#  Cascade of circulicity in compressible turbulence

O. Soulard<EMAIL_ADDRESS>A. Briard  CEA, DAM, DIF, F-91297 Arpajon,
France

###### Abstract

The purpose of this work is to investigate whether a cascading process can be
associated with the rotational motions of compressible three-dimensional
turbulence. This question is examined through the lens of circulicity, a
concept related to the angular momentum carried by large turbulent scales. By
deriving a Monin-Yaglom relation for circulicity, we show that an “effective”
cascade of this quantity exists, provided the flow is stirred with a force
having a solenoidal component. This outcome is obtained independently from the
expression of the equation of state. To supplement these results, a coarse-
graining analysis of the flow is performed. This approach allows to separate
the contributions of the transfer and production terms of circulicity and to
discuss their respective effects in the inertial range.

##  I Introduction

In incompressible turbulence, the behavior of small inertial scales is
commonly described by invoking the physical image of the Richardson cascade
richardson1922 ; davidson2011 ; alexakis2018 : under the stirring action of
velocity gradients, inertial eddies break down into smaller and less energetic
ones, that break down into even smaller and less energetic ones, and so on
until viscous dissipation occurs. This phenomenological description is
intimately linked with two sets of formal results stemming respectively from
Kolmogorov’s kolmogorov1941 ; frisch1995  and Onsager’s onsager1949 ;
eyink2006  works.

The first approach focuses on the statistical properties of the velocity field
taken at different points, a technique sometimes called “point-splitting”.
Kolmogorov showed that, in the inertial range of homogeneous turbulence,
kinetic energy flows on average from large to small scales with a constant
flux given by its mean dissipation rate $\overline{\varepsilon}$. When
turbulence is isotropic, this result takes the form of Kolmogorov’s
$4/5^{\mathrm{th}}$ law, while in anisotropic turbulence, it is expressed with
the Monin-Yaglom relation monin1975 . Both expressions relate the triple
correlation of the two-point increments of the velocity field to
$\overline{\varepsilon}$. They are both derived from the incompressible
Navier-Stokes equations under a crucial assumption : the mean dissipation
$\overline{\varepsilon}$ must remain finite when the Reynolds number
$\mathtt{Re}_{\mathtt{t}}$ tends to infinity. This hypothesis is sometimes
referred to as the zero${}^{\text{th}}$ law of turbulence and is related to
the notion of dissipative anomalies taylor1935 ; kolmogorov1941 ;
kolmogorov1941b ; onsager1949 ; john2021 .

Dissipative anomalies are also at the heart of the second set of formal
results, the one originating from Onsager’s work. The concept of anomaly
arises from the observation that, when the Reynolds number
$\mathtt{Re}_{\mathtt{t}}$ increases, dissipative events take place in thinner
and thinner structures while becoming more and more intense, in such a way
that their average $\overline{\varepsilon}$ is finite. In the limit
$\mathtt{Re}_{\mathtt{t}}\to\infty$, several authors, starting with Onsager
onsager1949 ; eyink2006 ; saw2016 ; dubrulle2019 , conjectured that this
phenomenology could be described in the form of a weak singular (“anomalous”)
solution of Euler equations. As opposed to Kolmogorov’s approach, Onsager’s
does not necessarily require statistical averages of two-point quantities.
Instead, it can be expressed with the local properties of a regularized
“coarse-grained” velocity field. Despite their inherent differences, Onsager’s
coarse-graining and Kolmogorov’s point-splitting visions share the same core
phenomelogical ideas. Accordingly, the coarse-graining approach has given rise
to a series of predictions closely related to Kolmogorov’s $4/5^{\mathrm{th}}$
law and to the idea that energy is transferred from scale to scale at a
constant rate eyink2006 . Both the “point-splitting” and “coarse-graining”
approaches sustain the idea that an energy cascade takes place in the inertial
range of incompressible turbulence.

When density inhomogeneities or compressible effects are present, the physical
image of the Richardson cascade is harder to uphold. In that case, eddies are
not only stirred and broken by velocity gradients, they are also affected by
baroclinic production and by local dilatations and compressions. In principle,
these additional processes may very well lead to an inverse transfer of
energy, at the very least locally and transiently. These processes also cast
their shadow on the point-splitting and coarse-graining approaches. Their
adaptations to compressible turbulence face several hurdles, some of which
have only been recently overcome. In particular, Aluie aluie2011 ; aluie2012 ;
aluie2013  and Eyink & Drivas eyink2018  have produced two seminal works based
on the coarse-graining approach. They have shown that in compressible
turbulence, an inertial range exists where a local conservative cascade of
kinetic energy takes place, despite the fact that kinetic energy is no longer
an inviscid invariant of the flow. This result is conditioned on the presence
of dissipative anomalies and on the sufficient roughness of the turbulent
fields.

From the point-splitting side, no equivalent predictions have so far been
derived. Still, several adaptations of the Monin-Yaglom relation to
compressible turbulence have been proposed falkovich2010 ; galtier2011 ;
wagner2012 ; banerjee2013 ; kritsuk2013 ; banerjee2014 ; banerjee2017 ;
banerjee2018 ; andres2018 ; andres2019 ; ferrand2020 ; simon2021 . But as
opposed to the coarse-graining approach, most these efforts were not aimed at
proving the existence of a cascading process. Instead, they were mostly driven
by a practical goal: being able to evaluate the dissipation and large-scale
injection of energy knowing two-point correlations. This information is indeed
useful for interpreting several astrophysical phenomena andres2019 ; simon2021
. As it turns out, one of the main difficulties for establishing Monin-Yaglom
relations in compressible flows comes from the pressure field and the way it
is coupled with other thermodynamical variables. To alleviate this difficulty,
most of the mentioned works falkovich2010 ; galtier2011 ; wagner2012 ;
banerjee2013 ; kritsuk2013 ; banerjee2014 ; banerjee2017 ; banerjee2018 ;
andres2018 ; andres2019 ; ferrand2020 ; simon2021  make a simplifying
assumption on the thermodynamical state of the flow: it is either isothermal
galtier2011 ; wagner2012 ; banerjee2013 ; kritsuk2013 ; banerjee2017 ;
banerjee2018 ; andres2018 ; andres2019 ; ferrand2020 , polytropic
falkovich2010 ; banerjee2014  or isentropic simon2021 . These assumptions are
not innocuous: they de facto prevent the existence of several physical
phenomena. For instance, the first two assumptions suppress the baroclinic
torque and the development of convective instabilities such as the Rayleigh-
Taylor and Richtmyer-Meshkov ones zhou2017a ; zhou2017b ; soulard2012b ;
soulard2017 . Besides, the third one is not compatible with the entropy
cascade predicted in eyink2018 . Therefore, despite their great practical use,
the Monin-Yaglom relations derived so far are in principle restricted to
particular flows. In this regard, it is worth highlighting that all the
mentioned Monin-Yaglom relations are coupled to the energetic and mixing
content of the flow, either through the presence of the pressure field, either
because they purposely consider the inertial transfer of the total energy.
None of them is purely dynamical, in the sense of involving only velocity and
density correlations. Whether such a relation exists remains an open question.
But its answer is tightly linked with the possibility of finding a Monin-
Yaglom equation that is not restricted to a particular thermodynamical
assumption. Besides, it can already be stressed that, if it exists, this
answer would require finding a cascading quantity different from the energy:
as shown with the coarse-graining approach in aluie2011 ; aluie2012 ;
aluie2013 ; eyink2018 , the pressure is indeed one of the components acting in
the interscale transfer of kinetic energy.

Before exploring further these issues, let us briefly return to the
incompressible case. In this context, the velocity field is purely solenoidal
and is directly connected to the vorticity field through a Helmholtz
decomposition sagaut2018 . In homogeneous turbulence, this connection is such
that the kinetic energy spectrum becomes equal to the circulicity spectrum.
Circulicity is a term coined by Kassinos et al. kassinos2001  in order to
refer to a tensor describing the “large-scale” structure of the vorticity
field. In this work, we will slightly tweak this definition and use
circulicity to refer to the “large-scale” structure of the angular momentum.
Indeed, when density is constant, vorticity becomes a proxy for measuring the
angular momentum of fluid particles, so that the two notions become
indistinguishable. Piecing these elements together, one arrives at the
conclusion that the energy cascade predicted by the point-splitting and
coarse-graining approaches also coincides with a cascade of circulicity. This
co-occurence agrees with the physical picture of the Richardson cascade in
which energy is transferred to smaller scales by the motion of turbulent
eddies, which can be thought of as coherent regions of angular momentum.

Coming back to compressible flows, the velocity field now possesses a
dilatational component and is not solely representative of vortical motions.
But most importantly, vorticity itself loses one of its key physical meaning.
It is not connected to the angular momentum of fluid particles any longer.
Indeed, angular momentum is defined with respect to the curl of the linear
momentum and is only related to vorticity when density is constant soulard2020
; soulard2022 . Therefore, contrary to the incompressible case, the spectral
properties of kinetic energy and circulicity dissociate in compressible
turbulence. In particular, in homogeneous turbulence, the identity between the
circulicity and kinetic energy spectra ceases to be valid. Instead, the
circulicity spectrum becomes equal to the spectrum of the solenoidal component
of the linear momentum. The latter is indeed the Helmholtz dual of the angular
momentum, just as the solenoidal velocity is with respect to vorticity. As a
consequence of this dissociation, the existence of the energy cascade
predicted with the coarse-graining approach in aluie2011 ; eyink2018  does not
automatically extend to circulicity : even if energy cascades to small scales,
it is not known whether this transfer goes along with the apparition of
smaller whirls. Thus, the question arises as to whether a circulicity cascade
exists in compressible turbulence. This question sends us back to the issues
we already raised when commenting existing Monin-Yaglom relations. Indeed, as
shown in soulard2020 ; soulard2022 , the evolution of the angular momentum and
of the solenoidal linear momentum are not directly impacted by the pressure
field. Hence, the study of their properties opens a door for deriving a Monin-
Yaglom relation that is not bound by the equation of state and that is
decoupled from the thermodynamics and mixing content of the flow.

Given these considerations, the purpose of this work is to study the transfer
of circulicity in the inertial range of a compressible flow. More
particularly, we aim to discuss whether a circulicity cascade exists and, if
it does, what it implies for inertial scales. This study is amenable to both
the point-splitting and coarse-graining approaches. For the sake of
completeness, we will use the two techniques. To begin with, we will derive a
Monin-Yaglom relation for the solenoidal component of the linear momentum, an
appellation hereafter abridged to solenoidal momentum. We will then discuss
how this relation depends on the different variables describing the flow. We
will also discuss how it may potentially provide information about the scaling
of the turbulent spectra. These different elements will also be examined
through the lens of the coarse-graining approach. To this end, spatial
filtering will be applied to the solenoidal momentum equation.

##  II Governing Equations

###  II.1 Compressible Navier-Stokes equations

We consider a flow which state is defined by its density $\rho$, velocity
$\boldsymbol{v}$, internal energy $e$ and species mass fractions
$\left\\{c_{\alpha},\;\alpha=1\cdots N_{s}\right\\}$. These variables evolve
according to the Navier-Stokes equations giovangigli1999 :

$\displaystyle\partial_{t}\rho+\partial_{j}(\rho v_{j})$
$\displaystyle=0\hskip 8.67424pt,\hskip 8.67424pt$ (1a)
$\displaystyle\partial_{t}(\rho v_{i})+\partial_{j}(\rho v_{i}v_{j})$
$\displaystyle=-\partial_{j}(p\delta_{ij}+\sigma_{ij})+f_{i}\hskip
8.67424pt,\hskip 8.67424pt$ (1b) $\displaystyle\partial_{t}(\rho
e)+\partial_{j}(\rho v_{j}e)$
$\displaystyle=\rho\varepsilon-p\partial_{j}v_{j}-\partial_{j}\sigma^{(e)}_{j}\hskip
8.67424pt,\hskip 8.67424pt$ (1c) $\displaystyle\partial_{t}(\rho
c_{\alpha})+\partial_{j}(\rho v_{j}c_{\alpha})$
$\displaystyle=-\partial_{j}\sigma^{(\alpha)}_{j}$ (1d) where $p$ is the
pressure, $\boldsymbol{f}$ a volumetric force, $\boldsymbol{\sigma}$ the
viscosity tensor, $\boldsymbol{\sigma}^{(e)}$ the molecular heat flux,
$\boldsymbol{\sigma}^{(\alpha)}$ the molecular diffusion flux of species
$\alpha$ and $\rho\varepsilon=-\sigma_{ij}\partial_{j}v_{i}$ the dissipation
rate of the kinetic energy. For simplicity, we assume that the viscosity
tensor is defined by: $\displaystyle\sigma_{ij}=-\mu
S_{ij}-\eta\partial_{k}v_{k}\delta_{ij}\hskip 8.67424pt\text{ with }\hskip
8.67424ptS_{ij}=\partial_{j}v_{i}+\partial_{i}v_{j}-\frac{2}{3}\partial_{k}v_{k}\delta_{ij}\hskip
8.67424pt,\hskip 8.67424pt$ (1e) where the shear viscosity $\mu$ and the bulk
viscosity $\eta$ are constant. The molecular heat and diffusion fluxes,
$\boldsymbol{\sigma}^{(e)}$ and $\boldsymbol{\sigma}^{(\alpha)}$, will not be
used in the remainder of this text and do not need to be detailed. Expressions
for these quantities can be found for instance in giovangigli1999 . To close
the flow description, we assume that the pressure is given by an equation of
state which is a function of density, energy and mass fractions:
$\displaystyle p\equiv p(\rho,e,\boldsymbol{c})\hskip 2.17021pt.\hskip
8.67424pt$ (1f) This function can be general and is not constrained by an
isothermal, polytropic or isentropic assumption. Similarly to
$\boldsymbol{\sigma}^{(e)}$ and $\boldsymbol{\sigma}^{(\alpha)}$, the equation
of state does not play any role in the forthcoming analysis and does not need
to be made explicit.

