# Compressible two-dimensional turbulence: cascade reversal and sensitivity to
imposed magnetic field

Itzhak Fouxon1,3<EMAIL_ADDRESS>Alexei G. Kritsuk2<EMAIL_ADDRESS>Michael Mond1<EMAIL_ADDRESS>1 Department of Mechanical Engineering, Ben-
Gurion University of the Negev, P.O. Box 653, Beer-Sheva 84105, Israel  2
University of California, San Diego, La Jolla, CA 92093-0424, USA  3
Department of Chemical Engineering, Technion, Haifa 32000, Israel

###### Abstract

We study the impact of compressibility on two-dimensional turbulent flows,
such as those modelling astrophysical disks. We demonstrate that the direction
of cascade undergoes continuous transition as the Mach number $Ma$ increases,
from inverse at $Ma=0$, to direct at $Ma=\infty$. Thus, at $Ma\sim 1$
comparable amounts of energy flow from the pumping scale to large and small
scales, in accord with previous data. For supersonic turbulence with $Ma\gg 1$
the cascade is direct, as in three dimensions, which results in multifractal
density field. For that regime ($Ma\gg 1$) we derive a Kolmogorov-type law for
potential forcing and obtain an explicit expression for the third order
correlation tensor of the velocity. We further show that all third order
structure functions are zero up to first order in the inertial range scales,
which is in sharp contrast with incompressible turbulence where the third
order structure function, that describes the energy flux associated with the
energy cascade is non-zero. The properties of compressible turbulence have
significant implications on the amplification of magnetic fields in conducting
fluids. We thus demonstrate that imposing external magnetic field on
compressible flows of conducting fluids allows to manipulate the flow
producing possibly large changes even at small Mach numbers. Thus Zeldovich’s
antidynamo theorem, by which at $Ma=0$ the magnetic field is zero in the
steady state, must be used with caution. Real flows have finite $Ma$ and,
however small it is, for large enough values of $I$, the magnetic flux through
the disk, the magnetic field changes the flow appreciably, or rearranges it
completely. This renders the limit $Ma\to 0$ singular for non-zero values of
$I$. Of particular interest is the effect of the density multifractality, at
$Ma\gg 1$ which is relevant for astrophysical disks. We demonstrate that in
that regime, in the presence of non-zero $I$ the magnetic field energy is
enhanced by a large factor as compared to its estimates based on the mean
field. Finally, based on the insights described above, we propose a novel two-
dimensional Burgers’ turbulence, whose three-dimensional counterpart is used
for studies of the large-scale structure of the Universe, as a model for
supersonic two-dimensional magnetohydrodynamic flows.

The nature as well as the effects of turbulence in astrophysical disks remain
not fully understood despite its significant relevance to the theory of
accretion. In this paper we provide theoretical considerations that lead to
deeper insight and understanding of the structure of compressible turbulence
as well as its impact on the amplification of magnetic fields in thin disks,
assuming that they can be modelled as two-dimensional flows.

The degree of flow compressibility is measured by the Mach number $Ma$,
defined here as the ratio of a typical velocity at the forcing scale and speed
of sound. In the limit of $Ma\to 0$, two-dimensional modeling has far reaching
implications. In that parameter regime the flow is effectively incompressible,
the density is constant and magnetic fields, possibly present in conducting
fluids, asymptotically decay, according to the famous Zeldovich antidynamo
theorem ll6 ; antidynamo . Thus, in the absence of external magnetic fields,
the fluid is non-magnetized in the steady state and the planar flow is purely
hydrodynamic. Immersing the system at $Ma=0$ in a constant vertical magnetic
field, however strong it is, has no impact on the flow. This state is stable –
small, possibly three-dimensional, magnetic field perturbations decay.
Furthermore, perturbations of arbitrary magnitude decay also, provided that
they do not excite significant vertical flow, which would turn the flow into
three-dimensional.

The incompressible limit of $Ma\to 0$ seems therefore inadequate for modelling
astrophysical disks by planar flows since, in contrast to the Earth’s
atmosphere, magnetic fields typically do play important role in astrophysical
context. These conclusions are further supported by recalling that $Ma$ is
finite in real flows, and in particular may reach very large values in
astrophysical disks. We show here that compressibility may lead to a dramatic
change in the structure and nature of the two-dimensional turbulence.

We first discuss pure hydrodynamics and demonstrate that, considering the flow
as a function of the Mach number $Ma$, a strong rearrangement occurs as $Ma$
increases. Thus, at $Ma\ll 1$, the flow is approximately incompressible and
has constant density. It exhibits inverse cascade of energy, where the flow
creates organized larger scale motions starting from the scale at which the
energy is injected. In contrast, at high Mach number the cascade is direct and
the flow is characterized by a multifractal (i.e. singular and having non-
trivial scaling) density field kritsuk ; frisch ; fm . At $Ma=0$ as well as at
$Ma=\infty$ the velocity decouples from the density.

For conducting fluids study of hydrodynamics might be insufficient, as
magnetic fields could (spontaneously) arise. Two types of two-dimensional
configurations have been studied extensively over the last few decades, mainly
numerically. One configuration is characterised by a net magnetic flux that
pierces the disk vertically while in the other the net flux is zero Hawley ;
cattaneo ; Lesur ; Latter ; pessah ; Cui ; Datta . Large-scale net magnetic
flux in the former case may be provided by the dipole magnetic field of a
central accreting object or by its companion, or may be advected into the disk
from the interstellar medium. Large-scale magnetic fields provide key
ingredients in the generation of outflows that play an important role on the
formation of stars and their protostellar disks cao ; Ray . The zero net flux
studies on the other hand aim at exploring the possibility of a self-
magnetisation of the disk where such properties as angular momentum transfer
do not depend on external magnetic fields.

In order to gain a deeper understanding of the role played by the magnetic
flux we establish a generalization of Zeldovich’s theorem to include the
compressibility of planar flows. We show that if the magnetic field threading
the disk, as measured by the flux $I$, vanishes, the magnetic field decays at
any $Ma$, so that the fluid is non-magnetized in the steady state. However, if
the system is immersed in a constant (large-scale) magnetic field, so that
$I\neq 0$, the situation can be very different. In that case a steady-state
inhomogeneous non-zero magnetic field is generated, whose impact on the flow
is determined by the magnitude of $I$. We further show that for any $Ma$,
there is a range of $I$ for which the back reaction of the magnetic field on
the flow is non-negligible. For instance, Zeldovich’s result does not apply
for arbitrarily small $Ma$, if the net flux $I$ is large enough obeying $I\gg
1/Ma$ (see refinements below). In that case even though the Mach number is
arbitrarily small, the flow may be strongly affected by the magnetic field due
to the high values of the magnetic flux. In the opposite limit of large $Ma$,
we show that the magnetic energy density is larger than the square of the
mean-field by a power of $Ma$ for those $I$ that cause negligible Lorentz
forces on the velocity.

The above insights may turn useful in progressing toward an understanding of
accretion. They also suggest possible terrestrial applications where $Ma$ is
typically not that large. Thus for a conducting fluid whose flow can be
considered as planar, changing the external magnetic field changes the flow.
These changes may include amounts of energy directed to large and small
scales. The ease with which external magnetic fields can be controlled makes
the study of such prospective of flow manipulation highly attractive.

We use for our study the simplest model that captures the relevant physics.
The flow is modelled as two-dimensional isothermal compressible turbulence.
Generally, two-dimensional fluid mechanics is expected to describe motions on
scales larger than the system (disk) thickness. For those scales, it seems
plausible that introduction of three-dimensional perturbations would not
change the main features uncovered by the two-dimensional study. This was
shown to be true at small Mach numbers, see guo  and references therein.
Furthermore, checking the validity of the isothermal approximation requires
first a thorough scrutiny of its consequences and implications. A step towards
achieving that goal in the highly compressible turbulence regime is proposed
in the current work.

We start by assuming a pure hydrodynamic framework where there is no magnetic
field. We study how the properties of two-dimensional isothermal compressible
turbulence change as the Mach number increases from zero to infinity (the
Reynolds number is assumed to be large and fixed). We observe that the
transfer of kinetic energy changes direction from inverse to direct cascade.
This extends the observation of fk  that as the Mach number increases, growing
portions of the kinetic energy, which at $Ma\ll 1$ is transferred
predominantly from small to large scales (inverse cascade), are transferred
from large to small scales. Thus at $Ma\sim 1$ the amount of energy
transferred down and up the energy injection scale are comparable fk . We
provide arguments that this trend continues and at $Ma=\infty$ all kinetic
energy is transferred down scales, which implies deep changes in the
properties of the velocity and density fields. Our results suggest that, in
contrast to the $Ma\ll 1$ case, supersonic turbulent flows in two and three
dimensions are rather similar in some respects.

The self-consistency of pure hydrodynamic framework for turbulent flows of
conducting fluids needs to be examined. This is due to the possible
instability, known as dynamo effect, by which small fluctuations of magnetic
fields that are ubiquitous in the Universe, are amplified by the flow ll8 . As
already said, Zeldovich’s theorem suggests that neglecting the magnetic field
is self-consistent in the limit $Ma\to 0$ where the flow could be thought to
good approximation to be incompressible. In the opposite limit of $Ma\gg 1$,
similarity of supersonic turbulence in two and three dimensional situations
raises the possibility that the flow has dynamo action as its three-
dimensional counterpart, i.e., amplifies small fluctuations of magnetic field.
Moreover, due to compressibility, not only fluctuations but also the mean
magnetic field can be relevant. This calls for studying the role of the
magnetic field which is done after exploring the pure hydrodynamics case.

The fundamental model and its properties—Turbulence is assumed to be driven by
external acceleration fields which do not depend on the flow (the external
volume force is proportional to the density) and have characteristic scale of
spatial variations $L$, the pumping scale. We use the simplest model which is
isothermal turbulence with simplified viscous dissipation term. There the
pressure $p$ is given by $\rho c_{s}^{2}$ where $\rho$ and $c_{s}$ are the
mass density of the disk and the constant speed of sound, respectively. We use
dimensionless variables where the velocity is rescaled by its characteristic
value $u$ at the scale $L$, the density is rescaled by its mean, time and
space coordinates are rescaled by $L/u$ and $L$, respectively. The flow is
governed therefore by the following equations

$\displaystyle\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\partial_{t}\rho+\nabla\cdot(\rho\bm{v})=0,$
(1)
$\displaystyle\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\partial_{t}\bm{v}+(\bm{v}\cdot\nabla)\bm{v}=-\frac{1}{Ma^{2}}\nabla\ln\rho+\bm{a}+\nu\nabla^{2}\bm{v},$

where $Ma\equiv u/c_{s}$, the dimensionless density is $\rho$, and
$\bm{v}\equiv\bm{u}/u$ with $\bm{u}$ the dimensional flow. Here $\bm{a}$ is
the dimensionless acceleration and we denote the inverse Reynolds number by
$\nu$, which is assumed to be constant. We use a simplified form of the
viscosity term since we are interested in the inertial range properties of
turbulence where its precise form is believed to be irrelevant frisch . We
stress that our definition of the Mach number uses velocity at $L$ and not the
actual velocity as, e.g., in fk . The definitions are similar at $Ma\gtrsim
1$, however differ at $Ma\ll 1$ where, due to inverse cascade, typical
velocity is much larger than the velocity at the pumping scale.

Eqs. (1) can be considered as a fundamental model of compressible turbulence.
In spite of including compressibility in the simplest possible way, they are
widely believed to have significant applications, see e.g. kritsuk  and
references therein.

In the two-dimensional case the vorticity $\omega$ (given by the only non-
vanishing, vertical, component of the velocity curl) obeys the advection-
diffusion equation with forcing

$\displaystyle\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\partial_{t}\omega+\nabla\cdot(\omega\bm{v})=\nu\nabla^{2}\omega+s.$
(2)

Here $s$ is the source of vorticity, given by the only non-vanishing,
vertical, component of $\nabla\times\bm{a}$. The absence in Eq. (2) of the
vorticity stretching term, present in the three-dimensional case, has far-
reaching consequences.

Potential forcing—One of the consequences of Eq. (2) is significant
simplifications in the case of potential forcing, such for instance as gravity
where $\bm{a}=-\nabla\psi$ with a certain $\psi$. We have then $s=0$ in Eq.
(2) and the equation is of the Fokker-Planck form. If the velocity were
independent of $\omega$ then, due to dissipative nature of the Fokker-Planck
equations, the steady state would have $\omega\equiv 0$, see fp . However,
vorticity is a so-called active scalar that reacts on the flow that advects
it. This could cause a difference. A well-known example is vorticity of three-
dimensional turbulence and small fluctuations of the magnetic field. Both obey
the same equation of first order in time, however their behavior is very
different review . In our case however, the existence of entropy functional
guarantees the asymptotic decay of $\omega$, that has zero average, see
detailed discussion in Ch. III of fm  and references therein, and also the
study of vertical magnetic field below.

