\delta
v\sqrt{4\pi\rho_{0}})/B_{0}\right\rangle$, with fluctuations coming from
$\delta v$, since
$\partial_{x_{i}}\mathrm{{\mn@boldsymbol{\mathit{B}}}}_{0}=\partial_{t}\mathrm{{\mn@boldsymbol{\mathit{B}}}}_{0}=0$.
Column (5): the mean magnetic field strength in units of
$c_{s}\rho_{0}^{1/2}$. Column (6): the volume-averaged square of the turbulent
magnetic field, proportional to the turbulent magnetic energy, in units of
thermal energy. Column (7): the volume-averaged root-mean-squared of the
magnetic coupling term
$\operatorname{\delta\mathrm{{\mn@boldsymbol{\mathit{B}}}}\cdot\operatorname{\mathrm{{\mn@boldsymbol{\mathit{B}}}}_{0}}}$,
in units of thermal energy. Column (8): the Alfvén Mach number of the
turbulent magnetic field, Equation 28. Column (9): the Alfvén Mach number of
the total magnetic field, Equation 29.

Table 1: Main simulation parameters and derived quantities used throughout this study. | Turbulence |  Large-scale $\mathrm{{\mn@boldsymbol{\mathit{B}}}}$-Field |  Fluctuating $\mathrm{{\mn@boldsymbol{\mathit{B}}}}$-Field |  Total $\mathrm{{\mn@boldsymbol{\mathit{B}}}}$-Field
---|---|---|---|---
Simulation ID | $\mathcal{M}$ | $\frac{\ell_{\text{cor},v}}{\ell_{0}}$ | $\operatorname{\mathcal{M}_{\text{A0}}}$ | $\frac{B_{0}}{c_{s}\rho_{0}^{1/2}}$ | $\left\langle\frac{\delta B^{2}}{c_{s}^{2}\rho_{0}}\right\rangle_{\operatorname{\mathcal{V}}}$ | $\left\langle\left(\frac{\delta\mathrm{{\mn@boldsymbol{\mathit{B}}}}\cdot\operatorname{\mathrm{{\mn@boldsymbol{\mathit{B}}}}_{0}}}{c_{s}^{2}\rho_{0}}\right)^{2}\right\rangle^{1/2}_{\operatorname{\mathcal{V}}}$ | $\operatorname{\mathcal{M}_{\text{A,turb}}}$ | $\operatorname{\mathcal{M}_{\text{A,total}}}$
(1) | (2) | (3) | (4) | (5) | (6) | (7) | (8) | (9)
M05MA001 | $0.567$ | ${}_{-0.07}^{+0.02}$ | $1.48$ | ${}_{-0.07}^{+0.05}$ | $0.0113$ | ${}_{-0.001}^{+0.0003}$ | $177.0$ | $0.000191$ | ${}_{-0.005}^{+0.004}$ | $2.32$ | ${}_{-0.5}^{+0.4}$ | $392.0$ | ${}_{-40.0}^{+20.0}$ | $0.0113$ | ${}_{-0.001}^{+0.0003}$
