# The Role of Helical and Non-Helical Drives on the evolution of Self-
Consistent Dynamos

Shishir Biswas  0000-0002-4879-8889<EMAIL_ADDRESS><EMAIL_ADDRESS>Institute for Plasma Research, Bhat, Gandhinagar,
Gujarat 382428, India  Homi Bhabha National Institute, Training School
Complex, Anushaktinagar, Mumbai 400094, India  Rajaraman Ganesh
0000-0002-5095-3067 Institute for Plasma Research, Bhat, Gandhinagar, Gujarat
382428, India  Homi Bhabha National Institute, Training School Complex,
Anushaktinagar, Mumbai 400094, India

###### Abstract

In the self-consistent dynamo limit, the magnetic feedback on the velocity
field is sufficiently strong to induce a change in the topology of the
magnetic field. Consequently, the magnetic energy reaches a state of non-
linear saturation. Here, we investigate the role played by helical and non-
helical drives in the triggering and the eventual saturation of a self-
consistent dynamo. Evidence of small-scale dynamo (SSD) activity is found for
both helical and non-helical forcing, driven at the largest possible scale.
Based on the spectrum analysis, we find that the evolution of kinetic energy
follows Kolmogorov’s $k^{-\frac{5}{3}}$ law while that of magnetic energy
follows Kazantsev’s $k^{\frac{3}{2}}$ scaling. Also, we have verified that the
aforementioned scalings remain valid for various magnetic Prandtl numbers
($P_{m}$). Statistical analysis is found to support our numerical finds.

##  I Introduction

A substantial portion of the universe is magnetized. Multi-scale magnetic
fields are detected in a variety of astrophysical entities such as the solar
system, planets, galaxy clusters, interstellar medium, star cores, accretion
disks, and spiral galaxies. One pertinent issue that arises is: what are the
sources of this multi-scale magnetic field? The turbulent dynamo mechanism,
which is the primary means of converting the kinetic energy of the turbulent
motion of conducting fluids into magnetic energy, is thought to amplify and
sustain magnetic fields in the majority of these systems. The dynamo process
naturally necessitates a weak seed field, which eventually becomes stronger
and comparable to the turbulent kinetic energy 1; 2; 3; 4. One possible
explanation for the formation of these seed magnetic fields is the Biermann
battery mechanism 5, wherein a fragile magnetic field is produced by the
separation of ions and electrons. This mechanism can occur during several
astrophysical events such as cosmic reionization in photogalaxies, and during
the development of large-scale structures. Alternatively, Weibel instability 6
is known to produce magnetic fields from counter-streaming plasmas, which are
also regarded the source of seed magnetic fields.

Dynamos are categorized into two distinct flavours based on the feedback
strength of the magnetic field onto the flow field. For example, 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
Navier–Stokes equation. A nonlinear dynamo or self-consistent dynamo is when
the nonlinear effects start to change the flow (once the magnetic field is
large enough) to stop further magnetic field growth, that is, the flow and
B-field “back react” on each other, typically leading to nonlinear saturation.
Several reviews discuss the theoretical, numerical, and observational aspects
of dynamo activity 7; 8; 9; 10; 11; 12; 13.

In addition to the classification previously described, dynamos are further
categorized into two subgroups based on the length scales involved. The first
type is the large scale dynamo (LSD) or mean field dynamo, which refers to the
increase of magnetic field on a scale that exceeds the forcing length scale of
fluid motion. Large scale dynamo activity necessitates symmetry breaking, such
as fluid helicity (mirror symmetry breaking), differential rotation, density
lamination, etc. Numerical investigations have shown that significant scale
separation between the energy-containing scale and the largest scale of the
flow can lead to the generation of large-scale dynamo activity through the
helical drive, which is not mirror symmetric. The Large Scale Dynamo (LSD)
activity is sometimes known as the $\alpha$-dynamo. It has been determined
that the $\alpha$-dynamo effect is proportional to the inverse of the magnetic
Reynolds number ($R_{m}$), expressed as $\alpha\propto\frac{1}{R_{m}}$. The
scale separation enhances the inverse cascade of magnetic helicity, leading to
the generation of large-scale magnetic fields 14; 15; 16; 17; 18; 19; 20; 21;
22; 23; 24; 25; 26; 27. Large-scale dynamo action is crucial for understanding
the magnetic field in the solar photosphere. The helical motion in the solar
convection zone creates the $\alpha$ effect, which can generate a poloidal
field that transforms into a toroidal field 28. It is a crucial element in the
11-year solar cycle, leading to the emergence of the solar butterfly diagram
29.

Here we focus on the second kind of dynamo, namely the small-scale or
fluctuation dynamo (SSD), which is defined as the growth of magnetic-
fluctuation energy at or below the forcing length scale of the fluid motion
30; 14; 31; 28; 32; 33; 34; 35; 36; 37; 38; 39; 40; 41; 42; 43; 44; 22; 45;
46; 47; 48; 49; 50; 51; 52; 53. Fluid helicity is believed to not to impose
any constraints for this type of dynamo action. Small scale dynamos (SSD) are
essential in numerous astrophysical scenarios such as spiral galaxies 9,
elliptical galaxies 54, galaxy clusters 55, and stars 28. In addition, small-
scale dynamos influence the dynamic processes of the Sun, such as angular
momentum transport, the formation of differential rotation, and contribute to
coronal heating. Accretion disk physics involves the presence of small-scale
magnetic fields, which introduce a new paradigm for angular momentum
transport. Therefore, it is evident that small-scale magnetic fields are
ubiquitous in nature, similar to turbulence.

The previous discussion indicates that homogeneous isotropic turbulence
lacking mirror symmetry leads to $\alpha$-dynamo. Additionally, flow shear and
differential rotation have a pervasive impact in a wide range of astrophysical
situations. The paradigm of rapid increase in magnetic field resulting from
the interplay between small-scale velocity fluctuations and a large-scale
velocity shear is known as the “shear dynamo problem” 56; 57. The interaction
of dynamo action with flow shear can create the so-called $\omega$-dynamo,
while helical turbulence combined with shear can result in an oscillatory
dynamo or a $\alpha-\omega$ dynamo. Additionally, dynamo instability is
affected by the Coriolis force as a consequence of differential rotation. In a
rapidly rotating system, the flow becomes effectively 2-dimensional because
the Coriolis force suppresses fluctuations in the direction of rotation 58;
59. The transformation of the flow into two dimensions modifies statistical
features of the background turbulence, such as altering the direction of the
energy cascade from forward to inverse.

Now let us return to small-scale or fluctuations dynamos. The fluctuation
dynamo produces small-scale magnetic fields through the random stretching of
field lines by turbulent velocity fields, as seen in Figure 1. The primary
process believed to be responsible for this turbulent amplification of the
magnetic field is the stretch-twist-fold (STF) mechanism 60. Small-scale
turbulence initially elongates magnetic field lines, enhancing the field
strength while maintaining a steady magnetic flux. The elongated field lines
are subsequently twisted. This stage is important to highlight as it involves
three spatial dimensions. Ultimately, the twisted field lines are folded,
enhancing the magnetic flux. It is evident that rapid growth in the strength
of the magnetic field would result from the repetition of this procedure. This
process is illustrated using a cartoon diagram (Refer to Fig. 2). Exponential
growth of the field is possible as long as the dynamo remains in the linear
(kinematic) regime, where the magnetic field is sufficiently weak that the
tension force of the field lines is significantly smaller than the force
contained in the small-scale turbulent eddies. As the field strength
increases, the tension force strengthens, causing the dynamo to shift to the
non-linear regime where the tension force is similar in magnitude to the
forces produced by turbulence. Therefore, the energy contained in turbulence
becomes insufficient to continue the process of folding field lines, causing
the dynamo to attain saturation.

Both helical and non-helical dynamos are significant in different
astrophysical scenarios. The evolution of both types of dynamos has been
studied numerically in the presence of a variety of flow drives, including
Arnold-Beltrami-Childress (ABC) flow 61, Galloway-Proctor (GP) flow 62,
Roberts flow 63; 40, Cats Eye flow 64, Taylor-Green (TG) flow 65, Archotis
flow 66; 67, Ponomarenko flow 68, interacting vortex tubes 69, Precession
driven flow 70, and Yoshida-Morrison (YM) flow 71; 52. These flow drives are
appreciated for their symmetry and significance in astrophysical scenarios and
liquid metal research in laboratory environments 72; 73; 74.

