#  Local fluxes in MHD Turbulence

Alexandros Alexakis 1 & Sergio Chibbaro 2<EMAIL_ADDRESS>Laboratoire de
Physique de l’Ecole Normale Supérieure, ENS, Université PSL, CNRS, Sorbonne
Université, Université Paris-Diderot, Sorbonne Paris Cité, Paris, France, 2
Université Paris-Saclay, CNRS, LISN, 91400 Orsay, France

###### Abstract

Using highly resolved direct numerical simulations we examine the statistical
properties of the local energy flux rate $\Pi_{\ell}(x)$ towards small scales
for three isotropic turbulent magnetohydrodynamic flows, which differ in
strength and structure of the magnetic field. We analyse the cascade process
both in the kinetic and magnetic energy, disentangling the different flux
contributions to the overall energy dynamics. The results show that the
probability distribution of the local energy flux develops long tails related
to extreme events, similar to the hydrodynamic case. The different terms of
the energy flux display different properties and show sensitivity on the type
of the flow examined. We further examine the joint pdf between the local
energy flux and the gradients of the involved fields. The results point out a
correlation with the magnetic field gradients, showing however a dispersion
much stronger than what is observed in hydrodynamic flows. Finally, it is also
shown that the local energy flux shows some dependence on the local amplitude
of the magnetic field. The present results have implications for subgrid scale
models that we discuss.

##  1 Introduction

Most stellar objects from stars to galaxies are made of hot gas in a ionized
state. The magneto-hydrodynamic (MHD) equations give the simplest description
of the dynamics involved (Goldstein et al., 1995; Battaner, 1996; Biskamp,
2003; Verma, 2004; McKee & Ostriker, 2007; Bruno & Carbone, 2013; Galtier,
2016) and give a general framework relevant for dynamo and many plasma
phenomena (Landau et al., 2013; Brandenburg & Subramanian, 2005; Davidson,
2002). At large Reynolds numbers (small dissipation parameters) the MHD
turbulent dynamics become turbulent sharing many features with hydrodynamic
turbulence. The key idea is the existence of cascade processes, meaning that
inviscid invariants are transferred across scales so that they are efficiently
dissipated at the smallest scales (Monin & Yaglom, 1975; Frisch, 1995;
Alexakis & Biferale, 2018). This idea sketched by Richardson, was at the basis
of the phenomenological statistical theory by Kolmogorov (Kolmogorov, 1941;
Kraichnan, 1971) and is the cornerstone of our understanding of turbulence.
Different theories have been proposed that attempt to describe quantitatively
the behavior of MHD turbulence (Iroshnikov, 1964; Kraichnan, 1965). Yet, MHD
turbulence displays an even more complex behaviour than Hydrodynamic one with
diverse regimes and recently new theories have been proposed (Goldreich &
Sridhar, 1995; Boldyrev, 2006; Beresnyak, 2011). A variety of methods have
been applied in MHD to test and constrain existing theories. Two point
correlation functions have led to some constraints on third order structure
functions (Politano & Pouquet, 1998; Galtier, 2009). Scale by scale analysis
in Fourier space that has been extensively used in hydrodynamics (Chen et al.,
2003; Domaradzki et al., 2009; Teaca et al., 2011; Alexakis et al., 2007;
Verma, 2004, 2019) has been applied with some success in MHD as well (Verma,
2021; Dar et al., 2001; Teaca et al., 2009; Alexakis et al., 2005; Mininni et
al., 2005).

New insights on the cascade physics have been obtained analysing the scale-by-
scale budgets of energy of filtered fields, including the fluctuations of the
flux (Eyink & Sreenivasan, 2006; Dubrulle, 2019). Notably, this kind of
complex tools have been used to get insights on the physical mechanisms in 2D
and 3D fluid turbulence (Piomelli et al., 1991; Eyink & Aluie, 2009; Eyink,
2005; Liu et al., 1994; Chen et al., 2006) and more recently, scale-by-scale
filtering analysis has been started to be applied also to MHD turbulence to
highlight and understand some specific processes (Eyink et al., 2013; Galtier,
2018; Bian & Aluie, 2019).  The interest of such tools is witnessed by its use
in many other problems such as two-fluid plasma models (Camporeale et al.,
2018), hybrid-kinetic models (Cerri & Camporeale, 2020), gyro-kinetics (Teaca
et al., 2021), and full-kinetic systems (Eyink, 2018; Yang et al., 2022). More
generally, this kind of approach seems promising to go beyond standard
statistical information and access to energetic processes, which are usually
the most important for applications, and are nowadays commonly used in a
variety of cases (Danaila et al., 2001; Casciola et al., 2003; Sorriso-Valvo
et al., 2007; Cimarelli et al., 2013; Valori et al., 2020; Innocenti et al.,
2021). Our work follows this path in order to get similar understanding for
MHD turbulence.

Such an understanding of the cascade process is paramount for astrophysical
and industrial applications for the following reason. While the numerical
solution of the fundamental equations is in principle possible, the vast range
of scales excited in such objects prohibits any direct numerical calculation
that resolves all the scales down to the dissipation ones. However, if all
scales are not properly resolved, the numerical approach is tantamount to
apply a coarse-graining to the initial problem discarding a part of the
information, related to the small scales not resolved. As a result all
simulations of astrophysical flows require some modeling of smaller unresolved
scales that are responsible for the energy dissipation. Such modeling will
thus have to compensate the energy transfers between resolved and unresolved
scales so that it correctly captures the energy dissipation of the flow. The
construction of such models requires a deep and quantitative understanding of
how turbulence transfers energy to small scales.

This practice to compute only large scales and to put forward approximate
models at small scales is referred to as Large-eddy simulations (LES), and has
a long history in fluid turbulence in particular for practical applications
(Leonard, 1975; Germano, 1992; Meneveau & Katz, 2000; Sagaut, 2001; Lesieur et
al., 2005; Pope, 2000). In hydrodynamic turbulence the development and testing
of such models is thus well advanced. A particular class of hydrodynamic
models quantify the dissipation energy by a an eddy-viscosity term whose
coefficient depends on the gradients of the resolved flow (Smagorinsky, 1963;
Germano et al., 1991; Germano, 1992), and many studies have been devoted in
quantifying this dependence (Vreman et al., 1994, 1997; Borue & Orszag, 1998;
Meneveau & Katz, 2000).