###  II.2 Helmholtz decomposition of the momentum

Using an Helmholtz decomposition, along with the fact that the domain
considered in this study is unbounded, the momentum
$\boldsymbol{j}=\rho\boldsymbol{v}$ can be split into a solenoidal and a
dilatational component respectively denoted by $\boldsymbol{s}$ and
$\boldsymbol{d}$ soulard2020 ; soulard2022 :

$\displaystyle\boldsymbol{j}=\rho\boldsymbol{v}=\boldsymbol{s}+\boldsymbol{d}\hskip
8.67424pt\text{ with }\hskip 8.67424ptd_{i}=-\partial_{i}\Phi\hskip
8.67424pt\text{ and }\hskip
8.67424pts_{i}=\epsilon_{ijk}\partial_{j}\Psi_{k}\hskip 8.67424pt,\hskip
8.67424pt$ (2)

where $\epsilon_{ijk}$ is the Levi-Civita tensor. The scalar and vector
potentials $\Phi$ and $\boldsymbol{\Psi}$ are given by:

$\displaystyle\partial^{2}_{jj}\Phi=-\Xi\hskip 8.67424pt\text{ and }\hskip
8.67424pt\partial^{2}_{jj}\Psi_{i}=-\Omega_{i}\hskip 8.67424pt,\hskip
8.67424pt$ (3)

where $\Xi$ is divergence of the momentum and $\boldsymbol{\Omega}$ is the
angular momentum of a fluid particle:

$\displaystyle\Xi=\partial_{j}(\rho v_{j})=-\partial_{t}\rho\hskip
8.67424pt\text{ and }\hskip
8.67424pt\Omega_{i}=\epsilon_{ijk}\partial_{j}(\rho v_{k})\hskip
2.17021pt.\hskip 8.67424pt$ (4)

It is important to stress that the angular momentum $\boldsymbol{\Omega}$ does
not carry the same information as the vorticity $\boldsymbol{\omega}$. The
latter is defined by:

$\omega_{i}=\epsilon_{ijk}\partial_{j}v_{k}={\Omega_{i}}/{\rho}-\epsilon_{ijk}v_{k}\partial_{j}\rho/\rho\hskip
2.17021pt.\hskip 8.67424pt$

Similarly, the momentum divergence $\Xi$ is different from the velocity
divergence $\Theta$ defined as

$\Theta=\partial_{j}u_{j}={\Xi}/{\rho}-u_{j}\partial_{j}\rho/\rho\hskip
2.17021pt.\hskip 8.67424pt$

Accordingly, the Helmholtz decomposition of the momentum $\rho\boldsymbol{v}$
is also different from the Helmholtz decomposition of the velocity field
$\boldsymbol{v}$, which can be written as:

$\displaystyle\boldsymbol{v}=\boldsymbol{v^{s}}+\boldsymbol{v^{d}}\hskip
8.67424pt\text{ with }\hskip 8.67424ptv^{d}_{i}=-\partial_{i}\varphi^{v}\hskip
8.67424pt,\hskip
8.67424ptv^{s}_{i}=\epsilon_{ijk}\partial_{j}\psi^{v}_{k}\hskip
8.67424pt\text{ and }\hskip
8.67424pt\partial^{2}_{jj}\varphi^{v}=-\Theta\hskip 8.67424pt,\hskip
8.67424pt\partial^{2}_{jj}\psi^{v}_{i}=-\omega_{i}\hskip 2.17021pt.\hskip
8.67424pt$ (5)

Except when density is constant, one generally has:

$\boldsymbol{\Omega}\neq\rho\boldsymbol{\omega}\hskip 8.67424pt,\hskip
8.67424pt\Xi\neq\rho\Theta\hskip 8.67424pt,\hskip
8.67424pt\Phi\neq\rho\varphi^{v}\hskip 8.67424pt\text{ and }\hskip
8.67424pt\boldsymbol{\Psi}\neq\rho\boldsymbol{\psi}^{v}\hskip 2.17021pt.\hskip
8.67424pt$

###  II.3 Circulicity

Given the Helmholtz decomposition (2)-(3), the solenoidal momentum
$\boldsymbol{s}$ characterizes the rotational motions of the flow,
“rotational” being used here as a reference to angular momentum and not
vorticity. This link between $\boldsymbol{s}$ and $\boldsymbol{\Omega}$ can be
formalized further with the notion of circulicity. More precisely, in this
work, we define the local value of the circulicity field $\mathcal{C}$ by:

$\displaystyle\mathcal{C}=\frac{1}{2}s_{i}s_{i}\hskip 2.17021pt.\hskip
8.67424pt$ (6)

The quantity $\mathcal{C}/\rho$ can be interpreted as the contribution of
rotational motions to the kinetic energy $\frac{1}{2}\rho v_{i}v_{i}$. But
$\mathcal{C}$ also has another interpretation that is not linked to energy and
that is more easily discussed in homogeneous turbulence. In this context, the
mean of $\mathcal{C}$, which is half the variance of $\boldsymbol{s}$, is
equal to the integral of the solenoidal momentum spectrum $E_{ss}$:

$\displaystyle\text{Homogeneous turbulence : }\hskip
21.68231pt\overline{\mathcal{C}}=\frac{1}{2}\overline{s_{i}s_{i}}=\int
E_{ss}\mathrm{d}k\hskip 8.67424pt,\hskip 8.67424pt$ (7)

where $\overline{\cdot}$ denotes the ensemble mean and where $k$ is the wave-
number. Besides, given the Poisson equations (3), the spectrum $E_{ss}$ of
$\boldsymbol{s}$ is linked to the spectrum $E_{\Omega\Omega}$ of
$\boldsymbol{\Omega}$ by :

$\displaystyle\text{Homogeneous turbulence : }\hskip
21.68231ptE_{ss}=\frac{E_{\Omega\Omega}}{k^{2}}\hskip 2.17021pt.\hskip
8.67424pt$ (8)

Equation (8) indicates that the solenoidal momentum carries the same two-point
information as the angular momentum, with the difference that a greater weight
is given to large scales. When combined with Eq. (7), this result shows that
the mean circulicity $\overline{\mathcal{C}}$ accounts for the large-scale
content of the angular momentum. This conclusion has been illustrated for
homogeneous turbulence, but the overall correspondence between circulicity and
large-scale angular momentum extends to the inhomogeneneous case.

Note also that in a constant-density flow, one would simply have
$\overline{\mathcal{C}}=\rho^{2}\overline{v_{i}v_{i}}/2$. Hence, circulicity
would be redundant with kinetic energy and there would be no need to study its
properties, nor those of the solenoidal momentum two-point correlations.
However, this is no longer the case when density is variable. As mentioned in
the introduction, the concepts of circulicity and energy dissociate in
compressible flows. Finally, it is worth stressing that the notion of
circulicity used here is only loosely based on the one proposed by Kassinos et
al. kassinos2001 . The latter introduced this concept to measure different
contributions of the Reynolds-stress tensor of an incompressible velocity
field. Here, we only preserved the idea that circulicity gives access to the
large-scale structure of the angular momentum.

###  II.4 Evolution of the solenoidal momentum $\boldsymbol{s}$

The solenoidal momentum $\boldsymbol{s}$ is the main focus of this work
because of its connection to angular momentum and circulicity. To derive its
evolution, we start from the linear momentum equation (1b) and note that
$\partial_{t}d_{i}=-\partial_{i}\partial_{t}\Phi$. This term can then be added
along with $\partial_{i}p$ into the gradient of a pseudo-pressure $\pi^{s}$,
which value is set by the fact that $\boldsymbol{s}$ is divergence-free. The
same can also be done for the dilatational parts of the forcing and viscous
terms. This procedure yields:

$\displaystyle\partial_{t}s_{i}+\partial_{j}\left(\rho
v_{i}v_{j}\right)=-\partial_{i}\pi^{s}-\partial_{j}\sigma^{s}_{ij}+f_{i}^{s}\hskip
8.67424pt,\hskip 8.67424pt$ (9a) where $\boldsymbol{\sigma}^{s}$ and
$\boldsymbol{f}^{s}$ are the solenoidal parts of the forcing and viscous
terms:
$\displaystyle\sigma_{ij}^{s}=-\mu\left(\partial_{j}v_{i}+\partial_{i}v_{j}\right)+2\mu\partial_{k}v_{k}\delta_{ij}\hskip
8.67424pt\text{ and }\hskip
8.67424ptf_{i}^{s}=\epsilon_{ijk}\partial_{j}\psi_{k}^{f}\hskip
8.67424pt\text{ with }\hskip
8.67424pt\partial^{2}_{jj}\psi_{i}^{f}=-\epsilon_{ijk}\partial_{j}f_{k}\hskip
2.17021pt.\hskip 8.67424pt$ (9b) The pseudo-pressure $\pi^{s}$ enforces the
divergence-free constraint of $\boldsymbol{s}$ for the advection term
$\partial_{j}(\rho v_{i}v_{j})$:
$\displaystyle\partial^{2}_{jj}\pi^{s}=-\partial^{2}_{ij}\left(\rho
v_{i}v_{j}\right)\hskip 2.17021pt.\hskip 8.67424pt$ (9c)

It is worth highlighting that $\pi^{s}$ is different from the actual pressure
$p$:

$\pi^{s}\neq p\hskip 2.17021pt.\hskip 8.67424pt$

The pressure $p$ is set by the equation of state (1f). By contrast, $\pi^{s}$
is the solution of a Poisson equation and is set by the dynamics of the flow.
This remains true whether the flow displays strong compressibility effects or
not. Another significant point is that $p$ does not appear explicitly in Eq.
(9a). Thus, even though the equation of state is important for determining the
overall evolution of the flow, its influence on the solenoidal momentum
$\boldsymbol{s}$ is only indirect. This is not the case for all solenoidal
fields. For instance, pressure appears explictly in the evolution equation of
the solenoidal velocity field $\boldsymbol{v}^{s}$.

###  II.5 Evolution of the circulicity $\mathcal{C}$

Starting from the evolution equation (9a) of the solenoidal momentum and using
the Poisson equations (3), the following evolution of the circulicity can be
deduced:

$\displaystyle\partial_{t}\mathcal{C}+\partial_{j}\mathcal{F}_{j}^{c}=\mathcal{P}_{c}-\varepsilon_{c}+s_{i}f_{i}^{s}\hskip
8.67424pt,\hskip 8.67424pt$ (10)

where $\boldsymbol{\mathcal{F}}^{c}$ is the flux of circulicity:

$\displaystyle\mathcal{F}_{i}^{c}=\frac{1}{2}|\boldsymbol{j}|^{2}v_{i}+\pi^{s}j_{i}+\Phi\partial_{j}\left(\rho
v_{i}v_{j}+\pi^{s}\delta_{ij}\right)+\mu
s_{j}(\partial_{j}v_{i}-\partial_{i}v_{j})\hskip 8.67424pt,\hskip 8.67424pt$
(11)

where $\mathcal{P}_{c}$ and $\varepsilon_{c}$ are terms associated with the
local production and dissipation of circulicity:

$\displaystyle\mathcal{P}_{c}=-\frac{1}{2}|\boldsymbol{j}|^{2}\Theta+\pi^{s}\Xi\hskip
8.67424pt\text{ and }\hskip
8.67424pt\varepsilon_{c}=\mu\Omega_{i}\omega_{i}\hskip 8.67424pt,\hskip
8.67424pt$ (12)

where we recall that $\boldsymbol{j}=\rho\boldsymbol{v}$,
$\Theta=\partial_{i}v_{i}$ and $\Xi=\partial_{i}(\rho v_{i})$.

This formulation of the circulicity equation allows to highlight the role
played by the dilatational components of the velocity and momentum: it is the
divergences of these fields that control the circulicity production term
$\mathcal{P}_{c}$. The other interest of this formulation is to allow for a
direct comparison with the evolution of the total momentum norm. Indeed, one
has:

$\displaystyle\partial_{t}\left(\frac{1}{2}|\boldsymbol{j}|^{2}\right)+\partial_{j}\left[\frac{1}{2}|\boldsymbol{j}|^{2}v_{j}+pj_{j}+\sigma_{ij}j_{i}\right]=-\frac{1}{2}|\boldsymbol{j}|^{2}\Theta+p\Xi+\sigma_{ij}\partial_{j}j_{i}+j_{i}f_{i}\hskip
2.17021pt.\hskip 8.67424pt$ (13)

The right-hand sides of equations (10) and (13) share the same structure, with
similar source, dissipation and forcing terms. The main differences are that,
in Eq. (10), the dynamic pressure $\pi^{s}$ replaces the actual pressure $p$
and that the solenoidal force replaces the total one. Besides, the dissipation
term $\varepsilon_{c}$ only accounts for shear viscosity effects and not bulk
ones. It is worth stressing that $\varepsilon_{c}$ is not necessarily
positive. The angular momentum and vorticity fields can indeed be misaligned.
This misalignment is linked to the density and velocity gradients as follows:

$\varepsilon_{c}=\mu\,\rho\omega_{i}\omega_{i}+\mu\,v_{i}\left(\partial_{j}v_{i}-\partial_{i}v_{j}\right)\partial_{j}\rho\hskip
2.17021pt.\hskip 8.67424pt$

However, if we assume that the density and velocity fields decorrelate at
small scales, the mean of $\varepsilon_{c}$ will be dominated by the mean of
the first term. In that case, the mean of $\varepsilon_{c}$ remains positive,
even though local values of $\varepsilon_{c}$ can be negative. In the
remaining of this work, we will assume that such an hypothesis holds and that:

$\overline{\varepsilon_{c}}>0\hskip 2.17021pt.\hskip 8.67424pt$

##  III Monin-Yaglom relation and transfer of circulicity in homogeneous
turbulence

Our purpose in this section is to study how circulicity is transferred from
scale to scale in the inertial range of a homogeneous compressible flow. Given
the relation between circulicity and solenoidal momentum (see Sec. II.3), we
start by deriving a Karman-Howarth equation for the solenoidal momentum
$\boldsymbol{s}$. Then, we use this equation to express a Monin-Yaglom
relation describing the interscale transfer of this quantity and by extension
of circulicity. This relation is compared against existing ones falkovich2010
; galtier2011 ; wagner2012 ; banerjee2013 ; kritsuk2013 ; banerjee2014 ;
banerjee2017 ; banerjee2018 ; andres2018 ; andres2019 ; ferrand2020 ;
simon2021  and is also used to discuss inertial range scalings. This section
only makes use of the point-splitting approach. The coarse-graining method
will be discussed in the next section.

###  III.1 Karman-Howarth equation for the solenoidal momentum

The first step in our study consists in deriving a Karman-Howarth equation for
the solenoidal momentum, or, in other words, an evolution equation for the
second-order structure function of the solenoidal momentum. The latter is
defined by:

$\overline{\Delta s_{i}\Delta
s_{i}}(\boldsymbol{r},t)=2\overline{s_{i}s_{i}}-2\overline{s_{i}(\boldsymbol{x})s_{i}(\boldsymbol{x}+\boldsymbol{r})}\hskip
8.67424pt,\hskip 8.67424pt$

where for any quantity $X$, $\Delta X$ refers to the difference between the
values of $X$ taken at two different points $\boldsymbol{x}$ and
$\boldsymbol{x}^{\prime}$, separated by the vector
$\boldsymbol{r}=\boldsymbol{x}^{\prime}-\boldsymbol{x}$. For later purposes,
it will also be useful to introduce the median value $\\{X\\}_{\frac{1}{2}}$of
$X$ at these two points. These two-point difference and median values are
defined by:

$\Delta
X(\boldsymbol{x},\boldsymbol{r},t)=X(\boldsymbol{x}^{\prime},t)-X(\boldsymbol{x},t)\hskip
8.67424pt\text{ and }\hskip
8.67424pt\\{X\\}_{\frac{1}{2}}(\boldsymbol{x},\boldsymbol{r},t)=(X(\boldsymbol{x}^{\prime},t)+X(\boldsymbol{x},t))/2\hskip
8.67424pt\text{ with }\hskip
8.67424pt\boldsymbol{r}=\boldsymbol{x}^{\prime}-\boldsymbol{x}\hskip
2.17021pt.\hskip 8.67424pt$

We would like to stress again that $\overline{\Delta s_{i}\Delta s_{i}}$ is
related to circulicity and that, as such, it accounts for the large-scale
content of the angular momentum. Indeed, one has: $\overline{\Delta
s_{i}\Delta s_{i}}(\boldsymbol{r},t)=4\overline{\mathcal{C}}-4\int
e^{\imath\boldsymbol{k}\cdot\boldsymbol{r}}\frac{E_{\Omega\Omega}}{2\pi
k^{4}}\mathrm{d}\boldsymbol{k}$. Therefore, studying the inertial range
transfer of $\overline{\Delta s_{i}\Delta s_{i}}$ is equivalent to studying
the transfer of circulicity.