M05MA01 | $0.566$ | ${}_{-0.07}^{+0.02}$ | $1.47$ | ${}_{-0.07}^{+0.05}$ | $0.113$ | ${}_{-0.01}^{+0.004}$ | $17.7$ | $0.0181$ | ${}_{-0.04}^{+0.03}$ | $2.23$ | ${}_{-0.4}^{+0.3}$ | $35.6$ | ${}_{-2.0}^{+2.0}$ | $0.113$ | ${}_{-0.01}^{+0.004}$
M05MA05 | $0.534$ | ${}_{-0.04}^{+0.004}$ | $1.44$ | ${}_{-0.07}^{+0.02}$ | $0.534$ | ${}_{-0.04}^{+0.004}$ | $3.54$ | $0.233$ | ${}_{-0.08}^{+0.05}$ | $1.16$ | ${}_{-0.1}^{+0.05}$ | $7.28$ | ${}_{-0.2}^{+0.3}$ | $0.544$ | ${}_{-0.05}^{+0.005}$
M05MA1 | $0.469$ | ${}_{-0.05}^{+0.01}$ | $0.953$ | ${}_{-0.005}^{+0.02}$ | $0.938$ | ${}_{-0.1}^{+0.03}$ | $1.77$ | $1.2$ | ${}_{-0.3}^{+0.1}$ | $1.0$ | ${}_{-0.07}^{+0.08}$ | $2.75$ | ${}_{-0.2}^{+0.1}$ | $0.974$ | ${}_{-0.08}^{+0.07}$
M05MA2 | $0.462$ | ${}_{-0.03}^{+0.03}$ | $0.783$ | ${}_{-0.03}^{+0.06}$ | $1.85$ | ${}_{-0.1}^{+0.1}$ | $0.886$ | $3.17$ | ${}_{-0.3}^{+0.3}$ | $0.871$ | ${}_{-0.07}^{+0.03}$ | $1.67$ | ${}_{-0.08}^{+0.2}$ | $1.59$ | ${}_{-0.09}^{+0.2}$
M05MA4 | $0.473$ | ${}_{-0.03}^{+0.06}$ | $0.838$ | ${}_{-0.03}^{+0.03}$ | $3.79$ | ${}_{-0.2}^{+0.5}$ | $0.443$ | $3.12$ | ${}_{-0.2}^{+0.2}$ | $0.415$ | ${}_{-0.009}^{+0.02}$ | $1.98$ | ${}_{-0.2}^{+0.2}$ | $2.1$ | ${}_{-0.2}^{+0.4}$
M05MA6 | $0.483$ | ${}_{-0.03}^{+0.06}$ | $0.904$ | ${}_{-0.03}^{+0.03}$ | $5.8$ | ${}_{-0.4}^{+0.8}$ | $0.295$ | $2.74$ | ${}_{-0.1}^{+0.09}$ | $0.256$ | ${}_{-0.006}^{+0.01}$ | $2.32$ | ${}_{-0.1}^{+0.4}$ | $2.51$ | ${}_{-0.2}^{+0.5}$
M05MA8 | $0.502$ | ${}_{-0.02}^{+0.06}$ | $0.907$ | ${}_{-0.02}^{+0.05}$ | $8.03$ | ${}_{-0.3}^{+1.0}$ | $0.222$ | $2.47$ | ${}_{-0.2}^{+0.1}$ | $0.192$ | ${}_{-0.01}^{+0.009}$ | $2.76$ | ${}_{-0.1}^{+0.3}$ | $2.99$ | ${}_{-0.2}^{+0.3}$
M05MA10 | $0.514$ | ${}_{-0.03}^{+0.06}$ | $0.934$ | ${}_{-0.03}^{+0.02}$ | $10.3$ | ${}_{-0.5}^{+1.0}$ | $0.177$ | $2.24$ | ${}_{-0.07}^{+0.1}$ | $0.145$ | ${}_{-0.006}^{+0.01}$ | $3.05$ | ${}_{-0.2}^{+0.4}$ | $3.18$ | ${}_{-0.1}^{+0.4}$
M05MA100 | $0.637$ | ${}_{-0.05}^{+0.09}$ | $0.889$ | ${}_{-0.03}^{+0.04}$ | $127.0$ | ${}_{-10.0}^{+20.0}$ | $0.0177$ | $0.721$ | ${}_{-0.3}^{+0.1}$ | $0.00852$ | ${}_{-0.001}^{+0.0008}$ | $7.9$ | ${}_{-1.0}^{+3.0}$ | $7.92$ | ${}_{-1.0}^{+3.0}$
