# Novel turbulence and coarsening arrest in active-scalar fluids

Nadia Bihari Padhan<EMAIL_ADDRESS>Centre for Condensed Matter Theory,
Department of Physics, Indian Institute of Science, Bangalore, 560012, India.
Kolluru Venkata Kiran<EMAIL_ADDRESS>Centre for Condensed Matter
Theory, Department of Physics, Indian Institute of Science, Bangalore, 560012,
India.  Rahul Pandit<EMAIL_ADDRESS>Centre for Condensed Matter Theory,
Department of Physics, Indian Institute of Science, Bangalore, 560012, India.

###### Abstract

We uncover a new type of turbulence – activity-induced homogeneous and
isotropic turbulence in a model that has been employed to investigate
motility-induced phase separation (MIPS) in a system of microswimmers. The
active Cahn-Hilliard-Navier-Stokes equations (CHNS), also called active model
H, provides a natural theoretical framework for our study. In this CHNS model,
a single scalar order parameter $\phi$, positive (negative) in regions of high
(low) microswimmer density, is coupled with the velocity field $\bm{u}$. The
activity of the microswimmers is governed by an activity parameter $\zeta$
that is positive for extensile swimmers and negative for contractile swimmers.
With extensile swimmers, this system undergoes complete phase separation,
which is similar to that in binary-fluid mixtures. By carrying out
pseudospectral direct numerical simulations (DNSs), we show, for the first
time, that this model (a) develops an emergent nonequilibrium, but
statistically steady, state (NESS) of active turbulence, for the case of
contractile swimmers, if $\zeta$ is sufficiently large and negative and (b)
this turbulence arrests the phase separation. We quantify this suppression by
showing how the coarsening-arrest length scale does not grow indefinitely,
with time $t$, but saturates at a finite value at large times. We characterise
the statistical properties of this active-scalar turbulence by employing
energy spectra and fluxes and the spectrum of $\phi$. For sufficiently high
Reynolds numbers, the energy spectrum $\mathcal{E}(k)$ displays an inertial
range, with a power-law dependence on the wavenumber $k$. We demonstrate that,
in this range, the flux $\Pi(k)$ assumes a nearly constant, negative value,
which indicates that the system shows an inverse cascade of energy, even
though energy injection occurs over a wide range of wavenumbers in our active-
CHNS model.

##  I Introduction

Active turbulence, spatiotemporal chaos in active-matter systems [see, e.g.,
Refs. [1, 2, 3, 4]], has garnered considerable attention over the past decade.
This intriguing form of turbulence manifests itself in various experimental
systems, including bacterial suspensions [5, 6, 1, 7, 8, 9, 10, 11],
suspensions of microtubles, and molecular motors [12, 13]. In classical-fluid
turbulence, a nonequilibrium statistically steady state (NESS) is reached when
the fluid is driven by an external force; by contrast, in active fluids, the
microscopic constituents drive the system by converting chemical sources of
energy into kinetic energy [5, 14]. Many studies have focused on understanding
emergent turbulence-type patterns by using continuum hydrodynamical models, in
which phenomenological parameters depend on the microscopic details of the
active fluid. An overview of the various models can be found in Ref. [3]. In
certain models, the energy spectrum of such turbulence exhibits universal
power-law behaviors [15, 4]. In contrast, there are instances in which power-
law exponents for these energy spectra depend on parameters in the model [16,
7, 11]. The elucidation of the statistical properties of these types of
emergent turbulent states continues to be an important challenge in active-
matter research. Recent studies have shown the importance of fluid inertia in
some systems that display active turbulence such as active polar and nematic
fluids [see, e.g., Refs. [17, 18, 19, 20]]. In addition, considerable
attention has been directed towards the study of scalar active fluids, in
which the intricate spatiotemporal evolution of an active fluid arises from
the interaction of a scalar order parameter $\phi$ with the fluid velocity
$\bm{u}$. Scalar active fluids are simpler than their polar or nematic
counterparts, yet they are rich enough to yield intriguing emergent NESSs [21,
22, 23]; and they have been used in studying active droplets [24] and active
stratified turbulence [25].