Starting from Eq. (9a) and using some of the relations detailed in App. A, we
find after some algebra that:

$\displaystyle\partial_{t}\overline{\Delta s_{i}\Delta
s_{i}}(\boldsymbol{r},t)=-\partial_{r_{j}}\mathscr{F}_{j}(\boldsymbol{r},t)-4\mathscr{E}_{\mathrm{inj}}(t)+2\mu\partial^{2}_{r_{j}r_{j}}\overline{\Delta
v_{i}\Delta s_{i}}(\boldsymbol{r},t)+2\overline{\Delta s_{i}\Delta
f_{i}^{s}}(\boldsymbol{r},t)\hskip 8.67424pt,\hskip 8.67424pt$ (14a) where the
circulicity flux $\boldsymbol{\mathscr{F}}$ and injection rate
$\mathscr{E}_{\mathrm{inj}}$ are defined by:
$\displaystyle\mathscr{F}_{j}(\boldsymbol{r},t)=\overline{\Delta v_{j}\Delta
j_{i}\Delta j_{i}}(\boldsymbol{r},t)-2\overline{\Delta
v_{j}^{d}\\{j_{i}j_{i}\\}_{\frac{1}{2}}}(\boldsymbol{r},t)-4\overline{\Delta{d}_{i}\\{\rho
v_{i}v_{j}\\}_{\frac{1}{2}}}(\boldsymbol{r},t)\hskip 8.67424pt,\hskip
8.67424pt$ (14b) and
$\displaystyle\mathscr{E}_{\mathrm{inj}}(t)=\overline{s_{i}f_{i}^{s}}-\partial_{t}\overline{\mathcal{C}}=\overline{\varepsilon_{c}}(t)-\overline{\mathcal{P}_{c}}(t)\hskip
2.17021pt.\hskip 8.67424pt$ (14c)

The first term in the right-hand side of Eq. (14a) describes the non-linear
transfer of circulicity from scale to scale. It is expressed as the divergence
of a flux. The second term accounts for the unforced evolution of the mean
circulicity $\overline{\mathcal{C}}=\overline{s_{i}s_{i}}/2$ and is scale-
independent. The third one expresses the transfer of circulicity induced by
viscous effects. The last one corresponds to the correlation between
$\boldsymbol{s}$ and the external force $\boldsymbol{f}$ or more precisely its
solenoidal component $\boldsymbol{f}^{s}$. Overall, the evolution equation of
$\overline{\Delta s_{i}\Delta s_{i}}$ retains the same form as the equation of
the second-order structure-function of the velocity field classically derived
in incompressible flows monin1975 ; frisch1995 . However, the flux and
dissipation terms of Eq. (14a) are affected by the presence of density
inhomogeneities and by compressibility.

More precisely, the non-linear transfer of circulicity involves three
different components. The first one, $\overline{\Delta v_{j}\Delta j_{i}\Delta
j_{i}}$, is linked to the advection of $\boldsymbol{s}$ by the velocity field
$\boldsymbol{v}$. It takes the form of a third-order structure-function. As
shown in App. C, its expression is identical to the one appearing in the
transfer flux of $\overline{\Delta j_{i}\Delta j_{i}}$. Hence, this first
component is representative of the transfer between scales not only of the
solenoidal momentum, but also of the total momentum. The remaining two
contributions involve the increments of dilatational components of the
momentum and velocity fields. Their presence is due to the fact that
circulicity is driven by a production term $\mathcal{P}_{c}$ and is not a
conserved quantity. However, equation (14a) shows that the effects of this
non-conservative production can still be cast in the form of a conservative
flux as far as the interscale transfer of circulicity is concerned. Note that
both of these contributions vanish in constant-density turbulence. In that
case, one is left with $\mathscr{F}_{j}=\rho^{2}\overline{\Delta v_{j}\Delta
v_{i}\Delta v_{i}}$, which is the flux obtained in the incompressible case
weighted by $\rho^{2}$.

Concerning $\mathscr{E}_{\mathrm{inj}}$, it has two components. The first one
is the mean dissipation of circulicity $\overline{\varepsilon_{c}}$, the
second its mean production rate $\overline{\mathcal{P}_{c}}$ (see Eq. (10)).
The latter has the same origin as the two dilatational contributions of the
non-linear transfer flux. In this regard, a balance between these two
dilatational terms and $\overline{\mathcal{P}_{c}}$ would exist in the limit
$r\to 0$ if the turbulent fields were smooth. For rough fields, this is not
case. Note also that for constant-density turbulence, ${\mathcal{P}_{c}}$
vanishes and one is left with
$\mathscr{E}_{\mathrm{inj}}(t)=\mu\,\rho\overline{\omega_{i}\omega_{i}}$,
which is the solenoidal dissipation weighted by the density. We would also
like to emphasize that the definition of $\mathscr{E}_{\mathrm{inj}}$ stems
from the unforced evolution of $\overline{\mathcal{C}}$, as given by the first
equality of Eq. (14c). This origin is important because in forced stationary
turbulence, one gets:

$\displaystyle\mathscr{E}_{\mathrm{inj}}=\overline{s_{i}f_{i}^{s}}\hskip
2.17021pt.\hskip 8.67424pt$ (15)

In that case, the value of $\mathscr{E}_{\mathrm{inj}}$ corresponds to the
injection rate of circulicity. It is then set by the solenoidal component of
the force and not by the total force. The implications of the latter property
will be commented further in Sec. III.3.

###  III.2 Monin-Yaglom relation

We now consider the limit $\mu\to 0$ and $r\to 0$ of the Karman-Howarth
equation (14a). The notation $\mu\to 0$ is used as a shorthand for referring
to the high Reynolds number limit. Then, provided the correlation of the force
$\boldsymbol{f}$ falls off exponentially, the only two terms that do not
vanish in this limit are the non-linear flux divergence and
$\mathscr{E}_{\mathrm{inj}}$. Thus, in the limit $\mu\to 0$, $r\to 0$, we
obtain:

$\displaystyle\partial_{r_{j}}\left(\overline{\Delta v_{j}\Delta j_{i}\Delta
j_{i}}-2\overline{\Delta
v_{j}^{d}\\{j_{i}j_{i}\\}_{\frac{1}{2}}}-4\overline{\Delta{d}_{i}\\{\rho
v_{i}v_{j}\\}_{\frac{1}{2}}}\right)=-4\mathscr{E}_{\mathrm{inj}}\hskip
2.17021pt.\hskip 8.67424pt$ (16)

This equation is one of the main results of this work. It constitutes the
Monin–Yaglom relation derived from the solenoidal momentum evolution Eq. (9).
Interestingly, this relation can be written in several alternative forms that
may help shed light on different aspects of the transfer of circulicity. For
instance, using the Helmholtz decomposition (2)-(3) and the Poisson equation
(9c), one may get rid of the dilatational components $v_{i}^{d}$ and $d_{i}$.
Equation (16) can indeed be rewritten as:

$\displaystyle\partial_{r_{j}}\left(\overline{\Delta v_{j}\Delta j_{i}\Delta
j_{i}}-2\overline{\Delta
v_{j}\\{j_{i}j_{i}\\}_{\frac{1}{2}}}+4\overline{\Delta{j}_{j}\\{\pi^{s}\\}_{\frac{1}{2}}}\right)=-4\mathscr{E}_{\mathrm{inj}}\hskip
2.17021pt.\hskip 8.67424pt$ (17)

In agreement with this formulation, one may also rewrite
$\mathscr{E}_{\mathrm{inj}}$ as
$\mathscr{E}_{\mathrm{inj}}=\overline{\varepsilon_{c}}+\frac{1}{2}\overline{j_{i}j_{i}\partial_{j}v_{j}}-\overline{\pi^{s}\partial_{j}(\rho
v_{j})}$. Another possibility is to forgo the flux divergence formulation and
allow for the presence of a source term. In particular, one may reformulate
Eq. (16) as:

$\displaystyle\partial_{r_{j}}\Big{(}\overline{\Delta v_{j}\Delta j_{i}\Delta
j_{i}}\Big{)}+\overline{\Delta\Theta\Delta(j_{i}j_{i})}-2\overline{\Delta\Xi\Delta\pi^{s}}=-4\overline{\varepsilon_{c}}\hskip
8.67424pt,\hskip 8.67424pt$ (18)

where we recall that $\Theta=\partial_{j}v_{j}$ and $\Xi=\partial_{j}(\rho
v_{j})$. This formulation is closer to the ones derived for instance in
galtier2011 ; banerjee2014 . As in these references, the source terms can be
associated with the dissipation term to provide a scale dependent effective
dissipation $\varepsilon_{\mathrm{eff}}$.

###  III.3 Circulicity cascade

The main physical interpretation of Eq. (16), and its variants Eqs. (17) and
(18), is similar to the one put forward in the classical incompressible case,
except for two important points. First, it applies to circulicity and not
energy, and second, the flux $\boldsymbol{\mathscr{F}}$ in Eq. (16) is an
“effective” flux that accounts not only for the scale-to-scale transfer of
circulicity but also for its production by the dilatational source term
$\mathcal{P}_{c}$. The two processes are actually intertwined and setting a
precise separation between the two bears some degree of arbitrariness. For
this reason, the fact that these two effects can be collected within a single
flux entity allows to get rid of this distinction and to put forward a simple
interpretation of the Monin–Yaglom equation (16). More precisely, the global
flux $\Pi$ of circulicity flowing from scales larger than $r$ to scales
smaller than $r$ is defined by podesta2007 :

$\Pi=-\frac{1}{4V_{r}}\int_{|\boldsymbol{r^{\prime}}|\leq
r}\partial_{r_{j}}\mathscr{F}_{j}(\boldsymbol{r}^{\prime})\mathrm{d}\boldsymbol{r}^{\prime}=-\frac{3}{4r}\langle{\mathscr{F}_{\parallel}}\rangle_{4\pi}(r)\hskip
8.67424pt,\hskip 8.67424pt$

where the subscript $\parallel$ denotes the coordinate of a vector in the
direction $\hat{\boldsymbol{r}}={\boldsymbol{r}}/{r}$ and where
$\langle{\cdot}\rangle_{4\pi}$ refers to the average over the surface of the
unit sphere: $\mathscr{F}_{\parallel}=\mathscr{F}_{j}\hat{r}_{j}\hskip
8.67424pt\text{ and }\hskip
8.67424pt\langle{\cdot}\rangle_{4\pi}=\frac{1}{4\pi}\oint\cdot\mathrm{d}\hat{\boldsymbol{r}}$.
From the definition of $\boldsymbol{\mathscr{F}}$, one can express this flux
as:

$\displaystyle\Pi=-\frac{3}{4r}\langle{\overline{\Delta v_{\parallel}\Delta
j_{i}\Delta j_{i}}-2\overline{\Delta
v_{\parallel}^{d}\\{j_{i}j_{i}\\}_{\frac{1}{2}}}-4\overline{\Delta{d}_{i}\\{\rho
v_{i}v_{\parallel}\\}_{\frac{1}{2}}}}\rangle_{4\pi}\hskip 2.17021pt.\hskip
8.67424pt$ (19)

In isotropic turbulence, the brackets $\langle{\cdot}\rangle_{4\pi}$ can be
dropped from this expression.

The Monin-Yaglom relation Eq. (16) states that the global flux of circulicity
is constant and equal to $\mathscr{E}_{\mathrm{inj}}$:

$\displaystyle\Pi=\mathscr{E}_{\mathrm{inj}}\hskip 2.17021pt.\hskip 8.67424pt$
(20)

Alternatively, one can write that:

$\langle{\mathscr{F}_{\parallel}}\rangle_{4\pi}=\langle{\overline{\Delta
v_{\parallel}\Delta j_{i}\Delta j_{i}}-2\overline{\Delta
v_{\parallel}^{d}\\{j_{i}j_{i}\\}_{\frac{1}{2}}}-4\overline{\Delta{d}_{i}\\{\rho
v_{i}v_{\parallel}\\}_{\frac{1}{2}}}}\rangle_{4\pi}=-\frac{4}{3}\mathscr{E}_{\mathrm{inj}}r\hskip
2.17021pt.\hskip 8.67424pt$

Now, let us assume that the flow is forced and stationary. Then, the value of
$\mathscr{E}_{\mathrm{inj}}$ is determined by the injection of solenoidal
momentum: equation (15) is valid and one has
$\mathscr{E}_{\mathrm{inj}}=\overline{s_{i}f_{i}^{s}}\geq 0$. From there, two
situations can be distinguished.

  * •

The first one occurs when the solenoidal component of the force is not null.
Then, we have :

$\Pi=\mathscr{E}_{\mathrm{inj}}>0\hskip 2.17021pt.\hskip 8.67424pt$

This condition indicates the existence of an effective circulicity cascade:
circulicity is transferred on average from large to small scales at a rate
$\mathscr{E}_{\mathrm{inj}}$ independent from the scale $r$. This description
is similar to the forward energy cascade associated with the Kolmogorov-
Obukhov theory in incompressible turbulence.

  * •

The second situation arises when $\boldsymbol{f^{s}}=0$ and turbulence is only
forced on dilatational modes. Then, the flux of circulicity becomes null:

$\Pi=\mathscr{E}_{\mathrm{inj}}=0\hskip 2.17021pt.\hskip 8.67424pt$

In that case, there is no effective circulicity cascade in the inertial range.
Instead, an effective non-linear equilibrium alexakis2018  takes place in the
inertial range, set by an exchange between the variances of the dilatational
and solenoidal components of the momentum and constrained by the condition
$\Pi=0$.

Thus, the existence of a circulicity cascade is not systematic and depends on
the way turbulence is forced at large scales. A parallel can be drawn between
this conclusion and several observations which have been made in simulations
of forced compressible turbulence federrath2010 ; federrath2013 ; wang2018 ;
donzis2020 ; john2021 . Depending on whether the forcing is solenoidal or
dilatational, significant differences were observed in the statistics of the
simulated flow. Even though the force is massic in federrath2010 ;
federrath2013 ; wang2018 ; donzis2020 ; john2021  and not volumetric as in
this study, a link between these distinct behaviors and the disappearance of
the circulicity cascade would be worth investigating.

We close this section as we started it, by stressing that circulicity is not a
conserved quantity and that the notions of its cascade and equilibrium state
are, to some extent, ill-posed. This is why we used the term “effective” to
qualify them. The flux $\boldsymbol{\mathscr{F}}$ and its integral $\Pi$
combine non-linear effects coming from conservative and non-conservative
processes. In section IV, we will use the coarse-graining approach to help
disentangle the roles of cascading and production effects for circulicity in
the inertial range.

