# Revisiting blackKinematic Fast Dynamo in 3-dimensional magnetohydrodynamic
plasmas: Dynamo transition from non-Helical to Helical flows

Shishir Biswas,1,2 Rajaraman Ganesh,1,2
1Institute for Plasma Research, Bhat, Gandhinagar, Gujarat 382428, India
2Homi Bhabha National Institute, Training School Complex, Anushaktinagar,
Mumbai 400094, India
E-mail<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
(Accepted XXX. Received YYY; in original form ZZZ)

###### Abstract

Dynamos wherein magnetic field is produced from velocity fluctuations are
fundamental to our understanding of several astrophysical and/or laboratory
phenomena. blackThough fluid helicity is known to play a key role in the onset
of dynamo action, its effect is yet to be fully understood. In this work, a
fluid flow proposed recently [Yoshida et al. Phys. Rev. Lett. 119, 244501
(2017)] is invoked such that one may inject zero or finite fluid helicity
using a control parameter, at the beginning of the simulation. Using a simple
blackkinematic fast dynamo model, we demonstrate unambiguously the strong
dependency of short scale dynamo on fluid helicity. blackIn contrast to
conventional understanding, it is shown that fluid helicity does strongly
influence the physics of short scale dynamo. To corroborate our findings, late
time magnetic field spectra for various values of injected fluid helicity is
presented along with rigorous “geometric” signatures of the 3D magnetic field
surfaces, which shows a transition from “untwisted” to “twisted” sheet to
“cigar” like configurations. It is also shown that one of the most studied ABC
dynamo model is not the “fastest” dynamo model for problems with lower
magnetic Reynolds number. This work brings out, for the first time, the role
of fluid helicity in moving from “non-dynamo" to “dynamo" regime
systematically.

###### keywords:

blackfluid simulation – plasmas – blackmagnetohydrodynamics – dynamo –
magnetic fields

††pubyear: 2022††pagerange: Revisiting blackKinematic Fast Dynamo in
3-dimensional magnetohydrodynamic plasmas: Dynamo transition from non-Helical
to Helical flows–B

##  1 Introduction

The theories of HydroDynamics (HD) (Kraichnan, 1965; Brachet et al., 1988) and
MagnetoHydroDynamics (MHD) (Biskamp, 2003; Beresnyak, 2019; Schekochihin,
2022) are often used to analyze the HD turbulence and magnetized plasma
turbulence respectively, which are fundamental to our understanding of the
behaviour of astrophysical plasmas present in the Sun or other young stars.
The presence of blacksmall scale, mean or large scale magnetic fields are very
much common in such astrophysical objects, planets, stars, interstellar
medium, galaxy, accretion disks and also in the Sun (Squire & Bhattacharjee,
2014a, b, 2015). Understanding the origin of these blackmulti scale magnetic
fields, which are in turn responsible for various complex phenomenon in the
universe, is of paramount importance. Naturally a relevant question arises
blackas to, what are the sources of these blackmulti scale magnetic energy? As
first pointed out by Parker, blacksuch magnetic fields are generated by the
motions of conducting fluids through a transfer of kinetic to magnetic energy,
that is, via Dynamo action (Parker, 1979).

blackDepending on the length scales involved, dynamos are broadly classified
into two categories : Small Scale Dynamo (SSD) and Large Scale Dynamo (LSD) or
mean field dynamo. For LSD, it is believed that a lack of reflectional
symmetry (or in other words, nonzero fluid helicity) to be a crucial element,
whereas for SSD, no correlation has been shown to exist between onset and
sustenance of SSD on fluid helicity (Rincon, 2019). Depending on the time
scale, dynamos may further divided into two categories: Fast dynamos (growth
rate can remain finite in the limit $R_{m}\to\infty$) and Slow dynamos
(magnetic diffusion plays a significant role) (Rincon, 2019). Depending on the
feedback strength of the magnetic field on to the fluid motion, dynamos are
categorized as linear or non-linear. A linear dynamo is one in which the
magnetic field dynamics does not “back react” with the velocity field and the
velocity field is either given or it obeys the NS equation (Rincon, 2019),
whereas a self-consistent dynamo or a nonlinear dynamo tends to change the
flow - as the magnetic field becomes large enough - to further impede magnetic
field growth. That is, the flow and the B-field “back react” on each other
leading to a nonlinear saturation (Rincon, 2019).

blackIn the present study, we focus on a kinematic fast dynamo model, wherein
velocity and magnetic fields are un-correlated. As indicated earlier, in a
fast dynamo model, below a certain value of resistivity, the growth rate
becomes insensitive to the magnitude of resistivity, i.e, becomes independent
of resistivity. (Childress & Gilbert, 1995). A popular example of fast dynamo
is the Solar dynamo (Choudhuri, 1990).

It is now well acknowledged that, to achieve fast dynamo, an blackcareful
selection of plasma flow profile is necessary, as the key mechanism behind the
fast dynamo action is that blackof stretching, twisting and folding (STF) of
magnetic field lines, “advected” in the fluid flow. Hence the flow is expected
to have “enough dynamcs” to amplify the magnetic field exponentially.

blackAs discussed before, depending on the length scale involved, dynamos are
largely classified into two broad categories : Small Scale Dynamo (SSD) and
Large Scale Dynamo (LSD). For a LSD or mean field dynamo, a lack of
reflectional symmetry is required, where as conventional understanding is
that, no such condition is expected to be satisfied for SSD (Rincon, 2019).
Among various astrophysical flows addressed, the Arnold-Beltrami-Childress
[ABC] flow is a well known prototype for fast dynamo action for its stretching
ability and chaotic transport properties. A detailed dynamo study using ABC
flow has been reported by various authors, eg. (Arnold & Korkina, 1983;
Galloway & Frisch, 1984; Dombre et al., 1986; Galanti et al., 1992; Dorch,
2000; Archontis, V. et al., 2003; Bouya & Dormy, 2013). Along with the
exponential growth of magnetic energy, the magnetic energy iso-surface
blackare known to exhibit special signature called “cigar” like (Galloway &
Frisch, 1986) or “ribbon” like (Galloway & Frisch, 1986; Archontis, V. et al.,
2003) structure. For ABC flow profile, blackthe vorticity field is parallel to
the velocity direction, i.e. $\vec{\nabla}\times\vec{u}$ has a finite parallel
component to $\vec{u}$ at all times. This naturally indicates that, finite
fluid helicity [$\int_{V}\vec{u}.(\vec{\nabla}\times\vec{u})dV$] is present in
the flow. The fluid helicity effect on dynamo action is often expressed as
$\alpha$ effect, and studied extensively using ABC flow family (Courvoisier et
al., 2006).

blackStudy of dynamo instability using a 3D flows becomes numerically
demanding and is perhaps relatively more complex. To reduce the complexity in
numerical methods often, time dependent two dimensional flows
[$\vec{u}(x,y,t)$] have been taken into consideration for the investigation of
fast dynamo action. Naturally, the presence of time dependency introduces
chaoticity in such flows. Galloway-Proctor flow, sometimes known as GP flow,
is one such well-known flow (Galloway & Proctor, 1992). In light of the fact
that the flow profile is entirely independent of one spatial coordinate, in
this case the z-coordinate, it is conceivable to seek monochromatic solutions
for the magnetic field of the form
$\vec{B}(x,y,z,t)=\vec{B}(x,y,t)e^{ik_{z}z}$, so that the problem becomes two-
dimensional for a given value of $k_{z}$. In past several authors (Cattaneo et
al., 1995, 1996; Hughes et al., 1996; Rädler & Brandenburg, 2003) have
investigated the dynamo instability using Galloway-Proctor (GP) flow. Recently
GP flow (Galloway & Proctor, 1992) has been taken into consideration for
studying large scale magnetic fields in the presence of velocity shear (Tobias
& Cattaneo, 2013).