Relaxation of $\omega$ to zero signifies that mathematically well-defined
potential solutions of Eqs. (1), with $\bm{v}=\nabla\phi$, given by gradient
of certain potential $\phi$, are stable. Thus for potential forcing the flow
is going to be potential, which is not necessarily so in the three-dimensional
case where instability is possible. This allows us to rewrite the momentum
equation as (cf. with Bernoulli’s equation)

$\displaystyle\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\partial_{t}\bm{v}=-\nabla\left(\frac{\ln\rho}{Ma^{2}}+\frac{v^{2}}{2}+\psi\right)+\nu\nabla^{2}\bm{v},$
(3)

so that the flow potential $\phi$ obeys

$\displaystyle\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\partial_{t}\phi+\frac{(\nabla\phi)^{2}}{2}=-\frac{\ln\rho}{Ma^{2}}-\psi+\nu\nabla^{2}\phi.$
(4)

Thus, the evolution reduces to two scalar fields, $\rho$ and $\phi$.

Scaling relation—We observe that Eq. (3) has the form of a local conservation
law. It can be written as $\partial_{t}v_{i}+\nabla\cdot\bm{j}^{i}=0$ where
the flux functions are given by
$(j^{i})_{k}=\delta_{ik}\left(Ma^{-2}\ln\rho+v^{2}/2+\psi\right)+\nu\nabla_{k}v_{i}$.
In the free (forceless) non-dissipative case the latter are given by
$(j^{i})_{k}=\delta_{ik}\left(Ma^{-2}\ln\rho+v^{2}/2\right)$, and give rise to
the following general scaling relation in steady state us :

$\displaystyle\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\left\langle
v_{i}(0)\left(\frac{\ln\rho(\bm{r})}{Ma^{2}}+\frac{v^{2}(\bm{r})}{2}\right)\right\rangle=\frac{\langle\bm{v}(0)\cdot\bm{a}(0)\rangle\bm{r}}{2}.$
(5)

This equation is obtained by using Eq. (3) in the stationarity condition
$\partial_{t}\langle v_{i}(0)v_{i}(\bm{r})\rangle=0$. Eq. (5) holds in the
inertial range of scales, which are below $L$, and yet not too small so that
the viscosity is negligible (here and below, for transparency we often use
dimensional quantities in qualitative considerations). In the case of inverse
cascade (see below) a sustainable steady state may require the presence of a
linear dissipation term in the momentum equation, see e.g. buhler  and cf. fk
where it is shown that steady state can also hold for Eqs. (1) as they are.
Here we refer to cascade in the sense of the transfer of energy. Statistical
isotropy is also assumed above, for a form that does not involve the isotropy
assumption see us1 .

We observe that by spatial homogeneity, which is assumed in our study, we have
$\langle\bm{v}(0)\cdot\bm{a}(0)\rangle=\langle\psi(0)\nabla\cdot\bm{v}(0)\rangle$
in the RHS of Eq. (5) where we use $\bm{a}=-\nabla\psi$. Here arbitrary
additive constant in the potential is irrelevant due to
$\langle\nabla\cdot\bm{v}(0)\rangle=0$. Eq. (5) can be rewritten in a standard
way as a relation for the difference frisch , which will be subject of future
investigations. In the limit $Ma\to 0$ it is equivalent to two-dimensional
version of the Kolmogorov relation and hence it can be considered as its
finite Mach number generalization us ; review .

We return to Eqs. (1) for generic, not necessarily potential $\bm{a}$, and
study the behavior of turbulence that they describe, as a function of $Ma$,
assuming that all other parameters in the equations are fixed. For this we
consider the two opposite limits, of $Ma\to 0$ and $Ma\to\infty$.

Low Mach number limit—In the limit of $Ma\to 0$ Eqs. (1) reduce to the well-
studied equations of incompressible turbulence (there are some delicate points
in the case of potential forcing that does not inject solenoidal component of
velocity. These are irrelevant for the discussion here), see e.g. ll6 . A
fundamental property of incompressible turbulence is that non-linear
interactions organize so as to provide a constant flux of energy $\int
v^{2}d\bm{x}/2$, see e.g. frisch . Thus in the inertial range, defined as the
range where both the forcing and the viscosity can be neglected, the energy is
transferred between the Fourier modes of the flow at constant total rate. It
can flow either from large to small scales (direct cascade) or from small to
large scales (inverse cascade) review . The direction of the cascade depends
on the number of quantities that have a positive density in Fourier space and
are conserved by the Navier-Stokes equations without forcing and dissipation
terms. Three-dimensional turbulent flow has only one such quantity, which is
energy. In this case the quadratic non-linearity of the equations gives rise
to a direct energy cascade. In contrast, two-dimensional incompressible
turbulence, that holds in the $Ma\to 0$ limit, has two such quantities -
energy and enstrophy (integral of squared vorticity). As a result the energy
flows from smaller to larger scales, from the injection scale to larger
scales, corresponding to the inverse cascade of energy, see e.g. frisch . This
direction of kinetic energy flow can be understood as arising from
interactions of planar vortices and their self-organization, see e.g. review .

Infinite Mach number limit—We turn now to the opposite limit of supersonic
turbulence, $Ma\to\infty$, and demonstrate that the energy cascade is direct
there, namely from large to small scales.

We observe that if all parameters besides $Ma$ are fixed, and $Ma\to\infty$,
the pressure term in Eqs. (1) becomes negligible at any given scale $l$ that
satisfies $(lMa^{2})^{-1}\ll 1$. Thus, for large enough $Ma$, pressure is
negligible at the pumping scale $L$. Large scales (here $\lesssim L$) are then
described by the equations

$\displaystyle\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\partial_{t}\rho+\nabla\cdot(\rho\bm{v})=0,$
(6)
$\displaystyle\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\partial_{t}\bm{v}+(\bm{v}\cdot\nabla)\bm{v}=\bm{a}+\nu\nabla^{2}\bm{v},$

obtained by setting $Ma=\infty$ in Eqs. (1). In the case of
$\bm{a}=-\nabla\psi$ the flow, as explained above, is potential and the above
equations reduce to the so-called Burgers’ turbulence review ; Burgulence .

The pressure is negligible at large scales because, for supersonic turbulence,
inertial degrees of freedom of large scale eddies are much faster than the
pressure degrees of freedom, propagating at the speed of sound. Nonetheless,
since smaller scale eddies have smaller velocities, then at scales where the
flow velocity is of order of the speed of sound or smaller, the pressure is
relevant again. Thus, Eqs. (6) apply only at scales larger than the so called
sonic scale $l_{s}$, defined as the scale where the eddy velocity and the
sound velocity are equal. For quantitative definition of $l_{s}$ the second
order structure function of the flow could be used sonic .

Density is smooth below the sonic scale—By definition the eddies below $l_{s}$
are subsonic, so that they act on the density as effectively incompressible
velocity. Thus, the density field below $l_{s}$ is smooth at any Mach number
sonic . For $Ma\ll 1$ the Kolmogorov scaling of the inverse cascade guo  gives
$l_{s}\sim L/Ma^{3}\gg L$ so the density is smooth up to scales much larger
than $L$.

Kolmogorov scaling can only be used if eddies at scales larger than $L$ are
really subsonic, which, e.g., is not the case of fk , where $k^{-2}$ spectrum
fits the data better than the Kolmogorov scaling, implying $l_{s}\sim
L/Ma^{2}$. The data for the case of fk  is shown in Fig. 1. To determine the
Mach number in the Figure, we observed that for parameters of fk  we have
$l_{s}\sim L10^{1.7}$ which results in $Ma\simeq\sqrt{L/l_{s}}\sim 0.1$ (we
remark that within the definition of the Mach number in fk  this case
corresponded to $Ma=0.62$). Fig. 1 demonstrates that the density field is
smooth over scales much larger than the pumping scale. The actual scale of
variations is the sonic scale that in this case coincided with the system
size. We also provide in Fig. 2 the transient state before the steady state is
reached. That shows much smaller scale structures that however disappear in
the steady state.

In the opposite limit of $Ma\gg 1$, using $k^{-2}$ spectrum of the Burgers’
turbulence (see below and fk ; boldyrev ; grs ) we find $l_{s}\sim L/Ma^{2}\ll
L$. This agrees with observations: at $Ma\simeq 4$ we have $l_{s}/L\simeq
1/16$ for simulation parameters in sonic . The range of scales $l_{s}\ll r\ll
L$, that exists at $Ma\gg 1$, was named in fm  supersonic inertial range.
Finally, by definition $l_{s}\sim L$ at $Ma\sim 1$, see Fig. 3 (here our
definition of $Ma$ and that of fk  coincide).

An approach to compressible turbulence—Similarity of two-dimensional
compressible turbulence with ”small pressure”, i.e. Eqs. (1) at $Ma\gg 1$, and
Burgers turbulence was mentioned previously in boldyrev . It is most
remarkable that both at $Ma=0$ and $Ma=\infty$ the velocity decouples from the
density and obeys closed equation. Correspondingly, the density becomes a
passively advected field. The decoupling allows to study first the properties
of velocity separately from the density, and then study the density as in a
passive scalar problem fm ; review . Since velocity-density coupling is a main
difficulty introduced by the compressibility then this opens the possibility
to describe finite $Ma$ cases by interpolation between the two limits.

Figure 1: A sample snapshot of the steady state density field from a
$4096^{2}$ simulation of near-isothermal ($\gamma=1.001$) turbulence at
$Ma\sim 0.1$ (case E from fk ). The grey-scale ramp follows $\ln\rho$, showing
lower densities with darker shades of grey, $\rho\in[0.58,1.60]$. A black
scale bar in the lower left indicates the forcing scale. Density varies mostly
at larger scales.

Figure 2: Same as in Fig. 1, but from a $2048^{2}$ simulation early in the
evolution — before the system reaches a steady state with $Ma\sim 0.1$;
$\rho\in[0.94,1.12]$.

Figure 3: Same as in Fig. 1, but from a $512^{2}$ simulation in a
statistically steady state at $Ma=1.0$; $\rho\in[0.20,5.11]$. It is seen that
density varies over the pumping scale.

Kolmogorov-type law for potential pumping—In the potential case, similarly to
Eq. (3) we can write the last of Eqs. (6) as

$\displaystyle\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\
\partial_{t}\bm{v}+\nabla\frac{v^{2}}{2}=\bm{a}+\nu\nabla^{2}\bm{v}.$ (7)

The potential equation becomes

$\displaystyle\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\partial_{t}\phi+\frac{(\nabla\phi)^{2}}{2}=-\psi+\nu\nabla^{2}\phi,$
(8)

cf. Eq. (4). This equation implies the following evolution law for the pair-
correlation function:

$\displaystyle\\!\\!\\!\\!\\!\\!\\!\partial_{t}\langle\phi(0)\phi(\bm{r})\rangle\\!+\\!\frac{\left\langle\phi(\bm{r})v^{2}(0)\right\rangle\\!+\\!\left\langle\phi(0)v^{2}(\bm{r})\right\rangle}{2}\\!=\\!-\langle\psi(0)\phi(\bm{r})\rangle$
$\displaystyle\\!\\!\\!\\!\\!\\!\\!-\langle\phi(0)\psi(\bm{r})\rangle\\!+\\!\nu\langle\phi(\bm{r})\nabla^{2}\phi(0)\rangle+\nu\langle\phi(0)\nabla^{2}\phi(\bm{r})\rangle,$
(9)

which is valid in the supersonic range of scales, $r\gg l_{s}$, where the
pressure is negligible. We assume that the viscosity is so small that it is
also negligible in this range, as discussed previously. We consider the above
equation in the steady state, where the time derivative term vanishes. We
have, assuming spatial homogeneity and isotropy, that at $r\ll L$

$\displaystyle\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\nabla^{2}\langle\phi(0)\psi(\bm{r})\rangle=\langle\bm{v}(0)\cdot\bm{a}(\bm{r})\rangle\\!\approx\\!\langle\bm{v}(0)\cdot\bm{a}(0)\rangle\\!=\\!\epsilon,$
(10)

where we introduced the pumping power
$\epsilon\equiv\langle\bm{v}(0)\cdot\bm{a}(0)\rangle$, and used
$\bm{a}=-\nabla\psi$ and $a(\bm{r})\approx a(0)$ at $r\ll L$. We conclude that
in isotropic turbulence Eq. (9) yields in the supersonic inertial range,
$l_{s}\ll r\ll L$, the following lowest order relation:

$\displaystyle\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\left\langle
v^{2}(0)\phi(\bm{r})\right\rangle\\!=\\!-\langle\psi(0)\phi(0)\rangle-\frac{\epsilon
r^{2}}{2d}.$ (11)

We introduced the space dimension $d$ as a useful parameter, in our case $d=2$
(if $d=3$, the consideration serves merely a reference purpose, since
potential solutions are probably unstable). The above relation holds at any
fixed $r$, which is much smaller than $L$, in the limit of infinite Mach and
Reynolds numbers.