The magnetic Prandtl number ($P_{m}$) is a crucial parameter in dynamo
instability, defined as the ratio of viscosity to resistivity or the ratio of
magnetic Reynolds number to kinetic Reynolds number
($P_{m}=\frac{\nu}{\eta}=\frac{R_{m}}{R_{e}}$). $P_{m}$ is defined as
$10^{-5}\frac{T^{4}}{n}$ for a fully ionized medium, where $T$ is measured in
Kelvin and $n$ is the particle concentration in $cm^{-3}$. Most astrophysical
systems exhibit either high or low values of $P_{m}$. The former limit is
suitable for low-density, warm and diffuse plasmas, including the intracluster
medium of galaxy clusters, protogalaxies, high-temperature phases of the
interstellar medium, the early universe, and so forth. The latter limit takes
place in dense environments such as liquid-metal cores of planets, star
convective zones, protostellar disks, and in liquid-metal (mercury, sodium,
gallium) investigations. The range of $P_{m}$ has a significant influence on
dynamo instability and is a crucial parameter for investigation. Therefore,
several numerical studies 32; 33; 34; 40; 41; 21; 22; 75; 76; 47; 48; 51; 53
have been conducted at various magnetic Prandtl values to simulate diverse
astrophysical and laboratory situations related to dynamo instability.

The present study examines the effects of helical and non-helical drives on
dynamos operating at small scales. By employing a self-consistent dynamo
model, we have examined the effect of helical and non-helical dynamos. We
simulate the velocity field in our model using the Navier-Stokes equation
along with a drive. We have incorporated a recently reported three-dimensional
Yoshida–Morrison flow (abbreviated YM flow) 71 into our simulation as the flow
driver. The mirror symmetry (kinetic helicity) of YM flow can be manipulated
by adjusting certain specific flow parameters 52. YM flow exhibits
similarities to the Arnold-Beltrami-Childress (ABC) flow in the maximal
helicity limit. In the non-helical limit, it is often referred to as EPI2D
flow 71. While the dynamo action using ABC flow drive has been extensively
studied, the usefulness of EPI2D flow drive for dynamo action is only starting
to be investigated.

A case study is carried out to evaluate the impact of controlled helicity
injection on small-scale dynamos in the nonlinear limit. Numerical
investigation is conducted on the impact of magnetic Prandtl numbers ($P_{m}$)
on small-scale helical and non-helical dynamos. In addition, we have developed
various diagnostic tools such as those utilized to monitor the variations in
kinetic and magnetic energy over time [$E_{u}(t)$ & $E_{B}(t)$], evaluate the
distribution of kinetic and magnetic energy across different scales
[$E_{u}(k)$ & $E_{B}(k)$], determine the coherence length scale [$l_{u}(t)$ &
$l_{B}(t)$], examine the probability distribution function (PDF) of velocity
and magnetic field fluctuations, evaluate the PDF of alignment angle between
different fields, calculate the Skewness and Kurtosis [$S(f)$ & $K(f)$],
determine the structure functions of different orders [$S^{p}(l)$], and
evaluate the hyper-flatness [$\mathcal{F}^{6}(l)$].

The organization of the paper is as follows. In Sec. II we present about the
dynamic equations. Our numerical solver and simulation details are described
in Sec. III. The initial conditions, parameter details are shown in Sec. IV.
Section V 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. VI.

Figure 1: Schematic illustrating the interaction between the seed magnetic
field and turbulent velocity field leading to the stretching of the magnetic
field. Figure 2: Schematic diagram illustrating the small-scale turbulent
dynamo, showing the stretching, twisting, and folding (STF) of field lines
caused by turbulence. Stretching the magnetic field increases its strength
while maintaining a steady flux. The stretched field lines become twisted,
necessitating movement into the third dimension. During the final stage, the
twisted field lines are folded, leading to an increase in the flux. The
graphic depicts symbols wherein $B$ denotes magnetic field, $S$ represents the
cross-sectional area of a flux tube, and $\phi$ symbolizes magnetic flux.

##  II Governing Equations

The governing equations for the single fluid MHD plasma are as follows,

$\displaystyle\frac{\partial\rho}{\partial
t}+\vec{\nabla}\cdot\left(\rho\vec{u}\right)=0$ (1)
$\displaystyle\frac{\partial(\rho\vec{u})}{\partial
t}+\vec{\nabla}\cdot\left[\rho\vec{u}\otimes\vec{u}+\left(P+\frac{B^{2}}{2}\right){\bf{I}}-\vec{B}\otimes\vec{B}\right]$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}=\frac{1}{R_{e}}\nabla^{2}\vec{u}+\vec{f}$
(2) $\displaystyle P=C_{s}^{2}\rho$ (3)
$\displaystyle\frac{\partial\vec{B}}{\partial
t}+\vec{\nabla}\cdot\left(\vec{u}\otimes\vec{B}-\vec{B}\otimes\vec{u}\right)=\frac{1}{R_{m}}\nabla^{2}\vec{B}$
(4)

for the above said system of equations, $\rho$, $\vec{u}$, $P$ and $\vec{B}$
represent the density, velocity, kinetic pressure and magnetic fields
respectively. $\vec{f}$ is the external drive available in the system. All
quantities are appropriately normalised as discussed below. Eqn. 1 to Eqn. 4
are solved by GMHD3D suite 77; 78; 79 of codes using high performance
computing. GMHD3D suite also provides a choice between utilizing the energy
equation or the equation of state. For all cases studied here, we have used
equation of state (see Eq. 3).

We define Alfven speed as, $V_{A}=\frac{u_{0}}{M_{A}}$, here $M_{A}$ is the
Alfven Mach number of the plasma flow and $u_{0}$ is a typical velocity scale.
Sound speed of the fluid is defined as $C_{s}=\frac{u_{0}}{M_{s}}$, where
$M_{s}$ is the sonic Mach number of the fluid flow and the dynamic sound speed
$C_{s}$ contains the inherent information regarding the temperature of the
system. The initial magnetic field present in the plasma is calculated from
relation $B_{0}=V_{A}\sqrt{\rho_{0}}$, $\rho_{0}$ is the initial density of
the flow. Thus in our simulations, time is normalized to Alfven times as
$t=t_{0}*t^{\prime}$, $t_{0}=\frac{L}{V_{A}}$ and length to a typical
characteristic length scale L.

The dimensionless numbers are defined as, $R_{e}=\frac{u_{0}L}{\mu}$,
$R_{m}=\frac{u_{0}L}{\eta}$ , here $R_{e}$ and $R_{m}$ are the kinetic
Reynolds number and magnetic Reynolds number, $\mu$ & $\eta$ are the kinematic
viscosity and magnetic diffusivity. Magnetic Prandtl number is also be defined
as, $P_{M}=\frac{R_{m}}{R_{e}}$. The symbol ’$\otimes$’ represents the dyadic
between two vector quantities.

For solving the above set of equations at high grid resolution, we have
developed a suite of GPU codes namely GMHD3D 77; 78; 79, which is briefly
described in the following Section.

##  III 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 recently upgraded
an already existing well bench-marked single GPU MHD solver 80, developed in
house at Institute for Plasma Research to multi-node, multi-card (multi-GPU)
architecture for better performance 78; 77; 79. This 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 solver 78; 77; 79. This suite
(GMHD3D) includes both 2-dimensional and 3-dimensional HydroDynamic (HD) and
MagnetoHydrodynamic (MHD) solvers. 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 81. OpenACC FFT library (AccFFT library 82) 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” 83 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 84 and
Paraview 85 is used to visualize the data. Further, several other benchmarking
studies have been performed at different grid resolutions 78; 77; 79. As will
be discussed in the coming Section, numerical simulations reported here are
performed in $256^{3}$ grid sizes.

As discussed in the Introduction, to study the self-consistent dynamo action,
an accurate selection of “driving” field is crucial, which we discuss in the
Section to follow.

##  IV Initial Condition

Recently Yoshida and Morrison 71 (YM) proposed a new intermediate class of
flow, which may be regarded as a topological bridge between quasi-2D and 3D
flow classes. The flow is formulated as follows:

$\vec{u}_{b}=u_{0}\alpha\vec{u}_{+}+u_{0}\beta\vec{u}_{-}$ (5)

with

$\displaystyle\vec{u}_{+}$
$\displaystyle=\begin{bmatrix}Bsin(k_{0}y)-Ccos(k_{0}z)\\\ 0\\\
Asin(k_{0}x)\\\ \end{bmatrix}$ (6)

and

$\displaystyle\vec{u}_{-}$ $\displaystyle=\begin{bmatrix}0\\\
Csin(k_{0}z)-Acos(k_{0}x)\\\ -Bcos(k_{0}y)\\\ \end{bmatrix}$ (7)

so that,

$\displaystyle u_{x}$ $\displaystyle=\alpha
u_{0}[B\sin(k_{0}y)-C\cos(k_{0}z)]$ (8) $\displaystyle u_{y}$
$\displaystyle=\beta u_{0}[C\sin(k_{0}z)-A\cos(k_{0}x)]$ $\displaystyle u_{z}$
$\displaystyle=u_{0}[\alpha A\sin(k_{0}x)-\beta B\cos(k_{0}y)]$

where ${\vec{u}}_{b}$ represents the base flow, $k_{0}$, $u_{0}$,
$\alpha,\beta$, A, B and C are arbitrary real constants. For the present
study, we consider the value of $\alpha$, A, B and C to be unity.