M05MA1000 | $0.672$ | ${}_{-0.05}^{+0.07}$ | $0.835$ | ${}_{-0.04}^{+0.03}$ | $1340.0$ | ${}_{-100.0}^{+100.0}$ | $0.00177$ | $0.36$ | ${}_{-0.3}^{+0.2}$ | $0.000613$ | ${}_{-0.0002}^{+0.0001}$ | $11.8$ | ${}_{-3.0}^{+8.0}$ | $11.8$ | ${}_{-3.0}^{+8.0}$
M2MA001 | $1.9$ | ${}_{-0.06}^{+0.04}$ | $1.34$ | ${}_{-0.09}^{+0.02}$ | $0.0095$ | ${}_{-0.0003}^{+0.0002}$ | $709.0$ | $0.000879$ | ${}_{-0.003}^{+0.02}$ | $18.7$ | ${}_{-2.0}^{+9.0}$ | $498.0$ | ${}_{-30.0}^{+20.0}$ | $0.00943$ | ${}_{-0.0003}^{+0.0002}$
M2MA01 | $1.85$ | ${}_{-0.02}^{+0.06}$ | $1.37$ | ${}_{-0.1}^{+0.03}$ | $0.0926$ | ${}_{-0.001}^{+0.003}$ | $70.9$ | $0.0989$ | ${}_{-0.03}^{+0.1}$ | $18.4$ | ${}_{-1.0}^{+4.0}$ | $41.1$ | ${}_{-3.0}^{+2.0}$ | $0.0917$ | ${}_{-0.001}^{+0.003}$
M2MA05 | $2.24$ | ${}_{-0.09}^{+0.07}$ | $1.26$ | ${}_{-0.06}^{+0.08}$ | $0.561$ | ${}_{-0.02}^{+0.02}$ | $14.2$ | $6.81$ | ${}_{-0.3}^{+0.3}$ | $20.6$ | ${}_{-2.0}^{+3.0}$ | $5.28$ | ${}_{-0.3}^{+0.3}$ | $0.575$ | ${}_{-0.03}^{+0.01}$
M2MA1 | $1.98$ | ${}_{-0.09}^{+0.1}$ | $0.845$ | ${}_{-0.02}^{+0.02}$ | $0.989$ | ${}_{-0.05}^{+0.05}$ | $7.09$ | $25.3$ | ${}_{-1.0}^{+0.3}$ | $16.3$ | ${}_{-1.0}^{+0.7}$ | $2.47$ | ${}_{-0.2}^{+0.2}$ | $1.01$ | ${}_{-0.06}^{+0.05}$
M2MA2 | $1.95$ | ${}_{-0.2}^{+0.2}$ | $0.815$ | ${}_{-0.04}^{+0.04}$ | $1.95$ | ${}_{-0.2}^{+0.2}$ | $3.54$ | $40.6$ | ${}_{-0.9}^{+0.6}$ | $12.3$ | ${}_{-1.0}^{+0.7}$ | $1.73$ | ${}_{-0.07}^{+0.4}$ | $1.54$ | ${}_{-0.1}^{+0.1}$
M2MA4 | $2.07$ | ${}_{-0.1}^{+0.2}$ | $0.92$ | ${}_{-0.04}^{+0.02}$ | $4.13$ | ${}_{-0.3}^{+0.5}$ | $1.77$ | $37.0$ | ${}_{-0.3}^{+0.3}$ | $5.88$ | ${}_{-0.3}^{+0.4}$ | $2.39$ | ${}_{-0.4}^{+0.1}$ | $2.39$ | ${}_{-0.4}^{+0.3}$
M2MA6 | $2.09$ | ${}_{-0.2}^{+0.2}$ | $0.946$ | ${}_{-0.05}^{+0.02}$ | $6.26$ | ${}_{-0.6}^{+0.6}$ | $1.18$ | $33.8$ | ${}_{-0.4}^{+0.5}$ | $3.83$ | ${}_{-0.1}^{+0.1}$ | $2.45$ | ${}_{-0.3}^{+0.5}$ | $2.68$ | ${}_{-0.5}^{+0.6}$
M2MA8 | $2.05$ | ${}_{-0.2}^{+0.09}$ | $0.945$ | ${}_{-0.02}^{+0.04}$ | $8.21$ | ${}_{-0.7}^{+0.4}$ | $0.886$ | $21.5$ | ${}_{-0.3}^{+0.3}$ | $2.39$ | ${}_{-0.1}^{+0.1}$ | $3.44$ | ${}_{-0.3}^{+0.2}$ | $3.91$ | ${}_{-0.3}^{+0.2}$
