Fluid Dynamics, Statistical Mechanics

A. Alexakis

#  $\lambda$-Navier-Stokes turbulence

A. Alexakis1 and L. Biferale2 1 Laboratoire de Physique de l’Ecole normale
supérieure, ENS, Université PSL, CNRS, Sorbonne Université, Université de
Paris, F-75005 Paris, France
2 Department of Physics and INFN University of Rome ‘Tor Vergata’, Via della
Ricerca Scientifica 1, 00133 Rome, Italy<EMAIL_ADDRESS>
###### Abstract

We investigate numerically the model proposed in [1] where a parameter
$\lambda$ is introduced in the Navier-Stokes equations such that the weight of
homochiral to heterochiral interactions is varied while preserving all
original scaling symmetries and inviscid invariants. Decreasing the value of
$\lambda$ leads to a change in the direction of the energy cascade at a
critical value $\lambda_{c}\sim 0.3$. In this work, we perform numerical
simulations at varying $\lambda$ in the forward energy cascade range and at
changing the Reynolds number $\mathrm{Re}$. We show that for a fixed injection
rate, as $\lambda\to\lambda_{c}$, the kinetic energy diverges with a scaling
law ${\mathcal{E}}\propto(\lambda-\lambda_{c})^{-2/3}$. The energy spectrum is
shown to display a larger bottleneck as $\lambda$ is decreased. The forward
heterochiral flux and the inverse homochiral flux both increase in amplitude
as $\lambda_{c}$ is approached while keeping their difference fixed and equal
to the injection rate. As a result, very close to $\lambda_{c}$ a stationary
state is reached where the two opposite fluxes are of much higher amplitude
than the mean flux and large fluctuations are observed. Furthermore, we show
that intermittency as $\lambda_{c}$ is approached is reduced. The possibility
of obtaining a statistical description of regular Navier-Stokes turbulence as
an expansion around this newly found critical point is discussed.

###### keywords:

Turbulence

##  1 Introduction

In a turbulent flow energy is injected at large scales and dissipated at much
smaller scales by viscosity [2]. A transfer of energy is thus required from
one scale to the other that is achieved by the energy cascade caused by the
non-linearity of the Navier-Stokes equations. The fundamental idea of the
energy cascade across scales was first introduced by Richardson (1922) [3] and
later quantified by Kolmogorov [4]. Under the assumption of scale-similarity,
Kolmogorov predicted a power-law behaviour for the energy spectrum
$E(k)\propto\epsilon^{2/3}k^{-5/3}$ and for the scaling of the moments of
velocity’s differences across a distance $r$:
$\langle|\delta_{r}u|^{p}\rangle\propto(\epsilon r)^{p/3}$. However,
overwhelming experimental and numerical evidence have shown that the process
of transferring energy from one scale to the other is not self-similar and
that there exist anomalous exponents such that
$\langle|\delta_{r}u|^{p}\rangle\propto(\epsilon r)^{\zeta_{p}}$ with
$\zeta_{p}\neq p/3$. This is true for all moments except for the third moment
of the longitudinal velocity difference where $\zeta_{3}=1$ is exact. This
departure from self-similarity has been phenomenologically explained in terms
of multifractal theory and intermittency [5]. There have been various attempts
to predict and explain the observed exponents, see e.g. [6, 7, 8, 9]. However,
all these attempts are based on simplified phenomenological assumptions and no
exact or systematic derivation of the anomalous corrections directly from the
Navier-Stokes equations has been proposed so far. As a result, the existence
of exactly solvable limits from where to develop perturbative or asymptotic
expansions has been long sought. The main theoretical obstacle to attack three
dimensional turbulent flows comes from being out-of-equilibrium, with
anomalous scaling laws and stronger and stronger non-Gaussian small-scales
statistics at increasing Reynolds numbers. In general no universal recipes
exist for the treatment of out-of-equilibrium problems. Equilibrium Gaussian,
or quasi-Gaussian systems have predictable statistics but are only met in
fluid dynamics for the truncated Euler equations where only a finite number of
Fourier modes are kept. In this case, energy is conserved exactly and no
finite energy flux through scales exists [10, 11, 12]. Although an expansion
from such a state to a weakly cascading case can be performed [13] it seems
unlikely to serve as a starting point to recover regular Navier-Stokes
turbulence. Solvable out-of-equilibrium states over which an expansion could
be carried out have been sought with the use of re-normalization group theory.
Some of such studies consider deviations from a power-law force spectrum
$F\propto k^{-d+(4-\epsilon)}$, where $\epsilon=d-2$ (with $d=3$ the
dimension) corresponds to a molecular background noise, representing
fluctuations in an equilibrium fluid at absolute temperature and $\epsilon=4$
corresponds to real turbulence [14, 15, 16, 17]. Other studies consider
expansions from critical dimensions $d\gg 3$ where turbulence is conjectured
to follow mean field dynamics, or from $d=4/3$ where the finite flux spectrum
coincides with thermal spectrum, or at changing the couplings among triads in
Fourier space [18, 19, 20, 21, 22, 23, 24, 25]. The effect of helicity has
also been investigated with these techniques and found to play a minor role
[26, 27, 28].  Higher or non-integer dimensions however are not physically
realisable and can be studied and tested with the use of numerical simulations
only [29, 30, 31, 32, 33].