We derive now a Kolmogorov-type relation from Eq. (11). For that purpose we
observe that the three-point correlation function of the velocity potential
$\phi$ obeys

$\displaystyle\\!\\!\\!\\!\\!\\!\\!\\!\\!\left\langle\phi(\bm{r}_{1})\phi(\bm{r}_{2}))\phi(\bm{r}_{3})\right\rangle\\!=\\!C((\bm{r}_{2}\\!-\\!\bm{r}_{1})^{2},(\bm{r}_{3}\\!-\\!\bm{r}_{1})^{2},(\bm{r}_{3}\\!-\\!\bm{r}_{2})^{2}),$

where $C(x,y,z)$ is a symmetric function of the three independent (dummy)
variables, $x$, $y$ and $z$. We assumed spatial homogeneity and isotropy,
which imply that the above correlations depend only on the geometry of the
triangle whose vertices are $\bm{r}_{i}$. Consider next the tensor

$\displaystyle\\!\\!\\!\\!\\!\\!\\!\\!\\!D_{kl}(\bm{r})\equiv\left\langle
v_{k}(0)v_{l}(0)\phi(\bm{r})\right\rangle$
$\displaystyle\\!\\!\\!\\!\\!\\!\\!\\!\\!=\nabla_{1k}\nabla_{2l}C(r_{21}^{2},r_{31}^{2},r_{32}^{2})|_{\bm{r}_{1}=\bm{r}_{2}=0,\
\ \bm{r}_{3}=\bm{r}},$ (12)

which is subsequently given by:

$\displaystyle\\!\\!\\!\\!\\!\\!\\!\\!\\!D_{kl}(\bm{r})=4r_{l}r_{k}C_{yz}(0,r^{2},r^{2})-2\delta_{kl}C_{x}(0,r^{2},r^{2}).$
(13)

Taking the trace of the above equation, introducing $q=r^{2}$, and using Eq.
(11) yields

$\displaystyle\\!\\!\\!\\!\\!\\!\\!\\!\\!2qC_{yz}(0,q,q)-dC_{x}(0,q,q)\\!=\\!-\frac{\langle\psi(0)\phi(0)\rangle}{2}-\frac{\epsilon
q}{4d},$ (14)

which is valid to linear order in $q$ since Eq. (11) holds to quadratic order
in $r$. Evaluating the above equation and its derivative with respect to $q$
at $q=0$, and employing the symmetry of $C(x,y,z)$, we obtain

$\displaystyle\\!\\!\\!\\!\\!\\!\\!\\!\\!C_{x}(0,0,0)\\!=\\!\frac{\langle\psi(0)\phi(0)\rangle}{2d},\
\ C_{xz}(0,0,0)\\!=\\!\frac{\epsilon}{8d(d\\!-\\!1)}.$ (15)

In addition, use has been made of the fact that $C(x,y,z)$ must have finite
cross derivatives of second order (in order to represent finite correlations
of the flow) so that asymptotic expansions to the considered order can be
obtained by differentiation. The above equation cannot be used at $d=1$ since
in that case the three-point correlation function of $\phi(\bm{r})$ depends on
two, rather than three variables, thus leading to different considerations.

We find that to linear order in $q$ (i.e. quadratic order in $r$)

$\displaystyle\\!\\!\\!\\!\\!\\!\\!\\!\\!C_{x}(0,q,q)=C_{x}(0,0,0)+q\left(C_{xy}(0,0,0)+C_{xz}(0,0,0)\right)$
$\displaystyle\\!\\!\\!\\!\\!\\!\\!\\!\\!=\frac{\langle\psi(0)\phi(0)\rangle}{2d}+\frac{\epsilon
q}{4d(d-1)}.$ (16)

We therefore conclude by using the above in Eq. (13) that to quadratic order
in $r$

$\displaystyle\\!\\!\\!\\!\\!\\!\\!\\!\\!D_{kl}(\bm{r})=\frac{\epsilon
r_{l}r_{k}}{2d(d-1)}-\frac{\langle\psi(0)\phi(0)\rangle\delta_{kl}}{d}-\frac{\epsilon
r^{2}\delta_{kl}}{2d(d-1)}.$ (17)

In the next step towards our goal, we find that the third order correlation
tensor of the velocity $T_{ikl}(\bm{r})$, which is symmetric in the last two
indices, is given by:

$\displaystyle\\!\\!\\!\\!\\!\\!T_{ikl}(\bm{r})\equiv\left\langle
v_{k}(0)v_{l}(0)v_{i}(\bm{r})\right\rangle=\nabla_{i}D_{kl}(\bm{r})=-\frac{\epsilon
r_{i}\delta_{kl}}{d(d-1)}$
$\displaystyle\\!\\!\\!\\!\\!\\!+\frac{\epsilon\left(r_{k}\delta_{il}+r_{l}\delta_{ik}\right)}{2d(d-1)}.$
(18)

This is a remarkable result that is the counterpart of a similar result for
incompressible turbulence. The only difference is in the numerical value of
the coefficients, cf. frisch . This difference however is qualitative and not
only quantitative. The above form of $T_{ikl}(\bm{r})$ gives

$\displaystyle\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\left\langle\left(v_{i}(\bm{r})-v_{i}(0)\right)\left(v_{k}(\bm{r})-v_{k}(0)\right)\left(v_{l}(\bm{r})-v_{l}(0)\right)\right\rangle$
$\displaystyle\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!=2\left(T_{lik}(\bm{r})+T_{kli}(\bm{r})+T_{ikl}(\bm{r})\right)=0,$
(19)

that holds at any $d$. This implies that all third order structure functions,
both longitudinal and transversal ones, are identically zero. This is in sharp
contrast with incompressible turbulence where the third order structure
function, that describes the energy flux associated with the energy cascade
frisch , is non-zero both for $d=2$ as well as $d=3$. Here however, the third
order structure function does not provide information on the cascade of the
energy $\int\rho v^{2}d\bm{r}/2$. It is rather a degenerate quantity that
vanishes due to the effective one-dimensionality of the shocks. The direction
of the shock formation is random, so that averaging over those directions the
odd third power of velocity difference yields zero.

Despite that the structure function vanishes, the third order correlation
tensor of the velocity obeys the non-trivial Eq. (18). That equation is
predicted to hold in the supersonic inertial range and can be used for
experimental testing of the reduction to Burgers turbulence.

The above considerations should be understood not as a statement on the
vanishing of third order correlations of the velocity difference. They rather
mean that the third order structure function of the velocity is of order
higher than linear in $r/L$. Furthermore, anisotropy, that is invariably
present in various applications and is neglected above, could produce in the
third order structure functions a contribution that would be larger than the
isotropic value. The study of this possibility is left for future work.

The above raises the issue of how to define the scaling exponents of the
velocity, given that the third order structure function vanishes. The quantity
that can be used for that purpose is the absolute value of the difference and
not its longitudinal component,

$\displaystyle\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\left\langle|\bm{v}(\bm{r})-\bm{v}(0)|^{p}\right\rangle\propto
r^{\zeta_{p}}.$ (20)

This is in contrast with the three-dimensional incompressible turbulence where
scaling exponents defined with the help of longitudinal, transversal structure
functions or the modulus of the difference would be identical, cf. Iyer .

Cascade direction at $Ma\gg 1$ in the potential case—Reduction to Eqs. (6) at
$Ma\gg 1$ allows to understand the direction of the cascade in this limit. We
first consider the case of potential Burgers’ turbulence. Free flow evolution,
obtained by discarding the acceleration term in Eqs. (6), is described by the
Burgers equation
$\partial_{t}\bm{v}+(\bm{v}\cdot\nabla)\bm{v}=\nu\nabla^{2}\bm{v}$. In the
case of potential flow this equation is integrable. It can be reduced to the
linear heat equation by the Cole-Hopf substitution, see e.g. Burgulence . The
solution demonstrates that in the small viscosity limit, the evolution
consists of steepening of large scale profiles of the flow, which forms shocks
with non-trivial geometry. This can be understood by observing that at small
viscosity, Burgers’ equation reduces to the Hopf equation
$\partial_{t}\bm{v}+(\bm{v}\cdot\nabla)\bm{v}=0$, that governs the evolution
of the velocity field of free particles. The evolution behaves according to
the well-known implicit solution
$\bm{v}(\bm{x}+\bm{v}_{0}(\bm{x})t,t)=\bm{v}_{0}(\bm{x})$ that express the
conservation of the initial velocity $\bm{v}_{0}(\bm{x})$ of a particle
located initially at $\bm{x}$. Generally, linear trajectories of free
particles would intersect at a finite time, making the velocity field multi-
valued. The intersection is preceded by the blowup of $\nabla\bm{v}$ at the
formation of an infinitely thin shock. Due to the singularity, the neglect of
the viscous, higher derivative term in the Burgers equation becomes
inconsistent near the blowup. Finite viscosity, however small it is,
regularizes the divergences, arrests overturning of the shocks, and keeps the
flow single-valued and smooth. The above implies that the direction of cascade
in the inertial range, where by definition the forcing and viscosity terms are
negligible frisch , is direct because the flow generates smaller scale,
infinitely thin, structures, that are eventually dissipated by the viscosity.
Thus shocks of Burgers turbulence exhibit dissipative anomaly with energy
dissipation remaining finite in the zero viscosity limit Burgulence . This is
in contrast to the lack of such anomaly in the case of the inverse cascade,
see e.g. review .

In the presence of forcing then, the steady state consists of continuous
injection by the force of flow profiles of scale of order $L$, which then
generate smaller scale structures until dissipated by viscosity at the
smallest scale of the flow. The cascade of the Burgers turbulence is direct in
any dimension, in contrast to the incompressible Navier-Stokes equations.

It must be remarked that here we speak of the cascade in the sense of
generation of higher wave numbers of the Fourier image of velocity field,
since $\int v^{2}d\bm{x}/2$ is not conserved and cannot be regarded as energy.
In the absence of external force and dissipative processes, Eqs. (6) imply
$\partial_{t}\rho v^{2}+\nabla\cdot\left(\rho v^{2}\bm{v}\right)=0$ so that
the energy which is conserved is the kinetic energy $\int\rho v^{2}d\bm{x}/2$.
Since the density $\rho$ is passive it is a functional of the velocity, so
that $\rho v^{2}$ must be considered as energy defined in terms of the
velocity field only.

Generalization to non-potential stirring—If the accelerations’ field is non-
potential then the flow has finite vorticity, generated by the source term,
$s$, in Eq. (2). In this case the flow at $Ma\to\infty$ still obeys the
Burgers equation, given by the last of Eqs. (6). However, the equation with
$\bm{a}=0$ is no longer integrable to the best of our knowledge. The same Hopf
equation of free particles arises at zero viscosity, which implies that the
cascade is still direct. This conclusion is the extension of observations of
fk  that were described in the introduction. Thus, as the Mach number
increases from zero the kinetic energy transfer to smaller scales from $L$
increases monotonously. Generally this process may be described by the order
parameter $R$ that measures the ratio of the kinetic energy fluxes from the
forcing scale to small and large scales. The results of fk  indicate that
$R(Ma)$, considered as a function of the Mach number, increases monotonously
and continuously from $R(0)=0$ to $R(\infty)=\infty$ with $R(Ma\sim 1)\sim 1$.
Thus the transition from inverse to direct cascade is not sharp, it is
continuous. The degree of universality of $R(Ma)$, i.e., its dependence on the
forcing is of interest.

Lower cutoff scale of the cascade—We recall that the direct cascade of
incompressible three-dimensional turbulence ends at the lower cutoff scale
determined by the viscosity frisch . In our case, the direct Burgers-type
cascade of velocity variance (specific kinetic energy) would typically end at
the sonic scale $l_{s}$. Comparing the material derivative and pressure terms
in the last of Eqs. (1), we see that the definition of $l_{s}$ implies that at
this scale the neglected ”small” pressure term in the last of Eqs. (1) is of
the same order as the material derivative term. Hence at scales smaller or of
order $l_{s}$ the pressure forces must be included. In contrast, typically
viscosity is irrelevant at $l_{s}$, becoming significant only at a much
smaller scale. Hence what changes the direct Burgers-type cascade downscales
from $L$ is the pressure at $l_{s}$.

The structure of the flow below $l_{s}$ can be understood by considering it as
an effective flow with Mach number of order one. (The local, scale-dependent
Mach number is defined as the typical velocity of eddies at that scale divided
by $c_{s}$, see sonic  and cf. with local Reynolds number frisch . By
definition, the local Mach number at $l_{s}$ is of order one.) Here all scales
larger than $l_{s}$ provide effective random forcing for the smaller scales’
flow (this is only a qualitative statement since the small scales react back
on the large scales). Below $l_{s}$ the eddies have subsonic velocities, and
for some aspects the turbulence is effectively incompressible. For instance,
as already said, the density $\rho$ is roughly constant below $l_{s}$, as in
the incompressible case sonic ; dust ; fm .

Kinetic energy transfer between Fourier modes at $Ma\lesssim 1$—We study the
nature of turbulence at $Ma\lesssim 1$ by considering how incompressible
turbulence, that holds at $Ma=0$, is changed by a small but finite $Ma$. For
finite $Ma$ the enstrophy is no longer an integral of motion of two-
dimensional Navier-Stokes equations without forcing and dissipation.
Turbulence then can be thought of as arising from complex interactions of
planar vortices and sound waves, where vortices tend to support inverse and
waves direct energy cascades (this way of thinking might need refinements when
temperature degree of freedom is present) fk . At small but finite $Ma$ the
vortices still produce inverse cascade of energy and the energy injected into
the flow at the pumping scale $L$ flows to larger scales. That inverse cascade
is associated with velocity magnitude growth as the cascade reaches larger and
larger scales. Characteristic velocity $v_{l}$ of eddies with scale $l$ obeys
the Kolmogorov scaling $v_{l}\sim u(l/L)^{1/3}$, that is valid for two-
dimensional inverse cascade review . That scaling however needs to be modified
to $v_{l}\sim u(l/L)^{1/2}$, that corresponds to $k^{-2}$ spectrum, for $l$ so
large that $v_{l}$ is comparable with the speed of sound fk . When the cascade
reaches the sonic scale $l_{s}\sim Ma^{-3}L$ (this estimate is done using
Kolmogorov scaling, the actual law depends on lengths of different scaling
ranges and can be readily worked out), by definition, the characteristic flow
velocity equals speed of sound. Here we assume that the system size is larger
than $l_{s}$ and observe that $l_{s}\gg L$ holds at even not so small $Ma$. At
$l_{s}$ “resonance” condition is obeyed and turbulence starts to emit sound
waves strongly (of course the emission happens also at intermediate scales
$L\lesssim l\lesssim l_{s}$, however in that range it is too weak to stop the
cascade). The waves then transfer energy to small scales, thereby creating an
energy flux loop observed in fk . Thus, the energy is predominantly contained
at scales $l_{s}$ whereas its dissipation occurs at smallest scales generated
by the shock waves of the direct cascade.