M2MA10 | $2.11$ | ${}_{-0.2}^{+0.1}$ | $0.987$ | ${}_{-0.03}^{+0.03}$ | $10.5$ | ${}_{-1.0}^{+0.7}$ | $0.709$ | $18.4$ | ${}_{-0.1}^{+0.3}$ | $1.74$ | ${}_{-0.04}^{+0.1}$ | $4.07$ | ${}_{-0.6}^{+0.3}$ | $4.65$ | ${}_{-0.9}^{+0.4}$
M2MA100 | $2.36$ | ${}_{-0.2}^{+0.1}$ | $1.01$ | ${}_{-0.02}^{+0.03}$ | $118.0$ | ${}_{-8.0}^{+6.0}$ | $0.0709$ | $3.3$ | ${}_{-0.5}^{+0.08}$ | $0.0729$ | ${}_{-0.006}^{+0.003}$ | $16.1$ | ${}_{-2.0}^{+3.0}$ | $17.0$ | ${}_{-3.0}^{+4.0}$
M2MA1000 | $2.37$ | ${}_{-0.1}^{+0.08}$ | $0.999$ | ${}_{-0.05}^{+0.01}$ | $1180.0$ | ${}_{-70.0}^{+40.0}$ | $0.00709$ | $0.276$ | ${}_{-0.3}^{+0.3}$ | $0.00214$ | ${}_{-0.0007}^{+0.0005}$ | $61.9$ | ${}_{-20.0}^{+40.0}$ | $62.8$ | ${}_{-20.0}^{+40.0}$
M4MA01 | $4.03$ | ${}_{-0.5}^{+0.06}$ | $1.39$ | ${}_{-0.09}^{+0.05}$ | $0.101$ | ${}_{-0.01}^{+0.002}$ | $142.0$ | $0.838$ | ${}_{-0.4}^{+0.2}$ | $106.0$ | ${}_{-20.0}^{+9.0}$ | $36.5$ | ${}_{-3.0}^{+3.0}$ | $0.1$ | ${}_{-0.01}^{+0.002}$
M4MA05 | $4.08$ | ${}_{-0.1}^{+0.04}$ | $1.27$ | ${}_{-0.1}^{+0.04}$ | $0.51$ | ${}_{-0.01}^{+0.005}$ | $28.4$ | $24.4$ | ${}_{-0.6}^{+0.4}$ | $74.1$ | ${}_{-7.0}^{+7.0}$ | $4.94$ | ${}_{-0.4}^{+0.3}$ | $0.513$ | ${}_{-0.01}^{+0.007}$
M4MA1 | $4.12$ | ${}_{-0.4}^{+0.5}$ | $0.832$ | ${}_{-0.04}^{+0.04}$ | $1.03$ | ${}_{-0.09}^{+0.1}$ | $14.2$ | $84.3$ | ${}_{-2.0}^{+2.0}$ | $61.9$ | ${}_{-10.0}^{+7.0}$ | $2.49$ | ${}_{-0.1}^{+0.3}$ | $1.02$ | ${}_{-0.1}^{+0.05}$
M4MA2 | $4.02$ | ${}_{-0.3}^{+0.2}$ | $0.846$ | ${}_{-0.02}^{+0.04}$ | $2.01$ | ${}_{-0.1}^{+0.08}$ | $7.09$ | $125.0$ | ${}_{-1.0}^{+1.0}$ | $43.4$ | ${}_{-6.0}^{+2.0}$ | $1.93$ | ${}_{-0.1}^{+0.2}$ | $1.49$ | ${}_{-0.1}^{+0.2}$
M4MA4 | $4.03$ | ${}_{-0.4}^{+0.2}$ | $0.916$ | ${}_{-0.03}^{+0.02}$ | $4.03$ | ${}_{-0.4}^{+0.2}$ | $3.54$ | $109.0$ | ${}_{-0.5}^{+0.2}$ | $20.3$ | ${}_{-0.5}^{+2.0}$ | $2.38$ | ${}_{-0.3}^{+0.1}$ | $2.41$ | ${}_{-0.3}^{+0.1}$
M4MA6 | $3.97$ | ${}_{-0.4}^{+0.2}$ | $0.959$ | ${}_{-0.02}^{+0.02}$ | $5.96$ | ${}_{-0.6}^{+0.3}$ | $2.36$ | $76.3$ | ${}_{-0.6}^{+0.3}$ | $11.7$ | ${}_{-0.4}^{+0.7}$ | $2.94$ | ${}_{-0.3}^{+0.4}$ | $3.29$ | ${}_{-0.2}^{+0.4}$