We uncover a new type of turbulence – activity-induced homogeneous and
isotropic turbulence in a model that has been employed to investigate
motility-induced phase separation (MIPS) [22, 24] in a system of
microswimmers. The active Cahn-Hilliard-Navier-Stokes equations (CHNS), also
known as the active model H [22], provides a natural theoretical framework for
our study. In this CHNS model, a single scalar order parameter $\phi$ [which
is positive (negative) in regions where the microswimmer density is high
(low)] is coupled with the velocity field $\bm{u}$. The activity of the
microswimmers is governed by an activity parameter $\zeta$ that is positive
for extensile swimmers and negative for contractile swimmers. With extensile
swimmers, this system undergoes complete phase separation, which is similar to
that in binary-fluid mixtures [22]. By carrying out pseudospectral direct
numerical simulations (DNSs) we show, for the first time, that this model
develops an emergent nonequilibrium, but statistically steady, state (NESS) of
active turbulence, for the case of contractile swimmers, if $\zeta$ is
sufficiently large and negative. This turbulence arrests the phase separation
into regions with positive and negative values of $\phi$, in much the same way
as conventional fluid turbulence leads to the suppression of phase separation
in a binary-fluid mixture [26, 27, 28]. We quantify this suppression by
showing how the coarsening-arrest length scale does not grow indefinitely,
with time $t$, but saturates at a finite value at large times. We then
characterise the statistical properties of this active-scalar turbulence by
employing the energy spectrum and fluxes, which are familiar from classical
fluid turbulence, and also the spectrum of $\phi$, which is used in studies of
phase separation. For sufficiently high Reynolds numbers, the energy spectrum
$\mathcal{E}(k)$ displays an inertial range, with a power-law dependence on
the wavenumber $k$. We demonstrate that, in this range, the flux $\Pi(k)$
assumes a nearly constant, negative value, which indicates that the system
shows an inverse cascade of energy that is similar to its counterpart in 2D
homogeneous and isotropic fluid turbulence, even though energy injection
occurs over a wide range of wavenumbers in our active-CHNS model.

The remaining part of this paper is organised as follows. Section II
introduces the active CHNS model, summarises the numerical methods we employ
to study it, and defines the statistical measures we use to characterise
active turbulence in this model. In Section III we present the results of our
study. We discuss the significance of our results in Section IV.

(c)(d)(a)(b)

Figure 1: Pseudo-gray-scale plots of the $\phi$ field [at representative times
in the nonequilibrium statistically steady state (NESS)] for the activity
parameter (a) $|\zeta|=0.01$ and (b) $|\zeta|=1.5$. Pseudocolor plots of the
vorticity field, normalized by the maximum of $|\omega|$, are shown in (c) and
(d) for the parameters in (a) and (b), respectively.

(a)(b)

Figure 2: (a) Plot of the coarsening length scale $\mathcal{L}(t)$ [Eq. (8)]
versus time $t$ for various values of $|\zeta|$; the plot for $\zeta=0$ shows
growth that is consistent with the Lifshitz-Slyozov form $\mathcal{L}(t)\sim
t^{1/3}$ (dashed line); $\mathcal{L}(t)$ saturates to a finite value for
$|\zeta|>0$. (b) Log-linear plots of the mean coarsening-arrest scale
$L_{c}=\langle\mathcal{L}(t)\rangle_{t}$ (red curve) and the integral-scale
Reynolds number $Re_{L_{I}}$ (blue curve) versus $|\zeta|$.

##  II Model, Methods, and Statistical Measures

We introduce the active-CHNS model in Subsection II.1. In Subsection II.2 we
describe the statistical measures we use to characterise active turbulence in
this model. Finally, in Subsection II.3 we give the details of our
pseudospectral DNS.