More recently, systems where one dimension is compactified were demonstrated
to result it a transition from three-dimensional behavior with a forward
energy cascade, towards a two-dimensional behavior where energy cascades
backward [34, 35, 36]. In this kind of transitions, the systems dimension $d$
does not vary continuously from $d=3$ to $d=2$ as in the previous
considerations but have the advantage of being physically realizable. Similar
transitions have been observed in a variety of physical systems at varying
some control parameters, including the rotation intensity, magnetic fields,
stratification and forcing properties. For a recent review see [37]. In most
of these cases the transition from forward to inverse cascade occurred through
a split state where both fluxes exist simultaneously. Recently, a variant of
the Navier-Stokes equations was introduced, with a dimensionless control
parameter, $\lambda$, weighing the relative importance of homochiral and
heterochiral triads and developing an abrupt transition from forward to
inverse cascade at a critical value $\lambda_{c}$ [1]. Right at the critical
value a new singular state exist where energy doesn’t cascade neither forward
or inverse without the flow being necessarily at equilibrium.

Here, we study in greater detail the behavior of the system when $\lambda$ is
close but above this critical value. We argue that as the critical point is
approached the flow is closer and closer to a flux-loop state where the mean
energy flux towards the small scales is subdominant and large turbulent
fluctuations that transfer energy both to large and small scales develop. We
also demonstrate that intermittency in our model is reduced as
$\lambda\to\lambda_{c}$. This could indicate that the flux-loop state at
$\lambda=\lambda_{c}$ has a more tractable statistics and could serve as
starting point for perturbative expansion towards real Navier-Stokes
turbulence at $\lambda=1$.

##  2 Formulation

###  2.1 Helical decomposition

Let ${\bf u}(t,{\bf x})$ be a zero-mean divergence-free vector field defined
in a cubic triple periodic domain of side $L$. Its Fourier transform
$\tilde{\bf u}_{\bf k}(t)$ is given by

$\tilde{\bf u}_{\bf k}(t)=\frac{1}{L^{3}}\int{\bf u}(t,{\bf x})\,e^{-i{\bf
k}\cdot{\bf x}}dx^{3}\quad\mathrm{and}\quad{\bf u}(t,{\bf x})=\sum_{{\bf
k}}\tilde{\bf u}_{\bf k}(t)e^{i{\bf k}\cdot{\bf x}},$ (1)

where the three component complex vector $\tilde{\bf u}_{\bf k}$ satisfies
${\bf k}\cdot\tilde{\bf u}_{\bf k}=0$ due to the divergence-free condition.
Thus, each $\tilde{\bf u}_{\bf k}$ has two independent degrees of freedom. A
convenient way to express these degrees of freedom is using the helical mode
decomposition [38, 39, 40], where $\tilde{\bf u}_{\bf k}(t)$ is decomposed in
two helical modes

$\tilde{\bf u}_{\bf k}(t)=\tilde{u}_{\bf k}^{+}(t){\bf h}_{\bf
k}^{+}+\tilde{u}_{\bf k}^{-}(t)(t){\bf h}_{\bf k}^{-}.$ (2)

Here, $\tilde{u}_{\bf k}^{\pm}(t)$ are two independent complex scalar
amplitudes and the orthogonal unit vectors ${\bf h}^{\pm}_{\bf k}$ are given
by:

${\bf h}^{\pm}_{\bf k}=\frac{{\bf k}\times{\bf k}\times\hat{\bf
e}}{\sqrt{2}k|{\bf k}\times\hat{\bf e}|}\pm i\frac{{\bf k}\times\hat{\bf
e}}{\sqrt{2}|{\bf k}\times\hat{\bf e}|}$ (3)

where $\hat{\bf e}$ is an arbitrary vector non-parallel to ${\bf k}$. The
vectors ${\bf h}^{\pm}_{\bf k}$ are eigen-functions of the curl satisfying
$i{\bf k}\times{\bf h}^{\pm}_{\bf k}=\pm k{\bf h}^{\pm}_{\bf k}$ with $k=|{\bf
k}|$ and ${\bf h}^{s_{1}}_{\bf k}\cdot({\bf h}^{s_{2}}_{{\bf k}})^{*}={\bf
h}^{s_{1}}_{\bf k}\cdot{\bf h}^{-s_{2}}_{{\bf k}}={\bf h}^{s_{1}}_{\bf
k}\cdot{\bf h}^{s_{2}}_{-{\bf k}}=\delta_{s_{1},s_{2}}$ (where $s_{i}=\pm 1$
and $\delta_{s_{1},s_{2}}$ is the Kronecker delta). Using this decomposition
we can split the real vector field ${\bf u}(t,{\bf x})$ in two helical fields
as ${\bf u}(t,{\bf x})={\bf u}^{+}(t,{\bf x})+{\bf u}^{-}(t,{\bf x})$ with
${\bf u}^{\pm}(t,{\bf x})$ given by

${\bf u}^{\pm}(t,{\bf x})=\sum_{\bf k}\tilde{u}^{\pm}_{\bf k}(t){\bf
h}^{\pm}_{\bf k}e^{i{\bf k}\cdot{\bf x}}$ (4)