Some of the interesting and relevant questions are: is the blacksmall-scale
kinematic fast dynamo action possible only for chaotic ABC flow blackand time
dependent GP flow? what is the exact role of fluid helicity in the context of
blacksmall-scale fast dynamo action? Is there flow field using which one can
systematically inject fluid helicity in the system and clearly demonstrate a
non-dynamo to dynamo transition when the fluid flow transits from non-helical
to helical flow? blackDoes the fluid helicity affect short-scale dynamo
action?

blackIn past several authors examined the influence of the flow helicity in
the context of kinematic fast dynamo action. For example, Hughes et al. (1996)
show that the small scale, fast dynamo will always exist regardless of the
flow helicity distribution using three different types of helicity
distribution (non zero local and global helicity, non zero local helicity but
zero global helicity, and zero local and global helicity). The flow considered
by Hughes et al. (1996) is a time dependent 2-dimensional flow
[$\vec{u}(x,y,t)$] (Galloway & Proctor, 1992). As previously mentioned, the
induction equation for magnetic field supports a monochromatic solution of the
form $\vec{B}(x,y,t)\times e^{ik_{z}z}$, for a given $k_{z}$ value, the
resulting problem for the magnetic field is two-dimensional. The most
significant finding was that the helicity distribution of the driving flow did
not significantly affect the small-scale fast dynamo action (Hughes et al.,
1996). Additionally, it was pointed out that as fluid helicity increases, the
dynamo growth rate noticeably increases. However, these Authors quickly
clarify that this is only coincidence (Hughes et al., 1996).

blackBy keeping these earlier ideas in mind, we study the blackfast dynamo
action using a newly proposed Yoshida-Morrison flow or YM flow (Yoshida &
Morrison, 2017) blackto investigate the role of helicity on SSD. blackIt is
important to indicate that the whole class of YM flow depends upon all three
spatial coordinates i.e, $\vec{u}(x,y,z)$ similar to ABC flow. We have
considered small homogeneous ambient initial magnetic field in all 3
directions as considered earlier by Galloway & Frisch (1984); Dorch (2000);
Archontis, V. et al. (2003). blackAnother interesting and useful aspect of YM
flow is that, it is possible to inject finite fluid helicity
[$\int_{V}\vec{u}.(\vec{\nabla}\times\vec{u})dV$] in the system, by
systematically varying certain physical parameter. More over this flow has a
special property that, it establishes a topological bridge between blackquasi
2-dimensional class and 3-dimensional class of flows. blackTherefore, our
present study focuses on various flows with varying kinetic energy, chaotic
characteristics, and helicity on the onset and sustenance of SSD.

In the present work, we propose a new possible route that connects non-dynamo
regime to dynamo regime via fluid helicity injection using Yoshida-Morrison
(YM) flow as a prototype. blackOur numerical observation demonstrates that
small scale dynamo is feasible for this type of initial condition. Our work
shows unambiguously that fluid helicity does affect the dynamics of small-
scale dynamos. This finding is in complete contrast to the earlier results
(Hughes et al., 1996). Our findings systematically connect most of the
previous works and brings in several new insights. blackTo corroborate our
findings, time dependent magnetic energy spectra, for various magnitudes of
injected fluid helicity is calculated. We also show that, how a “cigar” like
iso-surface, which is a signature of fast dynamo action, emerges naturally
starting from a non-dynamo iso-surface. Our numerical investigation also
suggests that, for lower magnetic Reynolds number ($R_{m}$) experiments, the
regular ABC dynamo model is not the best dynamo model.

The organization of the paper is as follows. In Sec. 2 we present about the
dynamic equations. About our numerical solver and simulation details are
described in Sec. 3. The initial conditions, parameter details are shown in
Sec. 4. Section 5 is dedicated to the simulation results on induction dynamo
action that we obtained from our code and finally the summary and conclusions
are listed in Sec. 6.

##  2 Governing Equations

The governing equations to study blackKinematic Fast dynamo action for the
single fluid MHD plasma are as follows,

$\displaystyle\frac{\partial\vec{B}}{\partial
t}+\vec{\nabla}\cdot\left(\vec{u}\otimes\vec{B}-\vec{B}\otimes\vec{u}\right)=\textcolor{black}{\frac{1}{R_{m}}}\nabla^{2}\vec{B}$
(1) $\displaystyle\vec{\nabla}\cdot\vec{B}=0$ (2)

blackwhere, $\vec{u}$, $B$ and $R_{m}$ represent the velocity, magnetic fields
and magnetic Reynolds number respectively. We define Alfven speed as,
$V_{A}=\frac{u_{0}}{M_{A}}$, here $M_{A}$ is the Alfven Mach number of the
plasma and $u_{0}$ is the maximum fluid speed. blackThe initial magnetic field
($B_{0}$) is determined as from Alfven speed as, $V_{A}\propto B_{0}$. Time is
normalized as $t=t_{0}\times t^{\prime}$, $t_{0}=\frac{L}{V_{A}}$. The symbol
“$\otimes$” represents the dyadic between two vector quantities.

For solving the above set of equations at high enough grid resolution, one
need a suitable scalable numerical solver.

##  3 Simulation Details: GMHD3D Solver

In this Section, we discuss the details of the numerical solver along with the
benchmarking of the solver carried out by us. In order to study the plasma
dynamics governed by MHD equations described above, we have very recently
upgraded an already existing well bench-marked single GPU MHD solver
(Mukherjee, 2019), developed in house at Institute For Plasma Research to
multi-node, multi-card (multi-GPU) architecture for better performance (Biswas
et al., 2022a). The newly upgraded GPU based magnetohydrodynamic solver
[GMHD3D] is now capable of handling very large grid sizes. GMHD3D is a multi-
node, multi-card, three dimensional (3D), weakly compressible, pseudo-
spectral, visco-resistive magnetohydrodynamic solver (Biswas et al., 2022a).
It uses pseudo-spectral technique to simulate the dynamics of 3D
magnetohydrodynamic plasma in a cartesian box with periodic boundary
condition. By this technique one calculates the spatial derivative to evaluate
non-linear term in governing equations with a standard $\frac{2}{3}$ de-
aliasing rule (Patterson & Orszag, 1971). OpenACC FFT library [AccFFT library
(Gholami et al., 2016)] is used to perform Fourier transform and Adams-
bashforth time solver, for time integration. For 3D iso-surface visualization,
an open source Python based data converter to VTK [Visualization Tool kit] by
“PyEVTK” (Paulo-herrera, 2021) is developed, which converts ASCII data to VTK
binary format. After dumping the state data files to VTK, an open source
visualization softwares, VisIt 3.1.2 (LLNL, 2020) and Paraview (Kitware, 2022)
is used to visualize the data. As mentioned earlier, we have upgraded a well
benchmarked single GPU solver to multi-node, multi-card (multi-GPU)
architecture, for this present work, the new solver’s accuracy with the single
GPU solver has been cross-checked and it is verified that the results match
upto machine precision. Further few more benchmarking studies have been
performed such as, the 3D induction dynamo effect by (Galloway & Frisch, 1986;
Dorch, 2000; Archontis, V. et al., 2003), have been reproduced with ABC flow
at grid resolution $64^{3}$ (Details of these are not presented here.).