###  II.1 The Active Cahn-Hilliard-Navier-Stokes Model

We consider the incompressible active CHNS equations (also called active model
H) to study active turbulence in systems of contractile swimmers [22, 24] in
two spatial dimensions (2D):

$\displaystyle\partial_{t}\phi+(\bm{u}\cdot\nabla)\phi$ $\displaystyle=$
$\displaystyle\text{M}\nabla^{2}\left(\frac{\delta\mathcal{F}}{\delta\phi}\right)\,;$
(1) $\displaystyle\partial_{t}\omega+(\bm{u}\cdot\nabla)\omega$
$\displaystyle=$
$\displaystyle\nu\nabla^{2}\omega+\frac{3}{2}\epsilon\nabla\times(\nabla\cdot\bm{\Sigma}^{A})-\alpha\omega\,;\;$
(2) $\displaystyle\nabla\cdot\bm{u}$ $\displaystyle=$ $\displaystyle 0\,;$ (3)

$\omega$ is the vorticity field; $\nu$, $\alpha$, and M are the kinematic
viscosity, bottom friction, and mobility, respectively. $\mathcal{F}$ is the
Landau-Ginzburg variational free-energy functional

$\displaystyle\mathcal{F}[\phi,\nabla\phi]=\int_{\Omega}\left[\frac{3}{16}\frac{\sigma}{\epsilon}(\phi^{2}-1)^{2}+\frac{3}{4}\sigma\epsilon|\nabla\phi|^{2}\right]\,,$
(4)

in which the first term is a double-well potential with minima at $\phi=\pm
1$. The scalar order parameter $\phi$ is positive (negative) in regions where
the microswimmer density is high (low); in the interfaces between these
regions, $\phi$ varies smoothly, over a width $\epsilon$. The free-energy
penalty for an interface is given by the bare surface tension $\sigma$. In the
inherently nonequilibrium active model H all terms in the stress tensor do not
follow from $\mathcal{F}$. In particular, we must include the stress tensor
$\bm{\Sigma}^{A}$, which has the form of a nonlinear Burnett term and has the
components [24, 22, 25, 29]

$\displaystyle\Sigma^{A}_{ij}=-\zeta\left[\partial_{i}\phi\partial_{j}\phi-\frac{\delta_{ij}}{2}|\nabla\phi|^{2}\right]\,,$
(5)

where $\zeta$, the activity coefficient, can take both positive and negative
values: $\zeta<0$ ($\zeta>0$) for contractile (extensile) swimmers [22].

###  II.2 Statistical Characterisation

To characterise the statistical properties of active-scalar turbulence, we
employ energy spectra and fluxes, which are familiar from classical fluid
turbulence, and the spectrum of $\phi$. These quantities not only help us to
understand, via DNS, the emergent turbulent like behaviour (characterized by
spatiotemporal fluctuations) in Eqs. (1-3), but they also aid us in
differentiating active scalar turbulence from classical 2D incompressible
fluid turbulence [see, e.g., Refs. [30, 31, 32, 33]] and also turbulent
patterns found in other active systems [see, e.g., Refs. [17, 18, 19, 20]]. We
define the shell-averaged energy and phase-field spectra, respectively, both
at time $t$ and averaged over time [the time average is denoted by
$\langle\cdot\rangle_{t}$]:

$\displaystyle\mathcal{E}(k,t)$ $\displaystyle\equiv$
$\displaystyle\sum_{k\leq
k^{\prime}<k+1}|\hat{\bm{u}}(\bm{k}^{\prime},t)|^{2}\,;~{}~{}~{}\Phi(k,t)\equiv\sum_{k\leq
k^{\prime}<k+1}|\hat{\phi}(\bm{k}^{\prime},t)|^{2}\,;$
$\displaystyle\mathcal{E}(k)$ $\displaystyle\equiv$
$\displaystyle\langle\mathcal{E}(k,t)\rangle_{t}\,;~{}~{}~{}\Phi(k)\equiv\langle\Phi(k,t)\rangle_{t}\,;$
(6)

(a)(b)