The velocity field ${\bf u}(t,{\bf x})$ is evolved in time based on the
Navier-Stokes equations (NSE)

$\partial_{t}{\bf u}+{\bf u}\cdot\nabla{\bf u}=-\nabla P+\nu\nabla^{2}{\bf
u}+\bf f,$ (5)

where $P$ is the pressure enforcing incompressibility $\bf\nabla\cdot{\bf
u}=0$, $\nu$ is the viscosity and $\bf f$ is an external body force. In
particular, using the helical decomposition the Navier-Stokes can be written
as

$\partial_{t}{\bf u}^{s_{1}}=\mathbb{P}^{s_{1}}\left[\sum_{s_{2},s_{3}}({\bf
u}^{s_{2}}\times{\bf w}^{s_{3}})\right]+\nu\nabla^{2}{\bf u}^{s_{1}}+\bf
f^{s_{1}},$ (6)

where $s_{i}=\pm 1$, ${\bf w}^{\pm}={\bf\nabla}\times{\bf u}^{\pm}$,
$\mathbb{P}^{\pm}$ is a projector to the helical base

$\mathbb{P}^{\pm}\left[{{\bf u}(t,{\bf x})}\right]=\sum_{\bf k}{\bf
h}^{\mp}_{\bf k}\cdot\tilde{\bf u}_{\bf k}(t)e^{i{\bf k}\cdot{\bf x}}$ (7)

and ${\bf f}^{s_{1}}=\mathbb{P}^{s_{1}}\left[{\bf f}\right]$. Note that the
helical base is an incompressible base and projecting in to it eliminates the
pressure. From the eight nonlinear terms that appear in eq.6, one for each
sign combination $s_{1},s_{2},s_{3}$, the six that involve different signs are
responsible for transferring energy to the small scales while the two
homochiral terms $s_{1}=s_{2}=s_{3}=\pm 1$ transfer energy in the large scales
[41, 42]. In the absence of forcing, viscosity (and in the absence of
singularities) the evolution of ${\bf u}(t,{\bf x})$ conserves two ideal
invariants the energy ${\mathcal{E}}$ and the helicity ${\mathcal{H}}$:

${\mathcal{E}}=\frac{1}{2}\int|{\bf u}|^{2}dx^{3}=\frac{1}{2}\sum_{s}\sum_{\bf
k}|\tilde{u}^{s}_{\bf
k}|^{2},\quad\mathrm{and}\quad{\mathcal{H}}=\frac{1}{2}\int{\bf u}\cdot{\bf
w}dx^{3}=\frac{1}{2}\sum_{s}\sum_{\bf k}sk|\tilde{u}^{s}_{\bf k}|^{2},$ (8)

It is worth noting that while ${\mathcal{E}}$ is a positive quantity, the
helicity can take either sign.

###  2.2  Homochiral Navier-Stokes

When only the homochiral terms are kept in the NSE the system reduces to

$\partial_{t}{\bf u}^{s}=\mathbb{P}^{s}\left[({\bf u}^{s}\times{\bf
w}^{s})\right]+\nu\nabla^{2}{\bf u}^{s}+\bf f^{s}.$ (9)

In this case the two helical fields ${\bf u}^{\pm}$ evolve independently, with
the non-linearity conserving their energy and helicity

${\mathcal{E}}^{\pm}=\frac{1}{2}\sum_{\bf k}|\tilde{u}^{\pm}_{\bf
k}|^{2},\qquad{\mathcal{H}}^{\pm}=\frac{1}{2}\sum_{\bf k}\pm
k|\tilde{u}^{\pm}_{\bf k}|^{2}.$ (10)

However, unlike the Navier Stokes equation, in the homochiral version the two
helicities are sign definite quantities with ${\mathcal{H}}^{+}>0$ and
${\mathcal{H}}^{-}<0$. The sign definiteness of the helicity has a profound
impact on the cascade. As shown in [43, 44], it leads to an inverse cascade.
In fact when ${\mathcal{H}}^{\pm}$ are sign definite one can show that a
simultaneous forward cascade of ${\mathcal{E}}^{s}$ and ${\mathcal{H}}^{s}$ is
incompatible (using similar arguments to Fjortoft [45] for the dual cascade of
energy and enstrophy in two dimensions, see [37] sec. 3.5). On the contrary,
for the original Navier-Stokes case ($\lambda=1$), and in the presence of a
large scale helical forcing, the Energy and Helicity cascades are observed to
be forward as originally proposed in [46] and later verified numerically [47,
42].