###  III.4 Comparison with previous relations and practical use of Eq. (16)

The Monin–Yaglom equation (16), and its variants (17)-(18), are valid for
compressible turbulence, whether in a highly compressible or weakly pseudo-
compressible regime. As opposed to other such relations, for instance those
derived in falkovich2010 ; galtier2011 ; wagner2012 ; banerjee2013 ;
kritsuk2013 ; banerjee2014 ; banerjee2017 ; banerjee2018 ; andres2018 ;
andres2019 ; ferrand2020 ; simon2021 , Eq. (16) does not involve the actual
pressure field and does not make any assumption on the equation of state, i.e.
there is no hypothesis on whether the flow is isothermal, isentropic or
polytropic. What is more, all the terms appearing in Eq. (16) can be evaluated
knowing only the density $\rho$ and the velocity field $\boldsymbol{v}$ at a
given time. In this restricted sense, it is entirely decoupled from the
energetic and mixing content of the flow. Besides, it is also worth stressing
that, in Eq. (16), the transfer between scales is written as the divergence of
a flux $\boldsymbol{\mathscr{F}}$. By contrast, in the relations proposed in
falkovich2010 ; galtier2011 ; wagner2012 ; banerjee2013 ; kritsuk2013 ;
banerjee2014 ; banerjee2017 ; banerjee2018 ; andres2018 ; andres2019 ;
ferrand2020 ; simon2021 , source terms are present which might not be easily
recast in the form of a flux.

Compared to previous Monin–Yaglom relations there is also a major difference.
Most of these relations have been derived with a practical goal in mind:
estimating the total injection and dissipation rates of energy. The relation
we derive cannot fulfill the same goal. Indeed, it can only provide
information about the injection rate of solenoidal momentum
$\mathscr{E}_{\mathrm{inj}}=\overline{j_{i}f_{i}^{s}}=\overline{s_{i}f_{i}^{s}}$.
Thus, from a practical point of view, the Monin–Yaglom equation Eq. (16)
cannot work as a substitute to the previously derived relations falkovich2010
; galtier2011 ; wagner2012 ; banerjee2013 ; kritsuk2013 ; banerjee2014 ;
banerjee2017 ; banerjee2018 ; andres2018 ; andres2019 ; ferrand2020 ;
simon2021 . Instead, Eq. (16) should be viewed as providing a complementary
information. When coupled with previous relations, it should help disentangle
the solenoidal and dilatational contributions of the dissipation and energy
injection.

###  III.5 Approximate Monin-Yaglom relation in the weakly dilatational regime

In the Monin-Yaglom equation (16), the different contributions of the flux
$\boldsymbol{\mathscr{F}}$ are respectively on the order of:

$\overline{\Delta v_{j}\Delta j_{i}\Delta j_{i}}=\mathcal{O}\left(\delta
v\,\delta(\rho v)\,\delta(\rho v)\right)\hskip 8.67424pt,\hskip
8.67424pt\overline{\Delta
v_{j}^{d}\\{j_{i}j_{i}\\}_{\frac{1}{2}}}=\mathcal{O}\left(\delta
v^{d}\,\delta(\rho^{2}v^{2})\right)\hskip 8.67424pt\text{ and }\hskip
8.67424pt\overline{\Delta{d}_{i}\\{\rho
v_{i}v_{j}\\}_{\frac{1}{2}}}=\mathcal{O}\left(\delta d\,\delta(\rho
v^{2})\right)\hskip 8.67424pt,\hskip 8.67424pt$

where the notation $\delta X(r)$ refers to the order of magnitude of the
increment $\Delta X$. Provided $\delta\rho/\rho\to 0$ when $r\to 0$, one may
approximate $\delta(\rho v)$ by $\rho\delta v$, $\delta(\rho^{2}v^{2})$ by
$\rho^{2}\delta(v^{2})$ and $\delta(\rho v^{2})$ by $\rho\delta(v^{2})$.
Therefore, the two dilatational contributions $\overline{\Delta
v_{j}^{d}\\{j_{i}j_{i}\\}_{\frac{1}{2}}}$ and $\overline{\Delta{d}_{i}\\{\rho
v_{i}v_{j}\\}_{\frac{1}{2}}}$ become negligible compared to the first one
whenever:

$\displaystyle\delta v^{d}\ll\delta v\frac{(\delta
v)^{2}}{\delta(v^{2})}\hskip 8.67424pt\text{ and }\hskip 8.67424pt\delta
d\ll\rho\delta v\frac{(\delta v)^{2}}{\delta(v^{2})}\hskip 2.17021pt.\hskip
8.67424pt$ (21)

These conditions define a particular regime that we call weakly dilatational.
We note that when $\boldsymbol{v}$ is solenoidal, $\delta(v^{2})$ should be on
the order of $(\delta v)^{2}$, as discussed in eyink_cours . Therefore, to
reach the weakly dilatational regime, it could possibly be sufficient to
verify :

$\displaystyle\delta v^{d}\ll\delta v\hskip 8.67424pt\text{ and }\hskip
8.67424pt\delta d\ll\delta j\hskip 2.17021pt.\hskip 8.67424pt$ (22)

An important point is that conditions (21) or (22) only need to be verified at
small scales: large scales can still possess large dilatational components.
Besides, conditions (21) and (22) are verified whenever $\delta v^{d}$ and
$\delta d$ decrease sufficiently fast when $r\to 0$, in particular when they
decrease faster than their total counterparts. In that case, the inertial
range may be divided in two ranges by a limit scale $r_{s}$: below $r_{s}$,
conditions (21) or (22) are satisfied while above $r_{s}$ they are not.

Simulations of subsonic and transsonic turbulence suggest that $\delta
v^{d}\ll\delta v$ on most of the inertial range wang2018 , with $\delta
v^{d}\sim r^{1/2}$ and $\delta v\sim r^{1/3}$. Simulations of supersonic
turbulence federrath2010 ; ferrand2020 ; federrath2021  may be compatible with
the idea that $\delta v^{d}\ll\delta v$ but only below the sonic scale. We are
not aware of simulation results showing the behaviour of the solenoidal and
dilatational components of the momentum.

In any case, whenever the weakly dilatational regime defined by Eq. (21) and
possibly by Eq. (22) is reached, the Monin-Yaglom equation (16) can be
simplified as:

$\displaystyle\partial_{r_{j}}\overline{\Delta v_{j}\Delta j_{i}\Delta
j_{i}}=\partial_{r_{j}}\overline{\Delta v_{j}^{s}\Delta s_{i}\Delta
s_{i}}=-4\mathscr{E}_{\mathrm{inj}}\hskip 2.17021pt.\hskip 8.67424pt$ (23)

This relation possibly applies to subsonic and transsonic turbulence and below
the sonic scale for supersonic turbulence.

Another approximation of Eq. (16) can also be formulated, always based on the
idea that dilatational contributions can be neglected. In aluie2011 ;
aluie2012 , Aluie discarded the correlation between the pressure and velocity
divergence by assuming that their co-spectrum decays as $k^{-\beta}$ with
$\beta>1$. In terms of increments, Aluie made the assumption that
$\overline{\Delta p\Delta\Theta}\propto r^{(\beta-1)/2}$, so that
$\overline{\Delta p\Delta\Theta}\to 0$ when $r\to 0$. The same hypothesis can
also be applied to the version (18) of the present Monin-Yaglom relation. As
in aluie2011 ; aluie2012 , we can assume that the dilatational source terms
$\overline{\Delta\Theta\Delta(j_{i}j_{i})}$ and
$\overline{\Delta\Xi\Delta\pi^{s}}$ go to $0$ when $r\to 0$. In that case, the
Monin-Yaglom relation (18) is approximated by:

$\displaystyle\partial_{r_{j}}\overline{\Delta v_{j}\Delta j_{i}\Delta
j_{i}}=-4\overline{\varepsilon_{c}}\hskip 2.17021pt.\hskip 8.67424pt$ (24)

The two approximations (23) and (24) coincide provided
$\mathscr{E}_{\mathrm{inj}}=\overline{\varepsilon_{c}}$. This occurs when
$\overline{\mathcal{P}_{c}}\ll\overline{\varepsilon_{c}}$. This condition can
be met when dilatational motions are negligible not only at small scales but
also on the whole spectrum and in particular, when the dilatational forcing is
weak. In that case, the circulicity injected at large scales is equal to the
circulicity dissipated at small scales. Equations (23) and (24) are then
indicative of a circulicity cascade.

###  III.6 Inertial range scalings in the weakly dilatational regime

In incompressible turbulence, the classical Monin-Yaglom relation can be used
to determine the scaling of velocity increments frisch1995 . To this end, it
is assumed that velocity increments obey a self-similar scaling such that
$\Delta v_{i}\stackrel{{\scriptstyle\text{law}}}{{\propto}}r^{n_{v}}$. In
particular, the velocity structure function of order $p$ verifies:

$\displaystyle\overline{|\Delta\boldsymbol{v}|^{p}}\propto r^{n_{v}p}\hskip
2.17021pt.\hskip 8.67424pt$ (25)

The incompressible Monin-Yaglom relation then constrains $n_{v}$ to be equal
to $1/3$. It also implies that the prefactor of equation (25) is the
solenoidal dissipation $\overline{\varepsilon}^{1/3}$. These predictions must
of course be corrected to account for internal intermittency when necessary
frisch1995 .

In Sec. III.5, we argued that below a sonic-like scale, under conditions (22),
the Monin–Yaglom relation (16) could be approximated by Eq. (23). In addition
to this approximation, we assume that the increments of the different flow
variables obey a self-similar scaling as in incompressible turbulence. For a
given variable $X$, we write:

$\displaystyle\Delta X\stackrel{{\scriptstyle\text{law}}}{{\propto}}r^{n_{X}}$
(26)

where $n_{X}$ is the power law exponent associated with the variable $X$.
Then, the Monin-Yaglom relation (23) forces the relation:

$n_{v}+2n_{j}={1}\hskip 2.17021pt.\hskip 8.67424pt$

Furthermore, we conjecture that $\Delta\rho\to 0$ as $r\to 0$ and more
specifically that $n_{\rho}\geq n_{v}$. Then, using the relation $\Delta(\rho
X)=\Delta\rho\\{X\\}_{\frac{1}{2}}+\\{\rho\\}_{\frac{1}{2}}\Delta X$, one
deduces that $n_{j}=n_{v}$. Therefore, with this set of assumptions, one
arrives at the conclusion that:

$n_{v}=n_{j}=\frac{1}{3}\hskip 2.17021pt.\hskip 8.67424pt$

This is the usual Kolmogorov-Obukhov scaling of the velocity increment,
corresponding to a $k^{-5/3}$ spectrum. Note that the self-similarity
assumption (26) does not allow to separate the exponents of the different
components of the velocity and momentum fields. Given the weak dilatational
assumption, which is summed by Eq. (22), the common value of $n_{j}$ and
$n_{v}$ is that of the principal components of these fields, it is to say,
their solenoidal components. The dilatational components can have a different
exponent as long as it is larger than $1/3$.

There is also another important conclusion that can be derived from the
approximate Monin-Yaglom relation in the weakly dilatational regime. The sole
dimensioning parameter that appears in this law is
$\overline{\varepsilon_{c}}$, which is a dissipation rate based on solenoidal
fields and on the sole shear viscosity $\mu$. Therefore, one can refine the
similarity assumption for the velocity and momentum fields and write that:

$\displaystyle\Delta
v_{i}\stackrel{{\scriptstyle\text{law}}}{{\propto}}\left((\overline{\varepsilon_{c}}/\overline{\rho}^{2})\,r\right)^{1/3}\hskip
8.67424pt\text{ and }\hskip 8.67424pt\Delta
j_{i}\stackrel{{\scriptstyle\text{law}}}{{\propto}}\left((\overline{\rho}\;\overline{\varepsilon_{c}})\,r\right)^{1/3}$
(27)

This prediction is reminiscent of several simulations where turbulent velocity
spectra with a Kolmogorov-Obukhov scaling are seen to depend on the solenoidal
dissipation rate of energy and not the total one wang2018 ; john2019 .

###  III.7 Inertial range scalings with shocks

When shocks are present in the flow, the self-similarity assumption (26)
cannot be upheld. In particular, Lindborg lindborg2019  showed that when
dissipation occurs exclusively in shocks, the velocity field does not verify
equation (25). Instead, one has:

$\displaystyle\overline{|\Delta\boldsymbol{v}|^{p}}=\langle{|\Delta\boldsymbol{v}|^{p}}\rangle_{\mathtt{sh}}\frac{r}{d_{\mathtt{sh}}}\hskip
8.67424pt,\hskip 8.67424pt$ (28)

where $d_{\mathtt{sh}}$ is the mean distance between shocks and
$\langle{|\Delta\boldsymbol{v}|^{p}}\rangle_{\mathtt{sh}}$ is the average jump
of $|\Delta\boldsymbol{v}|^{p}$ across shocks.

If we want to analyze scalings compatible with Eq. (16) we must consequently
account for these two distinct behaviours. To this end, we propose to separate
the average of two-point correlations in order to make shock contributions
explicit. For a given quantity $X(\boldsymbol{x},\boldsymbol{x}^{\prime})$
depending on the two positions $\boldsymbol{x}$ and
$\boldsymbol{x}^{\prime}=\boldsymbol{x}+\boldsymbol{r}$, we split its average
as:

$\displaystyle\overline{X}(\boldsymbol{r})=P_{\mathtt{sh}}(\boldsymbol{r})\left.\overline{X}\right|_{\mathtt{sh}}(\boldsymbol{r})+(1-P_{\mathtt{sh}}(\boldsymbol{r}))\left.\overline{X}\right|_{\mathtt{ns}}(\boldsymbol{r})\hskip
8.67424pt,\hskip 8.67424pt$ (29)

where $P_{\mathtt{sh}}$ is the probability of finding a shock between two
points separated by $\boldsymbol{r}$ and where
$\left.\overline{X}\right|_{\mathtt{sh}}$ and
$\left.\overline{X}\right|_{\mathtt{ns}}$ are the averages of $X$ conditioned
respectively on the event of finding or not a shock between two points
separated by $\boldsymbol{r}$.