Figure 3: Log-log plots versus the wavenumber $k$ of spectra for
$|\zeta|=0.001,0.01,0.1,1.5$: (a) The compensated energy spectrum
$k^{5/3}\mathcal{E}(k)$. For $|\zeta|=0.1,1.5$, the exponent is $-5/3$ (black
dotted line in the compensated spectrum). (b) The compensated phase-field
spectrum $k^{-7/6}\Phi(k)$ for $|\zeta|=0.001,0.01,0.1,1.5$. These spectra
show the power-law behaviour $\Phi(k)\sim k^{7/6}$ (black dotted line in the
compensated spectrum).

the caret denotes a spatial Fourier transform. Our CHNS study of active scalar
turbulence uses the Reynolds, Péclet, Weber, and the non-dimensional friction
numbers that are, respectively:

Re $\displaystyle=$
$\displaystyle\frac{\text{L}_{I}u_{rms}}{\nu}\,;\;\;Pe=\frac{\epsilon{L}_{I}u_{rms}}{M\sigma}\,;$
$\displaystyle We$ $\displaystyle=$
$\displaystyle\frac{L_{I}u_{rms}^{2}}{\sigma}\,;\;\;\alpha^{\prime}=\frac{\alpha
L_{I}}{u_{rms}}\,;$ (7)

$u_{rms}$ and $L_{I}$ are, respectively, the root-mean-squared velocity and
the fluid integral length scale; and $\mathcal{L}(t)$ and $L_{c}$ are the
time-dependent and mean coarsening length scales; these are defined as
follows:

$\displaystyle u_{rms}$ $\displaystyle=$
$\displaystyle\left[\sum_{k}\mathcal{E}(k)\right]^{1/2}\;;\;\;\;L_{I}=2\pi\frac{\sum_{k}\mathcal{E}(k)}{\sum_{k}k\mathcal{E}(k)}\;;$
$\displaystyle\mathcal{L}(t)$ $\displaystyle=$
$\displaystyle\frac{\sum_{k}\Phi(k,t)}{\sum_{k}k\Phi(k,t)}\;;\;\;\;L_{c}=\langle\mathcal{L}(t)\rangle_{t}\;.$
(8)

In Table 1, we provide the non-dimensional parameters from our direct
numerical simulations (DNSs) for various values of the activity parameter
$|\zeta|$. Furthermore, $\mathcal{E}(k,t)$ satisfies the following energy-
budget equation [28, 34, 35]:

$\partial_{t}\mathcal{E}(k,t)=T(k,t)+D_{\alpha}(k,t)+D_{\nu}(k,t)+S^{\phi}(k,t)\,,$
(9)

where

$\displaystyle T(k,t)$ $\displaystyle=$
$\displaystyle-\Re\left[\sum_{k\leq|\bm{k}^{\prime}|<k+1}[\hat{\bm{u}}(-\bm{k}^{\prime},t)\cdot\bm{P}(\bm{k}^{\prime})\cdot\widehat{(\bm{u}\cdot\nabla\bm{u})}(\bm{k}^{\prime},t)]\right],$
$\displaystyle D_{\alpha}(k,t)$ $\displaystyle=$
$\displaystyle-2\alpha\mathcal{E}(k,t),~{}~{}~{}~{}D_{\nu}(k,t)=-2\nu
k^{2}\mathcal{E}(k,t),$ $\displaystyle S^{\phi}(k,t)$ $\displaystyle=$
$\displaystyle\Re\left[\sum_{k\leq|\bm{k}^{\prime}|<k+1}[\hat{\bm{u}}(-\bm{k}^{\prime},t)\cdot\bm{P}(\bm{k}^{\prime})\cdot\widehat{(\nabla\cdot\bm{\Sigma}^{A})}(\bm{k}^{\prime},t)]\right]$
(10)

are, respectively, the energy transfer because of the inertial term, the
energy dissipations arising from friction and the viscosity, and the energy
transfer via the active-stress term; the transverse projector
$\bm{P}(\bm{k})$, which enforces the incompressibility condition, has the
components $P_{ij}\equiv(\delta_{ij}-{k_{i}k_{j}}/{k^{2}})$ . We also use the
following mean energy transfers from the inertial, friction, viscous, and
active-stress terms, and the associated kinetic-energy and active-stress
fluxes [35]:

$\displaystyle T(k)$ $\displaystyle=$ $\displaystyle\langle
T(k,t)\rangle_{t}\,;\;\;\;\;S^{\phi}(k)=\langle
S^{\phi}(k,t)\rangle_{t}\,;\;\;\;$ $\displaystyle D_{\alpha}(k)$
$\displaystyle=$ $\displaystyle\langle
D_{\alpha}(k,t)\rangle_{t}\,;\;\;\;\;D_{\nu}(k)=\langle
D_{\nu}(k,t)\rangle_{t}\,;$ $\displaystyle\Pi(k)$ $\displaystyle=$
$\displaystyle-\int_{0}^{k^{\prime}}T(k^{\prime})dk^{\prime}\,;\;\;\;\;\Pi^{\phi}(k)=-\int_{0}^{k^{\prime}}S^{\phi}(k^{\prime})dk^{\prime}\,.$
(11)

###  II.3 Direct Numerical Simulation

We carry out DNSs of the active-CHNS partial differential equations (1)-(5) by
using the pseudospectral method [36, 27, 37, 24] in a 2D periodic square
domain, $\mathcal{D}\equiv[0,L]^{2}$, with $L$ the length of the side of the
square. We evaluate spatial derivatives in Fourier space and the nonlinear
terms in physical space. For time integration, we use the semi-implicit
exponential-time-difference Runge-Kutta2 (ETDRK2) method [38]. We employ the
$1/2$-dealiasing scheme to remove the Fourier aliasing errors [36, 27, 37,
24]. To resolve the interface of width $\epsilon$, we ensure that there are
three grid points across the interface. We use a CUDA-C code that we have
developed and optimised for an NVIDIA A100 processor.

###  II.4 Initial conditions

We use the following initial conditions for the $\omega$ and $\phi$ fields:

$\omega(x,y,0)=0\,;~{}~{}~{}~{}\phi(x,y,0)=\phi_{0}+\xi(x,y)\,;$ (12)

$\xi(x,y)$, a random number distributed uniformly on the interval
$[-0.1,0.1]$, provides a random perturbation to the $\phi=\phi_{0}=0$ state.

##  III Results

In this Section we present a series of DNSs that we have designed to
demonstrate how the activity of contractile swimmers [$\zeta<0$ in Eq. (5)]
leads to active turbulence that is strong enough to suppress motility-induced
phase separation. Our results are the active-turbulence counterparts of the
suppression of phase separation (also called coarsening arrest) by fluid
turbulence [see, e.g., Refs. [26, 27]].

(a)(b)

Figure 4: Plots versus $k$ (log scale) of the contributions
$T(k),\;D_{\alpha}(k),\;D_{\nu}(k),\;\rm{and}\;S^{\phi}(k)$, in red, purple,
yellow, and blue, respectively, for (a) $|\zeta|=0.001$ and (b) $|\zeta|=1.5$.

(a)(b)(c)

Figure 5:  Plots for $|\zeta|=0.001,0.01,0.1,$ and $1.5$ of the normalized (a)
energy flux $\Pi(k)L_{I}/u^{3}_{rms}$ versus $k$ (log scale) and log-log plots
versus $k$ of (b) $|\Pi(k)|L_{I}/u^{3}_{rms}$ and (c)
$\Pi^{\phi}(k)L_{I}/u^{3}_{rms}$.