To study the induction dynamo action, an accurate selection of “drive”
velocity field is imperative, which we discuss in the Section to follow.

##  4 Initial Condition

The study of dynamo action or more precisely blackkinematic fast dynamo action
in the presence of chaotic 3D Arnold-Beltrami-Childress [ABC] flow has been
explored by various authors (Galloway & Frisch, 1984; Dombre et al., 1986;
Galanti et al., 1992; Dorch, 2000; Archontis, V. et al., 2003; Bouya & Dormy,
2013). The velocity field for ABC flow is as follows:

$\displaystyle u_{x}$ $\displaystyle=u_{0}[A\sin(k_{0}z)+C\cos(k_{0}y)]$ (3)
$\displaystyle u_{y}$ $\displaystyle=u_{0}[B\sin(k_{0}x)+A\cos(k_{0}z)]$
$\displaystyle u_{z}$ $\displaystyle=u_{0}[C\sin(k_{0}y)+B\cos(k_{0}x)]$

Recently Yoshida et al. (Yoshida & Morrison, 2017) proposed a new intermediate
class of flow, that establishes a topological bridge between quasi-2D and 3D
flow classes. The flow is formulated as follows:

$\vec{u}=\alpha\times u_{0}\times\vec{u}_{+}+\beta\times
u_{0}\times\vec{u}_{-}$ (4)

with

$\displaystyle\vec{u}_{+}$
$\displaystyle=\begin{bmatrix}Bsin(k_{0}y)-Ccos(k_{0}z)\\\ 0\\\
Asin(k_{0}x)\\\ \end{bmatrix}$ (5)

and

$\displaystyle\vec{u}_{-}$ $\displaystyle=\begin{bmatrix}0\\\
Csin(k_{0}z)-Acos(k_{0}x)\\\ -Bcos(k_{0}y)\\\ \end{bmatrix}$ (6)

so that,

$\displaystyle u_{x}$ $\displaystyle=\alpha
u_{0}[B\sin(k_{0}y)-C\cos(k_{0}z)]$ (7) $\displaystyle u_{y}$
$\displaystyle=\beta u_{0}[C\sin(k_{0}z)-A\cos(k_{0}x)]$ $\displaystyle u_{z}$
$\displaystyle=u_{0}[\alpha A\sin(k_{0}x)-\beta B\cos(k_{0}y)]$

where black$k_{0}$, $\alpha,\beta$, A, B and C are arbitrary real constants.
As indicated earlier, we dub this flow (Eq. 7) as Yoshida-Morrison flow or YM
flow. blackWe consider the value of $k_{0},\alpha$, A, B and C to be unity for
this present study. The variation of $\beta$ value in YM flow leads to new
classes of flows.

For example, for black$\beta=0$, Yoshida et al. (Yoshida & Morrison, 2017)
classifies it as EPI-2D flow [See Fig. 1(a)] which is given by :

$\textcolor{black}{\begin{aligned} u_{x}&=u_{0}[\sin(y)-\cos(z)]\\\
u_{y}&=0\\\ u_{z}&=u_{0}[\sin(x)]\end{aligned}}$ (8)

blackThis flow (i.e, Eq. 8) is dependent on all the 3 spatial coordinates
(i.e, $x,y,z$), whereas only two flow components are nonzero. Thus EPI-2D flow
is quasi-2D in nature.

As can be expected, for $\beta=1$ the above Eq. 7 becomes well known ABC like
flow [See Fig.1(f)],

$\textcolor{black}{\begin{aligned} u_{x}&=u_{0}[\sin(y)-\cos(z)]\\\
u_{y}&=u_{0}[\sin(z)-\cos(x)]\\\ u_{z}&=u_{0}[\sin(x)-\cos(y)]\end{aligned}}$
(9)

(The only mathematical difference between the flow described in Eq. 9 and the
ABC flow [Eq. 3] are that the two terms are being subtracted in the latter
while added in the former. Nevertheless, as will be shown below, the flow in
Eq. 9 is identical in its properties to ABC flow [Eq. 3].) As $\beta$ is
varied from $0$ to $1.0$, a whole set of intermediate class of flows emerge,
such that a normalized fluid helicity is exactly $0.0$ for $\beta=0$ and is
maximum for $\beta=1.0$ (i.e, ABC-like flows). For a comprehensive study, we
explore various values of $\beta$ between $0$ to $1$, keeping all other
parameters identical [See Fig. 1(b), 1(c), 1(d), 1(e)].

By increasing $\beta$ values we inject normalized fluid helicity in the system
[See. Appendix A for details]. In Fig. 2, normalized fluid helicity
[$\int_{V}\vec{u}.(\vec{\nabla}\times\vec{u})dV$] is calculated, for YM flow.
It is observed that for increasing $\beta$ values normalized fluid helicity
also increases. It is also interesting to note from YM flow velocity contour
visualization [See Fig. 1(a), 1(b), 1(c), 1(d), 1(e), 1(f)] that for
increasing $\beta$ values, the velocity contours become increasingly chaotic
and visible separatrix forms emerge, which is a crucial ingredient of chaotic
flows. blackThe maximum fluid velocity is $u_{0}=1.0$ and Alfven Mach number
is $M_{A}=10.0$. Following (Galloway & Frisch, 1984), the initial magnetic
field is considered as very small but uniform in nature. With these given
velocity fields, we perform our simulation, the details of which is given
next.

(a)

(b)

(c)

(d)

(e)

(f)

Figure 1: Driving velocity contours of Yoshida-Morrison (YM) flow. The
velocity contour visualization is shown for (a) $\beta=0.0$, (b) $\beta=0.2$,
(c) $\beta=0.4$, (d) $\beta=0.6$, (e) $\beta=0.8$, (f) $\beta=1.0$, where
$\beta=0.0$ is least chaotic and $\beta=1.0$ is most chaotic. It is important
to identify that for the increasing $\beta$ values the velocity field lines
are becoming chaotic and finally the visible separatrix has been formed for
most helical (i.e. $\beta=1.0$) case. Figure 2: Normalized fluid helicity as a
function of $\beta$ for Yoshida-Morrison (YM) flow. blackHere $\beta$ is real
constant. $\beta=0.0$ corresponds to non-helical (zero normalized fluid
helicity) flow and $\beta=1.0$ corresponds to maximum helical (maximum
normalized fluid helicity) flow. For detail calculation and normalization
[See. Appendix A].