Then, the Monin-Yaglom relation (16) can be written as:

$\displaystyle\partial_{r_{j}}\left(P_{\mathtt{sh}}\left(\left.\overline{\Delta
v_{j}\Delta(\rho v_{i})\Delta(\rho
v_{i})}\right|_{\mathtt{sh}}-2\left.\overline{\Delta
v_{j}\\{\rho^{2}v_{i}v_{i}\\}_{\frac{1}{2}}}\right|_{\mathtt{sh}}-4\left.\overline{\Delta
d_{i}\\{\rho v_{i}v_{j}\\}_{\frac{1}{2}}}\right|_{\mathtt{sh}}\right)\right)$
$\displaystyle+$
$\displaystyle\partial_{r_{j}}\left((1-P_{\mathtt{sh}})\left(\left.\overline{\Delta
v_{j}\Delta(\rho v_{i})\Delta(\rho
v_{i})}\right|_{\mathtt{ns}}-2\left.\overline{\Delta
v_{j}\\{\rho^{2}v_{i}v_{i}\\}_{\frac{1}{2}}}\right|_{\mathtt{ns}}-4\left.\overline{\Delta
d_{i}\\{\rho
v_{i}v_{j}\\}_{\frac{1}{2}}}\right|_{\mathtt{ns}}\right)\right)=-4\mathscr{E}_{\mathrm{inj}}$
(30)

This equation only makes explicit the presence of shocks but does not
introduce any additional assumption compared to the Monin-Yaglom relation (16)
from which it is derived. However, if we want to interpret it further, some
simplifications must be made. First, for any given quantity
$X(\boldsymbol{x},\boldsymbol{x}^{\prime})$ with $\boldsymbol{x}$ and
$\boldsymbol{x}^{\prime}=\boldsymbol{x}+\boldsymbol{r}$, we assume that
$\left.\overline{X}\right|_{\mathtt{sh}}$ is dominated by the shock
contribution. In that case, we can write that:

$\left.\overline{X}\right|_{\mathtt{sh}}(\boldsymbol{r})=\langle{X}\rangle_{\mathtt{sh}}\hskip
8.67424pt,\hskip 8.67424pt$

where $\langle{X}\rangle_{\mathtt{sh}}$ represents the contribution of the
shock jumps to $X$, averaged over all shocks. This contribution is independent
from $\boldsymbol{r}$. Second, we assume that non-shocked contribution to
increments obeys a power law. Thus, we reduce Eq. (26) to the non-shocked
intervals:

$\left.\Delta
X\right|_{\mathtt{ns}}\stackrel{{\scriptstyle\text{law}}}{{\propto}}r^{n_{X}}$

Finally, we consider that the spatial repartition of shocks follows an
isotropic Poisson distribution with a mean distance $d$. As a result, as in
lindborg2019 , one has $P_{\mathtt{sh}}\approx\frac{r}{d}$ for $r\ll d$ and
$1-P_{\mathtt{sh}}\approx 1$. With these approximations, the Monin-Yaglom
relation (III.7) becomes:

$\displaystyle\partial_{r_{j}}\left(\left.\overline{\Delta v_{j}\Delta(\rho
v_{i})\Delta(\rho v_{i})}\right|_{\mathtt{ns}}-2\left.\overline{\Delta
v_{j}\\{\rho^{2}v_{i}v_{i}\\}_{\frac{1}{2}}}\right|_{\mathtt{ns}}-4\left.\overline{\Delta
d_{i}\\{\rho
v_{i}v_{j}\\}_{\frac{1}{2}}}\right|_{\mathtt{ns}}\right)=-4(\mathscr{E}_{\mathrm{inj}}-\mathcal{E}_{\mathtt{sh}})$
(31) with
$\displaystyle\mathcal{E}_{\mathtt{sh}}=-\frac{1}{4d_{\mathtt{sh}}}\left(\langle{\Delta
v_{\parallel}\Delta(\rho v_{i})\Delta(\rho
v_{i})}\rangle_{\mathtt{sh}}-2\langle{\Delta
v_{\parallel}\\{\rho^{2}v_{i}v_{i}\\}_{\frac{1}{2}}}\rangle_{\mathtt{sh}}-4\langle{\Delta
d_{i}\\{\rho v_{i}v_{\parallel}\\}_{\frac{1}{2}}}\rangle_{\mathtt{sh}}\right)$

If we except the particular case
$\mathscr{E}_{\mathrm{inj}}=\mathcal{E}_{\mathtt{sh}}$, this formula can only
be fulfilled if at least one of the terms in the divergence on the left-hand
side scales as $r$. Given the scaling laws assumed in the non-shocked region,
this condition implies that at least one of this condition must be verified:

$\displaystyle n_{v}+2n_{\rho v}\leq 1\hskip 8.67424pt,\hskip
8.67424ptn_{v}+n_{\rho^{2}v^{2}}\leq 1\hskip 8.67424pt,\hskip
8.67424ptn_{d}+n_{\rho v^{2}}\leq 1\hskip 2.17021pt.\hskip 8.67424pt$ (32)

We note that, since $\overline{\Delta(\rho v_{i})\Delta(\rho
v_{i})}=\overline{\Delta s_{i}\Delta s_{i}}+\overline{\Delta d_{i}\Delta
d_{i}}$, one must have $n_{d}\geq n_{\rho v}$. Furthermore, if we assume that
$\Delta\rho\to 0$ as $r\to 0$ and more specifically that $n_{\rho}\geq n_{v}$,
then, using the relation $\Delta(\rho
X)=\Delta\rho\\{X\\}_{\frac{1}{2}}+\\{\rho\\}_{\frac{1}{2}}\Delta X$, one can
show that $n_{\rho v}=n_{v}$, $n_{\rho^{2}v^{2}}=n_{v^{2}}$ and $n_{\rho
v^{2}}=n_{v^{2}}$. Finally, the Schwartz relation
$\overline{\Delta(v_{i}v_{i})^{2}}\leq
2\sqrt{\overline{(\\{v_{j}\\}_{\frac{1}{2}}\\{v_{j}\\}_{\frac{1}{2}})^{2}}}\sqrt{\overline{(\Delta
v_{i}\Delta v_{i})^{2}}}$ implies that $n_{v^{2}}\geq n_{v}$.

From there, we finally arrive at the conclusion that the first inequality of
equation (32) can be verified if $n_{v}\leq 1/3$, while the two others require
$n_{v}\leq 1/2$. This can be summed up as:

$\displaystyle\text{Non-vanishing }\left.\overline{\Delta v_{j}\Delta(\rho
v_{i})\Delta(\rho v_{i})}\right|_{\mathtt{ns}}~{}~{}~{}~{}~{}$
$\displaystyle:~{}~{}~{}~{}n_{v}\leq\frac{1}{3}\hskip 8.67424pt,\hskip
8.67424pt$ (33a) $\displaystyle\text{Non-vanishing }-2\left.\overline{\Delta
v_{j}\\{\rho^{2}v_{i}v_{i}\\}_{\frac{1}{2}}}\right|_{\mathtt{ns}}+4\left.\overline{\Delta
d_{i}\\{\rho v_{i}v_{j}\\}_{\frac{1}{2}}}\right|_{\mathtt{ns}}~{}~{}~{}~{}$
$\displaystyle:~{}~{}~{}~{}n_{v}\leq\frac{1}{2}\hskip 2.17021pt.\hskip
8.67424pt$ (33b)

The upper bound of the first constraint corresponds to the usual Kolmogorov-
Obukhov scaling of the velocity increment, corresponding to a $k^{-5/3}$
spectrum. The upper bound of the second corresponds to a velocity spectrum
scaling as $k^{-2}$. The first scaling is traditionally observed in
simulations with moderate Mach numbers and with solenoidal forcing wang2018 .
By contrast, in simulations of supersonic turbulence, a $k^{-2}$ spectrum is
obtained for scales larger than the sonic scale. Below this scale, a shallower
spectrum with a scaling close to $k^{-5/3}$ is observed ferrand2020 ;
federrath2021 . We recall that these scalings are derived for the non-shocked
regions of the flow. Given the decomposition $\overline{\Delta v_{i}\Delta
v_{i}}=P_{\mathtt{sh}}\left.\overline{\Delta v_{i}\Delta
v_{i}}\right|_{\mathtt{sh}}+(1-P_{\mathtt{sh}})\left.\overline{\Delta
v_{i}\Delta v_{i}}\right|_{\mathtt{ns}}$, we note that the $k^{-2}$ spectrum
observed in simulations of supersonic turbulence can have two contributions.
The first comes from the presence of shocks, as in Burgers turbulence. The
other comes from the dilatational components of the velocity and momentum,
shock excluded. The role of this second contribution was also highlighted in
ferrand2020 .

To conclude on this topic, we would like to recall that without solenoidal
forcing, when $\mathscr{E}_{\mathrm{inj}}=\overline{j_{i}f_{i}^{s}}=0$, an
equilibrium takes place in the inertial range. Equation (31) shows that this
equilibrium can be understood as a balance between the shocked and non-shocked
regions of the flow. In particular, if $\mathcal{E}_{\mathtt{sh}}>0$, shocks
lead to a forward transfer of circulicity from large to small scales, while a
backward transfer occurs for non-shocked regions.

##  IV Coarse-graining approach

In this section, we return to discussing the existence of a circulicity
cascade. However, instead of the point-splitting methodology which led us to
derive the Monin-Yaglom equation (16), we now use the coarse-graining approach
which was initially propelled by Onsager’s work onsager1949 ; eyink2006 . This
approach does not require the flow to be homogeneous, and unless said
otherwise, we drop this hypothesis. As a whole, we closely follow the works of
Eyink & Drivas and Aluie aluie2011 ; aluie2012 ; aluie2013 ; eyink2018 .

###  IV.1 Resolved and subscale circulicities

The basic tool of the coarse-graining approach is spatial filtering. For a
given variable $X(\boldsymbol{x},t)$, one defines its filtered value by:

$\displaystyle{\left\langle{X}\right\rangle_{\ell}}(\boldsymbol{x},t)=\int
G_{\ell}(\boldsymbol{r})X(\boldsymbol{x}+\boldsymbol{r},t)\mathrm{d}\boldsymbol{r}\hskip
8.67424pt,\hskip 8.67424pt$ (34)

where $G_{\ell}(\boldsymbol{r})=G(\boldsymbol{r}/\ell)\ell^{-3}$ is a filter
function with a kernel $G$ rapidly decreasing at infinity and verifying the
normalization condition $\int G(\boldsymbol{r})\mathrm{d}\boldsymbol{r}=1$.
The idea is then to look at the properties of the filtered circulicity
${\left\langle{\mathcal{C}}\right\rangle_{\ell}}$. More precisely,
${\left\langle{\mathcal{C}}\right\rangle_{\ell}}$ can be split into a resolved
and subscale circulicity:

${\left\langle{\mathcal{C}}\right\rangle_{\ell}}={\left.{\mathcal{C}}\right|_{\ell}^{\mathrm{res}}}+{\left.{\mathcal{C}}\right|_{\ell}^{\mathrm{sub}}}\hskip
8.67424pt\text{ with }\hskip
8.67424pt{\left.{\mathcal{C}}\right|_{\ell}^{\mathrm{res}}}=\frac{1}{2}{\left\langle{s_{i}}\right\rangle_{\ell}}{\left\langle{s_{i}}\right\rangle_{\ell}}\hskip
8.67424pt\text{ and }\hskip
8.67424pt{\left.{\mathcal{C}}\right|_{\ell}^{\mathrm{sub}}}=\frac{1}{2}\tau(s_{i},s_{i})\hskip
8.67424pt,\hskip 8.67424pt$

where we used the notation $\tau(X,Y)$ to denote the subscale moment of given
quantities $X$ and $Y$:

$\tau(X,Y)={\left\langle{XY}\right\rangle_{\ell}}-{\left\langle{X}\right\rangle_{\ell}}{\left\langle{Y}\right\rangle_{\ell}}\hskip
2.17021pt.\hskip 8.67424pt$

The evolution of the resolved circulicity
${\left.{\mathcal{C}}\right|_{\ell}^{\mathrm{res}}}$ can be deduced by
filtering the evolution equation (9a) of $\boldsymbol{s}$ and multiplying it
by ${\left\langle{\boldsymbol{s}}\right\rangle_{\ell}}$. As for the subscale
circulicity, its evolution can be derived by filtering the circulicity
equation (10) and substracting the resolved part. By doing so, we obtain that:

$\displaystyle\partial_{t}{\left.{\mathcal{C}}\right|_{\ell}^{\mathrm{res}}}+\partial_{j}{\left.{\mathcal{F}_{j}^{c}}\right|_{\ell}^{\mathrm{res}}}=$
$\displaystyle-\Pi_{\ell}-\pi^{f\mu}_{\ell}+{\left.{\mathcal{P}_{c}}\right|_{\ell}^{\mathrm{res}}}+{\left\langle{s_{i}f_{i}^{s}}\right\rangle_{\ell}}\hskip
8.67424pt,\hskip 8.67424pt$ (35a)
$\displaystyle\partial_{t}{\left.{\mathcal{C}}\right|_{\ell}^{\mathrm{sub}}}+\partial_{j}{\left.{\mathcal{F}_{j}^{c}}\right|_{\ell}^{\mathrm{sub}}}=$
$\displaystyle\phantom{-}~{}\Pi_{\ell}+\pi^{f\mu}_{\ell}+{\left.{\mathcal{P}_{c}}\right|_{\ell}^{\mathrm{sub}}}-{\left\langle{\varepsilon_{c}}\right\rangle_{\ell}}\hskip
2.17021pt.\hskip 8.67424pt$ (35b)

In these equations,
${\left.{\boldsymbol{\mathcal{F}^{c}}}\right|_{\ell}^{\mathrm{res}}}$ and
${\left.{\boldsymbol{\mathcal{F}^{c}}}\right|_{\ell}^{\mathrm{sub}}}$ are the
resolved and subscale circulicity fluxes. We recall that the local circulicity
flux appearing in Eq. (10) is noted $\boldsymbol{\mathcal{F}^{c}}$. If we mark
symbolically the dependency of this flux upon the flow variables as
$\boldsymbol{\mathcal{F}^{c}}\equiv\boldsymbol{\mathcal{F}^{c}}[\boldsymbol{X}]$,
then we have :
${\left.{\mathcal{F}_{j}^{c}}\right|_{\ell}^{\mathrm{res}}}=\mathcal{F}^{c}_{j}[{\left\langle{\boldsymbol{X}}\right\rangle_{\ell}}]+{\left\langle{j_{i}}\right\rangle_{\ell}}\tau(j_{i},v_{j})$
and
${\left.{\mathcal{F}_{j}^{c}}\right|_{\ell}^{\mathrm{sub}}}={\left\langle{\mathcal{F}^{c}_{j}[{\boldsymbol{X}}]}\right\rangle_{\ell}}-\mathcal{F}^{c}_{j}[{\left\langle{\boldsymbol{X}}\right\rangle_{\ell}}]-{\left\langle{j_{i}}\right\rangle_{\ell}}\tau(j_{i},v_{j})$.
These two fluxes do not play any role in the forthcoming analysis. The term
$\pi_{\ell}^{f\mu}$ combines a filtered scale dissipation and the subscale
contribution of the force $\boldsymbol{f}^{s}$. It is defined as
$\pi_{\ell}^{f\mu}=\tau(s_{i},fi^{s})+\mu{\left\langle{\Omega_{i}}\right\rangle_{\ell}}{\left\langle{\omega_{i}}\right\rangle_{\ell}}$.
Like the fluxes, the role of $\pi_{\ell}^{f\mu}$ is inessential since it
vanishes in the high Reynolds limit and with the assumption that the forcing
is at large scale. The last three terms that need to be defined,
${\left.{\mathcal{P}_{c}}\right|_{\ell}^{\mathrm{res}}}$,
${\left.{\mathcal{P}_{c}}\right|_{\ell}^{\mathrm{sub}}}$ and $\Pi_{\ell}$, are
the ones playing a central role in the present analysis. The first two
correspond to the resolved and subscale contributions of the circulicity
production term $\mathcal{P}_{c}$. More precisely, we have:

$\displaystyle{\left\langle{\mathcal{P}_{c}}\right\rangle_{\ell}}={\left.{\mathcal{P}_{c}}\right|_{\ell}^{\mathrm{res}}}+{\left.{\mathcal{P}_{c}}\right|_{\ell}^{\mathrm{sub}}}$
(36a) with
$\displaystyle{\left.{\mathcal{P}_{c}}\right|_{\ell}^{\mathrm{res}}}=-\frac{1}{2}|{\left\langle{j_{i}}\right\rangle_{\ell}}{\left\langle{j_{i}}\right\rangle_{\ell}}|^{2}{\left\langle{\Theta}\right\rangle_{\ell}}+{\left\langle{\pi^{s}}\right\rangle_{\ell}}{\left\langle{\Xi}\right\rangle_{\ell}}$
(36b) and
$\displaystyle{\left.{\mathcal{P}_{c}}\right|_{\ell}^{\mathrm{sub}}}=-\frac{1}{2}\left(\tau(j_{i}j_{i},\Theta)+{\left\langle{\Theta}\right\rangle_{\ell}}\tau(j_{i},j_{i})\right)+\tau(\pi^{s},\Xi)\hskip
2.17021pt.\hskip 8.67424pt$ (36c)

The third term is defined by:

$\displaystyle\Pi_{\ell}=-\tau(v_{j},j_{i})\partial_{j}{\left\langle{j_{i}}\right\rangle_{\ell}}\hskip
2.17021pt.\hskip 8.67424pt$ (37)

This term consists in a subscale-stress multiplying the gradient of a filtered
quantity and marks the interaction between resolved and unresolved scales. It
appears as a sink in the resolved circulicity equation (35a) and as a source
in the subscale circulicity equation (35b). Thus, $\Pi_{\ell}$ can be
interpreted as a non-linear exchange term between the resolved and subscale
circulicities.