The variation of $\beta$ value in YM flow leads to new classes of base flows.
For example, for $\beta=0$, Yoshida et al. 71 classify this flow as EPI-2D
flow which is given by :

$\displaystyle u_{x}$ $\displaystyle=u_{0}[\sin(k_{0}y)-\cos(k_{0}z)]$ (9)
$\displaystyle u_{y}$ $\displaystyle=0$ $\displaystyle u_{z}$
$\displaystyle=u_{0}[\sin(k_{0}x)]$

This flow (i.e, Eq. 9) 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 $\beta=1$ Eq. 8 becomes the well known
Arnold–Beltrami–Childress flow [ABC] like flow,

$\displaystyle u_{x}$ $\displaystyle=u_{0}[\sin(k_{0}y)-\cos(k_{0}z)]$ (10)
$\displaystyle u_{y}$ $\displaystyle=u_{0}[\sin(k_{0}z)-\cos(k_{0}x)]$
$\displaystyle u_{z}$ $\displaystyle=u_{0}[\sin(k_{0}x)-\cos(k_{0}y)]$

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) 52. The
variation of $\beta$ value clearly leads to two distinguishable class viz
helical ($\beta>0$) and non-helical ($\beta=0$) class of base flows 52.

The external forcing function $\vec{f}$ in Equation 2 introduces both kinetic
energy and kinetic helicity. This study utilized various sets of forcing
functions to produce dynamos. The class of YM flows, denoted as $\vec{f}$, is
utilized in the following manner:

$\displaystyle\vec{f}=$ $\displaystyle
f_{0}\\{[\sin(k_{f}y)-\cos(k_{f}z)]\hat{x}$
$\displaystyle+\beta[\sin(k_{f}z)-\cos(k_{f}x)]\hat{y}$
$\displaystyle+[\sin(k_{f}x)-\beta\cos(k_{f}y)]\hat{z}\\}$

where $f_{0}$ is the forcing or drive amplitude and $k_{f}$ is the forcing or
drive length scale.

The helical form of this forcing function is determined by the parameter
$\beta$ by the equation
$\langle\vec{f}\cdot(\vec{\nabla}\times\vec{f})\rangle=-\beta
k_{f}\langle\vec{f}\cdot\vec{f}\rangle$. Here $\langle...\rangle$ represents
the ensemble average. Based on the preceding discussion, it is determined that
when the value of $\beta$ is non zero, the forcing is helical in character
(i.e, $\langle\vec{f}\cdot(\vec{\nabla}\times\vec{f})\rangle\neq 0$).
Consequently, the generated dynamo is classified as helical dynamo. For
$\beta=0$ (i.e, $\langle\vec{f}\cdot(\vec{\nabla}\times\vec{f})\rangle=0$),
the dynamos are classified as non-helical dynamos.

We look at the impact of forcing at the largest feasible scale, specifically
when $k_{f}=1$. The magnitude of the forcing is given by $f_{0}$, which is
decided by the average force balance.

A recent study 52 based on kinematic model has demonstrated that a non-helical
quasi-2-dimensional EPI2D flow by itself cannot generate fast dynamo action
because it lacks sufficient stretching ability. The influence of “magnetic
back-reaction” has not been address for this flow drive. We have used a random
perturbation as the starting seed magnetic field for our numerical studies. We
have conducted numerical studies using a uniform magnetic field and observed
that the dynamo’s features are mostly unaffected by the initial specifications
of the B-field in both scenarios. For the rest of the discussion, we will
provide results derived from random perturbations as an initial magnetic
field.

We conduct our simulations using the provided initial conditions along with
additional parameter spaces.

###  IV.1 Parameter Details

For the class of YM flow profile, we evolve the set of equations mentioned in
Section II in a triply periodic box of length $L_{x}=L_{y}=L_{z}=2\pi$, using
time stepping $(dt)=10^{-4}$. In our present investigation, the grid
resolution is considered as $256^{3}$. We have conducted grid size scaling
studies and determined that a grid resolution of $256^{3}$ is sufficient for
the current investigation. Our investigation has identified that growth rates
and spectral convergence remain consistent at a grid resolution of $256^{3}$.
Consequently, we have presented datum with $256^{3}$ resolution. In light of
these initial conditions and parameter spaces, we present the results of our
numerical simulation.

##  V Simulation Results

Dynamo action refers to the systematic and sustained generation of a magnetic
field through the nonlinear transfer of energy from kinetic to magnetic mode
by stretching, twisting, and folding magnetic field lines. First, we will
discuss our simulation results using the ABC flow drive, also known as the
helical drive. Fig. 3(a) shows that a robust magnetic field is produced from
an initial magnetic field via the dynamo process, which then begins to impact
the structure of the velocity field. Therefore, the velocity field has been
altered and is no longer the ABC field. A “back reaction” occurs when the
magnetic field receives feedback from the velocity field and vice versa. This
mutually reinforcing connection impacts the dynamics of the magnetic field and
results in non-linear saturation. Three regions are distinguished: the
kinematic regime where the magnetic field increases linearly up to a certain
point, followed by a short transition region where the magnetic influence on
the velocity field is significant enough to modify its form, and finally
leading to the non-linear saturation region (Refer to Fig. 3(a)). This model
is referred to as the “Self-Consistent Dynamo” model.

(a)

(b)

Figure 3: (a) Temporal evolution of kinetic
($E_{v}=\frac{1}{2}\int_{V}(u_{x}^{2}+u_{y}^{2}+u_{z}^{2})dxdydz$) and
magnetic ($E_{B}=\frac{1}{2}\int_{V}(B_{x}^{2}+B_{y}^{2}+B_{z}^{2})dxdydz$)
energies respectively. In the kinematic stage (light green shaded area),
magnetic energy increases exponentially as a result of dynamo action. As the
magnetic field increases, the dynamo goes through a transitional phase (light
gray shaded area) before reaching a stable saturated state (light yellow
shaded area) caused by magnetic feedback. This model is referred to as the
“Self-Consistent dynamo” model. (b) The temporal evolution of kinetic and
magnetic energy under two distinct initial conditions-uniform (green) and
random seed (red). The magnetic and kinetic energy evolution is observed to be
the same in both scenarios.

In order to illustrate the concept of initial condition independence, our
simulation is initiated under two distinct initial conditions. Both uniform
and random seed magnetic fields have been used as initial conditions. It can
be seen from Fig. 3(b) that the magnetic energy increases exponentially with a
very similar growth rate during the kinematic stage and reaches a
statistically steady state (non-linear saturated stage) with the same value
for both the initial condition. Therefore, any memory of the original seed
field is observed to be absent. After demonstrating that the time evolution of
the magnetic field is not influenced by the nature of seed magnetic field, we
investigate how the initial seed magnetic field impacts the structural
dynamics of the magnetic field.

We provide a cross-section (X-Y plane view at $Z=0$) of the magnetic field’s
time evolution for both scenarios in Figure 4. It is evident from Figure 4
that regardless of the initial condition (uniform or random seed), the
magnetic field structures in the kinematic stage (represented in the middle
column of Fig. 4) and self-consistent stage (last column of Fig. 4) are
analogues if the drive is identical (helical ABC drive in this case). It is
not dependent on the initial seed magnetic field.

Uniform

(a) Time = 0.0

(b) Time = 15.0

(c) Time = 100.0

Random

(d) Time = 0.0

(e) Time = 15.0

(f) Time = 100.0

Figure 4: A two-dimensional slice ($X-Y$ plane at $Z=0$) of the magnetic field
in the initial (shown in first column), kinematic (shown in second column),
and saturated (shown in third column) stages for two distinct seed fields:
uniform (shown in first row) and random (shown in second row). Qualitatively
speaking, the fields appear fairly similar in both the kinematic and
saturation stages.

In order to gain a more comprehensive understanding of the similarity between
magnetic field structures, we have developed a number of diagnostic
techniques. One of them is the time variation of the normalized magnetic field
coherence length ($\frac{l_{B}}{L}$) as described in Ref. 47. The magnetic
field coherence length, which is normalized to the size of the numerical
domain $L$, is defined as:

$\frac{l_{B}}{L}=\frac{\int k^{-1}B(k)dk}{\int B(k)dk}$ (11)

An estimation has been obtained for various time snapshots and initial
conditions, as illustrated in Figure 5(a). It is clear that the normalized
magnetic field coherence length is greater in the saturated phases compared to
the kinematic stage, and it remains unaffected by the initial condition (See
Figure 5(a)).