###  2.3 The $\lambda$-Navier-Stokes model

The different roles played by the homochiral and heterochiral interactions
lead Sahoo et al [1] to propose a model that transitions from the forward
cascading Navier-Stokes 6 to the inverse cascading homochiral Navier-Stokes 9
varying continuously a parameter $\lambda$. In detail, the model reads:

$\partial_{t}{\bf u}^{s}=\mathbb{P}^{s}\left[({\bf u}^{s}\times{\bf
w}^{s})\right]+\lambda\mathbb{P}^{s}\left[({\bf u}^{-s}\times{\bf
w}^{s})+({\bf u}^{s}\times{\bf w}^{-s})+({\bf u}^{-s}\times{\bf
w}^{-s})\right]+\nu\nabla^{2}{\bf u}^{s}+\bf f^{s},$ (11)

For $\lambda=1$ homochiral and heterochiral terms are balanced so that one
recovers the Navier-Stokes equation 6 where energy cascades forward. For
$\lambda=0$, the heterochiral terms are eliminated and the system reduces to
the homochiral NSE 9. For any finite value of $\lambda$ the system 11 has
exactly the same ideal invariants as the NSE
${\mathcal{E}}={\mathcal{E}}^{+}+{\mathcal{E}}^{-}$ and helicity
${\mathcal{H}}={\mathcal{H}}^{+}+{\mathcal{H}}^{-}$. As for the original NSE,
${\mathcal{H}}$ is not a sign definite quantity and thus it poses no
restriction in the direction of the energy cascade. One can not thus trivially
predict the direction of the energy transfer when $\lambda\neq 0$.

In [1] it was shown that as the parameter $\lambda$ was varied from $1$ to $0$
a change of the cascade direction was observed from a forward to an inverse
cascade. In the limit of infinite Reynolds number this transition was shown to
converge to a critical discontinuous transition at a critical value
$\lambda_{c}\simeq 0.3$ such that for $\lambda<\lambda_{c}$ all injected
energy cascades to large scales while for $\lambda>\lambda_{c}$ all energy
cascades to the small scales.

##  3 Numerical set-up

In the present work we are going to investigate the limit
$\lambda\to\lambda_{c}$ from above $\lambda>\lambda_{c}$. To do that, we
perform numerical simulations of the $\lambda$-Navier-Stokes system 11 in a
triple periodic cubic domain of size $2\pi L$. The velocity field ${\bf u}$ is
evolved using a pseudo-spectral code with 2/3 dialiasing and a second order
Runge-Kutta method for the time advancement. We use a uniform grid with $N$
grid points in each direction. The values of $N$ used varied from $N=128$ to
$N=1024$ depending on the Reynolds number used.

In the examined range of $\lambda>\lambda_{c}$ the cascade is forward so we
pick the forcing to act only on Fourier modes that lie inside a sphere of
radius $k_{f}=2/L$. The phases of the forced Fourier modes $\tilde{\bf f}_{\bf
k}$ are changed randomly at every time step so that the forcing is delta
correlated in time and injects energy on average at a fixed rate denoted here
by $\epsilon$, and with zero helicity injection. Given the input parameters of
our system, the only other non-dimensional number, besides $\lambda$, is the
$\epsilon$-based Reynolds number:

$\mathrm{Re}_{\epsilon}\equiv\frac{\epsilon^{1/3}L^{4/3}}{\nu}.$ (12)

Small resolution runs $N\leq 256$ started from random initial data and were
evolved until a statistically steady state is reached where all quantities
fluctuate around a mean value and the energy injection is balanced by the
energy dissipation. Larger resolution runs $N\geq 512$ started with initial
conditions obtained from smaller resolution runs extrapolated to the new grid.
They were then evolved until a statistically steady state is reached.

A table of the parameters of our runs is given in table 1. Let us also stress
that simulations with values of $\lambda$ on the other side of the transition
$\lambda<\lambda_{c}$ are hindered from the fact that in such a range one
would need to fully resolve also the inverse energy cascade range and this is
by far too demanding for the scopes of this work.

$\lambda=$ |  0.30 | 0.35 |  0.40 | 0.45 | 0.50 | 0.60 | 0.80 | 1.00
---|---|---|---|---|---|---|---|---
$\mathrm{Re}_{\epsilon}=500$ | $128$ | $128$ | $256$ | $\,256\,$ | $\,256\,$ | $\,256\,$ | $\,256\,$ | $\,256\,$
$\mathrm{Re}_{\epsilon}=840$ | $128$ | $256$ | $256$ | $256$ | $256$ | $256$ | $512$ | $512$
$\mathrm{Re}_{\epsilon}=2500$ | - | $512$ | $512$ | - | - | $512$ | $1024$ | $1024$
$\mathrm{Re}_{\epsilon}=6300$ | - | $1024$ | $1024$ | - | - | $1024$ | - | -
Table 1: Resolutions used for all the simulations performed. It is worth
noting that values of $\lambda$ closer to the critical value
$\lambda_{c}\simeq 0.3$ require less resolution for the same value of
$\mathrm{Re}_{\epsilon}$ but require longer computational time to converge to
a statistically steady state.

##  4 Results

###  4.1 Energy balance relations

Figure 1 shows the time evolution of the total energy, ${\mathcal{E}}$, for
the smallest value of $\mathrm{Re}_{\epsilon}=500$ examined and for different
values of $\lambda$. As the value of $\lambda$ approaches its critical value
$\lambda_{c}\simeq 0.3$ the mean (time averaged) energy is increased, as also
are the fluctuations around the mean value. This behavior could in part be
anticipated since by taking the limit $\lambda\to\lambda_{c}$ we reduce the
efficiency of the flow to transport energy to the small scales so the
amplitude of the turbulent fluctuations has to increase to compensate this
lack of efficiency and maintain the flux of energy to the small scales fixed
and equal to the injection rate $\epsilon$.