Note that if we introduce a density-weighted filter and density-weighted
subscale moments as ${\big{.}\widetilde{X}\big{|}_{\ell}}={\left\langle{\rho
X}\right\rangle_{\ell}}/{\left\langle{\rho}\right\rangle_{\ell}}$ and
$\tilde{\tau}(X,Y)={\big{.}\widetilde{XY}\big{|}_{\ell}}-{\big{.}\widetilde{X}\big{|}_{\ell}}{\big{.}\widetilde{Y}\big{|}_{\ell}}$,
then the exchange term $\Pi_{\ell}$ can also be written as:

$\displaystyle\Pi_{\ell}=-{\left\langle{\rho}\right\rangle_{\ell}}^{2}\tilde{\tau}(v_{i},v_{j})\,\partial_{j}{\big{.}\widetilde{v_{i}}\big{|}_{\ell}}-\tau(\rho,v_{j})\partial_{j}\big{(}{\left\langle{\rho}\right\rangle_{\ell}}\,{\big{.}\widetilde{v_{i}}\big{|}_{\ell}}{\big{.}\widetilde{v_{i}}\big{|}_{\ell}}\big{)}-\tau(v_{j},j_{i}){\big{.}\widetilde{v_{i}}\big{|}_{\ell}}\partial_{j}{\left\langle{\rho}\right\rangle_{\ell}}\hskip
2.17021pt.\hskip 8.67424pt$ (38)

Up to a density factor, the first term on the right-hand side is the
deformation work that plays a central role in the transfer of kinetic energy
between scales aluie2011 ; aluie2012 ; aluie2013 ; eyink2018 . Besides, the
second term is similar to the “baropycnal work” which also appears as a means
to transfer kinetic energy. However, the actual pressure $p$ which enters the
definition of the “baropycnal work” is here replaced with a dynamic one:
${\left\langle{\rho}\right\rangle_{\ell}}{\big{.}\widetilde{v_{i}}\big{|}_{\ell}}{\big{.}\widetilde{v_{i}}\big{|}_{\ell}}$.
Finally, the last term has no direct equivalent in the transfer of kinetic
energy. This comparison between $\Pi_{\ell}$ and the transfer of kinetic
energy reinforces the idea that $\Pi_{\ell}$ is an exchange term between
resolved and unresolved scales. Furthermore, it indicates that some of the
mechanisms at work in the transfer of energy and circulicity are similar.

To follow on this comparison, we note that the exchange term between the
resolved and subscale kinetic energies has been a staple of the analysis of
subgrid-scale models for large-eddy simulations (LES) for more than five
decades (see sagaut2006 ; garnier2009  and references therein). Its link with
the Kolmogorov-Obukhov spectrum and with the Monin-Yaglom relation has long
been discussed sagaut2006 ; garnier2009 . However, its use as a proper
theoretical tool able to demonstrate the existence of a cascading process
seems to have been only recognised more recently, most notably with Refs.
aluie2011 ; aluie2012 ; aluie2013 ; eyink2018 .

###  IV.2 Circulicity cascade

We now consider the high Reynolds limit of the resolved and subscale
circulicities, with the added assumption that the force only acts at large
scales. Taking the limit $\mu\to 0$ in Eqs. (35a) and (35b), and discarding
the subscale force contribution, we obtain that:

$\displaystyle\partial_{t}{\left.{\mathcal{C}}\right|_{\ell}^{\mathrm{res}}}+\partial_{j}{\left.{\mathcal{F}_{j}}\right|_{\ell}^{\mathrm{res}}}=$
$\displaystyle-\Pi_{\ell}^{*}-{\left.{\mathcal{P}_{c}}\right|_{\ell}^{\mathrm{sub}}}^{*}+{\left\langle{\mathcal{P}_{c}^{*}}\right\rangle_{\ell}}+{\left\langle{s_{i}f_{i}^{s}}\right\rangle_{\ell}}\hskip
8.67424pt,\hskip 8.67424pt$ (39a)
$\displaystyle\partial_{t}{\left.{\mathcal{C}}\right|_{\ell}^{\mathrm{sub}}}+\partial_{j}{\left.{\mathcal{F}_{j}}\right|_{\ell}^{\mathrm{sub}}}=$
$\displaystyle\phantom{-}~{}\Pi_{\ell}^{*}+{\left.{\mathcal{P}_{c}}\right|_{\ell}^{\mathrm{sub}}}^{*}-{\left\langle{\varepsilon_{c}^{*}}\right\rangle_{\ell}}\hskip
2.17021pt.\hskip 8.67424pt$ (39b)

where we substituted ${\left.{\mathcal{P}_{c}}\right|_{\ell}^{\mathrm{res}}}$
with
${\left\langle{\mathcal{P}_{c}}\right\rangle_{\ell}}-{\left.{\mathcal{P}_{c}}\right|_{\ell}^{\mathrm{sub}}}$
and where:

$\Pi_{\ell}^{*}=\lim_{\mu\to 0}\Pi_{\ell}\hskip 8.67424pt,\hskip
8.67424pt\varepsilon_{c}^{*}=\lim_{\mu\to 0}\varepsilon_{c}\hskip
8.67424pt,\hskip
8.67424pt{\left.{\mathcal{P}_{c}}\right|_{\ell}^{\mathrm{sub}}}^{*}=\lim_{\mu\to
0}{\left.{\mathcal{P}_{c}}\right|_{\ell}^{\mathrm{sub}}}\hskip 8.67424pt\text{
and }\hskip 8.67424pt\mathcal{P}_{c}^{*}=\lim_{\mu\to 0}\mathcal{P}_{c}\hskip
2.17021pt.\hskip 8.67424pt$

These limits are taken in a weak distributional sense. For instance, one has:

$\mathcal{P}_{c}^{*}=-\frac{1}{2}|\boldsymbol{j}|^{2}*\Theta+\pi^{s}*\Xi\hskip
8.67424pt,\hskip 8.67424pt$

where the operator $*$ stands for the distributional product in the limit
$\mu\to 0$. The reason for introducing this distributional meaning is
explained in eyink2018 . It comes from the fact that $\rho$ and
$\boldsymbol{v}$ become non-smooth when $\mu\to 0$. Hence,
$\Theta=\partial_{j}u_{j}$ and $\Xi=\partial_{j}(\rho u_{j})$ exhibit
divergences that can only be described by distributions. As a result, products
such as $|\boldsymbol{j}|^{2}\Theta$ and $\pi^{s}\Xi$ behave as the products
of Heaviside functions with Dirac distributions. Their values are ill-posed in
the sense that they depend on the particular regularizing path chosen to
describe them dalmaso1995 ; pares2006 . This regularizing path does not need
to be made explicit for our purpose. It is sufficient to know that it depends
on the physics of the problem and on the particular way the limit $\mu\to 0$
is approached.

To discuss the existence of a circulicity cascade, we make two crucial
hypotheses, similar to those made in eyink2018 . The first one is that
dissipative anomalies exist and that the dissipation of circulicity
$\varepsilon_{c}$ has a non-zero finite value in the infinite Reynolds limit:

$\displaystyle\varepsilon_{c}^{*}\neq 0\hskip 2.17021pt.\hskip 8.67424pt$ (40)

The second is that in the limit $\ell\to 0$, the evolution equation (39a) of
${\left.{\mathcal{C}}\right|_{\ell}^{\mathrm{res}}}$ tends, in the sense of
distributions, to the evolution equation of $\mathcal{C}$ in the ideal limit
$\mu\to 0$. The limit of Eq. (39a) can be written as follows:

$\displaystyle\partial_{t}\mathcal{C}+\partial_{j}\mathcal{F}_{j}^{c}=$
$\displaystyle\mathcal{P}_{c}^{*}+s_{i}f_{i}^{s}-\Pi_{0}^{*}-{\left.{\mathcal{P}_{c}}\right|_{0}^{\mathrm{sub}}}^{*}\hskip
8.67424pt,\hskip 8.67424pt$ (41)

with $\Pi_{0}^{*}=\lim_{\ell\to 0}\Pi_{\ell}^{*}$ and
${\left.{\mathcal{P}_{c}}\right|_{0}^{\mathrm{sub}}}^{*}=\lim_{\ell\to
0}{\left.{\mathcal{P}_{c}}\right|_{\ell}^{\mathrm{sub}}}^{*}$. The evolution
of $\mathcal{C}$ in the ideal limit is deduced from Eq. (10) and can be
written as:

$\displaystyle\partial_{t}\mathcal{C}+\partial_{j}\mathcal{F}_{j}^{c}=\mathcal{P}_{c}^{*}+s_{i}f_{i}^{s}-\varepsilon_{c}^{*}\hskip
2.17021pt.\hskip 8.67424pt$ (42)

The reason for the identity between these two limit equations is discussed in
eyink2018  (p. 10, second paragraph) and comes for the principle that
“objective physical facts such as the rate of decay of energy […] cannot
depend upon an arbitrary scale $\ell$”. One may expect that this “objectivity
principle” also applies to circulicity, its dissipation and injection rates.
Comparing Eqs. (42) and (41), one deduces that:

$\displaystyle\Pi_{0}^{*}+{\left.{\mathcal{P}_{c}}\right|_{0}^{\mathrm{sub}}}^{*}=\lim_{\ell\to
0}\lim_{\mu\to
0}\Pi_{\ell}+{\left.{\mathcal{P}_{c}}\right|_{\ell}^{\mathrm{sub}}}=\varepsilon_{c}^{*}\hskip
2.17021pt.\hskip 8.67424pt$ (43)

This equality is the main result of this section. It states that the
circulicity transferred non-linearly from resolved to subscales
($\Pi_{0}^{*}$) summed to the one produced at subscales
(${\left.{\mathcal{P}_{c}}\right|_{0}^{\mathrm{sub}}}^{*}$) does not vanish in
the limit $\mu\to 0,\ell\to 0$ : it tends to a non-zero finite value
independent from $\ell$ and equal, in the sense of distributions, to the
dissipation $\varepsilon_{c}^{*}$. Equation (43) does not assert the existence
of a circulicity cascade. Instead, it expresses a balance between transfer,
production and dissipation of circulicity in the subscale range. Since
$\varepsilon_{c}^{*}\neq 0$, this balance is anomalous and both the transfer
and subscale production of circulicity may be the source of the anomalous
dissipation $\varepsilon_{c}^{*}$. A similar result has already been obtained
for the kinetic energy: in eyink2018 , it was shown that the anomalous
dissipation of energy could have two origins, an energy cascade and a
pressure-dilatation defect akin to a subscale energy production. Both elements
are similar to the mechanisms identified in this work for circulicity.

Equation (43) has been cast in the form of a subscale balance. It can also be
used to express a balance for resolved scales. Indeed, using Eq. (36), one can
show that:

$\displaystyle{\left.{\mathcal{P}_{c}}\right|_{0}^{\mathrm{res}}}^{*}-\Pi_{0}^{*}=\lim_{\ell\to
0}\lim_{\mu\to
0}{\left.{\mathcal{P}_{c}}\right|_{\ell}^{\mathrm{res}}}-\Pi_{\ell}=\mathcal{P}_{c}^{*}-\varepsilon_{c}^{*}$
(44)

This equation shows that, in the limit $\mu\to 0$, $\ell\to 0$, the
circulicity transferred to small scales and produced at resolved scaled tends
to a finite value equal to the difference between the production and
dissipation of circulicity.

Equations (43) and (44) can be understood as the coarse-grained equivalents of
the Monin-Yaglom relations (18) and (16). However, as opposed to a Monin-
Yaglom relation, Eqs. (43) and (44) are not statistical and do not involve an
ensemble mean or, if ergodicity applies, an integration over a spatial domain.
Instead, they are written in a weak distributional sense. In this regard, it
is worth stressing that equation (43) does not imply that
$\Pi_{0}^{*}+{\left.{\mathcal{P}_{c}}\right|_{0}^{\mathrm{sub}}}^{*}=\varepsilon_{c}^{*}$
locally and instantly. As a weak equality, it involves spatial integrations
with test functions. When seen under this light, equality (43) has a meaning
which is indeed stronger that the one given to a Monin-Yaglom relation, but,
at the same time, which is maybe not so far removed from it. In the same line
of thought, it should also be mentioned that in a real flow, with finite $\mu$
and $\ell$, the measurable quantities which give rise to equality (43) are
$\Pi_{\ell}$, ${\left.{\mathcal{P}_{c}}\right|_{\ell}^{\mathrm{sub}}}$ and
${\left\langle{\varepsilon_{c}}\right\rangle_{\ell}}$ which appear in the
right-hand side of Eq. (35b). The left-hand side of this equation has no
reason to be equal to $0$ so that the values of
$\Pi_{\ell}+{\left.{\mathcal{P}_{c}}\right|_{\ell}^{\mathrm{sub}}}$ and
${\left\langle{\varepsilon_{c}}\right\rangle_{\ell}}$ have no reason to be
equal locally and instantaneously. Again, this is not what equation (43) is
meant to say. The weak distributional sense of this equation should always be
kept in mind in order to avoid over-interpreting it. The same also applies to
Eq. (44).