###  4.1 Parameter Details

We evolve the above set of equations discussed in Section 2, for class of YM
flow profile, in a triply periodic box of length $L_{x}=L_{y}=L_{z}=2\pi$ with
time steeping $(dt)=10^{-4}$ and grid resolution $256^{3}$. We have performed
grid size scaling study [See. Appendix B] using Arnold-Beltrami-Childress
(ABC) flow (Galloway & Frisch, 1984) blackfor different magnetic Reynolds
numbers. It is obvious that $256^{3}$ grid resolution is more than enough for
this problem [See. Appendix B]. With these initial conditions and parameter
spaces we present our numerical simulation results.

##  5 Simulation Results

We consider a recently proposed general class of flow named YM flow (Eq. 7)
(Yoshida & Morrison, 2017) for this present study, as indicated earlier and
evolve the homogeneous initial B-field in time, using Eq.1 for a given set of
values of $\beta$ and $u_{0}$ using our GPU solver GMHD3D. An interesting and
useful aspect of YM flow is that, it is possible to inject finite fluid
helicity [$\int_{V}\vec{u}.(\vec{\nabla}\times\vec{u})dV$] in the system [See
Fig. 2], by systematically varying certain physically meaningful parameter,
i.e. $\beta$.

We study the blackkinematic fast dynamo action for various $\beta$ value
starting from $0.0$ to $1.0$. For black$\beta=0.0$ value, YM flow leads to
EPI-2 dimensional flow (Yoshida & Morrison, 2017). EPI-2 dimensional flow
makes a topological bridge between blackquasi-2D and 3D class of flows
(Yoshida & Morrison, 2017). We perform numerical runs for a wide range of
magnetic Reynolds numbers ($R_{m}$). blackFrom Fig. 3(a) it is observed that,
for $\beta=0.0$ there is no prominent dynamo action, even at highest magnetic
magnetic Reynolds numbers ($R_{m}=5000$). blackAt late times, magnetic energy
curve is flat with respect to time, indicating that the growth rate is zero.
The key mechanism behind the blackfast dynamo action is believed to be
stretching, twisting and folding (STF) of magnetic field lines
black(Vainshtein & Zel’dovich, 1972), that generates blackexponential growth
of magnetic energy and hence it is necessary that, the flow should be able to
“twist” black(i.e., should contain enough chaoticity) and hence should contain
nonzero fluid helicity to generate exponential growth of magnetic energy. From
our earlier discussion it is already evident that, YM flow with $\beta=0.0$
has zero fluid helicity and hence is not able to amplify the magnetic field in
exponential manner i.e, not able to generate dynamo action.

blackIt is important to note that, there are examples of non-helical flows,
that can produce fast dynamo action, that is due to Cross-helicity effect or
the so-called Yoshizawa effect (Galloway & Proctor, 1992; Sur & Brandenburg,
2009). Also as discussed in the Introduction, several authors have reported
dynamo action using 2-dimensional time dependent non-helical flows. For
example, GP flow (Galloway & Proctor, 1992; Hughes et al., 1996), has been
used to study kinematic fast dynamo action. The flow profile goes as
$\vec{u}(x,y,t)$, so the initial magnetic field is modeled as $B(x,y,t)\times
e^{ik_{z}z}$, for a fixed $k_{z}$, the resulting problem for the magnetic
field is two-dimensional. Using these initial condition fast dynamo action has
been reported for non-helical flows.

Let us come back to YM flow with $\beta$ as the control parameter for initial
helicity injected. We study of magnetic energy iso-surfaces for the
$\beta=0.0$ case and observe that most of the energy is concentrated in
elongated almost two-dimensional un-twisted structures [See Fig. 3(b)]. To
obtain better visualization, we also provide volume rendered view [See Fig.
3(c)] of magnetic energy iso-surfaces and pseudo-color view [See Fig. 3(d)] of
magnetic energy iso-surfaces. blackThese several visualizations (iso-surface
view, volume rendering view, and pseudo-color view) aid in understanding the
nature of iso-surfaces. Such magnetic structures have been reported earlier
for kinematic dynamo simulations by Alexakis (2011) and are referred to as
“magnetic ribbons”. It is clear that the structures do not have any twisting,
which is due to the absence of fluid helicity in the flow field.

(a)

(b)

(c)

(d)

Figure 3: blackKinematic fast Dynamo effect for different magnetic Reynolds
number ($R_{m}$) using Yoshida-Morrison (YM) flow with black$\beta=0.0$. (a)
The magnetic energy $\left(\sum\limits_{V}\frac{B^{2}(x,y,z,t)}{2}\right)$
growth is seen to be unaffected with increase of $R_{m}$. The (b) three-
dimensional (3D) magnetic energy iso-surface black[Supplementary movie added],
(c) three-dimensional (3D) volume rendering view of magnetic energy iso-
surfaces and (d) three-dimensional (3D) pseudo-color view of magnetic energy
iso-surfaces (for magnetic Reynolds number $R_{m}=50$) is seen identical as a
untwisted “magnetic ribbon”. Simulation details: grid resolution $256^{3}$,
stepping time dt = $10^{-4}$, magnitude of fluid velocity $u_{0}=1.0$, Alfven
mach number $M_{A}=10.0$.

It was discussed earlier that, the injection of normalized fluid helicity in
the system is possible by systematically varying some meaningful physical
parameter, i.e. $\beta$ value for YM flow. In the following, we perform
numerical runs and present simulation results with $\beta=0.2$ for various
magnetic Reynolds numbers ($R_{m}$). The stretching, twisting and folding
(STF) of magnetic field lines black(Vainshtein & Zel’dovich, 1972) doubles the
magnetic field per cycle because of flux freezing. Presence of higher
diffusivity in the system makes the flux freezing condition to be violated,
which in turns results in, suppression of exponential growth of magnetic
energy. Hence, the fast dynamo action gets hindered for a higher diffusive
system. From Fig. 4(a) it is observed that, the YM flow (black$\beta=0.2$)
efficiently amplifies the magnetic field in exponential manner for a wide
range of $R_{m}$ value and exhibits dynamo action. A careful observation of
magnetic energy iso-surface shows that, the energy is concentrated in long and
flat structures or “ribbon” like structures [See Fig. 4(b), 4(c), 4(d)].
Although small scale structures are also present here, they appear to be more
“organized” to form blackspecial geometric patterns, which is reported earlier
(Alexakis, 2011). It is notable that, there is visible twisting in the
“ribbons” that indicates the presence of fluid helicity in the flow. With
respect to earlier case, it is identified that by varying $\beta$ value from
$0$ to $0.2$ we are able to produce blackfast dynamo action efficiently. For
$\beta=0.2$, there is fluid helicity present in the system and the presence of
fluid helicity twists the ribbon which produces dynamo action.