We note that, if the activity $\zeta=0$, the stress tensor (5) vanishes, so
the coupled Eqs. (1)-(5) decouple into Cahn-Hilliard equations or model B [39,
40] and the Navier Stokes equations [41, 42, 43] for a Newtonian fluid; the
advection term for the $\phi$ field is set to zero by virtue of the initial
conditions. Therefore, the domain growth or coarsening takes place solely via
diffusion, without hydrodynamical effects, and it follows the well-established
Lifshitz-Slyozov domain-growth form $\mathcal{L}(t)\sim t^{1/3}$ [see, e.g.,
Refs. [44, 40, 45]]; complete phase separation also occurs for extensile
swimmers [22] that lead to $\zeta>0$.

We concentrate on active-turbulence-induced suppression of phase separation
and the diffusive Lifshitz-Slyozov coarsening in our model (1)-(5) with
contractile swimmers, for which the activity parameter $\zeta<0$ 111At low
inertia motility-induced phase separation is also suppressed in this model
[22].. Active turbulence and coarsening arrest in the active-CHNS model
(1)-(5) can be visualized qualitatively by using pseudo-gray-scale plots of
the $\phi$ field as we show in Figs. 1 (a) and (b) for $|\zeta|=0.01$ and (b)
$|\zeta|=1.5$, respectively, at representative times in the nonequilibrium
statistically steady state (NESS). In Figs. 1 (c) and (d), we present
pseudocolor plots of the vorticity field, normalized by the maximum of
$|\omega|$, for the parameters in Figs. 1 (a) and (b), respectively. These
plots show clearly that the typical size of a single-phase domain decreases as
activity-induced turbulence is enhanced by an increase in the value of
$|\zeta|$.

We quantify active-turbulence-induced suppression of phase separation by
plotting the coarsening length scale $\mathcal{L}(t)$ versus time $t$ in Fig.
2(a) for various values of $|\zeta|$; the plot for $\zeta=0$ shows growth that
is consistent with the Lifshitz-Slyozov form $\mathcal{L}(t)\sim t^{1/3}$
(dashed line 222In any DNS in a finite domain, $\mathcal{L}(t)$ approaches a
finite value that is comparable to the linear size of the domain.). As $t$
increases, $\mathcal{L}(t)$ saturates to a finite value for $|\zeta|>0$, i.e.,
Eqs. (1)-(5) lead to coarsening-arrest because of active turbulence. In Fig.
2(b) we show how the mean coarsening-arrest scale
$L_{c}=\langle\mathcal{L}(t)\rangle_{t}$ decreases as $|\zeta|$ increases (red
curve); the attendant growth of the integral-scale Reynolds number
$Re_{L_{I}}$ (blue curve) signals the enhancement of activity-induced
turbulence.

We now characterise the statistical properties of activity-induced turbulence
in Eqs. (1)-(5). We begin with log-log plots of compensated energy and
scalar-$\phi$ spectra, $k^{5/3}\mathcal{E}(k)$ and $k^{-7/6}\Phi(k)$, versus
$k$, in Figs. 3 (a) and (b), respectively, for various values of $|\zeta|$.
These plots suggest that, as $\zeta$ increases, the activity-induced
turbulence in this systems leads to a nonequilibrium statistically steady
state (NESS) with an inertial range of scales in which the energy spectrum has
a power-law form that is consistent with $\mathcal{E}(k)\sim k^{-5/3}$. We
show below that this power-law spectrum arises because of an inverse cascade
of energy. Its power-law form can then be surmised as in statistically steady
homogeneous and isotropic 2D-fluid turbulence with an inverse energy cascade
[30, 31, 32, 33]. Note that this power-law region extends over nearly one-and-
a-half decades of $k$ at the largest value $|\zeta|(=1.5)$ that we consider.