Another remark worth making about equalities (43) and (44) comes from the fact
that a double limit is taken, first $\mu\to 0$ and then $\ell\to 0$. In doing
so, Eyink & Drivas eyink2018  noted that several distributional products
appear. For instance, the contribution $\tau(\pi^{s},\Xi)$ of
${\left.{\mathcal{P}_{c}}\right|_{\ell}^{\mathrm{sub}}}$ is equal to
${\left\langle{\pi\Xi}\right\rangle_{\ell}}-{\left\langle{\pi}\right\rangle_{\ell}}{\left\langle{\Xi}\right\rangle_{\ell}}$.
Taking the double limit, then leads to write:

$\lim_{\ell\to 0}\lim_{\mu\to
0}\tau(\pi^{s},\Xi)=\pi^{s}*\Xi-\pi^{s}\circ\Xi\hskip 2.17021pt.\hskip
8.67424pt$

In this equality, the symbol $\circ$ refers to the regularization of
${\left\langle{\pi}\right\rangle_{\ell}}{\left\langle{\Xi}\right\rangle_{\ell}}$
which is different from the one associated with the symbol $*$ and the
regularization of ${\left\langle{\pi\Xi}\right\rangle_{\ell}}$.

To conclude this discussion on Eqs. (43) and (44), we would like to mention
that a purely statistical version of these equalities can also be derived,
without invoking the finiteness of $\varepsilon_{c}^{*}$ and an “objectivity
principle”. Instead of these assumptions, one may assume that the flow is
homogeneous and statistically stationary, and that the mean of
$\varepsilon_{c}^{*}$ is finite, i.e. that $\lim_{\mu\to
0}\overline{\varepsilon_{c}}\neq 0$, which is a weaker statement than
$\varepsilon_{c}^{*}\neq 0$. Then, by averaging Eqs. (35b) and (35a), one
deduces that, in the limit $\mu\to 0$ and $\ell\to 0$, the following equality
stands:

$\displaystyle\lim_{\ell\to 0}\lim_{\mu\to
0}\overline{\Pi_{\ell}}+\overline{{\left.{\mathcal{P}_{c}}\right|_{\ell}^{\mathrm{sub}}}}=\overline{\varepsilon_{c}}\hskip
8.67424pt\text{ and }\hskip 8.67424pt\lim_{\ell\to 0}\lim_{\mu\to
0}\overline{\Pi_{\ell}}-\overline{{\left.{\mathcal{P}_{c}}\right|_{\ell}^{\mathrm{res}}}}=\overline{\varepsilon_{c}}-\overline{\mathcal{P}_{c}}=\mathscr{E}_{\mathrm{inj}}\hskip
2.17021pt.\hskip 8.67424pt$ (45)

These two relations can be obtained by averaging Eqs. (43) and (44). They can
also be derived by filtering the Monin-Yaglom relations (18) and (16).
Conditions under which
$\overline{{\left.{\mathcal{P}_{c}}\right|_{\ell}^{\mathrm{sub}}}}$ and
$\overline{{\left.{\mathcal{P}_{c}}\right|_{\ell}^{\mathrm{res}}}}$ could be
neglected compared to $\overline{\Pi_{\ell}}$ were given in Sec. III.5. They
are met in a so-called weakly dilatational regime. In that case, the mean
transfer of circulicity $\overline{\Pi_{\ell}}$ becomes independent of $\ell$
and tends to a finite non-zero value : a circulicity cascade takes place.
Otherwise, a mean transfer-production-dissipation equilibrium occurs in the
inertial range in lieu of this sole cascade. Still, in Sec. III.3, it was
shown that this mean equilibrium could be interpreted as an effective cascade
by recasting the source term in the form of a flux-like transfer term.

###  IV.3 Increment scalings and locality

There are two ways by which the anomalous dissipation $\varepsilon_{c}^{*}$
can be sustained : one is the transfer term $\Pi_{\ell}$, the other is the
subscale production ${\left.{\mathcal{P}_{c}}\right|_{\ell}^{\mathrm{sub}}}$.
The orders of magnitude of these terms are linked to the space increments of
the velocity, density and momentum fields. The order of magnitude of the
increment $\Delta X(\boldsymbol{x},\boldsymbol{r})$ of a quantity $X$ is noted
$\delta X(r)$. As discussed in aluie2011 ; eyink2018 ; eyink_cours , one has:

$\tau(X,Y)=\mathcal{O}(\delta X\delta Y)\hskip 8.67424pt,\hskip
8.67424pt\partial_{i}{\left\langle{X}\right\rangle_{\ell}}=\mathcal{O}\left(\frac{\delta
X}{\ell}\right)\hskip 8.67424pt,\hskip
8.67424ptX-{\left\langle{X}\right\rangle_{\ell}}=\mathcal{O}(\delta X)\hskip
2.17021pt.\hskip 8.67424pt$

Using these relations, it is straightforward to show from Eq. (37) that:

$\Pi_{\ell}=\mathcal{O}\left(\frac{\delta v(\delta j)^{2}}{\ell}\right)\hskip
2.17021pt.\hskip 8.67424pt$

One may also use Eq. (38) and the developments of Aluie aluie2011  to show
that this expression can be simplified as:

$\displaystyle\Pi_{\ell}=\mathcal{O}\left(\frac{(\delta
v)^{3}}{\ell}\right)\rho^{2}\left[1+\mathcal{O}\left(\frac{\delta\rho}{\rho}\right)+\mathcal{O}\left(\frac{\delta\rho^{2}}{\rho^{2}}\right)\right]\hskip
2.17021pt.\hskip 8.67424pt$ (46)

From this estimate, one deduces that the transfer term does not vanish in the
limit $\ell\to 0$ provided the velocity field is rough enough. More precisely,
let us assume that the order of magnitude $\delta X$ of the increment a
quantity $X$ obeys a power law:

$|\delta X(\ell)^{n}|\propto\ell^{\zeta_{n}^{X}}\hskip 2.17021pt.\hskip
8.67424pt$

Then, if $\zeta_{n}^{\rho}>0$, i.e. if $\delta\rho/\rho\to 0$ when $\ell\to
0$, Eq. (46) implies that $\Pi_{\ell}$ is non-vanishing provided:

$\displaystyle\zeta_{n}^{v}\leq\frac{n}{3}\hskip 8.67424pt\text{ for }\hskip
2.17021ptn\geq 3\hskip 2.17021pt.\hskip 8.67424pt$ (47)

This is a necessary condition for $\Pi_{\ell}$ to contribute to the anomalous
dissipation $\varepsilon_{c}^{*}$. This condition is identical to the one
derived in eyink2018  for the energy cascade. This is not entirely unexpected:
the resolved to unresolved exchange terms of circulicity and energy both
involve the deformation work as a common mechanism.

We now turn our attention to the subscale circulicity production
${\left.{\mathcal{P}_{c}}\right|_{\ell}^{\mathrm{sub}}}$ defined in Eq. (36).
This term has three components involving
$\Theta=\partial_{j}v_{j}=\partial_{j}v_{j}^{d}$ and
$\Xi=\partial_{j}j_{j}=\partial_{j}d_{j}$. By comparing the right- and left-
hand sides of the relation
$\partial_{i}{\left\langle{\Theta}\right\rangle_{\ell}}=\partial^{2}_{ij}{\left\langle{v_{j}^{d}}\right\rangle_{\ell}}$,
one deduces that $\delta\Theta/\ell\sim\delta v^{d}/\ell^{2}$ so that
$\delta\Theta\sim\delta v^{d}/\ell$. Similarly, one derives that
$\delta\Xi\sim\delta d/\ell$. Therefore, the three contributions of
${\left.{\mathcal{P}_{c}}\right|_{\ell}^{\mathrm{sub}}}$ lead to the following
order of magnitude:

${\left.{\mathcal{P}_{c}}\right|_{\ell}^{\mathrm{sub}}}=\mathcal{O}\left(\frac{\delta
v^{d}\delta(j^{2})}{\ell}\hskip 8.67424pt,\hskip 8.67424pt\frac{\delta
v^{d}(\delta j)^{2}}{\ell}\hskip 8.67424pt,\hskip
8.67424pt\frac{\delta\pi^{s}\delta d}{\ell}\right)\hskip 2.17021pt.\hskip
8.67424pt$

The Poisson equation for the pressure (9c) also leads to
$\delta\pi^{s}\sim\delta(vj)$. Thus, the conditions for the production to be
non-vanishing are:

$\zeta_{n}^{v^{d}}+\zeta_{n}^{j^{2}}\leq 1\hskip 8.67424pt,\hskip
8.67424pt\zeta_{n}^{v^{d}}+2\zeta_{n}^{j}\leq 1\hskip 8.67424pt,\hskip
8.67424pt\zeta_{n}^{d}+\zeta_{n}^{vj}\leq 1\hskip 8.67424pt\text{ for }\hskip
2.17021ptn\geq 3\hskip 2.17021pt.\hskip 8.67424pt$

If the dilatational fields $\boldsymbol{v}^{d}$ and $\boldsymbol{d}$ decay
sufficiently rapidly then these conditions are not fulfilled. In that case,
the transfer term $\Pi_{\ell}$ is the only one contributing to the anomalous
dissipation. Thus, sufficient conditions for the existence of a circulicity
cascade are:

$\displaystyle\zeta_{n}^{v^{d}}>1-\min(2\zeta_{n}^{j},\zeta_{n}^{j^{2}})\hskip
8.67424pt\text{ and }\hskip 8.67424pt\zeta_{n}^{d}>1-\zeta_{n}^{vj}\hskip
2.17021pt.\hskip 8.67424pt$ (48)

Other conditions may also lead to the fact that
${\left.{\mathcal{P}_{c}}\right|_{\ell}^{\mathrm{sub}}}$ vanishes in the limit
$\mu\to 0$, $\ell\to 0$. For instance, $\boldsymbol{d}$ and
$\boldsymbol{v}^{d}$ may simply be quantities belonging to the dissipative
range that vanish when $\mu\to 0$. This is the case for $\boldsymbol{v}^{d}$
in the pseudo-compressible limit studied in sandoval1995 ; livescu2008 ;
soulard2020 .

To conclude, the locality of the transfer term $\Pi_{\ell}$ obeys the same
constraints as the locality of the energy transfer term studied in aluie2011 .
We recall that locality requires the velocity and density exponents to
satisfy:

$\displaystyle\zeta_{n}^{v}<n\hskip 8.67424pt\text{ and }\hskip
8.67424pt\zeta_{n}^{\rho}>0.\hskip 2.17021pt.\hskip 8.67424pt$ (49)

If equations (47), (48) and (49) are verified then a local conservative
cascade of circulicity takes place in the inertial range.

##  V Conclusion

In eyink2018 , Eyink & Drivas noted that “it is not hard to find infinitely
many anomalous balance relations in the ideal limit of turbulence, but most of
them are not physically relevant and have no significant consequences”. In
this work, we started by raising a physical question: we asked whether a
cascade of rotational motions (whirls) exist in the inertial range of
compressible turbulence. This question was formulated in terms of a quantity
called circulicity which carries information about the large scale angular
momentum of the flow. We then used a point-splitting and a coarse-graining
approach to analyse the properties of the circulicity. The two approaches gave
similar results but allowed to highlight different aspects. Concerning the
point-splitting approach, we derived a Monin-Yaglom equation that differs from
previously derived expressions falkovich2010 ; galtier2011 ; wagner2012 ;
banerjee2013 ; kritsuk2013 ; banerjee2014 ; banerjee2017 ; banerjee2018 ;
andres2018 ; andres2019 ; ferrand2020 ; simon2021 . In particular, it only
involves quantities which can be reconstructed from the velocity and density
fields, $\rho$ and $\boldsymbol{v}$. It does not require any knowledge about
the energy or the pressure and is independent of the equation of state of the
flow. It is also valid whether the flow is fully compressible or is in a
quasi-incompressible limit. This Monin-Yaglom relation allowed to show that an
“effective” cascade of circulicity exists when the flow is stirred with a
solenoidal force, but not when the force is purely dilatational. Besides, it
also allowed to identify a weakly dilatational regime characterized by a
Kolmogorov-Obukhov scaling and a scaling parameter given by a solenoidal
dissipation. In the opposite regime, it was shown how this Monin-Yaglom
relation could be interpreted when strong shocks are present. Finally, the
practical relevance of this Monin-Yaglom was discussed.

The coarse-graining point of view brought additional information and allowed
to clarify several points. In particular, we were able to disentangle the role
played by the actual transfer of circulicity and its subscale production. This
allowed to propose conditions under which a conservative and local circulicity
cascade could take place.