(a)

(b)

(c)

(d)

Figure 4: blackKinematic fast Dynamo effect for different magnetic Reynolds
number ($R_{m}$) using Yoshida-Morrison (YM) flow with black$\beta=0.2$. (a)
The magnetic energy $\left(\sum\limits_{V}\frac{B^{2}(x,y,z,t)}{2}\right)$
growth is observed for different $R_{m}$ values. The (b) three-dimensional
(3D) magnetic energy iso-surface black[Supplementary movie added], (c) three-
dimensional (3D) volume rendering view of magnetic energy iso-surfaces and (d)
three-dimensional (3D) pseudo-color view of magnetic energy iso-surfaces (for
magnetic Reynolds number $R_{m}=50$) is seen to be twisted “ribbon” like
nature. Note that the twisting indicates the presence of fluid helicity in the
flow. Simulation details: grid resolution $256^{3}$, stepping time dt =
$10^{-4}$, magnitude of fluid velocity $u_{0}=1.0$, Alfven mach number
$M_{A}=10.0$.

To study the effect of fluid helicity in the context of blackkinematic fast
dynamo action in detail, we explore various $\beta$ values such as 0.4, 0.6,
0.8 and 1.0. For $\beta=0.4$, we observe that the generation of
blackexponential magnetic energy blackgrowth is possible for a substantial
range of magnetic Reynolds number ($R_{m}$) [See Fig. 5(a)]. Due to the
presence of finite fluid helicity in the system, blackthe flow is able to
produce dynamo action. From the magnetic energy iso-surface visualization,
energy is seen to be concentrated in elongated structures [See Fig. 5(b),
5(c), 5(d)]. These structures are part of sheetlike structures reported
earlier (Alexakis, 2011), much like the “ribbons” [See Fig. 5(b), 5(c), 5(d)]
observed for earlier cases as well.

(a)

(b)

(c)

(d)

Figure 5: blackKinematic fast Dynamo effect for different magnetic Reynolds
number ($R_{m}$) using Yoshida-Morrison (YM) flow with black$\beta=0.4$. (a)
The magnetic energy $\left(\sum\limits_{V}\frac{B^{2}(x,y,z,t)}{2}\right)$
growth is observed for different $R_{m}$ values. The (b) three-dimensional
(3D) magnetic energy iso-surface, (c) three-dimensional (3D) volume rendering
view of magnetic energy iso-surfaces and (d) three-dimensional (3D) pseudo-
color view of magnetic energy iso-surfaces (for magnetic Reynolds number
$R_{m}=50$) is seen to be twisted “ribbon” like nature like earlier. Note that
the twisting indicates the presence of helicity in the flow. Simulation
details: grid resolution $256^{3}$, stepping time dt = $10^{-4}$, magnitude of
fluid velocity $u_{0}=1.0$, Alfven mach number $M_{A}=10.0$.

For further higher $\beta$ value say $\beta=0.6$, it is observed that magnetic
energy grows exponentially with time for a wide range of $R_{m}$ values. We
provide the results for magnetic Reynolds number ($R_{m}$) $50$ to $5000$ in
small steps and observe that, for all the sets of numerical runs, there is
prominent dynamo action [See Fig. 6(a)]. This is due to the presence of fluid
helicity blackand twisting ability in the system. As in the earlier cases, the
presence of fluid helicity promotes the twisting ability of the flow and that
results in dynamo action. The magnetic energy is seen to concentrate in
elongated sheetlike or “ribbon” like structures [See Fig. 6(b), 6(c), 6(d)]
reported earlier as well (Alexakis, 2011). The magnetic energy iso-surfaces
form couple structures for some particular time. These structures are also
seen similar to the “double ribbon” structures similar to the “double cigar”
structures, reported earlier for ABC flow (Dorch, 2000). In the simulation
within the oscillatory regime, as the energy increases at the beginning of a
‘cycle’, the secondary ribbon increases in size until the two ribbon become
equal in size and strength midway through the cycle [See Fig. 6(b), 6(c),
6(d)]. The primary ribbon now becomes smaller and actually vanishes at the end
of the cycle, at which time the growth of the energy slows down. Towards the
end of a cycle, the primary ribbon begins to decay, because the supply of flux
to the primary ribbon is less than that to the secondary ribbon. At the end of
one cycle the secondary ribbon is the only remaining structure.

(a)

(b)

(c)

(d)

Figure 6: blackKinematic fast Dynamo effect for different magnetic Reynolds
number ($R_{m}$) using Yoshida-Morrison (YM) flow with black$\beta=0.6$. (a)
The magnetic energy $\left(\sum\limits_{V}\frac{B^{2}(x,y,z,t)}{2}\right)$
growth is observed for different $R_{m}$ values. The (b) three-dimensional
(3D) magnetic energy iso-surface black[Supplementary movie added], (c) three-
dimensional (3D) volume rendering view of magnetic energy iso-surfaces and (d)
three-dimensional (3D) pseudo-color view of magnetic energy iso-surfaces (for
magnetic Reynolds number $R_{m}=50$) is seen to be twisted “ribbon” like
nature. Due to the presence of oscillation in the magnetic energy “double
ribbon” structures are observed. Simulation details: grid resolution
$256^{3}$, stepping time dt = $10^{-4}$, magnitude of fluid velocity
$u_{0}=1.0$, Alfven mach number $M_{A}=10.0$.

Finally, we explore the only left $\beta$ value, i.e. $\beta=1.0$. As
discussed earlier, YM flow with black$\beta=1.0$ regenerates the classical
Arnold-Beltrami-Childress [ABC] flow. As in the earlier cases (viz.
$\beta=0.0,0.2,0.4,0.8$), we provide runs for a wide range of magnetic
Reynolds number ($R_{m}$) starting from $R_{m}=50$. As is expected, for
$\beta=1$, for all the sets of numerical runs, we observe significant magnetic
energy growth [See Fig. 7(a)]. The visualization of this maximum helical
($\beta=1.0$) flow leads to visible separatrix formation [See Fig. 1(f)], a
signature of chaotic flow which drives the exponential energy growth for
$\beta=1.0$, as the flow becomes maximum helical similar like ABC flow. This
exponential growth of magnetic energy using ABC flow via blackkinematic fast
dynamo action is well explored so far by various authors (Galloway & Frisch,
1986; Dorch, 2000; Archontis, V. et al., 2003; Bouya & Dormy, 2013). Probably
the ABC flow is the most explored flow, that has been used for dynamo action.
We also produce identical magnetic energy growth like ABC flow starting from
an initial negligible value, using YM flow with black$\beta=1$ [See Fig.
7(a)]. The study of magnetic energy iso-surface also leads to an interesting
observation. It is observed that, the exponentially growing nature of magnetic
energy generates blackrigorous geometric structures or likely to say “cigar
like” structures [See Fig. 7(b), 7(c), 7(d)], and it supported by literatures
(Galloway & Frisch, 1986; Dorch, 2000; Archontis, V. et al., 2003; Bouya &
Dormy, 2013). From our numerical experiments, we confirm that “cigar like”
structure is a stable structure and its appearance is a signature for the
onset of exponential growth in magnetic energy. Strong localization of the
current density structures co-existing with magnetic “cigars”, due to magnetic
reconnection, is also observed [See Fig. 7(e)].