The density associated with the above cascade is a large-scale field that has
significant variations only at scales of order $l_{s}$, see above and Fig. 1.
The smallest scales are also present in the density however their contribution
to the spatial density profile is negligible (this is not true for high-order
derivatives of the density however).

Physics at $Ma\sim 1$ can then be understood by asymptotic continuation of the
$Ma\ll 1$ case. When we consider larger, yet small $Ma$, the sonic scale
$l_{s}$ decreases and becomes of order $L$ at $Ma\sim 1$. This is the
situation where the characteristic velocity injected by the driving force
equals the speed of sound. The largest scale of the energy flux loop,
introduced at $Ma\ll 1$, is then of order of the pumping scale $L$. Vortices
still produce some kinetic energy flux up the hierarchy scals, however it does
not reach scales much larger than $L$, see fk . The waves take energy to
smaller scales from scales of order $L$, so that two-dimensional turbulence at
$Ma\sim 1$ exhibits essentially only the direct energy cascade, if not to
account for significant spectrum at wavenumbers somewhat larger than $L^{-1}$.
Small-scale turbulence in this case is still subsonic and contains vortices
which decelerate the direct cascade however do not stop it. Since subsonic
eddies of the small-scale turbulence are effectively incompressible, they do
not create significant spatial variations of $\rho$. Density is still a large-
scale field with variations at scale $l_{s}\sim L$, similarly to $Ma\ll 1$
case, see above and Fig. 2.

Density multifractality in supersonic inertial range—We turn now to discuss
cases with $Ma\gg 1$. The fact that in that regime the density field is a
passive scalar that is governed by Eqs. (6) has far reaching implications. It
was demonstrated that, provided that the scalings of solenoidal and
dilatational components of the flow are similar, the density field generated
by the continuity equation is multifractal fm . At Mach number values that are
much smaller than $1$ the scaling exponents of those velocity components are
not equal, however, as the Mach number grows, the difference between them
decreases fk ; K2018 . When it reaches small enough values a multifractal
density structure holds fm .

We first consider the potential case. The scaling condition above pertains to
scaling exponents, and not to the absolute magnitude of the velocity
components, so in the potential case it holds trivially. Then, the results of
fm  imply that in the potential case the density field is multifractal in the
supersonic inertial range while below the scale $l_{s}$ there are no
appreciable fluctuations of the density. We here confine ourselves with
implications for the density spectrum $E(k)$, that obeys for
$L^{-1}<k<l_{s}^{-1}$ the power-law $E(k)\propto k^{\delta-1}$ with
$\delta>0$. The so-called correlation dimension of the multifractal is defined
as $2-\delta$, and is strictly smaller than the full space dimension two, see
kritsuk  for observations in the three-dimensional case. The density
fluctuations obey in that case

$\displaystyle\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\left\langle\rho^{2}\right\rangle=\int_{0}^{\infty}E(k)dk\simeq
C_{0}\left(\frac{L}{l_{s}}\right)^{\delta}\simeq C_{0}C^{\delta}Ma^{2\delta},$
(21)

where $C_{0}$, $C$ are constants and we assumed that the leading order
behavior of the integral is determined by the power-law cutoff at $k\sim
l_{s}^{-1}$. We assumed the Burgers’ $k^{-2}$ spectrum in estimating $L/l_{s}$
as $CMa^{2}$ with a constant $C$, cf. above. The fluctuations diverge at
$Ma\to\infty$ where the cutoff scale of the multifractal $l_{s}$, at which the
density is supported, tends to zero.

In the case of non-potential acceleration field, scalings of the solenoidal
and dilatational components in the supersonic inertial range are identical,
for generic acceleration fields. This is because the evolution in that range
is describable by the Hopf equation
$\partial_{t}\bm{v}+(\bm{v}\cdot\nabla)\bm{v}=0$ that does not induce
difference in the components’ scalings, cf. above. Then we find again Eq.
(21).

Thus supersonic two-dimensional turbulence is not that different from its
three-dimensional counterpart, in sharp contrast to the incompressible limit
of small $Ma$, cf. dust ; fm . Both exhibit transfer of energy from large to
small scales and multifractal density in the supersonic inertial range.

Burgers’ model for accretion—The above considerations suggest the introduction
of a minimal model for accretion that is expected to capture the main physics
of supersonic thin disks. The model relies on the fact that the velocity is a
large-scale field and the density is smooth below $l_{s}$. We use as a model
the same Eqs. (6) however with the value of viscosity such that the viscous
scale is $l_{s}$. Thus the proposed velocity equation is

$\displaystyle\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\partial_{t}\bm{v}+(\bm{v}\cdot\nabla)\bm{v}=\bm{a}+\frac{b}{Ma^{3}}\nabla^{2}\bm{v},$
(22)

where $b$ is the adjustment numerical factor of order one which is considered
below. The coefficient of the Laplacian term is such that the ratio of the
non-linear and viscosity terms is of order one at $l_{s}$. Here for estimates
we assume velocity scaling that corresponds to $k^{-2}$ scaling of the
spectrum. The density obeys a modified continuity equation (diffusion equation
in moving fluid)

$\displaystyle\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\partial_{t}\rho+\nabla\cdot(\rho\bm{v})=\frac{b}{Ma^{3}}\nabla^{2}\rho.$
(23)

Here we observe that since the model omits the pressure term altogether, then
continuity equation without diffusivity could produce appreciable density
variations below $l_{s}$, as in the so called Batchelor regime review .
Introduction of the Laplacian term into Eq. (23) is a means to obtain the
required density smoothness below $l_{s}$.

It is expected that Eqs. (22)-(23) reproduce rather accurately the flow in the
supersonic inertial range, which is the main range of interest in the
accretion problem (this is the range where appreciable density variations
occur and also the range that determines the overall velocity and kinetic
energy). The advantage of this model is that in the case of potential
accelerations, which is relevant for gravity-driven accretion, the velocity
equation is mapped into a linear equation by the Cole-Hopf transformation
$\bm{v}=-2bMa^{-3}\nabla\ln f$. We have ($\bm{a}=-\nabla\psi$)

$\displaystyle\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\partial_{t}f=\frac{Ma^{3}}{2b}\psi
f+\frac{b}{Ma^{3}}\nabla^{2}f.$ (24)

It is much easier to perform simulations of the above (imaginary time
Schrodinger) linear equation than of the original non-linear equations. We
conjecture that the above model would faithfully reproduce the accretion rate
by an appropriate choice of $b$. On the theoretical side, the linear model
suggests that supersonic potential turbulence is much more stable than
turbulence usually is.

The model miscalculates possibly large fluctuations of the supersonic
cascade’s cutoff scale $l_{s}$ that exist due to intermittency. Thus the
original cutoff due to pressure is different from the model’s cutoff due to
viscosity. For $Ma\lesssim 10$ intermittency is moderate and this seems to be
not of much concern. However, for the highest Mach number of practical
relevance, $Ma\sim 100$ we have $L/l_{s}\sim 10^{4}$. The same length of
inertial interval in three-dimensional incompressible turbulence corresponds
to Reynolds numbers of order $10^{5}-10^{6}$ which are appreciably
intermittent frisch . The degree to which the proposed model is accurate must
be studied numerically and is left for future work.

Magnetohydrodynamics— Small fluctuations of magnetic field, invariably present
in astrophysical environments, may be significantly amplified by the
turbulence. In such cases the hydrodynamic equations need to be extended in
order to include the influence of the magnetic fields. This is the case for
three-dimensional turbulence, see e.g. ll8 , which is believed to be the
reason for the ubiquitous magnetic fields in astrophysics. Here we demonstrate
that in the two-dimensional case the situation is quite similar. However the
source of the magnetic field which is amplified by turbulence is not small
fluctuations, which in fact decay, but rather large-scale magnetic field.

It is often believed that the magnetic field is irrelevant in the two-
dimensional situation due to the antidynamo theorem antidynamo , which we
consider here in more detail. The theorem states that there is no dynamo
effect in incompressible two-dimensional turbulent flow as any magnetic field
inevitably decays to zero. Thus, it was observed that the vertical component
of the magnetic field obeys the heat conduction equation in a moving fluid
antidynamo . This implies that this component of the field relaxes to a
constant, similarly to how density of tracers becomes uniform due to mixing by
incompressible turbulence. Correspondingly, the vertical component disappears
from the condition of magnetic field’s solenoidality. The horizontal
components can then be described by a scalar potential, that again obeys the
heat conduction equation. Therefore, the potential decays to a constant, which
implies vanishing of the horizontal components. Thus, for incompressible flow
the steady state solution is a constant vertical field that generates no
Lorentz force and exerts no influence on the flow antidynamo .

Despite some doubts gk , the antidynamo theorem for incompressible flow has
been demonstrated to be valid (see for instance igor ) and appears today as a
solid result whose universality has far-reaching implications. Thus, in the
case of two-dimensional incompressible flows, the consideration that assumes
that there are no magnetic fields in the turbulent flow of conducting fluid is
self-consistent: small fluctuations would asymptotically decay with time.
These fluctuations can be three-dimensional: the Lorentz force is quadratic in
the magnetic field. Thus small three-dimensional fluctuations exert negligible
effect on the flow which remains two-dimensional with good approximation.
Still care is needed since significant transient growth is possible, see e.g.
igor ; tr , which could temporarily turn on the interaction of the magnetic
field with the flow.

In fact, the antidynamo theorem shows that any fluctuations of the magnetic
field, be they large or small, whether they interact with the flow for some
time or not, decay asymptotically on one condition: the Lorentz force due to
the magnetic field must keep the incompressible flow (approximately) two-
dimensional. Hence there remains the possibility for non-linear instability of
incompressible flow, where a large fluctuation of magnetic field would create
the third component of velocity that will prevent the fluctuation’s decay.
Summarizing, the antidynamo theorem for incompressible flows leads to the
conclusion that, if the flow can be considered two-dimensional, then the
steady state magnetic field is zero.

Any real flow, however, has finite compressibility, so the above anti-dynamo
theorem, at best, describes the case of $Ma\ll 1$, where the flow is
approximately incompressible. There is currently no result similar to
Zeldovich theorem that allows us to assess the relevance of compressibility to
sustaining non-trivial steady-state magnetic fields. Moreover, there is no
demonstration that at $Ma\ll 1$ the neglect of the magnetic field is indeed
valid (we will show that this is not always the case). In this section we fill
that gap and explore the role of flow compressibility in the magnetic field
amplification process.

We study the transport of a solenoidal magnetic field $\bm{B}$ by a two-
dimensional flow $\bm{v}$. We use the same framework as in the Zeldovich
antidynamo theorem antidynamo , relaxing the condition of incompressibility.
Hence the study below can be considered as compressible generalization of the
antidynamo theorem in two dimensions. The evolution of magnetic field $\bm{B}$
is described by the magnetohydrodynamic equations ll8 ,

$\displaystyle\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\partial_{t}\bm{B}+(\bm{v}\cdot\nabla)\bm{B}=(\bm{B}\cdot\nabla)\bm{v}-\bm{B}\nabla\cdot\bm{v}+\eta\nabla^{2}\bm{B},$
(25)

while the flow velocity $\bm{v}$ obeys

$\displaystyle\partial_{t}\bm{v}+(\bm{v}\cdot\nabla)\bm{v}=-\frac{1}{Ma^{2}}\nabla\ln\rho+\bm{a}+\nu\nabla^{2}\bm{v}$
$\displaystyle-\frac{1}{2\rho}\nabla
B^{2}+\frac{(\bm{B}\cdot\nabla)\bm{B}}{\rho},$ (26)

where the last two terms describe the Lorentz force, cf. Eqs. (1). The density
obeys the continuity equation given by the first of Eqs. (1). The velocity
field is considered to depend only on the horizontal coordinates $x$ and $y$
and to have zero vertical component, $v_{z}=0$. In contrast, we consider a
fully three-dimensional magnetic field that generally has a vertical component
$B_{z}$, and that depends on $x$, $y$ and $z$. We impose constraint on the
magnetic field that the vertical component of the flow and the $z-$dependence
of $\bm{v}$, as generated by the Lorentz force, are negligible. They must be
negligible to the extent that the magnetic field lines’ stretching term
$(\bm{B}\cdot\nabla)\bm{v}$ in the equation for $B_{z}$, which vanishes for
planar $\bm{v}$ identically, is negligible. This produces the following closed
equation for the vertical component of $\bm{B}$

$\displaystyle\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\partial_{t}B_{z}\\!+\\!\nabla\\!\cdot\\!(B_{z}\bm{v})\\!=\\!\eta\nabla^{2}B_{z}.$
(27)

We shortly recapture the Zeldovich antidynamo theorem that holds in the case
of incompressible flow.