In MHD, LES is less developed and tested than in fluid turbulence. At the same
time the complexity of the MHD is increased as more channels exist for the
energy to be transferred to the smaller scales and since there are two fields
involved there are more possibilities for the dependence of the eddy viscosity
term.  A convenient way to assess subgrid models is through direct numerical
simulations, which are however more difficult than in hydrodynamics and only
recently DNS with a sufficient resolution allowing for an inertial range to be
present have been made available. Yet, Smagorinski-like models have been
adapted through formal analogy in MHD since the 90’ (Theobald et al., 1994;
Müller & Carati, 2002; Verma & Kumar, 2004; Bian et al., 2021). These models
still represent the only available strategy for subgrid modelling (Miesch et
al., 2015), and they have been recently generalised even to compressible
(Chernyshov et al., 2014; Vlaykov et al., 2016; Grete et al., 2017) and
relativistic MHD (Viganò et al., 2019; Carrasco et al., 2020).

Although DNS has been used to partially assess the validity of such closures
(Agullo et al., 2001; Miesch et al., 2015; Kessar et al., 2016), a thorough
analysis is still needed, given the importance of the problem. Notably, in the
last years, new analysis of this kind of modelling has been made possible by
the use of scale-by-scale analysis (Linkmann et al., 2018; Biferale et al.,
2019; Buzzicotti et al., 2018; Alexakis & Chibbaro, 2020). In these works, it
has been pointed out that a deeper understanding of the cascade process,
including fluctuations is valuable also to improve modelling.

The purpose of this work is to shed some light on this direction by examining
different MHD turbulence flows and the statistical properties of the local
energy flux towards the smaller scales, bringing to MHD this original approach
already applied to hydrodynamic turbulence. In particular, we analyse the
scale-by-scale budgets of kinetic and magnetic energy of highly resolved DNS.
That allows to provide new information on the cascade process, including
fluctuations, and thus to go beyond usual investigation about the validity of
subgrid models. As a final introductory comment, it is worth emphasising that
our goal is not to put forward and/or test some specific model but rather to
get some insights on the modelling via DNS.

##  2 Theoretical formulation

###  2.1 Main Equations

We begin by considering the magneto-hydrodynamic equations in the
incompressible non-relativistic limit, given by (Biskamp, 2003)

$\displaystyle{\partial{\bf u}\over\partial t}+{\bf u}\cdot\nabla{\bf u}$
$\displaystyle=$ $\displaystyle-\nabla p+{\bf b}\cdot\nabla{\bf
b}+{\nu}\nabla^{2}{\bf u}$ (1) $\displaystyle{\partial{\bf b}\over\partial
t}+{\bf u}\cdot\nabla{\bf b}$ $\displaystyle=$ $\displaystyle{\bf
b}\cdot\nabla{\bf u}+{\eta}\nabla^{2}{\bf b},$ (2)
$\displaystyle\nabla\cdot{\bf v}=0,$ $\displaystyle\nabla\cdot{\bf b}=0,$ (3)

where the modified pressure $p$ is given by $p=P/\rho+b^{2}/2$, ${\bf b}={\bf
B}\sqrt{4\pi\rho}$, and $\rho$ is the mass density, ${\bf u}({\bf x},t)$ is
the flow velocity, $P({\bf x},t)$ is the thermal pressure, ${\bf B}({\bf
x},t)$ is the magnetic induction field $\nu$ is the viscosity and $\eta$ the
magnetic diffusivity.

###  2.2 Scale-By-scale Analysis

A considerable amount of work in analysing the transfer of energy among scales
has been performed in Fourier space with significant results (Domaradzki et
al., 2009; Alexakis et al., 2007; Teaca et al., 2011; Verma, 2004, 2019, 2021;
Teaca et al., 2009; Alexakis et al., 2005; Mininni et al., 2005). This kind of
approach while useful in understanding and quantifying the scale-by-scale
energy budget of turbulence, does not allow to simply link the related energy
transfers with local properties of the flow and thus limits their
applicability to subgrid scale models.

To analyse the local in space scale-by-scale dynamics, we use the local
filtering approach (Germano, 1992). We introduce a filter such that

$\widetilde{{\bf a}}_{\ell}({\bf x})=\int dr^{3}\,G_{\ell}({\bf r}){\bf
a}({\bf x}+{\bf r})$ (4)

where $G({\bf r})$ is a smooth filtering function, spatially localized and
such that $\int dr^{3}\ G({\bf r})=1$, and $\int dr^{3}\ |{\bf r}|^{2}G({\bf
r})\approx 1$. The function $G_{\ell}$ is rescaled with $\ell$ as
$G_{\ell}({\bf r})=\ell^{-3}G({\bf r}/\ell)$. Applying this filter to the MHD
equations one obtains (Aluie, 2017):

$\partial_{t}\widetilde{{\bf u}}_{\ell}+(\widetilde{{\bf
u}}_{\ell}\cdot\nabla)\widetilde{{\bf
u}}_{\ell}=-\nabla\widetilde{p}_{\ell}+(\widetilde{\bf
b}_{\ell}{\mbox{\boldmath$\cdot$}}\nabla)\widetilde{\bf
b}_{\ell}-\nabla\cdot({\mbox{\boldmath$\tau$}}_{\ell}^{uu}-{\mbox{\boldmath$\tau$}}_{\ell}^{bb})+\nu\nabla^{2}\widetilde{{\bf
u}}_{\ell}.$ (5)

$\partial_{t}\widetilde{{\bf b}}_{\ell}+(\widetilde{{\bf
u}}_{\ell}\cdot\nabla)\widetilde{{\bf b}}_{\ell}-(\widetilde{{\bf
b}}_{\ell}\cdot\nabla)\widetilde{{\bf
u}}_{\ell}=-\nabla\cdot({\mbox{\boldmath$\tau$}}^{ub}_{\ell}-{\mbox{\boldmath$\tau$}}^{bu}_{\ell})+\eta\nabla^{2}\widetilde{{\bf
b}}_{\ell}$ (6)

$\nabla\cdot\widetilde{{\bf b}}_{\ell}=\nabla\cdot\widetilde{{\bf
u}}_{\ell}=0.$ (7)

Here
${\mbox{\boldmath$\tau$}}^{uu}_{\ell},{\mbox{\boldmath$\tau$}}^{bb}_{\ell},{\mbox{\boldmath$\tau$}}^{ub}_{\ell}$
and ${\mbox{\boldmath$\tau$}}^{bu}_{\ell},$ are the subscale stress tensors
which describe the force exerted on scales larger than $\ell$ by fluctuations
at scales smaller than $\ell$. They are defined as