Figure 1:  Evolution of energy, ${\mathcal{E}}$, as a function of time for
$\mathrm{Re}_{\epsilon}=500$ and different values of $\lambda$.

In the left panel of figure 2 we show the time averaged energy as a function
of $\lambda$ for the different values of $\mathrm{Re}_{\epsilon}$. Different
symbols are used for the different values of $\mathrm{Re}_{\epsilon}$ as
indicated in the legend. For large values of $\lambda$ the amplitude of the
mean energy is practically independent on the value of
$\mathrm{Re}_{\epsilon}$ (in the range examined) and is weakly dependent on
$\lambda$. However as $\lambda$ is decreased close to $\lambda_{c}$ the mean
energy increases displaying a divergence at $\lambda_{c}$. Close to
$\lambda_{c}$ the mean energy strongly depends on the value of
$\mathrm{Re}_{\epsilon}$, increasing as $\mathrm{Re}_{\epsilon}$ is increased.
This implies that for values of $\lambda$ close to $\lambda_{c}$ we have not
yet reached the asymptotic state $\mathrm{Re}_{\epsilon}\to\infty$ where
energy saturation is independent on the value of viscosity.

Figure 2:  Left: Energy of the flow at steady state as a function of the
parameter $\lambda$. Different symbols correspond to different Reynolds
numbers. The dashed lines gives the prediction 14. The inset shows the same
data as a function of $\lambda-\lambda_{c}$ in a log-log scale. Right:
Normalized energy dissipation rate ${\mathcal{C}}_{\mathcal{D}}$ as a function
of $\lambda$.

An alternative way to plot the same data is to study the ratio

${\color[rgb]{0,0,0}{\mathcal{C}}_{\mathcal{D}}=\frac{2\pi
L\,\epsilon}{U^{3}}}$ (13)

where $U$ is the root mean square value of the velocity
$U=\sqrt{2{\mathcal{E}}}$. This ratio expresses the efficiency of turbulent
fluctuations of a given amplitude to cascade energy to the small scales and
sometimes it is called the normalised dissipation rate. It is a fundamental
property of turbulence that ${\mathcal{C}}_{\mathcal{D}}$ remains finite in
the $\mathrm{Re}_{\epsilon}\to\infty$ limit, resulting in finite dissipation
of energy in the zero viscosity limit. This quantity is plotted in the right
panel of fig. 2. For $\lambda=1$, ${\mathcal{C}}_{\mathcal{D}}\simeq 0.4$ that
is close to reported values ([48]) but is decreasing as $\lambda$ is
decreased. The data indicate that it linearly approaches zero as
$\lambda\to\lambda_{c}$, so that
${\mathcal{C}}_{\mathcal{D}}\propto(\lambda-\lambda_{c})$. Right at the
critical point $\lambda=\lambda_{c}$ the flow is inefficient to cascade the
energy to the small scales. At this critical point, the flow evolution is
limited only by viscous effects at the forcing scale to saturate the energy
injection and the amplitude of the fluctuations would diverge in the $\nu\to
0$ limit.

The linear approach to zero can be reinterpreted to find the divergence
observed in the energy in the left panel of 2 as

${\mathcal{E}}\propto\frac{\epsilon^{2/3}}{(\lambda-\lambda_{c})^{2/3}}.$ (14)

The dashed line in this figures shows that indeed this scaling is compatible
with the data.

There are a few comments that should follow the result in eq.14. First of all,
we should stress again that the increase in the energy as
$\lambda\to\lambda_{c}$ is approached is due a reduced efficiency of the flow
to cascade energy to smaller scales. This has some direct consequences. If we
define the Reynolds number based on the rms velocity of the flow

$\mathrm{Re}_{{}_{U}}\equiv\frac{UL}{\nu}$ (15)

the two definitions $\mathrm{Re}_{\epsilon},\mathrm{Re}_{{}_{U}}$ are not
equivalent but
$\mathrm{Re}_{{}_{U}}\propto(\lambda-\lambda_{c})^{-1/3}\mathrm{Re}_{\epsilon}$.

We also need to comment on the two limits $\lambda\to\lambda_{c}$ and
$\mathrm{Re}_{\epsilon}\to\infty$. For any value of $\lambda>\lambda_{c}$ the
normalized dissipation rate ${\mathcal{C}}_{\mathcal{D}}$ will remain strictly
positive in the $\mathrm{Re}_{\epsilon}\to\infty$ limit. On the other hand for
$\lambda=\lambda_{c}$, where there is no cascade to the small scales, velocity
fluctuations saturate with amplitudes such that $\epsilon\propto\nu
U^{2}/L^{2}$ leading to the estimate
${\mathcal{C}}_{\mathcal{D}}\propto\mathrm{Re}_{\epsilon}^{-3/2}$ which
becomes zero at infinite $\mathrm{Re}_{\epsilon}$.

###  4.2 Spectral properties

Modifying the efficiency of the flow to cascade the energy to small scales
will affect the turbulent scale-by-scale energy budget. Further understanding
can be obtained by looking at the spectral distribution of energy. The
spherically averaged energy spectrum $E(k)$ is defined as

$E(k)=\sum_{k<|{\bf q}|<k+1}\left[|\tilde{u}^{+}_{\bf
q}|^{2}+|\tilde{u}^{-}_{\bf q}|^{2}\right]$ (16)

and expresses the amount of energy in a spherical shell in Fourier space with
unit width. We note that the $\lambda$ model used here has exactly the same
scaling symmetries as the Navier-Stokes equations. Therefore, dimensional
analysis will imply again a Kolmogorov energy spectrum $E(k)\propto k^{-5/3}$.