(a)

(b)

(c)

(d)

(e)

Figure 7: blackKinematic fast Dynamo effect for different magnetic Reynolds
number ($R_{m}$) using Yoshida-Morrison (YM) flow with black$\beta=1.0$. (a)
The magnetic energy $\left(\sum\limits_{V}\frac{B^{2}(x,y,z,t)}{2}\right)$
growth is observed for different $R_{m}$ values. The (b) three-dimensional
(3D) magnetic energy iso-surface black[Supplementary movie added], (c) three-
dimensional (3D) volume rendering view of magnetic energy iso-surfaces and (d)
three-dimensional (3D) pseudo-color view of magnetic energy iso-surfaces (for
magnetic Reynolds number $R_{m}=50$) is seen to be “cigar like” in nature. (e)
Strong localization of the current density is also observed. Simulation
details: grid resolution $256^{3}$, stepping time dt = $10^{-4}$, magnitude of
fluid velocity $u_{0}=1.0$, Alfven mach number $M_{A}=10.0$.

By systematically varying the helicity parameter $\beta$ in the YM flow
(Yoshida & Morrison, 2017) or in the other words by injecting normalized fluid
helicity in the system, we have explored a systematic route that connects flat
“ribbon” like structures to “twisted ribbon” and finally towards classical
“cigar” like structures. The emergence of these blacksignificant structures
(“ribbon”, “twisted ribbon”, “cigar”) due to the injected normalized fluid
helicity are demonstrated via direct numerical simulation. Interestingly,
though all these structures have been reported earlier, but the significant
connection between these structures using a single fluid flow model, not
explored so far. Here in this present study we connect all these different
blackwell known structures via normalized fluid helicity injection using YM
flows.

Coming back to black$\beta=1.0$ case, as discussed, this class of flow is
identical to the well known Arnold-Beltrami-Childress [ABC] flow. It is shown
that, the fastest magnetic energy growth or the exponential energy growth for
this kind of flow is signified by generation of a special kind of structure
called “cigar” like structure. We study the magnetic Reynolds number ($R_{m}$)
effect on the “cigar” structure. Keeping all the parameters identical, we
perform numerical runs for various magnetic Reynolds number ($R_{m}$) and
visualize the magnetic energy iso-surfaces [See Fig. 8(a), 8(b), 8(c)]. From
figures 8(a), 8(b) and 8(c) it is identified that, the thickness of a “cigar”
($\delta$) decreases with the increase of magnetic Reynolds number ($R_{m}$).
We also calculate the the “cigar” thickness ($\delta$) for various
intermediate $R_{m}$ and plot it [See Fig. 8(d)]. From Fig. 8(d) it is
observed that, the “cigar” thickness ($\delta$) decreases with $R_{m}$ with a
strong scaling of $\frac{1}{\sqrt{R_{m}}}$. This $\frac{1}{\sqrt{R_{m}}}$
scaling is also identified by various authors (Galloway & Frisch, 1986;
Archontis, V. et al., 2003) for ABC flow and suggests Sweet-Parker scaling.
Here in this present study, we observe the identical scaling for YM flow
(black$\beta=1$). As the change of magnetic Reynolds number ($R_{m}$) only
changes the cigar thickness ($\delta$), hence to help visualize, we plot iso-
surfaces for magnetic Reynolds number ($R_{m}=50$) for all the case discussed
above.

(a)

(b)

(c)

(d)

Figure 8: Three-dimensional (3D) magnetic energy iso surface for (a)
$R_{m}=450$, (b) $R_{m}=1000$, (c) $R_{m}=3000$ using Yoshida-Morrison (YM)
flow with black$\beta=1.0$. (d) With the increase of $R_{m}$ value the
thickness of magnetic cigars ($\delta$) are seen to be decreased with a strong
scaling $\frac{1}{\sqrt{R_{m}}}$, which is observed earlier for ABC flow.

We also calculate the magnetic energy growth rate for all $\beta$ cases. It is
seen from Fig. 9 that, for $\beta=0$ as there is no fluid helicity present in
the system, blackadditionally, the flow does not have enough stretching
ability , so there is no dynamo excitation. The growth rate is also seen to be
flat (blue line in Fig. 9) for higher and higher magnetic Reynolds number
($R_{m}$). The growth rate for $\beta=0.2$ is seen to saturate over $R_{m}$,
(red line in Fig. 9) though the prominent dynamo action is observed for a
large range of magnetic Reynolds number ($R_{m}$) value. For higher $\beta$
value, i.e, $\beta=0.4,0.6,0.8$ it is noticed that, the magnetic energy growth
rate increases as $R_{m}$ increases, after some critical value. For all the
intermediate $\beta$ values, we observe significant magnetic energy growth
[See Fig. 9]. We estimate the energy growth rate for the well explored
$\beta=1.0$ case, which is homologous to the conventional ABC flow. For Fig. 9
it is observed that the the growth rate of magnetic energy for $\beta=1.0$
case increases significantly with $R_{m}$. For lower magnetic Reynolds number
($R_{m}$), there is a dip in the growth rate curve (purple line in Fig. 9),
which is reported earlier as well (Galloway & Frisch, 1986). It is also
interesting to note that, for lower magnetic Reynolds number ($R_{m}$), the
ABC dynamo model is not the best dynamo model [See Fig. 9] in the sense of
growth rate of magnetic energy. The magnetic energy growth rates for
$\beta=0.4$ & $\beta=0.6$ are seen to be larger than regular ABC flow or
$\beta=1.0$, flow for lower $R_{m}$ values. This observation suggests that for
the YM class of flows, maximum fluid helicity does not imply the fastest
dynamo action at lower $R_{m}$ values.

Figure 9: Magnetic energy growth rate for various class of Yoshida-Morrison
(YM) flow at different magnetic Reynolds numbers ($R_{m}$). Importantly, for
lower $R_{m}$ it is identified that ABC flow or Yoshida-Morrison (YM) flow
with black$\beta=1.0$ is not the best dynamo model.

blackFor each of the cases mentioned above, we have determined the time-
averaged magnetic energy spectrum [$\int_{0}^{\infty}B(k)dk$]. The spectra has
a peak at a higher mode number, which is the hallmark of small scale
dynamo(SSD), as can be seen in Fig. 10. Additionally, all of the spectra are
truncated at the mode value ($k_{max}=85$), and the peaks are in the range of
$k=15$ to $40$, indicating that it is well resolved from a spectral
perspective.

Figure 10: blackTime averaged (time average taken from t = 41.0 to 100.0)
magnetic energy spectral density $B(k)$ [such that $\int B(k,t)dk$ is the
total energy at time t] for different classes of Yoshida-Morrison (YM) flow at
fixed magnetic Reynolds numbers ($R_{m}=450.0$) are shown in linear and log-
log (inset view) scales. The spatial scale spectrum clearly indicates that the
dynamo is a smale-scale fast dynamo. Spectra are truncated at the mode number
value ($k_{max}=85$), and the peaks are in the range of $k=15$ to $k=40$,
indicating that it is well resolved from spectral point of view.