${\mbox{\boldmath$\tau$}}^{uu}_{\ell}=\widetilde{({\bf u}{\bf
u})}_{\ell}-\widetilde{\bf u}_{\ell}\widetilde{\bf
u}_{\ell},\qquad{\mbox{\boldmath$\tau$}}^{bb}_{\ell}=\widetilde{({\bf b}{\bf
b})}_{\ell}-\widetilde{\bf b}_{\ell}\widetilde{\bf b}_{\ell}$ (8)

are the subscale Reynolds stress and the subscale Maxwell stress respectively
and finally

${\mbox{\boldmath$\tau$}}^{ub}_{\ell}=\widetilde{({\bf u}{\bf
b})}_{\ell}-\widetilde{\bf u}_{\ell}\widetilde{\bf
b}_{\ell},\qquad{\mbox{\boldmath$\tau$}}^{bu}_{\ell}=\widetilde{({\bf b}{\bf
u})}_{\ell}-\widetilde{\bf b}_{\ell}\widetilde{\bf u}_{\ell}$ (9)

are cross-field tensors for which one is the transpose of the other
${\mbox{\boldmath$\tau$}}^{bu}_{\ell}=[{\mbox{\boldmath$\tau$}}^{ub}_{\ell}]^{T}$.

Using this notation we can write an equation for the large scale energy
densities $\tilde{\mathcal{E}}^{u}=\frac{1}{2}|\widetilde{\bf u}|^{2}$ and
$\tilde{\mathcal{E}}^{b}=\frac{1}{2}|\widetilde{\bf b}|^{2}$ as

$\displaystyle\partial_{t}\tilde{\mathcal{E}}_{\ell}^{u}+\nabla\cdot{\mathcal{J}}^{u}$
$\displaystyle=$
$\displaystyle-\Pi_{\ell}^{uu}-\Pi_{\ell}^{bb}+\mathcal{W}_{L}-\mathcal{D}_{u}$
(10)
$\displaystyle\partial_{t}\tilde{\mathcal{E}}^{b}_{\ell}+\nabla\cdot{\mathcal{J}}^{b}$
$\displaystyle=$
$\displaystyle-\Pi_{\ell}^{bu}-\Pi_{\ell}^{ub}-\mathcal{W}_{L}-\mathcal{D}_{b}$
(11)

where the currents $\mathcal{J}^{u},\mathcal{J}^{b}$ are given by

$\displaystyle\mathcal{J}^{u}_{j}$ $\displaystyle=$
$\displaystyle\left(\frac{1}{2}|\widetilde{\bf
u}|^{2}+\overline{p}\right)\widetilde{u}_{j}+(\tau^{uu}_{\ell,ij}-\tau^{bb}_{\ell,ij})\widetilde{u}_{i}-\nu\frac{1}{2}\partial_{j}|\widetilde{\bf
u}|^{2}$ (12) $\displaystyle\mathcal{J}^{b}_{j}$ $\displaystyle=$
$\displaystyle\left(\frac{1}{2}|\widetilde{\bf
b}|^{2}\right)\widetilde{u}_{j}+(\tau^{ub}_{\ell,ij}-\tau^{bu}_{\ell,ij})\widetilde{b}_{i}-(\widetilde{\bf
u}\cdot\widetilde{\bf b})\widetilde{b}_{j}-\eta(\widetilde{\bf
b}\times\nabla\times\widetilde{\bf b})$ (13)

and express the transport of energy in space. The rate of work done on the
flow originating from the Lorentz force is given by

$\mathcal{W}_{L}=\widetilde{u}_{i}\widetilde{b}_{j}\partial_{j}\widetilde{b}_{i}.$
(14)

This term is responsible for the transfer of kinetic energy to magnetic energy
in the large scales. The viscous and Ohmic energy dissipation rates are given
by

$\mathcal{D}_{u}=\nu(\partial_{j}\widetilde{u}_{i})(\partial_{j}\widetilde{u}_{i})\quad\mathrm{and}\quad\mathcal{D}_{b}=\eta(\nabla\times\widetilde{\bf
b})\cdot(\nabla\times\widetilde{\bf b})$ (15)

and express the rate energy is dissipated by viscous and Ohmic effects
respectively. These terms can be shown to be negligible if the filter scale is
large and the viscous/resistive coefficients are small (Aluie, 2017). Finally
the four fluxes in scale space
$\Pi_{\ell}^{uu},\Pi_{\ell}^{bb},\Pi_{\ell}^{ub},\Pi_{\ell}^{bu},$ explicitly
given by:

$\displaystyle\Pi_{\ell}^{uu}=-\tau_{\ell,ij}^{ub}\partial_{i}\widetilde{u}_{j}=-[\widetilde{(u_{i}u_{j})}_{\ell}-\tilde{u}_{\ell,i}\tilde{u}_{\ell,j}]\partial_{i}\widetilde{u}_{j}$
(16)
$\displaystyle\Pi_{\ell}^{bb}=+\tau_{\ell,ij}^{bb}\partial_{i}\widetilde{u}_{j}=+[\widetilde{(b_{i}b_{j})}_{\ell}-\tilde{b}_{\ell,i}\tilde{b}_{\ell,j}]\partial_{i}\widetilde{u}_{j}$
(17)
$\displaystyle\Pi_{\ell}^{ub}=-\tau_{\ell,ij}^{ub}\partial_{i}\widetilde{b}_{j}=-[\widetilde{(u_{i}b_{j})}_{\ell}-\tilde{u}_{\ell,i}\tilde{b}_{\ell,j}]\partial_{i}\widetilde{b}_{j}$
(18)
$\displaystyle\Pi_{\ell}^{bu}=+\tau_{\ell,ij}^{bu}\partial_{i}\widetilde{b}_{j}=+[\widetilde{(b_{i}u_{j})}_{\ell}-\tilde{b}_{\ell,i}\tilde{u}_{\ell,j}]\partial_{i}\widetilde{b}_{j}$
(19)

express the rate of gain (if $\Pi_{\ell}<0$) or loss (if $\Pi_{\ell}>0$) of
energy of the large scales to the small filtered-out scales. Their sum gives
the total energy flux

$\Pi_{\ell}({\bf x})=\Pi_{\ell}^{uu}({\bf x})+\Pi_{\ell}^{bb}({\bf
x})+\Pi_{\ell}^{ub}({\bf x})+\Pi_{\ell}^{bu}({\bf x})$ (20)

The fluxes are chosen so that they are invariant under Galilean
transformations ${\bf u}\to{\bf u}+{\bf U}_{0}$ and also under ${\bf b}\to{\bf
b}+{\bf B}_{0}$ for any flow realization $\bf u,b$, where $\bf U_{0},B_{0}$
are constant in space vector fields. The importance of Galilean invariance has
been noted in the past (Speziale, 1985; Aluie & Eyink, 2009, 2010). We note
however that while the defined fluxes are invariant under ${\bf b}\to{\bf
b}+{\bf B}_{0}$ the MHD equations are not. Therefore, the introduction of a
constant ${\bf B}_{0}$ in the dynamics of the MHD equations will alter the
statistical behavior of the fields $\bf u,b$ and as a result of the fluxes as
well.