## References

  * (1) L. F. Richardson.  Weather prediction by numerical process.  Cambridge university press, 1922.
  * (2) P. A. Davidson, Y. Kaneda, K. Moffatt, and K. R. Sreenivasan.  A voyage through turbulence.  Cambridge University Press, 2011.
  * (3) A. Alexakis and L. Biferale.  Cascades and transitions in turbulent flows.  Phys. Rep., 767:1–101, 2018.
  * (4) A. N. Kolmogorov.  Local turbulence structure in incompressible viscous fluid at very high Reynolds numbers.  Dokl. Akad. Nauk. SSSR, 30(4):299–303, 1941.
  * (5) U. Frisch.  Turbulence: the legacy of A. N. Kolmogorov.  Cambridge Univ. Press, 1995.
  * (6) L. Onsager.  Statistical hydrodynamics.  Il Nuovo Cimento (1943-1954), 6(2):279–287, 1949.
  * (7) G. Eyink and K. Sreenivasan.  Onsager and the theory of hydrodynamic turbulence.  Reviews of modern physics, 78(1):87, 2006.
  * (8) A.S. Monin and A.M. Yaglom.  Statistical fluid mechanics.  MIT Press, 1975.
  * (9) G.I. Taylor.  Statistical theory of turbulence.  Proc. R. Soc. London Series A, 151(873):421–444, 1935.
  * (10) A.N. Kolmogorov.  On the degeneration of isotropic turbulence in an incompressible viscous fluid.  Dokl. Akad. Nauk. SSSR, 31:538–541, 1941.
  * (11) J. John, D. Donzis, and K. Sreenivasan.  Does dissipative anomaly hold for compressible turbulence?  J. Fluid Mech., 920, 2021.
  * (12) E.-W. Saw, D. Kuzzay, D. Faranda, A. Guittonneau, F. Daviaud, C. Wiertel-Gasquet, V. Padilla, and B. Dubrulle.  Experimental characterization of extreme events of inertial dissipation in a turbulent swirling flow.  Nature communications, 7(1):1–8, 2016.
  * (13) B. Dubrulle.  Beyond Kolmogorov cascades.  J. Fluid Mech., 867:P1, 2019.
  * (14) H. Aluie.  Compressible turbulence: the cascade and its locality.  Phys. Rev. Lett., 106(17):174502, 2011.
  * (15) H. Aluie, S. Li, and H. Li.  Conservative cascade of kinetic energy in compressible turbulence.  Astr. J. Lett., 751(2):L29, 2012.
  * (16) H. Aluie.  Scale decomposition in compressible turbulence.  Physica D: Nonlinear Phenomena, 247(1):54–65, 2013.
  * (17) G. Eyink and T. Drivas.  Cascades and dissipative anomalies in compressible fluid turbulence.  Phys. Rev. X, 8(1):011022, 2018.
  * (18) G. Falkovich, I. Fouxon, and Y. Oz.  New relations for correlation functions in navier–stokes turbulence.  J. Fluid Mech., 644:465–472, 2010.
  * (19) S. Galtier and S. Banerjee.  Exact relation for correlation functions in compressible isothermal turbulence.  Phys. Rev. Lett., 107(13):134501, 2011.
  * (20) R. Wagner, G. Falkovich, A. Kritsuk, and M. Norman.  Flux correlations in supersonic isothermal turbulence.  J. Fluid Mech., 713:482–490, 2012.
  * (21) S. Banerjee and S. Galtier.  Exact relation with two-point correlation functions and phenomenological approach for compressible magnetohydrodynamic turbulence.  Phys. Rev. E, 87(1):013019, 2013.
  * (22) A. Kritsuk, R. Wagner, and M. Norman.  Energy cascade and scaling in supersonic isothermal turbulence.  J. Fluid Mech., 729, 2013.
  * (23) S. Banerjee and S. Galtier.  A kolmogorov-like exact relation for compressible polytropic turbulence.  J. Fluid Mech., 742:230–242, 2014.
  * (24) S. Banerjee and A. Kritsuk.  Exact relations for energy transfer in self-gravitating isothermal turbulence.  Phys. Rev. E, 96(5):053116, 2017.
  * (25) S. Banerjee and A. Kritsuk.  Energy transfer in compressible magnetohydrodynamic turbulence for isothermal self-gravitating fluids.  Phys. Rev. E, 97(2):023107, 2018.
  * (26) N. Andrés, F. Sahraoui, S. Galtier, L. Hadid, P. Dmitruk, and P. Mininni.  Energy cascade rate in isothermal compressible magnetohydrodynamic turbulence.  J. Plasma Phys., 84(4), 2018.
  * (27) N. Andrés, F. Sahraoui, S. Galtier, L. Hadid, R. Ferrand, and S. Huang.  Energy cascade rate measured in a collisionless space plasma with mms data and compressible hall magnetohydrodynamic turbulence theory.  Phys. Rev. Lett., 123(24):245101, 2019.
  * (28) R. Ferrand, S. Galtier, F. Sahraoui, and C. Federrath.  Compressible turbulence in the interstellar medium: new insights from a high-resolution supersonic turbulence simulation.  Astr. J., 904(2):160, 2020.
  * (29) P. Simon and F. Sahraoui.  General exact law of compressible isentropic magnetohydrodynamic flows: theory and spacecraft observations in the solar wind.  Astr. J., 916(1):49, 2021.
  * (30) Y. Zhou.  Rayleigh–Taylor and Richtmyer–Meshkov instability induced flow, turbulence, and mixing. I.  Phys. Rep., 720-722:1–136, 2017.
  * (31) Y. Zhou.  Rayleigh–Taylor and Richtmyer–Meshkov instability induced flow, turbulence, and mixing. II.  Phys. Rep., 723-725:1–160, 2017.
  * (32) O. Soulard.  Implications of the Monin-Yaglom relation for Rayleigh-Taylor turbulence.  Phys. Rev. Lett., 109:254501, 2012.
  * (33) O. Soulard and B.-J. Gréa.  Influence of zero-modes on the inertial-range anisotropy of Rayleigh-Taylor and unstably stratified homogeneous turbulence.  Phys. Rev. Fluids, 2:074603, 2017.
  * (34) P. Sagaut and C. Cambon.  Homogeneous turbulence dynamics.  Springer, 2018.
  * (35) S. Kassinos, W. Reynolds, and M. Rogers.  One-point turbulence structure tensors.  J. Fluid Mech., 428:213–248, 2001.
  * (36) O. Soulard, J. Griffond, B.-J. Gréa, and G. Viciconte.  Permanence of large eddies in decaying variable-density homogeneous turbulence with small mach numbers.  Phys. Rev. Fluids, 5(6):064613, 2020.
  * (37) O. Soulard and J. Griffond.  Permanence of large eddies in Richtmyer–Meshkov turbulence for weak shocks and high Atwood numbers.  Phys. Rev. Fluids, 7(1):014605, 2022.
  * (38) V. Giovangigli.  Multicomponent flow modeling.  Springer Science, 1999.
  * (39) J.J. Podesta, M.A. Forman, and C.W. Smith.  Anisotropic form of third-order moments and relationship to the cascade rate in axisymmetric magnetohydrodynamic turbulence.  Phys. Plasmas, 14:092305, 2007.
  * (40) C. Federrath, J. Roman-Duval, R.S. Klessen, W. Schmidt, and M.-M. Mac Low.  Comparing the statistics of interstellar turbulence in simulations and observations-solenoidal versus compressive turbulence forcing.  A&A, 512:A81, 2010.
  * (41) C. Federrath.  On the universality of supersonic turbulence.  MNRAS, 436(2):1245–1257, 2013.
  * (42) J. Wang, M. Wan, S. Chen, C. Xie, and S. Chen.  Effect of shock waves on the statistics and scaling in compressible isotropic turbulence.  Phys. Rev. E, 97(4):043108, 2018.
  * (43) D. Donzis and J. John.  Universality and scaling in homogeneous compressible turbulence.  Phys. Rev. F, 5(8):084609, 2020.
  * (44) G.L. Eyink.  Turbulence theory, course notes.  https://www.ams.jhu.edu/ẽyink/Turbulence/notes.html.
  * (45) C. Federrath, R. S. Klessen, L. Iapichino, and J. Beattie.  The sonic scale of interstellar turbulence.  Nature Astr., 5(4):365–371, 2021.
  * (46) J. John, D. Donzis, and K. Sreenivasan.  Solenoidal scaling laws for compressible mixing.  Phys. Rev. Lett., 123(22):224501, 2019.
  * (47) E. Lindborg.  A note on acoustic turbulence.  J. Fluid Mech., 874, 2019.
  * (48) P. Sagaut.  Large eddy simulation for incompressible flows: an introduction.  Springer Science & Business Media, 2006.
  * (49) E. Garnier, N. Adams, and P. Sagaut.  Large eddy simulation for compressible flows.  Springer Science & Business Media, 2009.
  * (50) G. Dal Maso, P.G. Lefloch, and F. Murat.  Definition and weak stability of nonconservative products.  Journal de mathématiques pures et appliquées, 74(6):483–548, 1995.
  * (51) Carlos Parés.  Numerical methods for nonconservative hyperbolic systems: a theoretical framework.  SIAM Journal on Numerical Analysis, 44(1):300–321, 2006.
  * (52) D.L. Sandoval.  The dynamics of variable density turbulence.  PhD thesis, Univ. Washington, 1995.
  * (53) D. Livescu and J.R. Ristorcelli.  Variable–density mixing in buoyancy–driven turbulence.  J. Fluid Mech., 605:145–180, 2008.

##  Appendix A Useful relationships for two-point correlations

For some given quantities $A$, $B$ and $C$ with homogeneous statistics, we
have:

$\displaystyle\overline{\Delta A\Delta B}=$ $\displaystyle
2\overline{AB}-\overline{AB^{\prime}}-\overline{A^{\prime}B}\hskip
8.67424pt,\hskip 8.67424pt$ (50) $\displaystyle\overline{\Delta A\Delta
B\Delta C}=$
$\displaystyle\overline{A^{\prime}(BC)}-\overline{A(BC)^{\prime}}+\overline{B^{\prime}(AC)}-\overline{B(AC)^{\prime}}+\overline{C^{\prime}(AB)}-\overline{C(AB)^{\prime}}\hskip
8.67424pt,\hskip 8.67424pt$ (51)
$\displaystyle\overline{A^{\prime}B}-\overline{AB^{\prime}}=$ $\displaystyle
2\overline{\Delta A\;\\{B\\}_{\frac{1}{2}}}=-2\overline{\Delta
B\;\\{A\\}_{\frac{1}{2}}}=\overline{\Delta
A\;\\{B\\}_{\frac{1}{2}}}-\overline{\Delta B\;\\{A\\}_{\frac{1}{2}}}\hskip
8.67424pt,\hskip 8.67424pt$ (52)
$\displaystyle\overline{\partial_{i}^{\prime}A^{\prime}B}-\overline{\partial_{i}AB^{\prime}}=$
$\displaystyle\partial_{r_{i}}(\overline{A^{\prime}B}+\overline{AB^{\prime}})=-\partial_{r_{i}}\overline{\Delta
A\Delta B}\hskip 2.17021pt.\hskip 8.67424pt$ (53)

If $v_{j}$ satisfies $\partial_{j}v_{j}=\partial_{j}a_{j}$, we have:

$\displaystyle\partial_{r_{j}}(\overline{v_{j}^{\prime}B^{\prime}C}-\overline{v_{j}BC^{\prime}})=$
$\displaystyle-\partial_{r_{j}}\overline{\Delta v_{j}\Delta B\Delta
C}-\partial_{r_{j}}(\overline{v^{\prime}_{j}BC^{\prime}}-\overline{v_{j}B^{\prime}C})+\partial_{r_{j}}(\overline{a^{\prime}_{j}BC}-\overline{a_{j}B^{\prime}C^{\prime}})\hskip
8.67424pt,\hskip 8.67424pt$ (54)
$\displaystyle\partial_{r_{j}}(\overline{v_{j}^{\prime}BB^{\prime}}-\overline{v_{j}BB^{\prime}})=$
$\displaystyle-\frac{1}{2}\partial_{r_{j}}\overline{\Delta v_{j}\Delta
B^{2}}+\frac{1}{2}\partial_{r_{j}}(\overline{a^{\prime}_{j}B^{2}}-\overline{a_{j}B^{\prime
2}})$ (55)

We also recall that homogeneity implies that:

$\displaystyle\overline{\partial^{\prime}_{j}(\cdot)}=-\overline{\partial_{j}(\cdot)}=\partial_{r_{j}}(\overline{\phantom{(}\cdot\phantom{)}})\hskip
2.17021pt.\hskip 8.67424pt$ (56)

##  Appendix B Evolutions of the angular momentum and of the dilatational part
of the linear momentum

The evolution of the angular momentum
$\boldsymbol{\Omega}=\epsilon_{ijk}\partial_{j}(\rho v_{k})$ is deduced from
Eq. (1b), knowing that $\partial_{j}(\rho_{j}v_{k})=v_{j}\partial_{k}(\rho
v_{j})+\rho v_{k}\partial_{j}v_{j}-\epsilon_{kpq}v_{p}\Omega_{q}$:

$\displaystyle\partial_{t}\Omega_{i}+v_{j}\partial_{j}\Omega_{i}=\Omega_{j}\partial_{j}v_{i}-\Omega_{i}\partial_{j}v_{j}+\epsilon_{ijk}\partial_{k}\left(\frac{\rho
v_{l}v_{l}}{2}\right)\frac{\partial_{j}\rho}{\rho}-\epsilon_{ijk}\partial_{j}\Big{(}\rho
v_{k}\partial_{l}v_{l}\Big{)}+\mu\partial^{2}_{jj}\omega_{i}+\epsilon_{ijk}\partial_{j}f_{k}^{s}\hskip
2.17021pt.\hskip 8.67424pt$ (57)

This equation can be compared against the vorticity equation:

$\displaystyle\partial_{t}\omega_{i}+v_{j}\partial_{j}\omega_{i}=\omega_{j}\partial_{j}v_{i}-\omega_{i}\partial_{j}v_{j}+\epsilon_{ijk}\frac{\partial_{k}p}{\rho}\frac{\partial_{j}\rho}{\rho}+\frac{\mu}{\rho}\partial^{2}_{jj}\omega_{i}+\epsilon_{ijk}\frac{\partial_{l}\sigma_{kl}}{\rho}\frac{\partial_{j}\rho}{\rho}+\epsilon_{ijk}\partial_{j}(f_{k}/\rho)\hskip
2.17021pt.\hskip 8.67424pt$ (58)

We note that similar stirring terms are present in both equations, while the
standard baroclinic torque acting in the vorticity equation is replaced by a
torque involving the dynamic pressure $\rho|\boldsymbol{v}|^{2}/2$ in the
angular momentum equation.

As for $\boldsymbol{d}$, its evolution is given by:

$\displaystyle\partial_{t}d_{i}=-\partial_{i}\pi^{d}-\partial_{j}\sigma_{ij}^{d}+f_{i}^{d}\hskip
8.67424pt,\hskip 8.67424pt$ (59) with $\displaystyle\pi^{d}=p-\pi^{s}\hskip
8.67424pt,\hskip
8.67424pt\sigma_{ij}^{d}=-(\eta+\frac{4}{3}\mu)\partial_{k}v_{k}\delta_{ij}\hskip
8.67424pt,\hskip 8.67424ptf_{i}^{d}=-\partial_{i}\phi^{f}\hskip
8.67424pt,\hskip 8.67424pt$

and $\partial_{jj}\phi^{f}=-\partial_{j}f_{j}$. This equation is readily
integrated and shows that the scalar potential $\phi$, such that
$d_{i}=-\partial_{i}\phi$, is directly related to the dilatational pressure
$\pi^{d}$ in addition to forcing and viscous terms:

$\displaystyle\phi(\boldsymbol{x},t)=\phi_{0}(t)+\int_{0}^{t}\pi^{d}(\boldsymbol{x},t^{\prime})\mathrm{d}t^{\prime}+\int_{0}^{t}\phi^{f}(\boldsymbol{x},t^{\prime})-(\eta+\frac{4}{3}\mu)\partial_{k}v_{k}(\boldsymbol{x},t^{\prime})\mathrm{d}t^{\prime}\hskip
8.67424pt,\hskip 8.67424pt$ (60)

with $\phi_{0}$ an arbitrary function of time.

##  Appendix C Karman-Howarth equation for the momentum

The Karman-Howarth equation for the linear momentum
$\boldsymbol{j}=\rho\boldsymbol{v}$ can be found in eyink2018 . Its
expression, derived from Eq. (1b), is recalled here:

$\displaystyle\partial_{t}\overline{\Delta j_{i}\Delta
j_{i}}(\boldsymbol{r},t)=-\partial_{r_{j}}\mathscr{F}^{j}_{j}(\boldsymbol{r},t)-4\mathscr{E}_{\mathrm{inj}}^{j}(t)+\partial^{2}_{r_{j}r_{k}}\mathcal{D}_{jk}^{j}(\boldsymbol{r},t)+2\overline{\Delta
j_{i}\Delta f_{i}}\hskip 8.67424pt,\hskip 8.67424pt$ (61a) with:
$\displaystyle\mathscr{F}^{j}_{j}(\boldsymbol{r},t)=\overline{\Delta
v_{j}\Delta j_{i}\Delta
j_{i}}+2\partial_{r_{j}}\Big{(}\overline{\Delta\Phi\Delta
p}-\frac{1}{2}\overline{\Delta\varphi^{v}\Delta(j_{i}j_{i})}\Big{)}\hskip
8.67424pt,\hskip 8.67424pt$ (61b)
$\displaystyle\mathscr{E}_{\mathrm{inj}}^{j}(t)=-\left(\overline{\sigma_{ij}\partial_{j}j_{i}}+\overline{p\partial_{j}j_{j}}-\frac{1}{2}\overline{j_{i}j_{i}\partial_{j}v_{j}}\right)\hskip
8.67424pt,\hskip 8.67424pt$ (61c)
$\displaystyle\mathcal{D}_{jk}^{j}(\boldsymbol{r},t)=2\mu\overline{\Delta
v_{i}\Delta j_{i}}\delta_{jk}+2(\eta+\frac{\mu}{3})\overline{\Delta
v_{j}\Delta j_{k}}\hskip 2.17021pt.\hskip 8.67424pt$ (61d)