We have already discussed the the effect of magnetic Reynolds number ($R_{m}$)
on different class of YM flow at different fluid helicity. We now present our
results keeping a particular $R_{m}$ fixed, we study the effect of $\beta$ or
the effect of fluid helicity on the flow. For $R_{m}=5000$, it is observed
that, by varying $\beta$ value from 0 to 1 magnetic energy generation rapidly
enhances [See Fig. 11(a)]. The reason behind this is explained via impact of
fluid helicity on dynamo instability. For $\beta=0.0$ there is no fluid
helicity in the system, where as for $\beta=1.0$ the flow is maximally
helical, that generates dynamo action. The exponential energy growth for
$\beta=1.0$ flow or the ABC flow is well known so far, but here we identify a
possible route that connects a non-dynamo regime ($\beta=0.0$) to a dynamo
regime ($\beta=1.0$) via fluid helicity injection. From Fig. 11(a) it is
verified that, the transition from non-dynamo to dynamo regime is not abrupt,
but is a continuous transition that is achieved via fluid helicity injection
in small steps.

To bring out the importance of our numerical findings, we examine magnetic
energy growth rate evolution with $R_{m}$ values $4000$ and $3000$ for all the
class of YM flows (viz. $\beta=0.0,0.2,0.4,0.6,0.8,1.0$). From Fig. 11(b),
11(c) the expected transition from non-dynamo to dynamo regime is identified
via fluid helicity injection, as before. For $R_{m}=2000$, a similar
transition (non-dynamo to dynamo) is observed. From Fig. 11(d) it is observed
that, there is a cross over of energy between $\beta=0.4$ and $\beta=0.6$
case. The energy growth is higher for a flow which has lower fluid helicity
for $R_{m}=2000$. To understand this “cross over” we further investigate for
few lower magnetic Reynolds numbers ($R_{m}$). In Fig. 11(e) & 11(f), we plot
the magnetic energy growth for $R_{m}=1000$ & $R_{m}=450$ and observe that,
the cross over is more prominent. It is also observed that, $\beta=0.4$ &
$\beta=0.6$ shows faster dynamo action than the regular ABC flow. This is
striking and from Fig. 11(e) & 11(f) one can conclude that, the YM flow with
black$\beta=1.0$ or the classical ABC flow is not the best dynamo model at
lower magnetic Reynolds numbers ($R_{m}$). This particular observation
suggests existence of a non-monotonous relationship between flow field
helicity and dynamo. It would be interesting to explore this parameter space
in greater detail, in the future.

(a)

(b)

(c)

(d)

(e)

(f)

Figure 11: Exponential growth of magnetic energy for (a) $R_{m}=5000$, (b)
$R_{m}=4000$, (c) $R_{m}=3000$, (d) $R_{m}=2000$, (e) $R_{m}=1000$, (f)
$R_{m}=450$ with various class of Yoshida-Morrison (YM) flow. A clear
transition from non-dynamo regime to dynamo regime is observed, at the cost of
fluid helicity for higher magnetic Reynolds numbers ($R_{m}$). These figures
significantly indicate the possible route that connects non-dynamo to dynamo
phase.

We also calculate the magnetic energy growth rate for the above discussed
cases. As discussed above, a clear transition from non-dynamo regime to dynamo
regime is observed in growth rate as a function of fluid helicity, for higher
magnetic Reynolds number ($R_{m}$) [See Fig. 12]. From Fig. 12 it is also
observed that the for lower magnetic Reynolds number ($R_{m}$) the ABC model
is not the best dynamo model.

blackIt is important to indicate that, for this present study addressed so
far, we have fixed $u_{0}=1.0$ and changed $\beta$, which basically changes
the kinetic energy of the driving flow. We have also considered another
possible approach and performed few test runs, in which we have kept the
average kinetic energy ($<E>$) constant and normalized the magnetic Reynolds
number using $<E>$. The same unambiguous shift from the non-dynamo to the
dynamo regime is seen in our numerical experiment using this initial condition
(not shown here). Thus, the major finding is unaffected by maintaining a
constant value for the average kinetic energy.

Figure 12: Magnetic energy growth rate for various class of Yoshida-Morrison
(YM) flow at fixed magnetic Reynolds numbers ($R_{m}$). A notable transition
from non-dynamo regime to dynamo regime is observed in growth rate, at the
cost of fluid helicity injection, for higher magnetic Reynolds number
($R_{m}$).

##  6 Summary and Conclusion

In this work, we have performed direct numerical simulations of 3-dimensional
magnetohydrodynamic plasmas at sufficiently high grid resolutions.

We have analyzed blackkinematic fast dynamo model using YM flow recently
proposed by (Yoshida & Morrison, 2017). An interesting and useful aspect of
this flow is that, it is possible to inject finite fluid helicity
[$\int_{V}\vec{u}.(\vec{\nabla}\times\vec{u})dV$] in the system, by
systematically varying certain physically meaningful parameter.

Our major findings are:

$\bullet$ Using a simple blackkinematic fast dynamo model, we demonstrate, a
fast dynamo action primarily due to normalized fluid helicity injection. By
injecting normalized fluid helicity in the system systematically, we have
explored a systematic route that connects “non-dynamo” to “dynamo” regime, via
direct numerical simulation.

$\bullet$ The study of magnetic field iso-surface is seen to exhibit untwisted
ribbon like structures, when there is no helicity injection, where as,
helicity injection is shown to introduce twisting in the iso-surfaces that
generates dynamo.

$\bullet$ We demonstrate as to, how an untwisted ribbon like non-dynamo iso-
surface gets converted into cigar like fast dynamo iso-surface with the
injection of fluid helicty.

$\bullet$ blackWe have calculated the time-averaged magnetic energy spectrum
and it is observed that, the spectra contain a discernible maxima at a higher
mode number, which is the distinguishing feature of small scale dynamo (SSD).

$\bullet$ blackThe conventional understanding is that a lack of reflectional
symmetry (e.g., non-zero fluid helicity) is necessary for large scale dynamo
action (LSD), whereas, for small scale dynamo (SSD), it is not. In our work,
for the flow considered here, we demonstrate how helical structure of the flow
does strongly controls the SSD growth and spectral structure. SSD. This is one
of the interesting findings of the present work.

$\bullet$ We also identify that the most studied ABC dynamo model is not the
best suited dynamo model for lower magnetic Reynolds number problems.

To conclude, with high enough resolution and at various magnetic Reynolds
number, we investigate a systematic route that connects non-dynamo to dynamo
regime via fluid helicity injection. Our observation is seen to conform with
the magnetic iso-surface dynamics. We have shown that, how an untwisted ribbon
like non-dynamo iso-surface slowly converts to twisted ribbon like dynamo iso-
surface and finally leads to cigar like fast dynamo iso-surface with injection
of fluid helicty. blackOur work also indicates that, the fluid helicity does
affect the dynamics of small-scale dynamos. We believe that, this work brings
out unambiguously, the role of fluid helicity blackon small-scale dynamos and
on the transition from “non-dynamo” to “dynamo” regime, via direct numerical
simulation.

In our present work, it is demonstrated that the role of fluid helicity is the
key factor to exhibit dynamo action. However, in the past, existence of non-
helical flows (i.e, flows with zero fluid helicity) to be able to generate
dynamo, in blackKinematic Fast and Saturation (with magnetic feedback) dynamos
models (Galloway & Proctor, 1992; Sur & Brandenburg, 2009), wherein cross-
helicity (also called Yoshizawa effect) is shown to play a crucial role.
However the investigation of the role of cross-helicity (also called Yoshizawa
effect) on YM flow and others, in the context of various dynamo action is
beyond the scope of the present work and will be addressed in future
communication. The effect of fluid helicity injection in the context of
recurrence (Mukherjee et al., 2019) and quasi-recurrence (Biswas et al.,
2022b) phenomenon is also an interesting study to investigate out, that will
also be reported soon. blackThe role of hydro-dynamically unstable shear flow
(Biswas & Ganesh, 2022) at the largest scale is interesting to investigate in
this context. We hope to attempt this problem in the near future.