The fluxes in (16)-(19) comprise the main object of the present work.

###  2.3 Filters

Although we formulate our filtering procedure in the physical space, since we
will be working in periodic domains it is useful to define the filters through
their Fourier transforms

$\hat{G}_{q}({\bf k})=\int G_{\ell}({\bf x})e^{i\bf k\cdot x}d{{\bf x}}$ (21)

where $q=1/\ell$ is the wavenumber corresponding to the filter length $\ell$,
not to be confused with the Fourier wavenumber $k$.  For the first filter we
consider a Gaussian kernel

$\hat{G}_{q}({\bf k})=\exp\left[-\frac{k^{2}}{2q^{2}}\right].$ (22)

For an infinite domain this filter corresponds to the Gaussian filter in real
space $G_{\ell}(r)=\exp(-\frac{1}{2}r^{2}/\ell^{2})/(2\pi\ell^{2})^{3/2}$. We
note that this filtering is not a projection and in general
$\widetilde{({\tilde{\bf u}_{\ell}})_{\ell}}\neq\tilde{\bf u}_{\ell}$. The
second filter we are going to consider is a sharp spectral filter such that

$\widetilde{{\bf u}}_{\ell}({\bf x},t)=\sum_{|{\bf k}|<q}\hat{{\bf u}}({\bf
k},t)e^{i{\bf k}\cdot{\bf x}}.$ (23)

This filtering is a projector $\widetilde{({\tilde{\bf
u}_{\ell}})_{\ell}}=\tilde{\bf u}_{\ell}$ and is based on a Galerkin
truncation for all wavenumbers larger than the given cutoff $q=1/\ell$. With
regard to the representation of energy fluxes, this filter has been shown in
the past not to be optimal as it leads to a wider fluctuations of the local
energy flux (Buzzicotti et al., 2018; Alexakis & Chibbaro, 2020), something
that as we will show also holds in MHD.

##  3 Sub-scale stress modeling

In this section, we provide some information about typical modelling strategy.
Our work focuses on fundamental issues, that allows us to build a precise
framework and to point to possible applications.

In a simulation for which only the large smoothed out fields $\widetilde{\bf
u},\widetilde{\bf b}$ fields are followed dynamically the effect of the small
unresolved scales on the resolved scales need to be captured by modeling the
sub-scale stress tensors
${\mbox{\boldmath$\tau$}}_{\ell}^{uu},{\mbox{\boldmath$\tau$}}_{\ell}^{ub},{\mbox{\boldmath$\tau$}}_{\ell}^{bu},{\mbox{\boldmath$\tau$}}_{\ell}^{bb}$.
In hydrodynamics, the simplest perhaps such model is formed by assuming that
${\mbox{\boldmath$\tau$}}^{uu}$ takes the form

${\mbox{\boldmath$\tau$}}^{uu}_{\ell,ij}=\nu^{e}_{\ell,ijkl}\nabla_{k}\widetilde{\bf
u}_{l}$ (24)

where the eddy viscosity tensor ${\bf\nu}^{e}$ is a function of space and time
and needs to be prescribed from the filtered field $\widetilde{\bf u}$ in
order for the filtered equations to be closed. It needs to satisfy
$\langle\nabla_{i}\widetilde{\bf
u}_{j}\nu^{e}_{\ell,ijkl}\nabla_{k}\widetilde{\bf u}_{l}\rangle=\epsilon>0$
and acts thus on average as a sink of energy. Furthermore, the divergence of
$\nabla\cdot{\mbox{\boldmath$\tau$}}^{uu}_{\ell}$ is not necessarily zero so
its projection to divergence free vector fields needs to be considered by
adding $p^{\prime}=\nabla^{-2}\nabla_{i}\nabla_{j}\tau^{uu}_{\ell,ij}$ to the
pressure. Since the system is Galilean invariant $\nu^{e}$ cannot depend on
the values of $\widetilde{\bf u}$ but only on its gradients
$\nabla\widetilde{\bf u},\nabla\nabla\widetilde{\bf u},\dots$, thus in the
simplest case $\nu_{\ell}^{e}$ is just a function of $\nabla\widetilde{\bf
u}$. If we further assume that isotropy is present at the small scales
$\nu^{e}[\nabla\widetilde{\bf u}]$ will only depend on the invariants (under
rotations and reflections) of the strain tensor $\nabla\widetilde{\bf u}$. In
the now classical Smagorinsky approach the sub-scale Reynolds stress tensor is
modeled as (Smagorinsky, 1963)

${\mbox{\boldmath$\tau$}}^{uu}_{\ell,ij}-\frac{1}{3}{\mbox{\boldmath$\tau$}}^{uu}_{\ell,kk}=\nu_{\ell}^{e}\widetilde{S}_{ij}=\frac{\nu^{e}}{2}\left(\partial_{i}\tilde{u}_{j}+\partial_{j}\tilde{u}_{i}\right)$
(25)

where $\widetilde{\bf S}$ is the symmetric part of the filtered strain tensor.
The $\nu^{e}$ is a scalar defined as (Smagorinsky, 1963)

$\nu_{\ell}^{e}=c\ell^{2}|\widetilde{\bf
S}|=c\ell^{2}\sqrt{\tilde{S}_{ij}\tilde{S}_{ij}}$ (26)

where $\ell$ is the filtering lengthscale and $c$ is the Smagorinsky constant,
an order one non-dimensional coefficient . Eq. (26) gives the only combination
of $\ell$ and $|\widetilde{\bf S}|$ with dimensions of viscosity. We note that
in general $\nu_{\ell}^{e}$ depends on space and needs to be taken in to
account for the pressure so that the divergence free condition for
$\widetilde{\bf u}$ is satisfied.