There is no magnetic field in steady state for incompressible flow—Eq. (27)
implies for incompressible flow

$\displaystyle\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\partial_{t}\int
B_{z}^{2}d\bm{x}=-2\eta\int\left(\nabla B_{z}\right)^{2}d\bm{x},$ (28)

where it it assumed that the boundary terms that are generated by integrating
by parts can be neglected as in antidynamo . Eq. (28) implies that magnetic
energy decays until the field vanishes (the possible non-zero constant field
can be eliminated by considering fluctuations around it). This conclusion on
the vanishing of a steady-state $B_{z}$ holds irrespective of whether $B_{z}$
reacts on $\bm{v}$ via the momentum equation, it rests solely upon the
assumption that $\bm{v}$ is two-dimensional antidynamo .

The vanishing of the horizontal components of the magnetic field for
incompressible flow is proved similarly. Due to $B_{z}=0$, the condition that
$\bm{B}$ is solenoidal yields $\partial_{x}B_{x}+\partial_{y}B_{y}=0$. We can
therefore describe these components with the help of scalar potential $A$
according to $B_{x}=\partial_{y}A$ and $B_{y}=-\partial_{x}A$. That potential
obeys an advection-diffusion equation

$\displaystyle\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\partial_{t}A\\!+\\!(\bm{v}\\!\cdot\\!\nabla)A\\!=\\!\eta\nabla^{2}A,$
(29)

which can be verified by taking derivatives and using
$\partial_{x}v_{x}+\partial_{y}v_{y}=0$. We see that for incompressible flow
$B_{z}$ and $A$ obey the same equation. By the same argument, $A$ relaxes to a
constant, implying $B_{x}=B_{y}=0$. Thus the magnetic field for steady state
incompressible planar flow is zero.

Any planar flow of conducting fluid is magnetised in the limit of strong
external field—Incompressible flow has a degeneracy - submitting the system to
an external magnetic field does not change the flow at all. The solution is
given by the superposition of that external field and the solution that holds
at zero field. Compressibility removes this degeneracy, inducing the flow to
change in response to the imposed external field. (We will see below, that the
impact of that field is determined by its total flux through the system.) As
discussed above, Eq. (27) is solved by a constant in the incompressible case
only. When the flow is compressible, writing $B_{z}$ as the sum of the
constant external field $B^{e}$ and the internal field $B^{i}$ gives

$\displaystyle\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\partial_{t}B^{i}\\!+\\!(\bm{v}\cdot\nabla)B^{i}\\!=\\!-B^{e}\nabla\cdot\bm{v}+\eta\nabla^{2}B^{i}.$
(30)

The constant external field generates an inhomogeneous source in the equation
for the internal field. In turn, that creates a non-trivial Lorentz force in
Eq. (26) that changes the velocity field. Thus imposing on the system an
external magnetic field can be used to manipulate the flow externally.

The above demonstrates that at $Ma\ll 1$ using a large external field, finite
or large effect on the flow can be created via $B^{i}$. In this case,
anticipating the vanishing of the horizontal components of the magnetic field
proved below, Eq. (26) becomes

$\displaystyle\\!\\!\\!\\!\\!\partial_{t}\bm{v}\\!+\\!(\bm{v}\\!\cdot\\!\nabla)\bm{v}\\!=\\!-\frac{\nabla\ln\rho}{Ma^{2}}\\!+\\!\bm{a}\\!+\\!\nu\nabla^{2}\bm{v}\\!-\\!\frac{B^{e}}{\rho}\nabla
B^{i},$ (31)

where we assume $B^{e}\gg B^{i}$. We consider as an example, the pseudosound
mode dominated regime of $Ma\ll 1$ where velocity divergence scales as
$Ma^{2}$, see e.g. pseudos ; levermore . We see then that taking $B^{e}\propto
Ma^{-1}$ we find $B^{i}\propto Ma$ from Eq. (30). This leads to magnetic
pressure force in Eq. (31) that is of order one and has a finite effect on the
flow (note that the assumption $B^{e}\gg B^{i}$ is correct). Furthermore,
$B^{e}\gg Ma^{-1}$ would produce a large Lorentz force, leading to complete
rearrangement of the flow.

A remark is in order. We assumed that at $Ma\ll 1$ the flow divergence is
$Ma^{2}$ times a field of order one. In fact, this is true in the three-
dimensional case pseudos ; levermore , however, might need refinements in two
dimensions. This is because of the inverse cascade that could cause
$\nabla\cdot\bm{v}$ to be a field whose dominant modes are of order of
$l_{s}^{-1}$. From the low-Mach number spectrum of the dilatational component
of the velocity, $E_{d}(k)$, presented in fk  this seems to be not the case.
It is seen by using that the spectrum of $\nabla\cdot\bm{v}$ is
$k^{2}E_{d}(k)$ that divergence is determined by eddies at the pumping scale.
Thus it appears that the above considerations hold also in the two-dimensional
case. However, the spectrum of $\nabla\cdot\bm{v}$ presented in K2018  seems
to indicate that the divergence is a small-scale field. Further studies are
necessary, however, even if another factor should be present in
$\nabla\cdot\bm{v}$, this does not change our main conclusion - any flow of a
conducting fluid in large enough magnetic field will generate appreciable
Lorentz force and hence evolve differently from the pure hydrodynamic case. In
other words, in the presence of non-zero magnetic flux compressibility is a
singular perturbation – even when it is small it can have a finite or large
effect on the flow when compensated by a large external magnetic field. We
leave detailed investigation of the critical value of $B^{e}$, above which the
flow changes at $Ma\ll 1$, for future work.

Stationary measure $n_{s}(t,\bm{x})$—We saw that inclusion of compressibility
might introduce a qualitative change in the flow by allowing for the magnetic
degrees of freedom to affect the steady state. Eq. (28) becomes

$\displaystyle\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\partial_{t}\int
B_{z}^{2}d\bm{x}=-\int B^{2}\nabla\cdot\bm{v}-2\eta\int\left(\nabla
B_{z}\right)^{2}d\bm{x},$ (32)

and we can no longer apply the argument of antidynamo , i.e. compressible
planar flow may sustain a finite magnetic field in the steady state. Insight
into the structure of the stationary magnetic field is gained by comparing it
to a passive concentration field $n(t,\bm{x})$ that obeys

$\displaystyle\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\partial_{t}n\\!+\\!\nabla\\!\cdot\\!(n\bm{v})\\!=\\!\eta\nabla^{2}n,\
\ \int nd\bm{x}=1.$ (33)

The difference from Eq. (27) is that $n$ is positive, does not react on
$\bm{v}$ and has finite unit mass. Despite these differences, it is
anticipated that $B_{z}$ in the steady state is proportional to the steady
state solution $n_{s}$ of the above equation, as will be proved rigorously
below.

Eq. (33) coincides with the equation governing the distribution of particles
that diffuse within a moving fluid. At zero diffusion, $\eta=0$, it has the
same form as the continuity equation whose density solution is steady. We will
assume below that Eq. (33) in cases of interest generates a steady state
solution. This solution, the so-called stationary measure $n_{s}(t,\bm{x})$,
can be derived as the infinite evolution time limit

$\displaystyle\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!n_{s}(t,\bm{x})\equiv\lim_{t_{0}\to-\infty}n(t,\bm{x}),\
\ n(t_{0})=n_{0}(\bm{x}).$ (34)

Above $t_{0}$ is the time at which the initial condition for Eq. (33) is set,
and it is assumed that $\int n_{0}(\bm{x})d\bm{x}=1$. It will be seen below
that the above equation defines a unique $n_{s}(t,\bm{x})$ that is independent
of $n_{0}(\bm{x})$.

Steady state for Eq. (33) can be realized by assuming that the flow is
confined to a finite volume, as when the velocity field vanishes or is
periodic on finite boundaries. Despite that these boundary conditions (b.c.)
are not necessarily true, they provide a useful framework within which Eq.
(33) generates a steady state (roughly the particles cannot escape and their
concentration reaches a steady state). These (artificial) b.c. will be
explained in more detail below.

Structure of $n_{s}(t,\bm{x})$ can be understood by considering first the case
of $\eta\to 0$ where $n_{s}(t,\bm{x})=\rho(t,\bm{x})$ could be anticipated. In
fact, $n_{s}(\eta=0)=\rho$ is typically untrue, since the continuity equation
is not dissipative and therefore the passive field $n$ and the active field
$\rho$ may differ, see detailed discussion in fm . Still, it is true that
$n_{s}(\eta=0)$ is qualitatively similar to $\rho$.

Finite $\eta$ smoothens $n_{s}(\eta=0)$ over the so-called resistive scale
$l_{\rm res}$. This is the scale so small that at $l_{\rm res}$ the resistive
term in Eqs. (27) or (33) is comparable with the advective term. Typical
velocity $u_{\rm res}$ of eddies at scale $l_{\rm res}$ is such that the local
magnetic Reynolds number $u_{\rm res}l_{\rm res}/\eta$ is of order one. The
actual value of $l_{\rm res}$ is determined by using velocity scaling at
larger scales, cf. the definition of the viscous scale frisch , and is not
necessary here. Thus roughly $n_{s}(\eta)$ is $n_{s}(\eta=0)$ smeared over the
scale $l_{\rm res}$. Combining this with the previous observation of the
similarity of $n_{s}(\eta=0)$ and $\rho$, we conclude that $n_{s}(t,\bm{x})$
is qualitatively similar to $\rho(t,\bm{x})$ coarse-grained over the scale
$l_{\rm res}$ (this only means that the statistics are similar, and does not
necessarily hold per realization).

We conclude that at $l_{\rm res}\lesssim l_{s}$, where $l_{\rm res}$ is not
larger than the smoothness scale of the density, the fields $n_{s}(t,\bm{x})$
and $\rho(t,\bm{x})$ have statistically similar properties. In contrast, if
$l_{res}\gg l_{s}$ then the fields are quite different. In this case
$n_{s}(t,\bm{x})$ is a large scale field, smooth over $l_{\rm res}$, at
$Ma\lesssim 1$. If $Ma\gg 1$ and $l_{s}\ll l_{\rm res}\ll L$ then
$n_{s}(t,\bm{x})$ is smooth below $l_{\rm res}$ and multifractal above it,
until the scale $L$. Finally, at $Ma\gg 1$ and $l_{\rm res}\gtrsim L$ the
field $n_{s}(t,\bm{x})$ is a large scale field, smooth over the scale $l_{\rm
res}$.

Magnetic field relaxes to $n_{s}(t,\bm{x})$—To study the magnetic field we
need to introduce boundary conditions for Eq. (27). To understand the role of
these conditions, we observe that Eq. (33) also arises in the study of the
motion of inertial particles in turbulence, already mentioned above (there
usually the diffusion is neglected). The main motivation for the study of the
particles’ problem is prediction of the rain formation, see e.g. shaw ; us .
We shortly review the relevant physics.

In warm clouds water droplets are located in a volume that does not have sharp
boundaries. Yet, to study their distribution in the bulk of the cloud, they
are usually represented as particles that move in a finite box. Due to the
confinement to a finite volume, such particles necessarily reach a steady
state distribution. Properties of this distribution are then applied to
droplets in clouds, despite that they do not obey the same boundary
conditions. The rationale for this approach is that relaxation to the steady
state is a local process. Thus, droplets in a cloud redistribute locally
according to the steady state in a box, and the actual effects of the boundary
conditions take much longer times. Hence the boundary conditions of a finite
box are used as merely a device to study the transient period when boundary
phenomena do not penetrate the bulk. Here we use the same approach, using the
boundary conditions that result in the steady state for Eq. (27) and then
applying the result to the actual magnetic field locally, far from the disk
boundaries.

We wish to show that in steady state $B_{z}(t,\bm{x})$ coincides with
$n_{s}(t,\bm{x})$. This is despite that, as already mentioned, generally,
$B_{z}$ is active field that reacts on the velocity, and $n_{s}$ is a passive
field that does not. The issue of whether an active and passive fields,
obeying the same first order in time equation, coincide is quite old in
turbulence. The most famous example is vorticity and passive magnetic field,
see fm  for a recent detailed discussion and also review . In our case,
however, the number of degrees of freedom is so constrained that the active
and passive fields obeying the same Eq. (27) do coincide in the long-time
limit. Our demonstration of this below is an adaptation of fm .