The four panels of figure 3 display $E(k)$ for different values of
$\mathrm{Re}_{\epsilon}$ and $\lambda$. The spectra are compensated by
$k^{5/3}$ so that a Kolmogorov spectrum would appear as flat. The $x$-axis has
been re-scaled by the Kolmogorov dissipation wavenumber

$k_{\nu}=\frac{\epsilon^{1/4}}{\nu^{3/4}}$ (17)

so that the spectra of different $\mathrm{Re}_{\epsilon}$ collapse together at
large wavenumbers.

Figure 3:  Energy spectra normalized by $k^{-5/3}$ for different Reynolds
numbers and for $\lambda$ as indicated in the legend. Horizontal lines are put
to guide the eyes and correspond to the Kolmogorov scaling.

The $\lambda=1$ case shows the typical large $\mathrm{Re}_{\epsilon}$ behavior
of a turbulent flow for which the power-law behavior $k^{-\alpha}$, (with
$\alpha\sim-5/3$), is followed by a bottleneck increase of the compensated
spectrum. The bottleneck behavior is well documented in the literature ([49,
50]) and is roughly explained as a pile up of the cascading energy when the
viscous cut-off is reached. In [51] it was argued that when a very high order
hyperviscosity is used the bottleneck is increased approaching a thermalized
state. Thermalized states manifest themselves in conservative systems like the
spectraly truncated Euler equations in which energy is equally distributed
among all modes leading to the energy spectrum $E(k)\propto k^{2}$ [12]. The
transition of hyperviscous runs to a thermalized state has been recently
demonstrated in [52]. For regular viscosity the bottleneck has thus been
interpreted as a partial thermalization.

As $\lambda$ is decreased this behavior starts to change. The bottleneck
appears to increase in amplitude and covers a wider range of wavenumbers. This
tendency is demonstrated in figure 4 where the spectra are plotted fro
different values of $\lambda$. For $\lambda=0.8$ one can still observe a
$k^{-5/3}$ range but for smaller values of $\lambda$ it is hard to observe a
power-law range with the present resolution. At the smallest values of
$\lambda$ examined $\lambda=0.35$ and $\lambda=0.40$ the bottleneck covers the
whole range of wavenumbers.Therefore as $\lambda\to\lambda_{c}$ the range of
wavenumbers which follow partial thermalisation increases. It is impossible,
within the given resolution limitations, to precisely estimate the scaling of
the extension of the bottleneck effects as a function of $\lambda$ and
Reynolds number. An understanding of why this excess of thermalisation occurs
as the critical point is approached is obtained by looking at the spectral
energy fluxes.

Figure 4:  Energy spectra normalized by $k^{-5/3}$ for the largest attained
Reynolds numbers overlapped for the different values of $\lambda$. Horizontal
lines are put to guide the eyes and corresponds to the Kolmogorov scaling.

The energy flux gives the rate that energy flows across a a particular scale.
It is defined as

$\Pi(k)=-\sum_{s}\langle{\bf u}^{<}_{k}\cdot{\bf u}\cdot\nabla{\bf u}\rangle$
(18)

where ${\bf u}^{<}_{k}$ stands for the velocity field filtered so that only
wavenumbers with norm $|{\bf k}|<k$ are kept. It has been shown [42] that can
be decomposed to a homochiral part stemming from same chirality interactions
and a heterochiral part stemming from cross-chirality interactions. The
homochiral flux is defined as

$\Pi^{homo}(k)=-\sum_{s}\langle{\bf u}^{<}_{k}\cdot\mathbb{P}^{s}\left[({\bf
u}^{s}\times{\bf w}^{s})\right]\rangle$ (19)

while the heterochiral flux is defined as

$\Pi^{hete}(k)=-\lambda\sum_{s}\langle{\bf
u}^{<}_{k}\cdot\mathbb{P}^{s}\left[({\bf u}^{-s}\times{\bf w}^{s})+({\bf
u}^{s}\times{\bf w}^{-s})+({\bf u}^{-s}\times{\bf w}^{-s})\right]\rangle.$
(20)

The total flux is equal to the sum of the two

$\Pi^{hete}(k)=\Pi^{homo}(k)+\Pi^{hete}(k).$ (21)

Figure 5: Fluxes: total time averaged (blue solid line), homochiral time
averaged (red dashed line), & heterochiral time averaged (purple dash-dot
line), instantaneous total fluxes (light blue)