This model implies that the flux of energy to the small scales is given by

$\Pi_{\ell}({\bf x})=c\ell^{2}|\widetilde{\bf S}|^{3}.$ (27)

After Smagorinsky’s work other models that also take into account the anti-
symmetric part of the stress tensor

$\widetilde{\Omega}_{ij}=\frac{1}{2}\left(\partial_{i}\tilde{u}_{j}-\partial_{j}\tilde{u}_{i}\right)$
(28)

have been developed. Approximating the sub-scale stress with its extreme local
expression, the nonlinear Clark model is obtained (Meneveau & Katz, 2000):

$\tau_{\ell}^{uu}\approx\dfrac{1}{3}C_{2}\ell^{2}\left(\widetilde{\bf
S}_{\ell}^{2}+\widetilde{\bf\Omega}_{\ell}^{2}+\widetilde{\bf\Omega}_{\ell}\widetilde{\bf
S}_{\ell}-\widetilde{\bf
S}_{\ell}\widetilde{\bf\Omega}_{\ell}\right)\leavevmode\nobreak\ ,$ (29)

where both strain and vorticity participate in the dynamics (Misra & Pullin,
1997; Borue & Orszag, 1998; Tennekes & Lumley, 1990) and leads to modeled
local energy flux:

$\Pi_{\ell}\approx\dfrac{1}{3}C_{2}\ell^{2}[-\mathrm{Tr}(\widetilde{\bf
S}_{\ell}^{3})+3\mathrm{Tr}(\widetilde{\bf
S}_{\ell}\widetilde{\bf\Omega}_{\ell}^{2})]\leavevmode\nobreak\ .$ (30)

The development and assessment of such models has been often guided by a
priori analysis (Piomelli et al., 1991; Meneveau, 1994; Borue & Orszag, 1998;
Meneveau & Katz, 2000), that is studying the filtered DNS field properties
rather than resorting to actual LES. Following this approach, in Alexakis &
Chibbaro (2020) we have demonstrated that there is indeed a strong correlation
between $\Pi_{\ell}$ and $|\widetilde{\bf S}|$ verifying the Smagorinsky
relation (27) for the mean value of $\Pi_{\ell}$ although strong fluctuations
in $\Pi_{\ell}$ were also measured making $\Pi_{\ell}$ not a strictly positive
quantity.

In MHD a similar type of modeling is considerably more difficult. First of all
there are three sub-scale turbulent stresses that need to be modeled
${\mbox{\boldmath$\tau$}}_{\ell}^{uu},{\mbox{\boldmath$\tau$}}_{\ell}^{ub},{\mbox{\boldmath$\tau$}}_{\ell}^{bb}$.
In the past literature these dependencies are modeled based on the symmetric
and anti-symmetric parts of the stress tensors of the two fields:

$\widetilde{S}_{ul,ij}=\frac{1}{2}\left(\partial_{i}\widetilde{u}_{j}+\partial_{j}\widetilde{u}_{i}\right),\quad\widetilde{\Omega}_{ul,ij}=\frac{1}{2}\left(\partial_{i}\widetilde{u}_{j}-\partial_{j}\widetilde{u}_{i}\right),\quad$
(31)
$\widetilde{S}_{bl,ij}=\frac{1}{2}\left(\partial_{i}\widetilde{b}_{j}+\partial_{j}\widetilde{b}_{i}\right),\quad\widetilde{\Omega}_{bl,ij}=\frac{1}{2}\left(\partial_{i}\widetilde{b}_{j}-\partial_{j}\widetilde{b}_{i}\right).$
(32)

An in depth discussion of different models used can be found in Müller &
Carati (2002); Miesch et al. (2015). Furthermore, although the system is still
Galilean invariant, it is not invariant under transformation $\bf b\to
b+B_{0}$ where $\bf B_{0}$ is a constant in space magnetic field. The
statistics of the sub-filter scale fields can depend on the local value of
$\widetilde{\bf b}$ and as a result so will the relevant eddie-viscosities
that attempt to model their effect. Finally, in the presence of helicity the
small scales are known to transfer energy to the large scales in a mean way by
the alpha dynamo mechanism that modeling needs to take into account.

Given the large number of the properties of the smoothed fields that the
modeling of the sub-scale turbulent stresses can depend on, it is important to
try to limit the possibilities identifying the parameters that play the most
important role. This is what we are attempting to do in the following sections
using direct numerical simulations for which the turbulent stresses can be
directly measured.  Thus, we need to stress that we do not test a posteriori
any proposed model. Instead we are trying a priori to find relations that hold
between the local energy flux and the gradients of the flow so that they can
be used in the construction of new models.

##  4 Numerical Simulations

###  4.1 Numerical Set-up

To investigate the local energy fluxes described before we use the results
from direct numerical simulations DNS using the pseudo-spectral code ghost
(Mininni et al., 2011) evolving the MHD equations (1)-(3) in a cubic triple
periodic domain of side $L=2\pi$ so that $|\bf k|=1$ give the smallest non-
zero wavenumber . The forcing is limited only to wave numbers with $|{\bf
k}|\leq k_{f}=2$ in all cases and was random and delta correlated in time so
that the mean energy injection rate $\epsilon=\left\langle{\bf u\cdot
f_{u}}\right\rangle+\left\langle{\bf b\cdot f_{b}}\right\rangle=1$ is
independent of the flow state. All runs have the same energy injection rate
and unit Prandtl number with $\nu=\mu=0.0001$.  The involved Reynolds number
based on the input parameters $\epsilon,L$ and $\nu$ is given by
$Re=\epsilon^{1/3}k_{f}^{4/3}/\nu$ that is thus fixed to $Re=3968$. The
resolution was fixed for all runs at $N=1024$ grid points in each direction
that was sufficient for the flow to be well resolved so that an exponential
decrease of the energy spectrum is observed at large $k$. No artificial
dissipation was used.