Long-time properties of $B_{z}(t,\bm{x})$ can be obtained by studying the
Green’s function $G(t,\bm{x}|\bm{x}^{\prime})$ that obeys,

$\displaystyle\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\partial_{t}G(t,\bm{x}|\bm{x}^{\prime})\\!+\\!\nabla\\!\cdot\\!\left(G(t,\bm{x}|\bm{x}^{\prime})\bm{v}(t,\bm{x})\right)\\!=\\!\eta\nabla^{2}G(t,\bm{x}|\bm{x}^{\prime}),$
(35)

where $G(t=0,\bm{x}|\bm{x}^{\prime})=\delta(\bm{x}-\bm{x}^{\prime})$. We have
$B_{z}(t,\bm{x})=\int
G(t,\bm{x}|\bm{x}^{\prime})B_{z}(0,\bm{x}^{\prime})d\bm{x}^{\prime}$ where it
is assumed that the Green’s function obeys proper boundary conditions that
guarantee that $B_{z}(t,\bm{x})$ also obeys them. We assume that finite box
boundaries whose vertical dimension is much smaller than the horizontal
dimensions can be used. The assumed details of the boundary conditions are
simple to see from how they appear in the calculations. We set the volume of
the domain of the flow to one by choice of units of length (here our
dimensionless variables differ from those used in the study of velocity). We
assume that, similarly to antidynamo , natural boundary conditions for
$G(t,\bm{x}|\bm{x}^{\prime})$ can be used fp  so that the spatial integral of
$G(t,\bm{x}|\bm{x}^{\prime})$, and thus also of $B_{z}(t,\bm{x})$, is
conserved. Then $G(t,\bm{x}|\bm{x}^{\prime})$ can be interpreted as the
probability density function (PDF) of the coordinate $\bm{x}(t)$ of the
Brownian particle given that $\bm{x}(0)=\bm{x}^{\prime}$ (transition
probability). The particle’s motion is driven by combination of transport by
the flow and diffusion with diffusion coefficient $\eta$. The main observation
is that $G(t,\bm{x}|\bm{x}^{\prime})$ becomes independent of $\bm{x}^{\prime}$
at large times. For time-independent flow $\bm{v}$ this is the consequence of
the PDF’s relaxation to the time-independent stationary solution. For time-
dependent random stationary flows, of interest here,
$G(t,\bm{x}|\bm{x}^{\prime})$ becomes at large times a stationary random
distribution $n_{s}(t,\bm{x})$ that describes the concentration of ensemble of
large number of independently moving Brownian particles (or the PDF of one
particle), introduced previously.

We demonstrate that at large times
$G(t,\bm{x}|\bm{x}_{1})=G(t,\bm{x}|\bm{x}_{2})$ holds for any $\bm{x}_{1}$ and
$\bm{x}_{2}$. We use the entropy functional approach fp  which here serves as
the counterpart of studying the evolution of magnetic energy in antidynamo ,
see Eq. (28). We observe that $G(t,\bm{x}|\bm{x}^{\prime})$ is strictly
positive at any $t>0$ as can be seen, e.g., from the path integral
representation of the transition probability, provided below. Thus at $t>0$ we
can define the distance between $n_{i}(t,\bm{x})\equiv G(t,\bm{x}|\bm{x}_{i})$
as fp ,

$\displaystyle H(t)=\int
n_{1}(t,\bm{x})\ln\frac{n_{1}(t,\bm{x})}{n_{2}(t,\bm{x})}d\bm{x}.$ (36)

We observe that $H(t)$ is a non-negative function. To see that we recall that
$\int n_{i}(\bm{x},t)d\bm{x}=1$ and rewrite $H$ as

$\displaystyle
H=\int\left[n_{1}\ln\frac{n_{1}}{n_{2}}-n_{1}+n_{2}\right]d\bm{x}$
$\displaystyle=\int n_{2}\left[R\ln R-R+1\right]d\bm{x},$ (37)

where we introduced $R\equiv n_{1}/n_{2}$. The last term in the above equation
is always non-negative as it follows from

$\displaystyle R\ln R-R+1=\int_{1}^{R}\ln xdx\geq 0,$ (38)

that holds for any $R\geq 0$. We therefore see that $H=0$ only at
$n_{1}(t,\bm{x})=n_{2}(t,\bm{x})$. Thus $H(t)$ is non-negative and it vanishes
only if $n_{i},i=1,2$ are equal, thus providing a good definition of the
distance between $n_{i}$. Moving on, the rate of change of $H$ with time is
given by:

$\displaystyle{\dot{H}}=\int\left[\frac{n_{1}}{n_{2}}\nabla\cdot(\bm{v}n_{2})-\ln\frac{n_{1}}{n_{2}}\nabla\cdot(\bm{v}n_{1})\right.$
$\displaystyle\left.-\eta\frac{n_{1}}{n_{2}}\nabla^{2}n_{2}+\eta\ln\frac{n_{1}}{n_{2}}\nabla^{2}n_{1}\right]d\bm{x},$
(39)

where we used the conservation constraint $\int n_{1}(\bm{x},t)d\bm{x}=1$.
Integrating now by parts yields:

$\displaystyle\\!\\!\\!\\!\\!\\!\\!\\!{\dot{H}}=\int\left[n_{1}\bm{v}-n_{1}\bm{v}\ln\frac{n_{1}}{n_{2}}-\eta\frac{n_{1}}{n_{2}}\nabla
n_{2}+\eta\ln\frac{n_{1}}{n_{2}}\nabla n_{1}\right]d\bm{S}$
$\displaystyle\\!\\!\\!\\!\\!\\!\\!\\!-\eta\int
n_{1}\left(\nabla\ln(n_{1}/n_{2})\right)^{2}d\bm{x},$ (40)

where the surface integral is over the boundary of the domain of the flow. We
assume that the boundary term can be discarded due to boundary conditions or
because it is negligible in comparison with the volume term. We find,

$\displaystyle\\!\\!\\!\\!\\!\\!\\!\\!{\dot{H}}\approx-\eta\int
n_{1}\left(\nabla\ln(n_{1}/n_{2})\right)^{2}d\bm{x}\leq 0,$ (41)

where equality holds for $n_{1}=n_{2}$ only. This equation implies relaxation
to $n_{1}=n_{2}$ since $H\geq 0$, cf. Eq. (28). We conclude that
$G(t,\bm{x}|\bm{x}^{\prime})$ becomes independent of $\bm{x}^{\prime}$ at
$t\to\infty$ and thus:

$B_{z}(t,\bm{x})\approx G(t,\bm{x}|\bm{x}_{0})\int
B_{z}(0,\bm{x}^{\prime})d\bm{x}^{\prime},$ (42)

which leads to the following remarkable relationship:

$B_{z}(t,\bm{x})=In_{s}(t,\bm{x}),\ \ I\equiv\int B_{z}(0,\bm{x})d\bm{x}$ (43)

where $\bm{x}_{0}$ is irrelevant. Here $n_{s}$ is defined in Eqs. (33)-(34),
and $I$ denotes the (approximately) conserved spatial integral of $B_{z}$. For
thin disks it is appropriate to call this the magnetic flux through the disk,
disregarding difference in the definition that here is minor. Thus in our case
of passive and active fields, there is no difference between them due to
finite dissipation, $\eta>0$, see detailed discussion in fm . We observe that
the stationary measure is $z-$independent since in vertical dimension we have
diffusion on a finite interval. Thus $B_{z}$ is $z-$independent in the steady
state.

We observe from Eq. (43) that $B_{z}$ is sign-definite in the steady state.
This is due to the decay of fluctuations as described by the case $I=0$.
Moreover, with no loss, we may assume that $I\geq 0$, as can be guaranteed by
the choice of the direction of $z-$axis, or by observing that we could use in
the magnetohydrodynamic equations $-B_{z}$ instead of $B_{z}$ (indeed $-B_{z}$
and $B_{z}$ obey the same equation and generate the same Lorentz force). We
conclude that the magnetic field endows the fluid with an effective density
$|I|n_{s}(t,\bm{x})$. The Lorentz force generated by the magnetic field
resembles then the pressure force generated by this density.

We see that $B_{z}$ decays to zero if $I=0$ and is non-zero for $I\neq 0$.
This can be understood by introducing the decomposition of $B_{z}$ into the
sum of the constant mean (recalling that we use unit volume) and the
fluctuation field $B^{\prime}_{z}(t)$ that obeys,

$\displaystyle\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\partial_{t}B^{\prime}_{z}\\!+\\!\nabla\\!\cdot\\!(B^{\prime}_{z}\bm{v})\\!=\\!\eta\nabla^{2}B^{\prime}_{z}-I\nabla\cdot\bm{v},$
(44)

cf. Eq. (30). If $I=0$ then $B^{\prime}_{z}$ that has zero spatial integral
decays due to mixing. The non-zero value of $I$ creates a stationary source of
fluctuations and thus a non-trivial statistics of finite $B^{\prime}_{z}$, cf.
Eq. (30).

The above considerations, despite that they have a more complex form, are
generalization of antidynamo . They also hold for any planar velocity field,
irrespective of whether the Lorentz force creates interaction of velocity and
magnetic field. In the compressible case the (Lyapunov) functional of $\bm{B}$
that demonstrates convergence to a unique solution at large times, is
significantly more complex than $\int B^{2}d\bm{x}$ considered in antidynamo ,
and is given by the ”entropy” $H(t)$. However the approach is similar and so
is that to the study of the horizontal components.

Horizontal magnetic field—We consider next the horizontal components of the
magnetic field. As discussed above, under the current conditions the latter is
derived from a scalar potential $A$ according to $B_{x}=\partial_{y}A$ and
$B_{y}=-\partial_{x}A$. That potential obeys the advection-diffusion equation
(29). That equation coincides with with its incompressible counterpart
antidynamo  and, as in that case, it has constant solutions. However, to prove
the long-time convergence to those solutions, a different approach must be
devised. We use the properties of the Green’s function to demonstrate that
$A(t,\bm{x})$ becomes constant at large times. Thus, we demonstrate that the
Green’s function ${\tilde{G}}(t,\bm{x}|\bm{x}^{\prime})$ defined by,

$\displaystyle\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\partial_{t}{\tilde{G}}(t,\bm{x}|\bm{x}^{\prime})\\!+\\!\bm{v}(t,\bm{x})\\!\cdot\\!\nabla{\tilde{G}}(t,\bm{x}|\bm{x}^{\prime})\\!=\\!\eta\nabla^{2}{\tilde{G}}(t,\bm{x}|\bm{x}^{\prime}),$
(45)

and ${\tilde{G}}(0,\bm{x}|\bm{x}^{\prime})=\delta(\bm{x}-\bm{x}^{\prime})$
becomes independent of $\bm{x}$ at $t\to\infty$. This implies that
$A(t,\bm{x})=\int{\tilde{G}}(t,\bm{x}|\bm{x}^{\prime})A(0,\bm{x}^{\prime})d\bm{x}^{\prime}$
tends to a constant at large times.

We consider the dependence of ${\tilde{G}}(t,\bm{x}|\bm{x}^{\prime})$ on the
time $t_{0}$ of setting the initial condition by defining
$G(t,\bm{x}|t_{0},\bm{x}^{\prime})$ as the solution of Eq. (45) that obeys
${\tilde{G}}(t_{0},\bm{x}|t_{0},\bm{x}^{\prime})=\delta(\bm{x}-\bm{x}^{\prime})$.
As a function of the initial coordinate and time the Green’s function obeys
the Hermitian conjugate equation fp  so that,

$\displaystyle\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\partial_{t^{\prime}}{\tilde{G}}^{\prime}(t^{\prime},\bm{x}^{\prime}|\bm{x})\\!-\\!\nabla^{\prime}\\!\cdot\\!\left({\tilde{G}}^{\prime}(t^{\prime},\bm{x}^{\prime}|\bm{x})\bm{v}(t-t^{\prime},\bm{x}^{\prime})\right)$
$\displaystyle\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!=\\!\eta\nabla^{\prime
2}{\tilde{G}}^{\prime}(t^{\prime},\bm{x}^{\prime}|\bm{x}),\ \
{\tilde{G}}^{\prime}(t^{\prime},\bm{x}^{\prime}|\bm{x})\equiv{\tilde{G}}(t,\bm{x}|t-t^{\prime},\bm{x}^{\prime}),$
(46)

where
${\tilde{G}}^{\prime}(t^{\prime}=0,\bm{x}^{\prime}|\bm{x})=\delta(\bm{x}-x^{\prime})$
and we designated derivative with respect to $\bm{x}^{\prime}$ by prime. We
observe that ${\tilde{G}}^{\prime}(t^{\prime}=0,\bm{x}^{\prime}|\bm{x})$ obeys
Eq. (52) at large times and thus becomes independent of $\bm{x}$ at large
times as demonstrated previously. This implies independence of
${\tilde{G}}(t,\bm{x}|\bm{x}^{\prime})$ of $\bm{x}$ at $t\to\infty$. Thus $A$
becomes constant at large times which by taking derivatives implies
$B_{x}=B_{y}=0$.

Summary—We conclude from the above that, provided that the contribution of the
boundary contributions is negligible on the considered times, at large times
the magnetic field becomes a stationary vertical field. It is proportional to
the stationary measure $n_{s}(t,\bm{x})$ which has a unit space integral, with
proportionality coefficient given by the constant magnetic flux. That measure
provides long time behavior of the solutions to Eq. (33). Thus, there is no
indefinite growth of the field, such as that occurring in three dimensions
(the growth can occur in a transient manner, cf. igor ) but rather settling
into a steady state field.

Zeldovich’s theorem generalizes in the zero flux case—If the flux vanishes,
$I=0$, then the above implies that the magnetic field relaxes to zero. Thus we
can formulate the statement: planar flows with zero magnetic flux do not
support magnetic fields in the steady state, irrespective of compressibility,
cf. antidynamo ; ll8 .