We have computed the normalized probability density functions (PDFs) for the
$z$-component of the magnetic field $\frac{B_{z}}{B_{rms}}$ and the magnetic
field strength $\frac{B}{B_{rms}}$ (where $B$ is defined as
$\sqrt{B_{x}^{2}+B_{y}^{2}+B_{z}^{2}}$ and $B_{rms}$ is defined as
$\sqrt{\langle B^{2}\rangle}$) at both the kinematic and saturated stages,
considering two distinct initial conditions. During the kinematic stage, the
magnetic field exhibits more intermittency, even when accounting for
statistical fluctuations. This is evident from the longer tails observed for
larger values of $\frac{B_{z}}{B_{rms}}$ and $\frac{B}{B_{rms}}$ in Figures
5(b) and 5(c), respectively. The extended tail is a clear indicator of
intermittency 86.

The probability density function (PDF) of the normalized magnetic field
strength ($\frac{B}{B_{rms}}$) during the kinematic stage has a log-normal
distribution (See Fig. 5(c)). Furthermore, it is observed that the tails of
these distributions are longer when compared to those of the saturated
magnetic field, as depicted in Figure 5(c). The saturated stage is
characterized by a higher degree of Gaussian distribution or volume filling
compared to the kinematic stage (refer to Figure 5(b)). Nevertheless, it has
been determined that the PDFs exhibit statistical similarity regardless of the
two distinct initial conditions of the magnetic fields. Therefore, it can be
inferred that the initial configuration of the seed magnetic field has no
impact on the statistical structure of the field in the subsequent kinematic
and saturation stages of the small-scale dynamo.

(a)

(b)

(c)

Figure 5: (a) The magnetic field coherence length ($\frac{l_{B}}{L}$),
normalized to the size of the numerical domain ($\frac{l_{B}}{L}=\frac{\int
k^{-1}B(k)dk}{\int B(k)dk}$) as a function of time $t$ . (b & c) The
probability density functions (PDFs) for the normalized $z$-component of the
magnetic field, $\frac{B_{z}}{B_{rms}}$, and the magnetic field strength,
$\frac{B}{B_{rms}}$, for two distinct starting seed magnetic fields in both
the kinematic and saturation stages. Here $B$ is defined as
$\sqrt{B_{x}^{2}+B_{y}^{2}+B_{z}^{2}}$ and $B_{rms}$ is defined as
$\sqrt{\langle B^{2}\rangle}$.

After verifying the independence of properties of dynamo on the nature of the
initial seed magnetic field, we have investigated the impact of Alfven speed
on dynamo instability. We conduct our simulation by varying the Alfven Mach
numbers (i.e., $M_{A}=10000,1000,100,10$), while keeping all other parameters
constant. For the two scenarios, when $M_{A}$ equals $10000$ and $1000$, three
unique slopes have been identified as shown in Figure 6(a). The magnetic
energy ($E_{B}=\frac{1}{2}\int_{V}(B_{x}^{2}+B_{y}^{2}+B_{z}^{2})dxdydz$)
initially grows exponentially over time. After undergoing several orders of
magnitude amplification, the exponent of increment decreases for $M_{A}=1000$
and $10000$. Subsequently, it abruptly begins to rise with a greater exponent
(refer to Fig. 6(a)). It is interesting to note that the initial growth rate
of magnetic energy is the same for both scenarios, but they diverge at a later
stage. Figure 6(a) shows that the saturation values of kinetic and magnetic
energies are the same for all situations, regardless of the Alfven speed.
Based on the aforementioned observations, it can be deduced that the dynamo
effect becomes more feasible in super-Alfvenic regimes (refer to Fig. 6(a))
when the velocity field is helically forced, irrespective of the seed magnetic
field. The dynamo action is quite strong, resulting in a final magnetic energy
that is comparable in magnitude to the kinetic energy.

Studying the impact of magnetic Prandtl number ($P_{m}$) on self-consistent
dynamo activity is crucial to the overall understanding of the dynamo problem.
We study the impact of $P_{m}$ by keeping the kinetic Reynolds number
($R_{e}$) constant at a specific value and conduct numerical experiments for
various magnetic Reynolds numbers ($R_{m}$). Fig. 6(b) shows that the increase
rate of magnetic energy decreases when the magnetic Prandtl number ($P_{m}$)
decreases. Therefore, it is determined that the dynamo action is inhibited at
lower $P_{m}$ values when a helical drive is present (Refer to Appendix B for
more details).

(a)

(b)

Figure 6: (a) Alfven speed influences helically forced dynamo action. Dynamo
action is more pronounced in super Alfvenic regimes when driven by helical
forces. (b) This study also examines the impact of magnetic Prandtl number
($P_{m}=\frac{R_{m}}{R_{e}}$) on helical dynamo action while keeping the
kinetic Reynolds number ($R_{e}$) constant. Helical dynamo activity is
inhibited at lower values of the magnetic Prandtl number ($P_{m}$). Here the
dotted lines indicate the kinetic energy.

Following an investigation regarding the temporal evolution of magnetic energy
across distinct $P_{m}$ limits, we have also investigated the dynamics of the
magnetic field iso-surface (IsoB) at various $P_{m}$ ranges. Figure 7
illustrates the iso-surfaces of magnetic fields during the kinematic and
saturation stages for various $P_{m}$ values. The magnetic field iso-surfaces
exhibit smaller scale structure during the kinematic stage and larger
structures at the saturated stage, as shown in Figure 7. The magnetic field
iso-surfaces’ structure increases in size when $P_{m}$ decreases throughout
both the kinematic and saturation periods, as seen in Fig. 7. A lower value of
$P_{m}$ at a fixed kinetic Reynolds number ($R_{e}$) leads to a high resistive
plasma framework, which suppresses magnetic energy fluctuations, resulting in
the formation of these larger scale structures.

Kinematic

(a) $P_{m}=2.0$

(b) $P_{m}=1.0$

(c) $P_{m}=0.25$

Saturated

(d) $P_{m}=2.0$

(e) $P_{m}=1.0$

(f) $P_{m}=0.25$

Figure 7:  Visualization of three-dimensional magnetic field iso-surfaces for
a helical dynamo in the kinematic stage (top row) and saturated stage (bottom
row) at magnetic Prandtl numbers of (a & d) $2$, (b & e) $1$, and (c & f)
$0.25$. The magnetic fields in the saturated stages exhibit structures larger
than those in the kinematic stages. The magnetic field structures increase in
size when the magnetic Prandtl number ($P_{m}$) decreases in both the
kinematic and saturation stages. Visualization is conducted using a
logarithmic scale.

An additional noteworthy observation is that, relative to the flow drive’s
length scale, the magnetic field iso-surfaces are predominantly composed of
small-scale structures for various $P_{m}$ value ranges, as illustrated in
Fig. 7. To verify this further, the spectral characteristics of the magnetic
fields are examined. We have computed the magnetic energy spectral density
$B(k)$ for a range of $P_{m}$ values, namely $P_{m}=0.25,1.0$, and $2.0$, such
that $\int|B(k,t)|^{2}dk$ is the total energy at time t and
$k=\sqrt{k_{x}^{2}+k_{y}^{2}+k_{z}^{2}}$. Our spectral analysis shows that
most of the power is concentrated on higher modes, indicating smaller length
scales, in all situations (refer to Fig. 8). Our spectral analysis indicates
that the dynamos are small scale dynamos (SSD). Furthermore, it has been
determined that the magnetic field exhibits a Kazantsev spectrum 30 with a
scaling of $B(k)\propto k^{\frac{3}{2}}$ which is unaffected by the parameter
$P_{m}$ (refer to Figure 8).

(a)

(b)

(c)

Figure 8:  Determination of the spectral density of magnetic energy $B(k)$ for
the following $P_{m}$ values: (a) $2.0$, (b) $1.0$, and (c) $0.25$, such that
$\int|B(k,t)|^{2}dk$ is the total energy at time t and
$k=\sqrt{k_{x}^{2}+k_{y}^{2}+k_{z}^{2}}$. The dynamos are small-scale dynamos
(SSD) and are independent of $P_{m}$. The magnetic field exhibits a well-
established Kazantsev $k^{\frac{3}{2}}$ scaling 30.

PDFs of normalized single components of the magnetic field
$\frac{B_{z}}{B_{rms}}$ and magnetic field strength $\frac{B}{B_{rms}}$ in the
kinematic and saturated dynamo stages for various $P_{m}$ have also been
computed (albeit with the z-component being the only one displayed here; we
have calculated PDFs for other components as well and determined that they are
identical). Figure 9(a) clearly indicates that the probability density
function (PDF) of the magnetic field is not Gaussian. Furthermore, it has been
observed that the magnetic field structure during the kinematic stage exhibits
greater intermittency compared to the self-consistent stage. The validity of
this observation has been confirmed by testing with various $P_{m}$ values,
and consistent results have been obtained. This intermittency is rationalized
by the extended tails illustrated in Figures 9(b) and 9(a). It is worth
mentioning that the probability density function (PDF) of the kinematic stage
follows a log-normal distribution, as seen in Figure 9(b).