Four different cases were considered. In the first, no magnetic forcing was
introduced and the magnetic field was set identically to zero $\bf b=0$ so
that the flow reduced to a hydrodynamic run, repeating essentially the results
of Alexakis & Chibbaro (2020). This run is used as a reference to identify the
differences of MHD from hydrodynamic runs and referred in the subsequent
figures as ‘Hydro’. The remaining three runs were designed to have different
levels of magnetic energy as in Alexakis (2013). The second run was thus a
dynamo run. No magnetic forcing was used but an initial small magnetic field
was introduced that was amplified by the turbulent motions until a steady
state was reached where the magnetic energy fluctuated around a mean value. We
note that the forcing was mirror symmetric so no helicity is injected in the
system. A result of the absence of helicity and the randomness of the flow is
that magnetic energy is concentrated in the small scales. This run is referred
to in the figures as ‘dynamo’. In the third run a magnetic forcing was also
used of equal magnitude as the mechanical flow. The resulting flow at steady
state has magnetic energy at equipartition with the kinetic energy at all
scales. The helicity and magnetic helicity injection rate for this flow is
also set to zero. Results from this run are referred to as ‘moderate’. Finally
the fourth flow has also both mechanically and magnetically forced but in this
case the magnetic helicity was weak but not zero. This has lead to the
formation of a slowly growing large scale helical magnetic field. As a result
this flow has magnetic energy that exceeds the kinetic energy of the flow.
This run is referred to in the figures as ‘strong’.

It is worth noting that all the statistics we present in the following are
obtained through time-averaging over a large amount of samples during the
steady state to assure statistical convergence. Whenever possible, several
field realizations have been used also to average over space, thanks to
spatial homogeneity.

Figure 1: Energy Spectra for the 4 different cases examined.

###  4.2 Energy Spectra and Magnetic field

Figure 1 shows the resulting time-averaged energy spectra for the four runs at
the steady state. For the dynamo run in particular analysis is made at the
state such that magnetic energy does not grow anymore.  All runs show energy
spectra compatible with a $k^{-5/3}$ power-law exponent. The magnetically
forced runs ‘moderate’ and ‘strong’ have equipartition magnetic energy across
wave numbers except for $k=1$ in the ’strong’ case for which magnetic energy
is larger. For the dynamo run the magnetic energy is weaker than kinetic at
small wave numbers but the reverse is true for the large wavenumbers. This
behavior is typical for randomly forced non-helical small scale dynamos
(Schekochihin et al., 2004; Moll et al., 2011; Brandenburg et al., 2012).
Dynamo flows with steady non-helical forcing, like Taylor-Green for example
(Ponty et al., 2008), produce spectra closer to the ‘moderate’ run. This
figure points out also that a quasi-inertial range is roughly displayed over
about a decade between $k=4$ and $k=64$, as standard with present resolution.

Figure 2:  Top panels: Visualisations of the magnetic energy density for the
dynamo (top left) the moderate case (top right) and the strong case (bottom
left). The bottom panel shows the PDF of $|{\bf b}|$ for the 3 different MHD
flows examined.

In figure 2, it is possible to get some qualitative insights on the different
cases studied. Notably, the visualization of the magnetic energy permits to
observe the structure of the large scales, through its space distribution. The
dynamo case displays very little dispersion of the magnetic field amplitude,
and energy is concentrated in flux tubes. The moderate case shows a larger
distribution and some regions of intense field. Finally the strong case shows
a much more widely distributed energy, with some regions characterised by
large values of the magnetic field, and in particular large-scale structures
are visible. To give a quantitative picture, we also show in figure 2 the
probability distribution of the magnetic amplitude $|\bf b|$ for the three MHD
cases examined. In the dynamo case the maximum is at small $|{\bf b}|$, with a
peaked distribution. The width of the distribution increases in the other two
cases, with a broad distribution for the strong case. In this last case, the
tails appear substantial.

##  5 Statistics of the local energy flux

Figure 3: Pdf of the total nonlinear flux $\Pi_{\ell}(x)$ at $q=32$ for a
sharp spectral filter on the left and a Gauss filter on the right, for the 4
different runs.

Figure 4:  Pdf of the Energy flux for different $q$ for the three MHD cases.
In the first panel, the hydro case is also plotted for comparison.

The local energy fluxes $\Pi_{\ell}^{uu}(x)$, $\Pi_{\ell}^{ub}(x)$,
$\Pi_{\ell}^{bu}(x)$, $\Pi_{\ell}^{bb}(x)$ and $\Pi_{\ell}(x)$ were calculated
for the four different runs at their steady state. Figure 3 shows this pdf for
the four different runs using a sharp-spectral filter (left) and Gaussian
filter (right) for a filter length $q=32$, which corresponds to a value at
around the end of the inertial range.

As in hydrodynamics, in MHD the sharp-spectral filter leads to a wider and
more symmetric pdf of $\Pi_{\ell}$. Interestingly, the pure hydro curve
presents the strongest tails, while with the Gaussian filter the profiles were
found always similar. Considering the representation of the fluxes, the sharp-
filter is thus found to enhance fluctuations displaying many negative events,
making it more difficult to use directly in LES. The reason for this behavior
is the fact that the sharp filter is not localised in space and not positive-
definite. For this reason, it appears less useful for the analysis of the
energy fluxes and it is abandoned in the following. In this sense, the present
analysis confirms previous results obtained in the pure hydrodynamic case
(Buzzicotti et al., 2018; Alexakis & Chibbaro, 2020).

The Gaussian filter leads for all four cases to a skewed pdf with non-Gaussian
(stretched exponential) tails. The tails for all cases extend to values
hundred times more than the mean value $\langle\Pi_{\ell}\rangle=1$, implying
that, although rare, these events can play a significant role in the dynamics.
The most notable result when comparing the four different cases is that the
dynamo run has weaker tails than the other flows. It turns out hence that the
dynamo dynamics suppresses efficiently extreme energy flux fluctuations. Small
differences are experienced between the moderate and strong cases, indicating
that fluid fluctuations are not affected by the large helical magnetic field,
which is present only in the strong case. It is possible to observe just a
slight increase in the probability of negative events with respect to positive
ones. Still, not much difference is found neither comparing the
moderate/strong cases to the hydro case.

The statistical behaviour of the energy flux $\Pi_{\ell}$ is further studied
in Figure 4. Here we present the probability density function of the flux
$\Pi_{\ell}$ at different scales $\ell=1/q$, for the 4 different cases. The
hydro results are given for comparison. The first remark is that as said
above, the dynamo case is quite different from the others, with more shrunk
distributions, though the distribution remains tailed and skewed. All the
cases displays similar changes with going towards smaller scales. The profiles
are much less wide in the large-scale range $q=8,16$, notably for $q=8$. Then,
the distributions are basically indistinguishable for $q=32,64$, which are
more or less in the far inertial range, and finally the tails are again
reduced for $q=128$, which is in the dissipation range. As expected,
intermittency is maximum at the end of the inertial range. Slight differences
between the moderate and the strong case can be detected, yet not significant.