##  7 ACKNOWLEDGMENTS

The simulations and visualizations presented here are performed on GPU nodes
and visualization nodes of Antya cluster at the Institute for Plasma Research
(IPR), INDIA. One of the author S.B is thankful to Dr. Rupak Mukherjee at
Central University of Sikkim (CUS), Gangtok, Sikkim, India for providing an
initial version of GMHD3D code. S.B thanks N. Vydyanathan, Bengaluru and B. K.
Sharma at NVIDIA, Bengaluru, India, for extending their help with basic GPU
methods. S.B is grateful to Mr. Soumen De Karmakar at IPR, India for many
helpful discussions regarding GPUs, and HPC support team of IPR for extending
their help related to ANTYA cluster. The authors would also like to thank Dr.
Jugal Chowdhury from IPR for careful reviewing the manuscript and for his
valuable suggestions.

##  8 DATA AVAILABILITY

The data underlying this article will be shared on reasonable request to the
corresponding author.

##  9 Conflict of Interest

The authors have no conflicts to disclose.

##  10 SUPPORTING INFORMATION

Supplementary movies:

  * •

YM_beta_0p0_B_Field_IsoB.mp4

  * •

blackYM_beta_0p2_B_Field_IsoB.mp4

  * •

YM_beta_0p6_B_Field_IsoB.mp4

  * •

YM_beta_1p0_B_Field_IsoB.mp4

are added.

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##  Appendix A Fluid helicity calculation for Yoshida-Morrison (YM) flow

Fluid helicity is defined as,

$H_{f}=\int_{V}\vec{u}.(\vec{\nabla}\times\vec{u})dV.$

Here,

$\vec{u}=\hat{i}\left(\alpha
u_{0}[B\sin(k_{0}y)-C\cos(k_{0}z)]\right)+\hat{j}\left(\beta
u_{0}[C\sin(k_{0}z)-A\cos(k_{0}x)]\right)+\hat{k}\left(u_{0}[\alpha
A\sin(k_{0}x)-\beta B\cos(k_{0}y)]\right).$

Taking curl and putting the real constant value, $\alpha=A=B=C=1.0$,
$k_{0}=1.0$ , we get,

$\vec{\nabla}\times\vec{u}=\hat{i}\left(\beta u_{0}[\sin y-\cos
z]\right)+\hat{j}\left(u_{0}[\cos x-\sin
z]\right)+\hat{k}\left(u_{0}[\beta\sin x-\cos y]\right)$

Taking $\vec{u}\cdot$ on both side,

$\displaystyle\vec{u}\cdot\vec{\nabla}\times\vec{u}=$
$\displaystyle\left(u_{0}[\sin y-\cos z]\right)\left(\beta u_{0}[\sin y-\cos
z]\right)+\left(\beta u_{0}[\sin z-\cos x]\right)\left(u_{0}[\cos x-\sin
z]\right)$ $\displaystyle+\left(u_{0}[\sin x-\beta\cos
y]\right)\left(u_{0}[\beta\sin x-\cos y]\right).$

or,

$\displaystyle\vec{u}\cdot\vec{\nabla}\times\vec{u}=\beta u_{0}^{2}\left[(\sin
y-\cos z)^{2}\right]-\beta u_{0}^{2}\left[(\cos x-\sin
z)^{2}\right]+u_{0}^{2}\left[(\sin x-\beta\cos y)(\beta\sin x-\cos y)\right]$

or,

$\displaystyle\vec{u}\cdot\vec{\nabla}\times\vec{u}=$ $\displaystyle\beta
u_{0}^{2}\left[(\sin^{2}y+\cos^{2}z-2\sin y\cos z)\right]-\beta
u_{0}^{2}\left[(\cos^{2}x+\sin^{2}z-2\cos x\sin z)\right]$
$\displaystyle+u_{0}^{2}\left[(\beta\sin^{2}x-\sin x\cos y-\beta^{2}\sin x\cos
y+\beta\cos^{2}y)\right]$

Integrating over total volume,

$\displaystyle\int_{V}\vec{u}.(\vec{\nabla}\times\vec{u})dV=$
$\displaystyle\beta
u_{0}^{2}\left[(\int_{0}^{2\pi}\int_{0}^{2\pi}\int_{0}^{2\pi}\sin^{2}ydxdydz+\int_{0}^{2\pi}\int_{0}^{2\pi}\int_{0}^{2\pi}\cos^{2}zdxdydz-\int_{0}^{2\pi}\int_{0}^{2\pi}\int_{0}^{2\pi}2\sin
y\cos zdxdydz)\right]$ $\displaystyle-\beta
u_{0}^{2}\left[(\int_{0}^{2\pi}\int_{0}^{2\pi}\int_{0}^{2\pi}\cos^{2}xdxdydz+\int_{0}^{2\pi}\int_{0}^{2\pi}\int_{0}^{2\pi}\sin^{2}zdxdydz-\int_{0}^{2\pi}\int_{0}^{2\pi}\int_{0}^{2\pi}2\cos
x\sin zdxdydz)\right]$ $\displaystyle+\beta
u_{0}^{2}\left[(\int_{0}^{2\pi}\int_{0}^{2\pi}\int_{0}^{2\pi}\sin^{2}xdxdydz\right]-u_{0}^{2}\left[\int_{0}^{2\pi}\int_{0}^{2\pi}\int_{0}^{2\pi}\sin
x\cos ydxdydz\right]$
$\displaystyle-\beta^{2}u_{0}^{2}\left[\int_{0}^{2\pi}\int_{0}^{2\pi}\int_{0}^{2\pi}\sin
x\cos ydxdydz\right]+\beta
u_{0}^{2}\left[\int_{0}^{2\pi}\int_{0}^{2\pi}\int_{0}^{2\pi}\cos^{2}ydxdydz\right]$

Finally after simplification one obtains,

$\int_{V}\vec{u}.(\vec{\nabla}\times\vec{u})dV=8\pi^{3}\beta u_{0}^{2}$

Considering velocity to be normalized,

$H_{f}=\beta$

So normalized fluid helicity for YM flow depends upon constant $\beta$ value
of the flow.

##  Appendix B Grid size scaling study for ABC flow

Grid size scaling study have been performed using Arnold-Beltrami-Childress
(ABC) flow [ Eq. 3 ] at different magnetic Reynolds numbers. It is seen that
$256^{3}$ grid resolution is more than enough for this problem [See Fig.
13(a), 13(b)].

(a)

(b)

Figure 13: blackFast Dynamo effect following Galloway et al. (Galloway &
Frisch, 1984) using Arnold-Beltrami-Childress (ABC) flow for magnetic Reynolds
numbers (a) $R_{m}=50.0$ (b) $R_{m}=450.0$ at different grid resolutions.