During both the kinematic and saturated stages, the magnetic field exhibits
characteristics of typical intermittency. However, as the field approaches
saturation, the degree of intermittency reduces. These observations align with
previous studies conducted using varying parameter spaces and starting
conditions. This ensures that the aforementioned discovery remains unaffected
by minor adjustments to the model configuration and parameters.

(a)

(b)

Figure 9: The probability density functions (PDFs) for the (a) normalized
$z$-component of the magnetic field, $\frac{B_{z}}{B_{rms}}$, and (b) the
magnetic field strength, $\frac{B}{B_{rms}}$, are determined for a helically
driven dynamo in both the kinematic and saturated stages. Here $B$ is defined
as $\sqrt{B_{x}^{2}+B_{y}^{2}+B_{z}^{2}}$ and $B_{rms}$ is defined as
$\sqrt{\langle B^{2}\rangle}$.

In addition, we have calculated the probability density function (PDF) of a
specific component of the velocity field, $\frac{u_{z}}{u_{rms}}$ (where
$u_{rms}$ is defined as $\sqrt{\langle
u_{x}^{2}+u_{y}^{2}+u_{z}^{2}\rangle}$), during both the kinematic and
saturated dynamo stages. The results of our numerical study for various
$P_{m}$ values are presented in Figure 10. Based on Figure 10, it can be
observed that the probability density functions (PDFs) have a close
resemblance to Gaussian distributions in both the kinematic and saturation
stages. This pattern holds true for the various $P_{m}$ instances investigated
in this study. Thus, despite the non-Gaussian nature of the magnetic PDFs, the
velocity field PDFs are nearly Gaussian. For homogeneous turbulence, this
Gaussian characteristic holds true in general.

Figure 10: The probability density functions (PDFs) for the normalized
$z$-component of the velocity field $\frac{u_{z}}{u_{rms}}$ and for a helical-
driven dynamo at various values of $P_{m}$ in both the kinematic and saturated
phases are presented. The PDFs exhibit a close resemblance to a Gaussian
distribution. Here $u_{rms}$ is defined as $\sqrt{\langle
u_{x}^{2}+u_{y}^{2}+u_{z}^{2}\rangle}$.

The previous discussion already demonstrated that the probability density
functions (PDFs) of the magnetic field exhibit a non-Gaussian distribution,
while the velocity field shows a close approximation to Gaussianity. A
deviation from Gaussianity in a distribution can be characterized by two
important parameters: Skewness and Kurtosis. In the present work, Kurtosis is
defined statistically as follows,

$K(f)=\frac{1}{N}\sum_{1}^{N}\left(\frac{f_{i}-\bar{f}}{\sigma}\right)^{4}-3$
(12)

Here, $f_{i}$ is the statistical quantity of interest, $N$ is the number of
grid points, and $\sigma$ represents the standard deviation. Similarly, the
Skewness of a distribution can be defined as follows:

$S(f)=\frac{1}{N}\sum_{1}^{N}\left(\frac{f_{i}-\bar{f}}{\sigma}\right)^{3}.$
(13)

Based on these definitions, the values of $K(f)$ and $S(f)$ are both 0 for a
distribution that is entirely Gaussian. We have computed the Skewness and
Kurtosis for both the velocity and magnetic field. The specific information
can be found in Table 1 and Table 2.

$P_{m}$ |  $\mathbf{K(B_{z})}$ [Kin] |  $\mathbf{K(B_{z})}$ [Sat] |  $\mathbf{K(B)}$ [Kin] |  $\mathbf{K(B)}$ [Sat] |  $\mathbf{K(u_{z})}$ [Kin] |  $\mathbf{K(u_{z})}$ [Sat]
---|---|---|---|---|---|---
2 | 3.560 | 2.783 | 8.777 | 3.397 | -0.709 | -0.109
1 | 5.935 | 3.260 | 11.480 | 2.875 | -0.768 | -0.069
0.25 | 19.087 | 4.208 | 34.944 | 6.568 | -0.751 | -0.645

Table 1:  Kurtosis $K$ for the velocity and magnetic field distribution in the
case of helical dynamo. High Kurtosis value for the magnetic field in the
kinematic stage indicates existence of intermittency (see Figure 9) . However,
the Kurtosis value for the velocity field is close to $0$ (less than 1),
indicating a high degree of Gaussianity (Refer to Fig. 10)

.

Based on the calculated value, it is observed that the Kurtosis and Skewness
of the magnetic field are larger in the kinematic stage compared to the
saturated stage. This suggests that the kinematic stage is more intermittent.
We have computed the Kurtosis and Skewness for the velocity field, and it is
evident that both values are close to 0 (less than 1), indicating a clear sign
of Gaussian distribution.

$P_{m}$ |  $\mathbf{S(B)}$ [Kin] |  $\mathbf{S(B)}$ [Sat] |  $\mathbf{S(u_{z})}$ [Kin] |  $\mathbf{S(u_{z})}$ [Sat]
---|---|---|---|---
2 | 2.111 | 1.586 | 0.096 | -0.152
1 | 2.517 | 1.470 | -0.173 | -0.0264
0.25 | 4.427 | 2.024 | -0.024 | -0.002
Table 2:  The Skewness $S$ of the magnetic and velocity field distribution for
the helically driven scenario. A high skewness value observed for the magnetic
field during the kinematic stage indicates the presence of intermittency, as
seen in Figure 9. However, the Skewness of the velocity field is close to 0
(less than 1), indicating a strong indication of Gaussian distribution (see
Figure 10). A positive Skewness indicates that the distribution is skewed
towards the right from the origin, whereas a negative Skewness indicates that
the distribution is skewed towards the left from the origin.

To understand the key mechanism behind the saturation of helical dynamo, we
have calculated PDF of these following quantities:

$\displaystyle\cos(\theta)_{u,B}=\frac{\vec{u}\cdot\vec{B}}{|\vec{u}||\vec{B}|}$
(14)
$\displaystyle\cos(\theta)_{j,B}=\frac{\vec{j}\cdot\vec{B}}{|\vec{j}||\vec{B}|}$
(15)
$\displaystyle\cos(\theta)_{\omega,u}=\frac{\vec{\omega}\cdot\vec{u}}{|\vec{\omega}||\vec{u}|}$
(16)

The variables $\vec{u}$, $\vec{B}$, $\vec{j}$, and $\vec{\omega}$ represent
velocity, magnetic field, current density, and vorticity, respectively.

Based on our numerical analysis, it has been determined that the probability
of the cosine of the angle between velocity and magnetic field, denoted as
$|\cos(\theta)_{u,B}|$, is significantly greater during the saturated stage
compared to the kinematic stage. A stronger alignment between $\vec{u}$ and
$\vec{B}$ results in a drop in the induction term
$\vec{\nabla}\times(\vec{u}\times\vec{B})$, leading to a reduction in the
amplification of the magnetic field. This relationship is illustrated in
Figure 11(a). Additionally, it has been documented that the probability the
cosine angle formed by the current density and magnetic field
($|\cos(\theta)_{j,B}|$) is significantly greater by an order of magnitude
during the saturated stage. Therefore, when the alignment between the
$\vec{j}$ and $\vec{B}$ increases, the Lorentz force (or the effect of the
velocity field on the flow, represented by $\vec{j}\times\vec{B}$) decreases.
This means that the field approaches a force-free state when it reaches
saturation (refer to Figure 11(b)). In other words, a higher degree of
alignment between $\vec{u}$ and $\vec{B}$ results in a reduced efficiency of
magnetic induction. Similarly, a greater degree of alignment between vector
$\vec{j}$ and $\vec{B}$ causes a decrease in the Lorentz force, meaning that
the field becomes more force-free.

We have computed the probability density function (PDF) of the absolute value
of the cosine of the angle $\theta$ between the velocity and vorticity field,
denoted as $|\cos(\theta)_{u,\omega}|$. The amplification of the vorticity
field is dependent upon the angle formed by the vectors $\vec{u}$ and
$\omega$. In the saturated stage, it is noticed that these two fields are
aligned with each other (refer to Figure 11(c)), similar to the alignment of
$\vec{u}$ and $\vec{B}$, and $\vec{j}$ and $\vec{B}$.

(a)

(b)

(c)

Figure 11:  PDFs are provided for the absolute value of the cosine of the
angle between (a) velocity and magnetic field ($|\cos(\theta)_{u,B}|$), (b)
current density and magnetic field ($|\cos(\theta)_{j,B}|$), and (c) the
velocity and vorticity ($|\cos(\theta)_{u,\omega}|$) for various values of
$P_{m}$ in the presence of helical drive.