Figure 5: The Different components of the Energy flux for $q=32$. The total
flux is also plotted for information. The three MHD cases are considered.

Now, we turn our attention to the four different components of the total
energy flux $\Pi_{\ell}(x)$. In Figure 5. We compare all the terms
$\Pi_{\ell}^{uu}(x)$, $\Pi_{\ell}^{ub}(x)$, $\Pi_{\ell}^{bu}(x)$,
$\Pi_{\ell}^{bb}(x)$ composing $\Pi_{\ell}$ as given by Eqs. (16)-(19). The
fluxes show different behavior in the three different cases examined.

The first striking observation is that in all cases the pure hydrodynamic
flux, is the smallest one. Most notably, it shows the least important negative
fluctuations. It was not possible to infer that from looking only at the total
flux $\Pi_{\ell}$, which is found in Fig. 2 to be similar in the hydro and all
the MHD cases. In the moderate and strong cases, the hydrodynamic flux
$\Pi_{\ell}^{uu}$ is basically the same as the cross magnetic-hydro flux
$\Pi_{\ell}^{ub}$, while the two are different for the dynamo case. As a
consequence, in all cases the terms found to be dominant are $\Pi^{bu}_{\ell}$
and $\Pi^{bb}_{\ell}$, that is the blue and red lines. Therefore, the coupling
magnetic term contributing to the kinetic energy flux dominates the pure hydro
term. Since the total flux in the hydro case is much similar, except for the
dynamo case, it appears that the cascade readjusts to distribute much of the
flux in this term, although the total kinetic flux remains unchanged. It is
interesting to remark also that in the strong case, where both the velocity
and the magnetic fields are fully developed, there is a nice symmetry between
the two fluxes, with $\Pi_{\ell}^{uu}\approx\Pi_{\ell}^{ub}$, and
$\Pi_{\ell}^{bu}\approx\Pi_{\ell}^{bb}$. That points out to two very similar
cascade processes  and could be related to the equipartition of the magnetic
and kinetic energy cascade recently conjectured in Bian & Aluie (2019). While
for the moderate case this symmetry appears also holding apart from the
extreme events, the dynamo case breaks it underlying different physics of the
cascade process.

Finally, it is useful to note that all the terms, and hence the total flux,
are production terms, that is the flux is in average positive towards smaller
scales, even though there may be a significant local negative flux. We note
also that in the dynamo case there is a mean transfer of kinetic energy to
magnetic energy both by the $\Pi^{bb}_{\ell}$ flux and the $\mathcal{W}_{L}$
term in the large scales (that we do not examine here). This may alter the
transfer balance between the different cases. Concerning the negative fluxes,
it is important to remark that, unlike the positive tails, the negative parts
of the pdfs are basically the same for all components. This is most notably
true for the strong case, but also in the other two cases the differences are
limited to the region of extremely rare events, where statistical errors may
be substantial. That shows that a change in large scale amplitude of the
magnetic field can affect the smaller scale forward cascade process.

##  6 Joint pdfs

###  6.1 Field Gradients

(a) Hydro q=32

(b) Dynamo q=32

(c) Moderate q=32

(d) Strong q=32

Figure 6: Joint pdfs of $\Pi_{\ell}$ (for $q=32$) with the modulus of the
different strain rates for (a) hydrodynamic case (b) dynamo case (c) moderate
case and (d) strong case. From left to right the examined strain is
$\Omega_{u,\ell}^{2},S_{u,\ell}^{2},\Omega_{b,\ell}^{2},S_{b,\ell}^{2}$.
Bright colors indicate high probability. The yellow dashed line indicates the
Smagorinsky scaling $\Pi_{\ell}\propto\Omega_{\ell}^{3/2}$ and
$\Pi_{\ell}\propto S_{\ell}^{3/2}$.

In order to help the construction of subgrid scale models that are based on
the gradients of the resolved fields we need to reveal what correlation exists
between the gradients and the local energy flux. In figure 6 we present the
joint pdf between $\Pi_{\ell}$ and the modulus squared of the symmetric and
anti-symmetric stress tensors
$\Omega_{u,\ell}^{2},\Omega_{b,\ell}^{2},S_{u,\ell}^{2},S_{b,\ell}^{2}$ The
results are for a given wavenumber in the inertial range $q=32$.

The results obtained for the pure hydrodynamic configuration are shown for
comparison in the top panel. In this case, the energy flux is essentially
uncorrelated with the anti-symmetric part of the strain $\Omega_{\ell}$ with
high probability events for a given $\Omega_{\ell}$ are concentrated around
$\Pi_{\ell}=0$. On the other hand, a visible correlation is observed with the
modulus of the symmetric part of the strain tensor $S_{\ell}$, with high
probability events for a given $S_{\ell}$ concentrated around a non zero value
of $\Pi_{\ell}$ that increases with $S_{\ell}$. The yellow line corresponds to
the Smagorinsky scaling of eq. (27). The flux $\Pi_{\ell}$ is concentrated
around this line giving support to the Smagorinsky model.

Such correlations are less clear for the MHD results. For the dynamo case in
the $\Pi_{\ell}-\Omega_{u\ell}$ diagram high-probability events are
concentrated around $\Pi_{\ell}=0$ again. For the $\Pi_{\ell}-S_{u\ell}$
diagram a much weaker correlation is observed compared to the hydro case. The
probability density shows long tails with large $\Pi_{\ell}$ for small
$S_{u\ell}$ indicating that $S_{u\ell}$ is not dominant in driving the
cascade. A stronger correlation is observed with $\Omega_{b,\ell}^{2}$ and
$S_{b,\ell}^{2}$. High probability events are aligned with the Smagorinsky
scaling (yellow dashed line) but the spread is much higher than in the hydro
case. The fact that the correlation with $\Omega_{b,\ell}^{2}$ and
$S_{b,\ell}^{2}$ look very similar indicates that neither of the two alone is
an optimal proxy to estimate the energy flux to the small scales.