Run | $|\zeta|$ | $Re_{L_{I}}$ | $\alpha^{\prime}$ | $Pe$ | $We$
---|---|---|---|---|---
R1 | 0 | 0 | $-$ | 0 | 0
R2 | 0.001 | $1.1\times 10^{0}$ | $5.6\times 10^{-2}$ | $2.8\times 10^{-1}$ | $1.6\times 10^{-4}$
R3 | 0.01 | $3.2\times 10^{0}$ | $7.9\times 10^{-3}$ | $8.5\times 10^{-1}$ | $2.3\times 10^{-3}$
R4 | 0.03 | $1.3\times 10^{1}$ | $6.6\times 10^{-3}$ | $3.5\times 10^{0}$ | $2\times 10^{-2}$
R5 | 0.05 | $4.3\times 10^{1}$ | $7.8\times 10^{-3}$ | $1.2\times 10^{1}$ | $1.1\times 10^{-1}$
R6 | 0.1 | $6.9\times 10^{1}$ | $7.0\times 10^{-3}$ | $1.9\times 10^{1}$ | $2.3\times 10^{-1}$
R7 | 0.5 | $1.1\times 10^{2}$ | $5.8\times 10^{-3}$ | $3.2\times 10^{1}$ | $5.3\times 10^{-1}$
R8 | 1 | $1.2\times 10^{2}$ | $6.8\times 10^{-3}$ | $3.3\times 10^{1}$ | $5.8\times 10^{-1}$
R9 | 1.5 | $1.3\times 10^{2}$ | $5.7\times 10^{-3}$ | $3.4\times 10^{1}$ | $6.1\times 10^{-1}$
Table 1:  The values of various parameters in our DNS runs R1-R9. The
following parameters are fixed in all these runs: $N=1024,\;\text{grid
size}\;\;dx=2\pi/N,\;\epsilon=3dx,\;L=2\pi,\;\text{Cn}=3dx/L,\;M=\epsilon^{2}/2,\;\sigma=1,\;\nu=5\times
10^{-3}$, $\alpha=0.01$.

We examine the energy-transfer mechanisms in the NESS of activity-induced
turbulence in Eqs. (1)-(5) by using the energy-budget equation (9) and
evaluating the relevant scale-by-scale energy contributions [28, 34, 35]. In
Figs. 4(a) and (b) we present, for the illustrative values $|\zeta|=0.001$ and
(b) $|\zeta|=1.5$, respectively, plots versus $k$ (log scale) of the inertial,
friction, viscous, and active-stress contributions
$T(k)\;({\rm{red}}),\;D_{\alpha}(k)\;({\rm{purple}}),\;D_{\nu}(k)\;({\rm{yellow}})$
and $S^{\phi}(k)\;({\rm{blue}}),$ which we have defined in Eq. (11). Such
plots indicate that, for low values of $|\zeta|$, the contributions of $T(k)$
and $D_{\alpha}(k)$ are negligible [Fig. 4(a)], so, in the NESS with
$\langle\partial_{t}\mathcal{E}(k,t)\rangle_{t}=0$, dominant balance yields
$D_{\nu}(k)+S^{\phi}(k)=0$. As $|\zeta|$ increases, both $T(k)$ and
$D_{\alpha}(k)$ increase in magnitude [Fig. 4(b)], so a four-term balance is
required in the NESS.

To show that activity-induced turbulence exhibits a bona fide inertial range
we present plots for $|\zeta|=0.001,0.01,0.1,$ and $1.5$ of the energy flux
$\Pi(k)$ [Eq. 11] versus $k$ (log scale) [Fig. 5(a)]. We also present log-log
plots versus $k$ of $|\Pi(k)|$ [Fig. 5(b)] and $\Pi^{\phi}(k)$ [Fig. 5(c)].
These plots show constant fluxes over at least one decade of the wavenumber
$k$, so we have a well-defined inertial range, that has a remarkable
similarity with fluid turbulence [48]. The sign of $\Pi(k)$ in this range of
scales indicates that activity-induced turbulence in Eqs. (1)-(5) yields an
inverse cascade of energy that is reminiscent of a similar cascade in 2D
statistically steady homogeneous and isotropic fluid turbulence [30, 31, 32,
33]. By comparing the different plots in Fig. 5(a), we see that this inverse
cascade is suppressed as $|\zeta|$ decreases; this is reminiscent of similar
cascade suppression in instability-driven 2D turbulence [49].