Structure function of various orders is an additional essential quantity that
must be computed in this context. The structure function of field $\vec{f}$
with an order of $p$ is formally defined as:

$S^{p}(l)=\langle\left[(\vec{f}(\vec{r}+\vec{l})-\vec{f}(\vec{r}))\cdot\left(\frac{\vec{l}}{l}\right)\right]^{p}\rangle$
(17)

where $\langle....\rangle$ represents the ensemble average, and $\vec{l}$ is
used to measure the separation distance between two points in the volume.

For extremely high Reynolds numbers, Kolmogorov 87 determined that the
structure function should exhibit the following scaling relationship:

$S^{p}(l)\sim l^{\frac{p}{3}}$ (18)

According to Kolmogorov’s phenomenological theory 87, the second-order
structure function scales as,

$S^{2}(l)\sim l^{\frac{2}{3}}$ (19)

In addition, there exists a precise relationship for the third-order structure
function in homogeneous, isotropic turbulence, denoted as,

$S^{3}(l)=-\frac{4}{5}\epsilon l$ (20)

easily obtained by employing the Kármán-Howarth-Monin relation 88 for the
energy flux. This relationship is alternatively referred to as the fourth-
fifths law. Here $\epsilon$ denotes the energy-dissipation rate.

We have computed the second-order ($S^{2}(l)$), third-order ($S^{3}(l)$), and
sixth-order ($S^{6}(l)$) structure functions of velocity and magnetic field
for a helical dynamo. Figure 12 (top row) displays the second-order, third-
order, and sixth-order structure functions for the magnetic field, whereas
figure 12 (bottom row) showcases the corresponding structure functions for the
velocity field. Our numerical calculation reveals that the predictions made by
Kolmogorov 87 are evident for both lower and higher order calculations (i.e,
$p=2,3$, and $6$). Additionally, it is noteworthy that the magnetic Prandtl
number ($P_{m}$) has minor influence on the dependency of $S^{2}(l)$,
$S^{3}(l)$, and $S^{6}(l)$. Nevertheless, the $P_{m}$ does not alter the slope
of the structural functions.

(a)

(b)

(c)

(d)

(e)

(f)

Figure 12: The plots display the structure functions of three distinct orders,
namely $p=2,3$, and $6$, as a function of $l$. These structure functions
represent the velocity and magnetic field for various $P_{m}$ values in the
context of a helical dynamo. The numerical analysis confirms that Kolmogrov’s
scalings 87, namely $S^{2}(l)\sim l^{\frac{2}{3}}$, $S^{3}(l)\sim l$, and
$S^{6}(l)\sim l^{2}$, are valid for both the velocity and magnetic field.

The ratio of the sixth-order structure function to the second-order structure
function for velocity and magnetic field serves as a reliable indicator of the
intermittency present in turbulence. The concept being referred to is hyper-
flatness, which has the following mathematical definition:

$\mathcal{F}^{6}(l)=\frac{S^{6}(l)}{(S^{2}(l))^{3}}$ (21)

For the helical dynamo, we plot hyper-flatness ($\mathcal{F}^{6}(l)$) versus
$l$ for the magnetic field and velocity field (refer to Fig. 13). This graphic
demonstrates that the hyper-flatness ($\mathcal{F}^{6}(l)$), increases as the
length scale, $l$, approaches zero ($l\to 0$). This suggests the presence of
small-scale intermittency at the scales of dissipation range.

(a)

(b)

Figure 13:  The hyper-flatness $\mathcal{F}^{6}(l)$ is calculated as a
function of $l$ for two separate cases: (a) magnetic field and (b) velocity
field. These calculations are performed at various values of $P_{m}$ for a
helical dynamo.

Following an extensive investigation of the influence of helical drive on the
activity of self-consistent dynamos, the present study investigates the fate
of self-consistent dynamos in the presence of non-helical drive. As previously
mentioned, our drive is YM flow drive, and its helical character can be
controlled using specific parameter spaces. Therefore, we begin by examining
the independence of the initial condition when a non-helical drive is present
(refer to Appendix A for further information). Additionally, we conduct a
comparison of the time evolution of magnetic energy
($E_{B}=\frac{1}{2}\int_{V}(B_{x}^{2}+B_{y}^{2}+B_{z}^{2})dxdydz$) under two
distinct drives: helical (ABC) and non-helical (EPI-2D). Fig. 14 shows that
the magnetic energy growth rate ($\gamma=\frac{d}{dt}(\ln E_{B}(t))$) is
significantly greater for the helically driven case compared to the non-
helically driven case. This difference is quantified by a linear fit shown by
the dotted lines, and the slopes of these fits are provided in the figure
legend. In the kinematic stage, the impact of the magnetic field on the
velocity field is minimal due to the small strength of the magnetic field.
When the magnetic field reaches a certain strength that affects the flow, non-
linearity develops earlier in the helical case, causing the exponential growth
rate to slow down earlier compared to the non-helical case (as illustrated in
Fig. 14). Hence, the helically forced turbulent dynamo saturates earlier than
the non-helically forced scenario as observed in Fig. 14.

(a)

(b)

Figure 14: Comparing helically driven dynamo with non-helically driven dynamo.
The growth rate of the dynamo is significantly greater in the helically driven
case compared to the non-helically driven condition, as determined by a linear
fit. Therefore, the helical dynamo saturates more quickly because of the
varying strength of the “back reaction”. Numerical experiments were conducted
for $P_{m}$ values of (a) $1.0$ and (b) $2.0$.

Additionally, the impact of Alfven speed is examined in the case of non-
helically driven dynamos. We have conducted simulations with various Alfven
Mach numbers ($M_{A}=10000,1000,100,10$) and found that the most effective
dynamo activity occurs in the super Alfvenic regime when the velocity field is
driven by non-helical forcing (Refer to Fig. 15(a)). Additionally, it is
noteworthy to mention that the saturation value of kinetic and magnetic energy
remains constant regardless of the value of $M_{A}$; this suggests that it is
not influenced by Alfven speed (refer to Figure 15(a)). This observation
aligns with the previous helically forced instance.

Following this, the impact of the magnetic Prandtl number ($P_{m}$) on these
non-helical dynamos is investigated. We maintain a constant kinetic Reynolds
number ($R_{e}$) while adjusting the magnetic Reynolds number ($R_{m}$) to
study the impact of $P_{m}$. Thus, we can also view this as a depiction of the
magnetic Prandtl number ($P_{m}$) relationship. Figure 15(b) indicates that
the dynamo action is inhibited at lower values of magnetic Prandtl number
($P_{m}$) with a non-helical drive (refer to Appendix B for details), similar
to the helical situations mentioned earlier.

(a)

(b)

Figure 15: (a) The non-helical forced dynamo action is being influenced by the
Alfven speed. Dynamo activity is more pronounced in super Alfvenic regimes
when driven by non-helical forces. (b) The impact of the magnetic Prandtl
number ($P_{m}=\frac{R_{m}}{R_{e}}$) on the action of a non-helical dynamo is
examined in this study, keeping the kinetic Reynolds number ($R_{e}$)
constant. At lower $P_{m}$ limit, the non-helical dynamo activity is
suppressed. Here the dotted lines indicate the kinetic energy.

Based on the preceding discussion, it has been determined that the time
evolution of magnetic energy under helical or non-helical drive behaves
identically across a range of $P_{m}$ values. The iso-surface dynamics of
magnetic fields (IsoB) for non-helical dynamos at various $P_{m}$ values are
now presented. The magnetic field in the kinematic stage exhibits smaller
scale structures compared to the saturated phases, as observed in Fig. 16. For
all values of $P_{m}$, the magnetic field structures in saturated phases
appear visibly larger in size (Refer to Fig. 16). This observation is found to
be identical as the helical dynamos addressed previously.

Kinematic

(a) $P_{m}=2.0$

(b) $P_{m}=1.0$

(c) $P_{m}=0.25$

Saturated

(d) $P_{m}=2.0$

(e) $P_{m}=1.0$

(f) $P_{m}=0.25$

Figure 16:  Visualization of three-dimensional magnetic field iso-surfaces
(Iso-B surfaces) for a non-helical dynamo in the kinematic stage (top row) and
saturated stage (bottom row) for (a & d) $P_{m}=2$, (b & e) $P_{m}=1$, and (c
& f) $P_{m}=0.25$. The magnetic fields in the saturated stages exhibit
structures that are larger than those in the kinematic stages. In the
kinematic stage, the magnetic field structures increase in size as the
magnetic Prandtl number ($P_{m}$) decreases. However, in the saturated stage,
the opposite occurs, with larger structures forming as $P_{m}$ is increased.
This observation is the complete opposite of the one made for the helical
example. The log scale is utilized for visualization.