Finite flux and effective equations of the magnetohydrodynamics—In the case of
$I\neq 0$ (recall that $I$ represents the magnetic flux that is associated
with the $z$-direction, see Eq. (43)), the magnetic field is non-trivial in
the steady state. The momentum equation (26) takes the form

$\displaystyle\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\partial_{t}\bm{v}+(\bm{v}\cdot\nabla)\bm{v}=-\frac{\nabla\ln\rho}{Ma^{2}}+\bm{a}+\nu\nabla^{2}\bm{v}-\frac{I^{2}}{2\rho}\nabla
n_{s}^{2},$ (47)

where $n_{s}$ is the steady state solution of Eq. (33). Thus the action of the
magnetic field on the velocity boils down to the magnetic pressure which is
similar to ordinary pressure in a gas with a polytropic index $2$. The role of
the Mach number is played by $I^{-1}$, where the role of sound velocity is
played by the (group) velocity of Alfvén waves ll8 .

Role of the boundary conditions—The above conclusions rely on boundary
conditions that probably are not realistic. For instance for the spatial
integral of $B_{z}$ (the magnetic flux) we have from Eq. (27) that

$\displaystyle\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\dot{I}=\frac{d}{dt}\int
B_{z}d\bm{x}\\!=\\!\eta\int\hat{\bm{n}}\cdot\nabla B_{z}ds,$ (48)

where the last integral is over the disk boundary which normal is designated
by ${\hat{n}}$. The other boundary integral appearing in the calculations,
$\int dsB_{z}\hat{\bm{n}}\cdot\bm{v}$, is assumed to vanish. This is true if
there is no flux of matter through the boundary ($\hat{\bm{n}}\cdot\bm{v}=0$)
or $B_{z}$ vanishes at the boundary as it was assumed in antidynamo . We see
that the rate of change of the magnetic flux, which by itself is proportional
to the area, is proportional to the disk’s perimeter. Thus in the large area
limit the non-conservation of the flux is negligible. Similar conclusions hold
for other boundary effects. In other words, the conclusions, obtained using
artificial finite box conditions, hold over a finite time interval, that
becomes infinite in the thermodynamic limit. In applications, this signifies
that adiabatic approach must be used where $I$ is considered as a variable
that slowly varies in time, and possibly also space.

Conditions for neglecting the magnetic field in steady state—We saw that at
$I=0$ there is no magnetic field in the steady state, whereas in the limit of
strong field, $I\to\infty$, the magnetic field generates Lorentz force that
influences the flow. In between these opposite limits there is a critical
amplitude of the flux $I_{cr}$, defined by the condition that we can neglect
the magnetic field at $I\ll I_{cr}$, but cannot otherwise.

In order to determine $I_{cr}$ we use self-consistency argument. We first
assume that the magnetic field is negligible. Then it is a passive field so
that the flow is purely hydrodynamic, as considered previously. Recalling that
$n_{s}$ is roughly $\rho$ coarse-grained over the scale $l_{\rm res}$, we can
employ the insight into $\rho$ gained from the analysis of pure hydrodynamic
compressible turbulence laid out in the previous sections, in order to
estimate $n_{s}=B_{z}/I$. Finally we plug the obtained $n_{s}$ into Eq. (47),
evaluate the resulting Lorentz force and check the assumption that it is
negligible. The value of $I_{cr}$, obtained in this way depends on the Mach
number and is considered below.

Magnetic field at $Ma\ll 1$—The considerations here resemble those of the
strong field limit above, where the role of external field is played by the
average magnetic field $I$. Under the assumption that magnetic field is
negligible, in pseudo-sound regime, the density modulation is of order
$Ma^{2}$, i.e. $\rho=1+O(Ma^{2})$, see e.g. levermore ; pseudos . It seems
inevitable that in all cases of practical interest we have $n_{s}=1+\delta n$
with $\delta n=\sigma Ma^{\beta}+o(Ma)$, where the actual value of the
positive exponent $\beta$, that most probably is equal to $2$, is irrelevant
below. Here the field $\sigma$ is independent of $Ma$ and is of order one,
unless there is a significant numerical factor, that is beyond the parameteric
estimates performed here. We find $B_{z}=I+I\delta n$ and Eq. (47) becomes

$\displaystyle\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\partial_{t}\bm{v}\\!+\\!(\bm{v}\\!\cdot\\!\nabla)\bm{v}\\!=\\!-\frac{\nabla\ln\rho}{Ma^{2}}\\!+\\!\bm{a}\\!+\\!\nu\nabla^{2}\bm{v}-I^{2}Ma^{\beta}\nabla\sigma,$
(49)

where the magnetic pressure is written to the leading order. We see that the
magnetic pressure term (Lorentz force) is not negligible if
$IMa^{\beta/2}\gtrsim 1$. For instance, for $\beta=2$ we find that the
magnetic field is relevant if $I\gtrsim 1/Ma$, cf. the discussion of external
magnetic field above. In other words, however small $Ma$ is, for large enough
$I$, the compressibility changes significantly the flow which acquires the
magnetic degree of freedom that interacts appreciably with the velocity.
Moreover, if $I\gg Ma^{-\beta}$ then the magnetic pressure makes a large
change in the flow - the assumption of order one change would not be
consistent with Eq. (49).

A remark similar to that made in the discussion of behavior in external field
must be made. The low Mach number density field was not studied much in the
two-dimensional case. The inverse cascade demands care in considering the
magnitude of $\nabla\sigma$ at different scales and its relevance at a given
scale. We leave this for future work. The above considerations seem to make it
inevitable that in the low Mach number limit $I_{cr}\propto Ma^{-\gamma}$ with
$\gamma>0$.

The above considerations reaffirm that care is necessary in using the
Zeldovich antidynamo theorem. This theorem, that could be hoped to apply at
$Ma\ll 1$, fails if there is a large magnetic field threading the disk.

Magnetic field at $Ma\sim 1$—In the case of $Ma\sim 1$, the density is a
smooth large-scale field with order one fluctuations, see Fig. 3. The
structure of $n_{s}=B_{z}/I$ is similar - the difference of passive and active
fields in this case is less relevant fm . We see from Eq. (47) that for $I\ll
1$ the magnetic field is negligible and for $I\sim 1$ it has order of one
effect on the flow. For $I\gg 1$ the magnetic field leads to significant
rearrangement of the velocity field. In other words, $I_{cr}\sim 1$. We
assumed that $l_{\rm res}\lesssim L$, the case which seems most practical.

Magnetic field at $Ma\gg 1$—Finally, we turn to the case of large Mach
numbers. In this regime the reversal of the energy cascade direction discussed
in the sections above comes into force and plays a most significant role in
the magnetic field amplification process. Indeed, as discussed above, the
direction of the cascade is now almost fully direct, just as in the three
dimensional case. As a result, the density field that is generated by the
continuity equation is multifractal, in a stark contrast to the smooth density
fields in the $Ma\lesssim 1$ case. Furthermore, at the large Mach numberss
regime the density field is passive, see Eq. (6). Therefore, making our probe
assumption that the impact of the magnetic field on the velocity is
negligible, $B_{z}/I$ at $\eta=0$ and $\rho$ coincide. Thus, we have from Eq.
(21) that at $\eta\to 0$ (here $\left\langle
B^{2}\right\rangle\approx\left\langle B_{z}^{2}\right\rangle$)

$\displaystyle\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\left\langle
B^{2}\right\rangle\simeq I^{2}C_{0}C^{\delta}Ma^{2\delta},\ \
\frac{\left\langle B^{2}\right\rangle}{\left\langle
B_{z}\right\rangle^{2}}\simeq C_{0}C^{\delta}Ma^{2\delta}.$ (50)

For other aspects of (multifractal) statistics of $B_{z}$ at $\eta=0$, see fm
. The smoothness scale of $B_{z}$ at $\eta=0$ is the sonic scale $l_{s}$,
which we assume here to obey $l_{s}\gtrsim l_{\rm res}$. Then Eq. (50) holds
also at finite $\eta$. We find that magnetic energy is increased by a power of
$Ma$, as compared to the mean field estimate.

Magnetic energy however is determined by the sonic scale and it cannot serve
to decide whether the magnetic field is relevant at large scales. For studying
these scales, we assume the condition $l_{\rm res}\ll l_{s}$, which implies
$n_{s}=\rho$. Then the reduced Eq. (47) becomes at scales larger than $l_{s}$

$\displaystyle\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\partial_{t}\bm{v}+(\bm{v}\cdot\nabla)\bm{v}=\bm{a}+\nu\nabla^{2}\bm{v}-I^{2}\nabla\rho.$
(51)

The equations have the form of the shallow water model. The relevance of
magnetic pressure at large scales is determined by $I$, as in $Ma\sim 1$ case.
This is because at large scales the density is of order one and has
fluctuations of order one, as at $Ma\sim 1$. Thus if $I\ll 1$ the magnetic
field is irrelevant at the injection scale, for $I\sim 1$ it changes the
large-scale flow, and at $I\gg 1$ the magnetic field brings a large change in
the flow at scale of the pumping.

If $I\ll 1$, the magnetic pressure term is irrrelevant at large scales.
However, as in the previous discussion of ordinary thermodynamic pressure, it
becomes relevant at smaller scales. Thus the last term in Eq. (51) is
comparable with the advective term at the “magnetosonic scale” $l_{m}$ where
eddy velocity and (group) velocity of Alfvén waves are of the same order.
Thus, at $l_{m}$ the squared eddy velocity is of order of $I^{2}$ times the
density. Using for estimates $k^{-2}$ spectrum for the velocity, and
$(L/l_{m})^{\delta/2}$ for density at the scale $l_{m}$, see Eq. (21) and fm ,
we find $l_{m}\sim I^{2/(1+\delta)}L$. By including the thermal pressure,
considered previously, we conclude that if $I^{2/(1+\delta)}>1/Ma^{2}$, then
the cascade is stopped by magnetic pressure at the scale $l_{m}$, and
otherwise it is stopped by the thermal pressure at $l_{s}$.

We comment on the assumption that $n_{s}=\rho$ under the condition $l_{res}\ll
l_{s}$. It could be that the limits of vanishing $\eta$ and infinite evolution
time do not commute per realization. Then $n_{s}=\rho$ is only an equality in
law and not an ordinary equality. This would make Eqs. (51) valid
qualitatively only. This fits our purposes here and we leave more detailed
study for future work.

Time of validity of neglecting the details of the boundary conditions—The time
of formation of the level of fluctuations described by Eq. (50) is determined
by the typical time for the cascade process to reach $l_{s}$ when starting
from $L$, see fm . This time is of order $L/u$ and is independent of $l_{s}$,
as in incompressible turbulence frisch . For instance, if we start with
uniform initial $B_{z}$ (which is the simplest case of $I\neq 0$), then Eq.
(50) is valid at times larger than $L/u$ and holds until times where the
boundary effects become relevant (this is true as long as the reaction of the
magnetic field on the flow is negligible). Thus our treatment of the boundary
conditions is self-consistent for the energy calculation, provided that the
characteristic time at which the boundary phenomena penetrate the bulk of the
disk is much larger than $L/u$.

Numerical calculation of the Green’s function—We conclude this section by
remarking on the numerical calculation of the evolution of $B_{z}$ as
described by $G(t,\bm{x}|\bm{x}^{\prime})$. The Green’s function can be found
as the PDF of $\bm{x}(t)$ that is obtained by solving the stochastic ODE
$\dot{\bm{x}}=\bm{v}(t,\bm{x}(t))+\sqrt{\eta}\bm{\Gamma}$ with the initial
condition $\bm{x}(0)=\bm{x}^{\prime}$. Here $\bm{\Gamma}$ is a white noise
with the dispersion
$\langle\Gamma_{i}(t)\Gamma_{k}(t^{\prime})\rangle=2\delta_{ik}\delta(t-t^{\prime})$,
see details in fp . Solving many realizations of the ODE necessary for the
statistics is simpler than solving the PDE. Alternatively, the solution is
given by the Feynman-Kac formula fp ,

$\displaystyle\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!G(t,\bm{x}|\bm{x}^{\prime})=\int_{\bm{x}(0)=\bm{x}^{\prime},\
\ \bm{x}(t)=\bm{x}}{\cal D}\bm{x}$
$\displaystyle\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\exp\left(-\frac{1}{4\eta}\int_{0}^{t}\left(\dot{\bm{x}}-\bm{u}(t,\bm{x}(t^{\prime}))\right)^{2}dt^{\prime}\right),$
(52)

see fp . The solution is valid also when the velocity field interacts with the
magnetic field so it can be used for simulations of the full system of
magnetohydrodynamic equations (MHD). Eq. (52) allows to see that at large
times $G(t,\bm{x}|\bm{x}^{\prime})$ as a function of $\bm{x}$ is independent
of $z$, assuming that $z$ belong to a fixed finite interval. Thus also the
magnetic field is independent of $z$ as already said. This is readily seen
from the decoupling
$\left(\dot{\bm{x}}-\bm{u}(t,\bm{x}(t^{\prime}))\right)^{2}=\sum_{k=1}^{2}\left(\dot{x}_{k}-u_{k}(t,\bm{x}(t^{\prime}))\right)^{2}+{\dot{z}}^{2}$
in the weight function in Eq. (52).