M4MA8 | $4.05$ | ${}_{-0.5}^{+0.08}$ | $0.979$ | ${}_{-0.03}^{+0.02}$ | $8.1$ | ${}_{-1.0}^{+0.2}$ | $1.77$ | $68.5$ | ${}_{-0.4}^{+0.6}$ | $8.38$ | ${}_{-0.3}^{+0.3}$ | $3.41$ | ${}_{-0.6}^{+0.3}$ | $4.15$ | ${}_{-1.0}^{+0.4}$
M4MA10 | $3.91$ | ${}_{-0.2}^{+0.2}$ | $0.994$ | ${}_{-0.01}^{+0.03}$ | $9.78$ | ${}_{-0.6}^{+0.5}$ | $1.42$ | $59.7$ | ${}_{-0.2}^{+0.2}$ | $6.27$ | ${}_{-0.3}^{+0.2}$ | $3.69$ | ${}_{-0.1}^{+0.3}$ | $4.23$ | ${}_{-0.3}^{+0.9}$
M4MA100 | $4.21$ | ${}_{-0.08}^{+0.2}$ | $1.02$ | ${}_{-0.02}^{+0.03}$ | $105.0$ | ${}_{-2.0}^{+5.0}$ | $0.142$ | $6.29$ | ${}_{-0.4}^{+0.4}$ | $0.203$ | ${}_{-0.01}^{+0.01}$ | $21.0$ | ${}_{-2.0}^{+3.0}$ | $24.2$ | ${}_{-2.0}^{+5.0}$
M4MA1000 | $4.31$ | ${}_{-0.2}^{+0.1}$ | $1.05$ | ${}_{-0.01}^{+0.02}$ | $1080.0$ | ${}_{-50.0}^{+30.0}$ | $0.0142$ | $0.391$ | ${}_{-0.3}^{+0.4}$ | $0.00519$ | ${}_{-0.001}^{+0.001}$ | $115.0$ | ${}_{-40.0}^{+20.0}$ | $128.0$ | ${}_{-50.0}^{+30.0}$
M6MA01 | $6.96$ | ${}_{-0.8}^{+0.5}$ | $1.35$ | ${}_{-0.04}^{+0.05}$ | $0.116$ | ${}_{-0.01}^{+0.008}$ | $213.0$ | $3.86$ | ${}_{-0.7}^{+0.3}$ | $325.0$ | ${}_{-60.0}^{+20.0}$ | $29.0$ | ${}_{-3.0}^{+4.0}$ | $0.115$ | ${}_{-0.01}^{+0.008}$
M6MA05 | $6.44$ | ${}_{-0.1}^{+0.2}$ | $1.21$ | ${}_{-0.04}^{+0.02}$ | $0.536$ | ${}_{-0.01}^{+0.02}$ | $42.5$ | $58.0$ | ${}_{-0.7}^{+1.0}$ | $169.0$ | ${}_{-10.0}^{+10.0}$ | $5.02$ | ${}_{-0.3}^{+0.3}$ | $0.535$ | ${}_{-0.01}^{+0.01}$
M6MA1 | $6.01$ | ${}_{-0.8}^{+0.8}$ | $0.844$ | ${}_{-0.02}^{+0.04}$ | $1.0$ | ${}_{-0.1}^{+0.1}$ | $21.3$ | $157.0$ | ${}_{-4.0}^{+1.0}$ | $121.0$ | ${}_{-20.0}^{+10.0}$ | $2.73$ | ${}_{-0.2}^{+0.2}$ | $0.937$ | ${}_{-0.2}^{+0.1}$
M6MA2 | $5.81$ | ${}_{-0.3}^{+0.3}$ | $0.844$ | ${}_{-0.03}^{+0.03}$ | $1.94$ | ${}_{-0.09}^{+0.1}$ | $10.6$ | $217.0$ | ${}_{-0.5}^{+2.0}$ | $83.7$ | ${}_{-5.0}^{+9.0}$ | $2.1$ | ${}_{-0.1}^{+0.2}$ | $1.58$ | ${}_{-0.09}^{+0.09}$
M6MA4 | $6.23$ | ${}_{-0.4}^{+0.1}$ | $0.94$ | ${}_{-0.01}^{+0.02}$ | $4.15$ | ${}_{-0.3}^{+0.08}$ | $5.32$ | $202.0$ | ${}_{-2.0}^{+0.8}$ | $42.8$ | ${}_{-5.0}^{+2.0}$ | $2.55$ | ${}_{-0.2}^{+0.2}$ | $2.53$ | ${}_{-0.2}^{+0.3}$