In the four panels of figure 5 we show the fluxes for four different values of
$\lambda=0.35,0.4,0.6,1.0$ for the largest resolutions attained. The time
averaged total energy flux $\Pi(k)$ is shown with a dark blue solid line. It
has been decomposed to its homochiral (red dashed line) and a heterochiral
component (purple dashed-dot line). With a light blue lines the instantaneous
total energy flux is shown for several different times. In all cases the total
energy flux is positive and equal to the energy injection/dissipation rate,
while the homochiral flux is negative and the heterochiral flux is positive.
For the Navier-Stokes case $\lambda=1$ the negative homochiral flux
constitutes a small fraction of the total flux (about 10%) so that
$-\Pi^{homo}\ll\Pi^{hete}$. Small fluctuations around the time averaged value
are observed in the instantaneous fluxes. As $\lambda$ approaches the critical
value $\lambda_{c}$ the amplitude of negative homochiral flux and the positive
heterochiral flux both increase, keeping of course their sum fixed to the
injection rate. As a result, the two competing processes for the transfer of
energy to smaller and larger scales come closer together in amplitude making
their relative difference smaller and smaller. This leads to also an increase
in the amplitude of the fluctuations around the mean value observed. On one
hand, the increase of fluctuations with respect to the mean could be a
potential indication that the system is driven toward a sort of quasi-thermal
state, that could interpret the large bottleneck as partial thermalization as
it [51]. On the other hand, the existence of two counter-directional fluxes
one driving energy toward small scales and the other oppositely with clear
non-zero average and amplitudes that becomes larger and larger by approaching
$\lambda_{c}$ is an indication that the fluid remains out of equilibrium but
in a flux-loop state where finite fluxes exist that however cancel each other
[37].

###  4.3 Intermittency

A key property of Navier-Stokes turbulent cascade is the presence of
intermittency manifesting itself as an breaking of scale-similarity, with
stronger “events" appearing as smaller scales are examined. Such deviations
from scale similarity are measured by examining the scaling behavior of
structure functions of different order $S_{n}(r)=\left\langle(\delta
u_{r})^{n}\right\rangle$ where $\delta u_{r}$ stands for either the
longitudinal increment, i.e. when ${\bf r}$ is parallel to ${\bf u}({\bf
x},t)-{\bf u}({\bf x}+{\bf r},t)$ or for the transverse, when ${\bf r}$ it is
perpendicular to the velocity increment, and the brackets stand for a space
and time average. It is well known that in Homogeneous and Isotropic Navier-
Stokes three-dimensional turbulence, structure functions enjoys anomalous
scaling, $S_{n}(r)\sim r^{\zeta_{n}}$, with power-law behaviours and exponents
that depart from the Kolmogorov mean-field prediction, $\zeta_{n}\neq n/3$
[31]. This is a signature of intermittency, i.e. that normalized and
standardized probability distribution functions (PDF) of velocity increments
cannot be superposed at changing the distance $r$ and develop a stronger and
stronger departure from Gausssian statistics. As a result, by simultaneously
decreasing the scale and increasing Reynolds one can obtain turbulent states
that are further and further away from a quasi-equilibrium distribution [31].
In our simulations, the limitation on the numerical resolution, imposed by the
need to perform many different investigations for different $\lambda$ values,
and the appearance of a strong bottleneck by approaching $\lambda_{c}$ result
in the absence of a well developed scaling range. As a result, we refrain from
giving any quantitative measurements on the scaling exponents $\zeta_{n}$. On
the other hand, in figure 6 we show a dimensionless measurement of the
departure from Gaussianity by plotting the Kurtosis of the velocity increments
PDF at changing $r$ and for different $\lambda$ for the highest Reynolds
number:

$\mathcal{K}_{{}_{\delta u}}(r)\equiv\frac{\left\langle(\delta
u_{r})^{4}\right\rangle}{\left\langle(\delta u_{r})^{2}\right\rangle^{2}}.$
(22)

As one can see in the left panel, the NSE case for $\lambda=1$ shows the
classical behavior of $\mathcal{K}_{{}_{\delta u}}$ increasing by decreasing
$r$, going form the Gaussian value $\sim 3$ for $r\sim L$ to the highly non-
Gaussian and intermittent plateau at $\sim 7$ inside the viscous range, $r\to
0$ . On the other range by decreasing $\lambda$ and approaching $\lambda_{c}$
we have a strong reduction in the inertial range values and also a
corresponding reduction for the viscous plateau, indicating that the flow is
approaching a closer and closer Gaussian distribution at all scales when
$\lambda\to\lambda_{c}$. Similarly, on the right panel of the same figure we
re-plot the data by showing the values of the Flatness at changing the
distance $r/L$ and for different $\lambda$. From the latter plot there is a
clearer tendency toward the mean-field Gaussian value, $\sim 3$, by
approaching $\lambda_{c}$ even though the behaviour is not extremely well
developed. Higher Reynolds numbers are probably needed in order to enhance the
critical behaviour.

Figure 6: Left: The Kurtosis $\mathcal{K}_{{}_{\delta u}}$ of the parallel
velocity difference as a function of $r$ for different values of $\lambda$.
Right: $\mathcal{K}_{{}_{\delta u}}-3$ as a function of $\lambda-\lambda_{c}$
for different values of $r$ in a log log scale. Data are obtained from the
highest $\mathrm{Re}_{\epsilon}$ runs.  Figure 7:  Kurtosis of the vorticity
PDF $(\mathcal{K}_{{}_{w_{z}}}-3)$ as a function of the different values of
$\lambda$. The inset shows $\mathcal{K}_{{}_{w_{z}}}-3$ as a function of
$\lambda-\lambda_{c}$ in a log-log scale. Same symbols are used as in 2:  red
diamonds $\mathrm{Re}_{\epsilon}=500$, green triangles
$\mathrm{Re}_{\epsilon}=840$, blue squares $\mathrm{Re}_{\epsilon}=2500$,
purple discs $\mathrm{Re}_{\epsilon}=6200$.