The lack of correlation of $\Pi_{\ell}$ with the velocity gradients is even
more clear for the moderate and strong case. For these cases no correlation is
seen neither with $\Omega_{u,\ell}^{2}$ nor with $S_{u,\ell}^{2}$. Although
there is some correlation with the magnetic field gradients
$\Omega_{b,\ell}^{2}$ and $S_{b,\ell}^{2}$ with the maximum of probability
(for fixed $\Omega_{b,\ell}^{2}$ or $S_{b,\ell}^{2}$) following the
Smagorinsky scaling (yellow dashed line), the spread is very large with highly
probable negative flux events that become more frequent for the strong field
case. This indicates that although there is some correlation with the magnetic
field gradients, modeling the subscale stresses with $\Omega_{b,\ell}^{2}$ and
$S_{b,\ell}^{2}$ alone is still missing important ingredients.

###  6.2 Local magnetic field

We pursue the analysis of the statistical phenomenology of the cascade process
looking at correlations between the flux and the magnetic field. Figure 7
shows the joint PDF of $\Pi_{\ell}$ and $\widetilde{\bf b}$ for the different
cases examined. The three cases display different distributions, with yet
similar characteristics. Generally speaking, maximum probability of the flux
as well as most extreme values of it are obtained in the region around the
average value of the magnetic field. Furthermore, the larger the average
magnetic field, the wider the distribution. In this concern, it is interesting
to note that the isolines are almost flat over a large range, indicating that
in such large region the flux distribution, and therefore the cascade process
is basically independent of the value of the magnetic field. That is
particularly true for the strong case.

Figure 7:  Joint pdf of the local flux $\Pi$ and the local amplitude of the
magnetic field $b$ for the three cases; dynamo (left), moderate (center) and
strong (right).

Figure 8: Conditioned pdf ie Probability of having flux $\Pi_{\ell}$ given the
magnetic field amplitude $|{\bf b}|$. In dynamo large $|{\bf b}|$ enhances
large fluctuations.

More insight is gained by looking at the pdf of the energy flux conditioned on
the knowledge of the magnetic field amplitude. This corresponds at looking at
“slices” of Figure 7 for fixed values of $\widetilde{\bf b}$ and allows to
have a more clear look at the extreme events. The results are displayed in
Fig. 8. Seen this way some differences can be noted between the three
different cases. For the Dynamo there is a monotonic increase of the tails of
the distribution with respect to increasing the magnetic field. Both negative
and positive tails are wider when related to a larger magnetic field
amplitude. In this case large values of $\widetilde{\bf b}$ indicate more
extreme events.

In the moderate case, the probability distribution seems to be almost
insensitive to changes in the magnetic field amplitude. The distribution would
point to an energy cascade somewhat independent of the magnetic field
amplitude. Some small differences are actually present in the distributions,
but they are not much more important than statistical errors.

For the strong case, the negative tails increase monotonically with the
amplitude of the magnetic field. The positive tails, on the other hand are
monotonically decreased by increasing the magnetic field. Therefore, in
strongly magnetized flows, large $|{\bf b}|$ reduces extreme direct cascade
fluctuations, while increases inverse cascade fluctuations.  In this sense,
the skewness of the distribution seems to be affected by the amplitude of the
magnetic field. It appears that in regions of very large magnetic field the
distribution is made symmetric. Possibly this behavior occurs because locally
in regions of large $\widetilde{\bf b}$ turbulence transitions to a wave
turbulence regime that has weaker forward energy flux or by rendering the flow
locally quasi-2D. This effect however would require further investigation to
be fully understood, and to assess possible finite-size effects.

##  7 Conclusions

In this work we have analysed the statistics of the local energy flux for
three different magneto-hydrodynamic flows. A scale-by-scale analysis in
physical space has been carried out that allows us to access relevant
information on the energy cascade process that could help the construction of
sub-grid scale models.

First, we have found that a Gaussian filter should be preferred to a spectral
one with regard to the representation of the Energy fluxes, a result
previously obtained for fluid turbulence. Then, the results of the total
energy flux showed some similarities with the hydrodynamic case with
$\Pi_{\ell}$ showing a skewed distribution with large tails. However, the
dynamo case displays clear differences with respect to the other two cases.
This is actually a general feature detected for all observables. On the other
hand, moderate and strong cases display overall similar trends and profiles,
as exposed by the fluxes computed at different scales.

The analysis of the different components of the energy flux has highlighted
that the magnetic field plays a dominant role in the forward cascade.
Specifically, the term related to the work made against the sub-scale Reynolds
stress is negligible. This is a key information for modelling. The negative
part of the flux, that is the back-scatter flux, is instead basically the same
for all components. Moreover, the study of the correlation between the energy
flux and the different strain and rotation tensors has indicated that: (i)
while results give some support to Smagorinski-like closure, at variance with
pure hydrodynamic turbulence MHD energy flux cannot be globally recovered with
a simple formula and several eddy viscosities should be used; (ii) as pointed
out in previous works, the vorticity tensor has little correlation and can be
omitted from modelling, at least as a first approximation; (iii) contrarily to
what is commonly used, we have found that the magnetic strain tensor
$S_{b,\ell}$ contribution is not negligible with respect to the anti-symmetric
rotation $\Omega_{b,\ell}$ one, with regard to the entire flux, so both
$S_{b,\ell}$ and $\Omega_{b,\ell}$ need to be used to model subscale stresses.
Finally (iv) we have found that the presence of a large magnetic field has an
impact on the local shape of the energy flux, and the flux in regions of very
high magnetic field has reduced forward fluctuations.

Present results indicate new directions for optimizing subgrid scale models in
MHD turbulent flows. Many open questions still remain that future research
should address. In particular the correlation between the local energy flux
$\Pi_{\ell}$ and physically motivated combinations of the tensors
$\Omega_{u,\ell},\Omega_{b,\ell},S_{u,\ell},S_{b,\ell}$ would be a next step.
Theoretical work in the same spirit as in the Clark model (Meneveau & Katz,
2000) can also help in this direction. We plan to address these issues in our
future work.

###### Acknowledgements.

This work was granted access to the HPC resources of MesoPSL financed by the
Région Ile de France and the project Equip@Meso (reference ANR-10-EQPX-29-01)
of the programme Investissements d’Avenir supervised by the Agence Nationale
pour la Recherche and the HPC resources of TGCC & CINES (allocations No.
A0090506421 A0110506421 & No. A0062B10759) attributed by GENCI (Grand
Equipement National de Calcul Intensif) where the present numerical
simulations were performed. This work was also supported by the Agence
nationale de la recherche (ANR DYSTURB project No. ANR-17-CE30-0004)

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