Burgers’ model for magnetohydrodynamic accretion at large $Ma$ and $I\ll 1$—We
consider now the implications of the above for the structure of solutions of
the full system of magnetohydrodynamic equations. We focus our attention on
the $Ma\gg 1$ case. We assume that $I\ll 1$ so that the magnetic field is
negligible at the pumping scale. We demonstrated previously that the cascade
is stopped by magnetic pressure at the scale $l_{m}$, and otherwise it is
stopped by the thermal pressure at $l_{s}$. Taking these considerations into
account, the following equations provide a generalisation of the Burgers’
model for accretion that is given by Eq. (22), by including the effects of
magnetic fields:

$\displaystyle\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\partial_{t}\bm{v}+(\bm{v}\cdot\nabla)\bm{v}=\bm{a}+\left(\frac{b}{Ma^{3}}+b^{\prime}I^{3/(1+\delta)}\right)\nabla^{2}\bm{v},$
(53)

where $b^{\prime}$ is a constant of order one and the cofficient of the
Laplacian term guarantees that the smoothness scale of the flow is of order
$max[l_{m},l_{s}]$. The density and magnetic field then obey the corresponding
equations

$\displaystyle\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\partial_{t}\rho+\nabla\cdot(\rho\bm{v})=\left(\frac{b}{Ma^{3}}+b^{\prime}I^{3/(1+\delta)}\right)\nabla^{2}\rho,$
$\displaystyle\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\partial_{t}B_{z}\\!+\\!\nabla\\!\cdot\\!(B_{z}\bm{v})\\!=\\!\left(\frac{b}{Ma^{3}}+b^{\prime}I^{3/(1+\delta)}\right)\nabla^{2}B_{z},$
(54)

with reasoning similar to that of Eq. (23). We do not consider the case where
the resistive scale is larger than $l_{s}$ since it seems of little practical
interest and can be treated readily. The advantage of the above model is
similar to that of the Burgers’ model for accretion without magnetic field,
considered previously. We leave for future work applications of the above
framework.

Conclusions—As the Mach number of the flow in thin disks increases, the sonic
scale $l_{s}$ decreases from a value much larger than the integral scale $L$,
for $Ma\ll 1$, to a value of order $L$ at $Ma\sim 1$, and, finally, to a value
much smaller than $L$ at $Ma\gg 1$. This decrease of $l_{s}/L$ is accompanied
by the reversal of the direction of energy cascade from inverse in the case of
small Mach numbers to direct cascade for supersonic turbulence. We can use the
ratio of the kinetic energy fluxes to large and small scales, cf. fk  as an
order parameter for this transition. That ratio changes continuously as a
function of the Mach number so that the transition is smooth.

Supersonic turbulence occurs at $Ma\gg 1$ in an interval of scales between
$l_{s}$ and $L$, the supersonic inertial range. This range widens indefinitely
as $Ma\to\infty$ and is characterised by multifractal fluctuations of the
density inside it, just as in supersonic three-dimensional turbulence dust ;
fm .

In order to gain deeper insight into compressible hydrodynamic turbulence we
have derived an explicit expression for the third order velocity correlation
tensor in the case of potential driving force. A significant result of that
derivation is that to lowest order all third order structure functions are
zero. As discussed above this reflects the nature of the physical processes
that take place in highly compressible turbulence and may have significant
implications for observations as well as numerical simulations.

The possibility of the emergence of magnetic fields in compressible turbulent
conducting fluids has been considered. Remarkably, we demonstrate that the
limit of incompressible flow, $Ma\to 0$, has singular features in the presence
of mean vertical magnetic field (flux) $I$. The limits of $Ma\to 0$ and
$I\to\infty$ do not commute. Thus for however small $Ma$, fields with
$I\gtrsim Ma^{-\gamma}$ cause the flow to differ appreciably from
incompressible purely hydrodynamic flows, that hold at $Ma=0$ according to the
antidynamo theorem. Incompressible flow is an idealization that is valid only
if $I\ll Ma^{-\gamma}$.

We performed a systematic study of the role played by the magnetic field in
different Mach number regimes. We provided a generalisation of the antidynamo
theorem to compressible flows. We demonstrated that the horizontal components
of initially small magnetic field fluctuations decay in time, as in the
incompressible case. The antidynamo theorem generalizes fully at $I=0$, where
the vertical component decays also. The result of decaying magnetic
fluctuations in the zero mean field is compatible with numerical simulations
of magnetohydrodynamic flows in astrophysical disks (see e.g. cattaneo  and
pessah ).

The situations with finite flux can be understood by introducing a
$Ma-$dependent threshold $I_{cr}$. The case of $I\ll I_{cr}$ is quite similar
to $I=0$. Here the magnetic field is stationary and non-trivial. However, its
steady state fluctuations generate weak Lorentz forces so the magnetic field
does not change velocity appreciably. The fluctuations are significantly
amplified at $Ma\gg 1$ where the magnetic field energy is enhanced
significantly in comparison with the square of the mean field values, due to
the multifractal nature of the density. Care is needed since magnetic field
can be negligible at some scales and relevant at others.

In the limit of $I\gtrsim I_{cr}$ the Lorentz forces are appreciable and we
have a steady state characterized by strong coupling of magnetic and
hydrodynamic degrees of freedom. This state is much simpler, though than in
three dimensions. The Lorentz force reduces to magnetic pressure forces which
are similar to pressure in the shallow water model. The resemblance is
reinforced by the observation that with no loss the steady state magnetic
field is equivalent to a positive density.

Our results can be used to provide predictions for observations of thin
astrophysical disks. A robust implication of our study is that for planar flow
horizontal magnetic field is zero in the steady state. Thus, observations of
horizontal components in thin astrophysical disks might be highly informative.
Their presence would signify that two-dimensional modelling is inaccurate and
there is some $z-$dependence of $B_{z}$.

Another property of much observational interest is the ratio $B_{z}/\rho$. We
recall that in ideally conducting fluid the ratio of magnetic field and the
density is simply transported along the Lagrangian (fluid particle)
trajectories of the fluid ll8 . Here it might have simpler properties, say at
$Ma\gg 1$ both $B_{z}$ and $\rho$ are multifractal yet their ratio must be
much smoother (if not a constant).

Finally, we proposed Burgers’ model of accretion as a way to provide effective
description of the flow at large $Ma$. Since in the case of gravity driven
turbulence the flow in two dimensions is potential, then the model is
equivalent to a linear equation, after a Cole-Hopf transformation. This is a
very promising direction for numerical studies since the solutions to the
equations are then much more stable and moreover can be obtained via path
integral solution. Considerable interaction between theory and observations is
necessary to ensure steady progress.

This work was supported by the BSF grant no. 8773312. The simulations utilized
a resource allocation MCA07S014 of the ACCESS program, which is supported by
NSF Grants #2138259, #2138286, #2138307, #2137603, and #2138296. A.K. was
supported in part by the NASA Grant 80NSSC22K0724.

## References

  * (1) L. D. Landau, and E. M. Lifshitz, Fluid Mechanics Course of Theoretical Physics, Volume 6 (Elsevier, 2013).
  * (2) Ya. B. Zel’dovich, A Magnetic Field in the Two-Dimensional Motion of a Conducting Turbulent Fluid, Zh. Eksp. Teor. Fiz. 31, 154 (1956) [Sov. Phys. JETP 4, 460 (1957)J.
  * (3) A. G. Kritsuk, M. L. Norman, P. Padoan, and R. Wagner, The statistics of supersonic isothermal turbulence, Astr. J. 665, 416 (2007).
  * (4) I. Fouxon and M. Mond, Density and. tracer statistics in compressible turbulence: phase transition to multifractality, Phys. Rev. E, 100(2), 02311 (2019).
  * (5) U. Frisch, Turbulence: the legacy of A. N. Kolmogorov, (Cambridge university press, 1995).
  * (6) X. Cao and H. C. Spruit, The large-scale magnetic fields of thin accretion disks, Astr. J. 765, 149 (2013).
  * (7) R.E. Pudritz, and T.P. Ray, The Role of magnetic fields in protostellar outflows and star formation, Front. Astron. Space Sci., Sec. Stellar and Solar Physics 6, 54 (2019).
  * (8) G. Boffetta, and R. E. Ecke, Two-dimensional turbulence, Ann. Rev. Fl. Mech. 44, 427 (2012).
  * (9) H. Risken, The Fokker-Planck equation, (Springer, Berlin, Heidelberg, 1996).
  * (10) G. Falkovich, I. Fouxon, and Y. Oz, New relations for correlation functions in Navier-Stokes turbulence, J. Fl. Mech. 644, 465 (2010).
  * (11) J. H. Xie, and O. BŸhler, Exact third-order structure functions for two-dimensional turbulence, J. Fl. Mech. 851, 672 (2018).
  * (12) G. Falkovich, K. Gawedzki, and M. Vergassola, Rev. Mod. Phys. 73, 913 (2001).
  * (13) U. Frisch and Bec, J., 2001. Burgulence. In New trends in turbulence Turbulence: nouveaux aspects (pp. 341-383). Springer, Berlin, Heidelberg.
  * (14) C. Federrath, R. S. Klessen, L. Iapichino, and J. R. Beattie, The sonic scale of interstellar turbulence, Nat. Astr. 5, 365 (2021).
  * (15) S. Boldyrev, Kolmogorov-Burgers model for star-forming turbulence, Astr. J. 569, 841 (2002).
  * (16) P. F. Hopkins and H. Lee, The fundamentally different dynamics of dust and gas in molecular clouds, Month. Not. Royal Astr. Soc. 456, 4174 (2016).
  * (17) G. Falkovich, and A. G. Kritsuk, How vortices and shocks provide for a flux loop in two-dimensional compressible turbulence, Phys. Rev. Fluids 2, 092603 (2017).
  * (18) L. D. Landau and E. M. Lifshitz, Electrodynamics of continuous media, Course of Theoretical Physics, Volume 8, (Elsevier, 2013).
  * (19) V. E. Zakharov, V. S. L’vov, and G. Falkovich, Kolmogorov spectra of turbulence I: Wave turbulence, (Springer Science and Business Media, 2012).
  * (20) K. P. Iyer, K. R. Sreenivasan, and P. K. Yeung, Reynolds number scaling of velocity increments in isotropic turbulence, Phys. Rev. E 95, 021101 (2017).
  * (21) V. R. Kogan, I. V. Kolokolov, and V. V. Lebedev, Kinematic magnetic dynamo in a random flow with strong average shear, J. Phys. A: Math. Theor. 43, 182001 (2010).
  * (22) I. V. Kolokolov, Evolution of magnetic field fluctuations in two-dimensional chaotic flow, J. Phys. A: Math. Theor. 50, 155501 (2017).
  * (23) R. A. Shaw, Particle-turbulence interactions in atmospheric clouds, An. Rev. Fl. Mech. 35, 183 (2003).
  * (24) G. Falkovich, A. Fouxon, and M. G. Stepanov, Acceleration of rain initiation by cloud turbulence, Nature 419, 151 (2002).
  * (25) M. Chertkov, G. Falkovich, I. Kolokolov, and M. Vergassola, Small-scale turbulent dynamo, Phys. Rev. Lett. 83, 4065 (1999).
  * (26) D. Ruelle, Smooth dynamics and new theoretical ideas in nonequilibrium statistical mechanics, J. Stat. Phys. 95, 393 (1999).
  * (27) B. J. Bayly, C. D. Levermore, and T. Passot, Density variations in weakly compressible flows, Phys. Fluids A 4, 945 (1992).
  * (28) J.F Hawley, C.F. Gammie, and S.A. Balbus, Local three-dimension magnetohydrodynamic simulations of accretion disks, Ap. J. 440, 742 (1995).
  * (29) G. Bodo, F. Cattaneo, A. Ferrari, A. Mignone, P. Rossi, Symmetries, scaling laws, and convergence in shearing-box simulations of magneto-rotational instability driven turbulence, Astr. J., 739, 82 (2011).
  * (30) N. Scepi, G. Lesur, and M. Flock, Turbulent and wind-driven accretion in dwarf novae threaded by a large-scale magnetic field, Astron. Astophys. 620, A49 (2018).
  * (31) H. Deng, L. Mayer, H. Latter, P.F. Hopkins, and X-N Bai, Local Simulations of MRI turbulence with meshless methods, A. J. Suppl., 241, 26 (2019).
  * (32) G. Mamatsashvili, G. Chagelishvili, M. Pessah, F. Stefani, G. Bodo, Zerto net flux MRI turbulence in disks: Sustenance scheme and magnetic Prandtl number dependence., Ap. J., 904, 47 (2020).
  * (33) C. Cui, and X-N Bai, Global three-dimensional simulations of outer protoplanetary discs with ambipolar diffusion, MNRAS, 507, 1106 (2021).
  * (34) S.R. Datta, T. Mondal, and B. Mukhopadhyay, Angular momentum transport and thermal stabilization of optically thin, advective accretion flows through large-scale magnetic fields, MNRAS, 513, 204 (2022).
  * (35) J. Wang, T. Gotoh, and T. Watanabe, Spectra and statistics in compressible isotropic turbulence, Phys. Rev. Fluids 2, 013403 (2017).
  * (36) A. G. Kritsuk, Energy Transfer and Spectra in Simulations of Two-dimensional Compressible Turbulence, In Turbulent Cascades II: Proceedings of the Euromech-ERCOFTAC Colloquium 589 (pp. 61-70). Springer International Publishing (2019).