Finally, because intermittency depends also on $\mathrm{Re}_{\epsilon}$ we
verify that the decrease of intermittency observed in the previously examined
figures is a trend that persists for all $\mathrm{Re}_{\epsilon}$ examined. In
figure 7 we plot the Kurtosis of the $z$ vorticity component

$\mathcal{K}_{{}_{w_{z}}}\equiv\frac{\left\langle
w_{z}^{4}\right\rangle}{\left\langle w_{z}^{2}\right\rangle^{2}}$ (23)

as a function of $\lambda$ for all examined runs. The symbols for the
different $\mathrm{Re}_{\epsilon}$ used are the same as in figure 2. All data
indicate that as $\lambda_{c}$ is approached intermittency tends to decrease.
In particular, for large values of $\lambda$ the Kurtosis is increased as
$\mathrm{Re}_{\epsilon}$ is increased, something well known in turbulence
theory due to intermittency [53, 54] while as $\lambda_{c}$ is approached, the
values appear to weakly depend on $\mathrm{Re}_{\epsilon}$ and tend to the
Gaussian value, $\to 3$ for all $\mathrm{Re}_{\epsilon}$.

##  5 Conclusions

Numerical results for a variant of the Navier-Stokes equations have been
presented. The model is characterized by a dimensionless control parameter
$\lambda$ that weights homochiral and heterochiral interactions such as to
link in a continuous way the forward cascading Navier-Stokes case for
$\lambda=1$ and the inverse cascading homochiral Navier-Stokes for
$\lambda=0$. In the investigation both $\lambda$ and the Reynolds number
$\mathrm{Re}_{\epsilon}$ were varied in order to investigate the behavior of
the system for the different values of $\lambda$ as
$\mathrm{Re}_{\epsilon}\to\infty$.

We have focused on the transition from the $\lambda=1$ case to case observed
at $\lambda_{c}\sim 0.3$ where the direct energy transfer stops and a highly
complex statistical state develops. We have shown that by approaching
$\lambda_{c}$ from above we have a tendency to develop a more and more intense
spectral viscous bottleneck. We should also note here that the limits
$\lambda\to\lambda_{c}$ and $\mathrm{Re}_{\epsilon}\to\infty$ do not
necessarily commute. I.e., when the $\lambda$ limit is taken first the
bottleneck would occupy all scales, while when
$\mathrm{Re}_{\epsilon}\to\infty$ at fixed $\lambda$ a power-law inertial
range is present. Furthermore, in the $\lambda\to\lambda_{c}$ limit larger and
large fluctuations around the mean energy flux exist whose energy is diverging
like $(\lambda-\lambda_{c})^{2/3}$, accompanied by larger and larger
heterochiral and homochiral opposite contributions to the energy cascade. As a
result, we are closer and closer to a turbulent state where the finite
counter-directing fluxes cancel each other leading to a subdominant mean
energy flux. We refer to this state as flux-loop. Such flux-loop states have
been met before in the literature in different contexts [55, 56, 57, 58, 59]
but are inadequately explored, and it is not known if methods from equilibrium
dynamics could be applied to them.

An other direction that could be followed would be the use of renormalization
group techniques for the $\lambda$-Navier-Stokes system in $d$ dimensions.
This could lead to a $\lambda$-dependence of the renormalized transport
coefficients that could change sign or the appearance of new
$\lambda$-dependent terms. Ideally an optimal path in the $(\lambda,d)$ could
exist where the $d=3$, $\lambda=1$ case could be solved for.

In the present investigation, the statistics is observed to have a decreasing
Kurtosis for both velocity increments in the inertial range and vorticity
components as the flux loop state is approached, $\lambda\to\lambda_{c}$. The
results thus indicate the possibility that the flux-loop state could have
Gaussian or quasi-Gaussian statistics. The entangled presence of non-zero
fluxes and reduction of non-Gaussian contributions opens the way to
perturbative approaches of intermittency, offering also a unique testing bed
for any new theory of turbulence because of the possibility to change the
Navier-Stokes statistics as a function of a free control parameter.

All authors participated in the analytical computations. AvK drafted the
manuscript and performed the numerical simulations. All authors read, edited
and approved the manuscript.

The authors declare that they have no competing interests.

This work was supported by Agence nationale de la recherche (ANR DYSTURB
project No. ANR-17-CE30-0004). This project has received funding also from the
European Research Council (ERC) under the European Union’s Horizon 2020
research and innovation programme (grant agreement No 882340).

LB acknowledges useful discussions with Roberto Benzi. This work was granted
access to HPC resources of MesoPSL financed by Region Ile de France and the
project Equip@Meso (reference ANR-10-EQPX-29-01) of the programme
Investissements d’Avenir supervised by Agence Nationale pour la Recherche and
HPC resources of GENCI-TGCC & GENCI-CINES (Projects No. A0080511423,
A0090506421).

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