It is interesting to observe that the magnetic field iso-surface (IsoB)
structure increases in kinematic stage as the $P_{m}$ value decreases (See
Fig. 16). This is caused by the increase in magnetic resistivity, which
subsequently minimizes smaller fluctuations in the kinematic phase of non-
helical dynamos. The magnetic field iso-surface structures exhibit a reversal
nature with $P_{m}$ in the saturated stage. During the saturated stage, it is
observed that the structures become increasingly larger as the parameter
$P_{m}$ increases (refer to Fig. 16). We have implemented an alternative
visualization tool to comprehend this unique behavior and do cross
verification. We used volume metric rendering to depict the magnetic fields at
saturation stage in a linear scale and observed the reversal nature more
prominently (See Fig. 17). Figure 17 clearly demonstrates that the magnetic
field structures’ scales increase when the magnetic Prandtl number ($P_{m}$)
increases for non-helical dynamos. This is a noteworthy and diametrically
opposed observation to the helical case.

Saturated

(a) $P_{m}=2.0$

(b) $P_{m}=1.0$

(c) $P_{m}=0.25$

Figure 17: Volume rendering visualization of magnetic field for non-helical
dynamo in saturated stage at (a) $P_{m}=2$ , (b) $P_{m}=1$ , and (c)
$P_{m}=0.25$. The magnetic field structures are getting larger as $P_{m}$ is
increased in saturated stage. Using this volume rendering visualization
technique larger scale structures are clearly identified for higher $P_{m}$
values. Visualization is done in linear scale.

It can be observed from Figures 16 and 17 that the structures of the magnetic
fields are predominantly composed of shorter scales. We have conducted
spectral analysis to confirm this further. The spectral density of magnetic
energy $B(k)$ is represented in Figure 18 as a function of $k$ for $P_{m}$
values of $0.25$, $1.0$, and $2.0$, where $\int|B(k,t)|^{2}dk$ represents the
total energy at time t and $k=\sqrt{k_{x}^{2}+k_{y}^{2}+k_{z}^{2}}$. Our
spectral estimate strongly indicates that the majority of power is
concentrated at smaller length scales, namely on higher modes. Based on our
comprehensive spectral analysis, it is obvious that these dynamos operate as
small scale dynamos (SSD). The magnetic field precisely adheres to the
Kazantsev $k^{\frac{3}{2}}$ spectral scaling 30, which remains consistent
across a variety of $P_{m}$ values as seen in Fig. 18.

Therefore, based on our numerical analysis, it has been determined that the
influence of both helical and non-helical drives results in the formation of
small scale dynamos (SSD) with a substantial Kazantsev scaling of
$k^{\frac{3}{2}}$ 30. Furthermore, our analysis demonstrates that this
conclusion holds true across a variety of $P_{m}$ values.

(a)

(b)

(c)

Figure 18:  Determination of the spectral density of magnetic energy $B(k)$
for the following $P_{m}$ values: (a) $2.0$, (b) $1.0$, and (c) $0.25$, such
that $\int|B(k,t)|^{2}dk$ is the total energy at time t and
$k=\sqrt{k_{x}^{2}+k_{y}^{2}+k_{z}^{2}}$. The dynamos are unambiguously
designated as small scale dynamos (SSD), which is not dependent on $P_{m}$.
The magnetic field exhibits a well-established Kazantsev $k^{\frac{3}{2}}$
scaling 30.

In order to have a clearer understanding of the dynamics of the magnetic
field, we have computed the coherence length of the magnetic field as defined
in Equation 11. The normalized magnetic field coherence length
($\frac{l_{B}}{L}$) is found to be greater in the saturated stages than in the
kinematic stage. Furthermore, the ratio of $l_{B}$ to $L$ is shown to rise
over time as $P_{m}$ increases in the presence of a non-helical drive (refer
to Figure 19). This result aligns with our findings about the dynamics of the
magnetic field, as depicted in figures 16 & 17.

Figure 19: The magnetic field coherence length, normalized to the size of the
numerical domain ($\frac{l_{B}}{L}=\frac{\int k^{-1}B(k)dk}{\int B(k)dk}$),
has been computed for a non-helically driven dynamo for various values of
magnetic Prandtl number ($P_{m}$). Refer to Fig. 16 & 17 for comparison.

We have computed the probability density functions (PDFs) of the normalized
$z$-component of the magnetic field, denoted as $\frac{B_{z}}{B_{rms}}$, as
well as the magnetic field strength, denoted as $\frac{B}{B_{rms}}$ (where $B$
is defined as $\sqrt{B_{x}^{2}+B_{y}^{2}+B_{z}^{2}}$ and $B_{rms}$ is defined
as $\sqrt{\langle B^{2}\rangle}$), during both the kinematic and saturated
dynamo stages. We have performed these calculations for various values of the
magnetic Prandtl number ($P_{m}$), similar to our previous analyses.
Additionally, we have confirmed that the PDFs for the other components of the
magnetic field exhibit identical behavior. Figure 20 unambiguously indicates
that the distribution is non-Gaussian. Additionally, it is observed that the
kinematic stage exhibits a more intermittent distribution of the magnetic
field compared to the self-consistent stage. The intermittent nature of this
phenomenon is supported by the presence of a larger tail, which conforms to a
log-normal distribution, as depicted in Figures 20(b) & 20(a).

It is crucial to note that as the value of $P_{m}$ increases, the coherence
length scale ($\frac{l_{B}}{L}$) of the magnetic field in the saturated regime
also increases (refer to Figure 19). This suggests that for higher $P_{m}$
cases, the structure should be more volume-filling or less intermittent,
resulting in earlier truncation of PDF (PDF is truncated in earlier B values).
This observation is precisely derived from the PDF computation, where the
green curve is terminated at an earlier point (See Fig. 20).

(a)

(b)

Figure 20: (a) The probability density functions (PDFs) for the normalized
$z$-component of the magnetic field $\frac{B_{z}}{B_{rms}}$ and (b) the
magnetic field strength $\frac{B}{B_{rms}}$ are calculated for a non-helically
driven dynamo in both the kinematic and saturated stages. Here $B$ is defined
as $\sqrt{B_{x}^{2}+B_{y}^{2}+B_{z}^{2}}$ and $B_{rms}$ is defined as
$\sqrt{\langle B^{2}\rangle}$.

Based on our previous discussion, it has been determined that in the saturated
regime of non-helical dynamos, larger magnetic field structures are evident at
higher $P_{m}$ values (refer to Figures 16 & 17). In order to provide clarity,
we have employed a volume metric representation scheme to represent the
velocity field in linear scale (refer to Figure 21). It is evident from Fig.
21 (upper row) that the velocity field structures exhibit a high degree of
similarity in kinematic phase across various magnetic $P_{m}$ ranges. This is
due to the fact that the kinetic Reynolds numbers ($R_{e}$) are constant
across all cases, and the magnetic fields in the kinematic phase are not
strong enough to alter the velocity fields, resulting in the formation of
identical structures (refer to the upper row of Figure 21). The strength of
magnetic fields in the saturated regime is sufficient to alter the velocity
field dynamics. As a result, distinct velocity field structures are observed
at various $P_{m}$ limits (refer to the lower row of Figure 21), despite the
fact that $R_{e}$ remains constant in all cases. It is crucial to mention that
as the value of $P_{m}$ increases, the structures of the velocity field
increase. This phenomenon precisely reflects the characteristics of the
magnetic field (See the lower row of Figure 21).

Kinematic

(a) $P_{m}=2.0$

(b) $P_{m}=1.0$

(c) $P_{m}=0.25$

Saturated

(d) $P_{m}=2.0$

(e) $P_{m}=1.0$

(f) $P_{m}=0.25$

Figure 21:  Volume rendering representation of the velocity field for a non-
helical dynamo in the kinematic stage (upper row) and saturated stage (bottom
row) with magnetic Prandtl number values of (a & d) $P_{m}=2$, (b & e)
$P_{m}=1$, and (c & f) $P_{m}=0.25$. Velocity field structures increase in
size as the magnetic Prandtl number ($P_{m}$) increases during the saturated
stage. Therefore, the velocity field dynamics precisely mimic the magnetic
field dynamics. Visualization is conducted using a linear scale.

After doing a more in-depth analysis of the velocity field data, we have
computed the temporal evolution of the coherence length scale
($\frac{l_{u}}{L}$) of the velocity field and the probability density function
(PDF) of one component of the velocity field. Figure 22(a) clearly
demonstrates that the velocity structures have a high level of symmetry during
the kinematic stage, but they are noticeably different during the saturated
stage. The PDF calculation reveals that the velocity field PDF closely follows
a Gaussian distribution (refer to Figure 22(b)), indicating the presence of
homogeneous turbulence. Significantly, in the case of greater $P_{m}$
($P_{m}=2$), the intermittency is more pronounced in saturated stages,
resulting in a non-Gaussian distribution (refer to Figure 22(b)). This is a
noteworthy and entirely dissimilar observation compared to the previous one.
The precise cause for this is not fully understood yet and necessitates
additional inquiry.