##  IV Conclusions

We have uncovered activity-induced homogeneous and isotropic turbulence in the
active Cahn-Hilliard-Navier-Stokes equations (CHNS), which provides a natural
theoretical framework for our study, in which a single scalar order parameter
$\phi$ [positive (negative) in regions where the microswimmer density is high
(low)] is coupled with the velocity field $\bm{u}$. The activity of the
microswimmers is governed by an activity parameter $\zeta$ that is positive
for extensile swimmers and negative for contractile swimmers [see Eqs.
(1)-(5)]. With extensile swimmers, this system undergoes complete phase
separation, as in binary-fluid mixtures [22]. By carrying out extensive
pseudospectral direct numerical simulations (DNSs) we have shown that this
model develops an emergent nonequilibrium, but statistically steady, state
(NESS) of active turbulence, for the case of contractile swimmers, if $\zeta$
is sufficiently large and negative. This turbulence arrests the phase
separation into regions with positive and negative values of $\phi$, as in
conventional fluid turbulence leads to the suppression of phase separation in
a binary-fluid mixture [26, 27, 28].

We have quantified this suppression by showing how the coarsening-arrest
length $\mathcal{L}(t)$ scale does not grow indefinitely, with time $t$, but
saturates at a finite value at large times. We have then characterised the
statistical properties of this active-scalar turbulence by employing the
energy spectrum $\mathcal{E}(k)$ and the fluxes $\Pi(k)$ and $\Pi^{\phi}(k)$.
We have also obtained the spectrum of $\phi$, which is used in studies of
phase separation. For sufficiently high Reynolds numbers, we have shown that
the energy spectrum $\mathcal{E}(k)$ displays an inertial range, with a power-
law dependence on the wavenumber $k$. We have demonstrated that, in this
range, the flux $\Pi(k)$ assumes a nearly constant, negative value, which
indicates that the system shows an inverse cascade of energy that is similar
to its counterpart in 2D homogeneous and isotropic fluid turbulence, even
though energy injection occurs over a wide range of wavenumbers in our active-
CHNS model.

Our statistical characaterization of active-CHNS turbulence shows that it is
fundamentally different from conventional 2D fluid turbulence [30, 31], forced
2D CHNS turbulence [27], and other types of active-fluid turbulence,
discussed, e.g., in Refs. [1, 2, 3, 4, 5, 6, 7, 8, 9, 25, 10, 11]. For large
values of $|\zeta|$, active-CHNS turbulence has some similarities with
conventional 2D fluid turbulence and 2D forced CHNS turbulence, inasmuch as it
shows an inverse-cascade region with $E(k)\sim k^{-5/3}$. The
$|\zeta|$-dependent, small-$k$, power-law regime in $E(k)$ is qualitatively
reminiscent of parameter-dependent small-$k$ power-law regimes in energy-
spectra in some minimal models for bacterial turbulence [16, 3, 4, 7, 11, 8,
9]. The scalar spectrum $\Phi(k)$ of active-CHNS turbulence shows a
substantial power-law regime which is different from that in conventional
forced 2D CHNS turbulence [27]. The fluxes and energy budgets in active-CHNS
turbulence are also markedly different from their counterparts in other types
of 2D turbulence. Our study investigates the effects of active-CHNS turbulence
that is distinct from the suppression of motility-induced phase separation in
the active model H [22] with external noise.

We hope, therefore, that our active-CHNS study will lead to investigations of
experimental realisations of this system. Our results are of potential
relevance to systems of contractile swimmers, e.g., Chlamydomonas reinhardtii
[50, 51] (C. reinhardtii) and synthetic active colloids [52, 53]. We look
forward to the experimental verification of our results, especially in the
former system, where it should be possible to control the activity by changing
the oxygen concentration in low-light conditions.

## Acknowledgments

We thank the Science and Engineering Research Board (SERB) and the National
Supercomputing Mission (NSM), India for support, and the Supercomputer
Education and Research Centre (IISc) for computational resources.

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