M6MA6 | $5.96$ | ${}_{-0.6}^{+0.4}$ | $0.946$ | ${}_{-0.01}^{+0.02}$ | $5.96$ | ${}_{-0.6}^{+0.4}$ | $3.54$ | $168.0$ | ${}_{-1.0}^{+1.0}$ | $25.8$ | ${}_{-2.0}^{+3.0}$ | $2.82$ | ${}_{-0.2}^{+0.3}$ | $3.04$ | ${}_{-0.2}^{+0.3}$
M6MA8 | $5.96$ | ${}_{-0.5}^{+0.2}$ | $0.979$ | ${}_{-0.01}^{+0.01}$ | $7.95$ | ${}_{-0.6}^{+0.2}$ | $2.66$ | $139.0$ | ${}_{-0.9}^{+0.3}$ | $17.4$ | ${}_{-0.6}^{+1.0}$ | $3.39$ | ${}_{-0.4}^{+0.2}$ | $3.73$ | ${}_{-0.4}^{+0.2}$
M6MA10 | $5.99$ | ${}_{-0.5}^{+0.2}$ | $0.99$ | ${}_{-0.02}^{+0.02}$ | $9.99$ | ${}_{-0.8}^{+0.4}$ | $2.13$ | $124.0$ | ${}_{-1.0}^{+0.5}$ | $12.6$ | ${}_{-0.4}^{+1.0}$ | $3.82$ | ${}_{-0.6}^{+0.8}$ | $4.11$ | ${}_{-0.7}^{+1.0}$
M8MA01 | $8.7$ | ${}_{-0.9}^{+0.09}$ | $1.49$ | ${}_{-0.1}^{+0.02}$ | $0.109$ | ${}_{-0.01}^{+0.001}$ | $284.0$ | $4.53$ | ${}_{-0.7}^{+0.4}$ | $488.0$ | ${}_{-90.0}^{+40.0}$ | $32.0$ | ${}_{-4.0}^{+4.0}$ | $0.108$ | ${}_{-0.01}^{+0.001}$
M8MA05 | $8.35$ | ${}_{-0.3}^{+0.2}$ | $1.14$ | ${}_{-0.02}^{+0.02}$ | $0.522$ | ${}_{-0.02}^{+0.01}$ | $56.7$ | $90.2$ | ${}_{-0.6}^{+1.0}$ | $278.0$ | ${}_{-30.0}^{+20.0}$ | $5.01$ | ${}_{-0.4}^{+0.3}$ | $0.505$ | ${}_{-0.03}^{+0.01}$
M8MA1 | $8.16$ | ${}_{-1.0}^{+0.6}$ | $0.837$ | ${}_{-0.03}^{+0.04}$ | $1.02$ | ${}_{-0.1}^{+0.08}$ | $28.4$ | $257.0$ | ${}_{-4.0}^{+2.0}$ | $205.0$ | ${}_{-20.0}^{+10.0}$ | $2.74$ | ${}_{-0.3}^{+0.3}$ | $0.92$ | ${}_{-0.1}^{+0.07}$
M8MA2 | $8.15$ | ${}_{-0.5}^{+0.4}$ | $0.863$ | ${}_{-0.03}^{+0.03}$ | $2.04$ | ${}_{-0.1}^{+0.1}$ | $14.2$ | $395.0$ | ${}_{-1.0}^{+1.0}$ | $148.0$ | ${}_{-8.0}^{+7.0}$ | $2.25$ | ${}_{-0.2}^{+0.2}$ | $1.59$ | ${}_{-0.2}^{+0.1}$
M8MA4 | $7.99$ | ${}_{-0.4}^{+0.3}$ | $0.933$ | ${}_{-0.03}^{+0.01}$ | $4.0$ | ${}_{-0.2}^{+0.1}$ | $7.09$ | $344.0$ | ${}_{-2.0}^{+0.8}$ | $72.7$ | ${}_{-8.0}^{+4.0}$ | $2.41$ | ${}_{-0.1}^{+0.2}$ | $2.3$ | ${}_{-0.2}^{+0.2}$
M8MA6 | $7.97$ | ${}_{-0.9}^{+0.4}$ | $0.938$ | ${}_{-0.02}^{+0.02}$ | $5.98$ | ${}_{-0.7}^{+0.3}$ | $4.73$ | $278.0$ | ${}_{-1.0}^{+2.0}$ | $42.4$ | ${}_{-1.0}^{+7.0}$ | $2.84$ | ${}_{-0.5}^{+0.2}$ | $2.93$ | ${}_{-0.6}^{+0.3}$