(a)

(b)

Figure 22:  (a) The coherence length of the velocity field, normalized to the
size of the numerical domain ($\frac{l_{u}}{L}=\frac{\int k^{-1}E(k)dk}{\int
E(k)dk}$), has been computed for non-helically driven dynamo for various
values of $P_{m}$. (b) The probability density function (PDFs) for the
normalized $z$-component of the velocity field $\frac{u_{z}}{u_{rms}}$ is
examined in both the kinematic and saturation stages of a non-helically driven
dynamo. Here $u_{rms}$ is defined as $\sqrt{\langle
u_{x}^{2}+u_{y}^{2}+u_{z}^{2}\rangle}$.

Based on our previous discussion, it has been determined that while the
velocity probability density functions (PDFs) exhibit a close resemblance to a
Gaussian distribution, the PDFs of the magnetic field do not. In order to
measure this departure from a Gaussian distribution, we have calculated the
Kurtosis and Skewness parameters. The specific information can be found in
Table 3 and Table 4.

$P_{m}$ |  $\mathbf{K(B_{z})}$ [Kin] |  $\mathbf{K(B_{z})}$ [Sat] |  $\mathbf{K(B)}$ [Kin] |  $\mathbf{K(B)}$ [Sat] |  $\mathbf{K(u_{z})}$ [Kin] |  $\mathbf{K(u_{z})}$ [Sat]
---|---|---|---|---|---|---
2 | 5.579 | 2.557 | 8.621 | -0.207 | -0.051 | 2.761
1 | 10.581 | 2.398 | 14.209 | 1.185 | -0.243 | -0.323
0.25 | 5.395 | 9.581 | 6.030 | 13.259 | -0.370 | 0.106
Table 3: Kurtosis $K$ for the distribution of magnetic and velocity fields in a non-helically driven scenario. A high Kurtosis value for the magnetic field during the kinematic stage indicates the presence of intermittency, as seen in Figure 20. On the other hand, Kurtosis for the velocity field is almost $0$ (except from the $P_{m}=2$ scenario), which is an excellent indication of Gaussianity (See Fig. 22(b)). The Kurtosis for $P_{m}=2$ has a significant value. $P_{m}$ |  $\mathbf{S(B)}$ [Kin] |  $\mathbf{S(B)}$ [Sat] |  $\mathbf{S(u_{z})}$ [Kin] |  $\mathbf{S(u_{z})}$ [Sat]
---|---|---|---|---
2 | 2.183 | 0.666 | 0.103 | 0.310
1 | 2.485 | 1.095 | -0.009 | -0.015
0.25 | 1.928 | 2.529 | $-3.378\times 10^{-10}$ | -0.094
Table 4: Skewness $S$ for the velocity and magnetic field distribution in the
absence of helical drive. A high Skewness value observed for the magnetic
field during the kinematic stage indicates the presence of intermittency, as
seen in Figure 20. However, the Skewness of the velocity field is close to $0$
(less than $1$), indicating a strong indication of Gaussian distribution (See
Figure 22(b)). A positive Skewness indicates that the distribution is skewed
towards the right, whereas a negative Skewness indicates that the distribution
is biased towards the left.

We have computed the probability density function (PDF) for different
alignment angles, specifically the PDF for $|\cos(\theta)_{u,B}|$,
$|\cos(\theta)_{j,B}|$, and $|\cos(\theta)_{u,\omega}|$, as I did before. Our
numerical study reveals that in the saturated stage, the alignment between the
magnetic field and the velocity field is greater, resulting in a reduction of
the induction effects (i.e., $\vec{\nabla}\times(\vec{u}\times\vec{B})\to 0$),
compared to the kinematic stage (refer to Fig. 23(a)). We have observed that
the current density ($\vec{j}$) and the magnetic field ($\vec{B}$) are aligned
to a high degree in the saturated stage. This alignment results in a minimum
Lorentz force, as shown in Figure 23(b). In other words, when the alignment
between vector $\vec{u}$ and vector $\vec{B}$ is improved, it results in a
reduction in the transfer of energy from the flow to the magnetic field. On
the other hand, when vector $\vec{j}$ and vector $\vec{B}$ are aligned, it
leads to a drop in the magnetic feedback. All of these factors ultimately lead
to the dynamo becoming saturated. In the saturated stage, it is noticed that
the $\vec{u}$ and $\omega$ fields are parallel to each other (refer to Figure
23(c)).

(a)

(b)

(c)

Figure 23:  PDFs are provided for the absolute value of the cosine of the
angle between (a) velocity and magnetic field ($|\cos(\theta)_{u,B}|$), (b)
current density and magnetic field ($|\cos(\theta)_{j,B}|$), and (c) velocity
and vorticity ($|\cos(\theta)_{u,\omega}|$) for various values of $P_{m}$ in
the presence of a non-helical drive.

Similar to the helical example, we have computed the second ($S^{2}(l)$),
third ($S^{3}(l)$), and sixth ($S^{6}(l)$) order structure function of the
magnetic field (refer to Fig. 24 [upper row]) and velocity (refer to Fig. 24
[bottom row]) for the non-helical dynamo as well (refer to Fig. 24). Our
numerical analysis reveals that the predictions made by 87 are evident in both
lower and higher order computations (i.e, $p=2,3$, and $6$), similar to a
helical dynamo. Interestingly, $S^{2}(l)$, $S^{3}(l)$, and $S^{6}(l)$ are
found to be independent of the magnetic Prandtl number ($P_{m}$).

(a)

(b)

(c)

(d)

(e)

(f)

Figure 24:  The plots display the structure functions of three distinct
orders, namely $p=2,3$, and $6$, as a function of $l$ for velocity and
magnetic field at various $P_{m}$ values in the context of a non-helical
dynamo. The numerical analysis confirms that Kolmogrov’s scalings 87,
specifically $S^{2}(l)\sim l^{\frac{2}{3}}$, $S^{3}(l)\sim l$, and
$S^{6}(l)\sim l^{2}$, hold true for both the velocity and magnetic field.

For a non-helical dynamo, we illustrate the hyper-flatness
$\mathcal{F}^{6}(l)$ versus $l$ for both the magnetic field and velocity field
(refer to Fig. 25). This plot demonstrates that the hyper-flatness function,
denoted as $\mathcal{F}^{6}(l)$, increases as the length scale $l$ approaches
zero. This suggests the presence of intermittent behavior at small scales
within the dissipation range.

(a)

(b)

Figure 25:  The hyper-flatness $\mathcal{F}^{6}(l)$ is calculated as a
function of $l$ for both (a) magnetic field and (b) velocity field at various
values of $P_{m}$ for a non-helical dynamo.

Additionally, it is mentioned earlier that the Yoshida-Morrison (YM) flow
family is utilized as the driving force in our simulation. An essential
feature of this YM flow is that finite fluid helicity injection into the
system is possible by regulating a specific physical parameter ($\beta$) prior
to simulation initiation. The influence of driven controlled fluid helicity
injection on self-consistent dynamo activity is now being examined. We compute
the magnetic field spectral density ($B(k)$) for various $\beta$ values and
plot it against $k$ over a variety of $P_{m}$ values. It is evident from Fig.
26 that the dynamos observed for each form of drive are, in essence, small-
scale dynamos (SSD). Additionally, the magnetic field is observed to adhere to
the widely recognized Kazantsev $k^{\frac{3}{2}}$ scaling 30 . Furthermore, it
remains unaffected by the helicity strength of the drive (refer to Figure 26).
The helical structure of the flow seems to impact the SSD spectral structure
throughout a variety of $P_{m}$ values, as seen in Fig. 26 (Also see Appendix
C). This observation is found to coincide with a recent study on kinematic
dynamos 52.

(a)

(b)

(c)

Figure 26:  Magnetic energy spectral density estimation for various classes of
Yoshida-Morrison (YM) flow drives with $P_{m}$ values of (a) $2.0$, (b) $1.0$,
and (c) $0.25$, such that $\int|B(k,t)|^{2}dk$ represents the total energy at
time t and $k=\sqrt{k_{x}^{2}+k_{y}^{2}+k_{z}^{2}}$. It is evident that the
dynamos are small scale dynamos (SSD) suitable for all kinds of drives.
Undoubtedly, the SSD spectral structure is ultimately impacted by the helical
configuration of the drive. Plots are represented on both a linear and log-log
scale (inset view).

Additionally, we have computed the spectral density of kinetic energy denoted
as $E(k)$, where $\int|E(k,t)|^{2}dk$ represents the total energy at time t
and $k=\sqrt{k_{x}^{2}+k_{y}^{2}+k_{z}^{2}}$. Figure 27 illustrates the
relationship between $E(k)$ and $k$ for various $\beta$ values and $P_{m}$
values of $0.25$, $1.0$, and $2.0$. The kinetic energy spectrum closely
resembles the Kolmogorov spectrum, with $E(k)\propto k^{-\frac{5}{3}}$ scaling
(See Fig. 27). The observed scaling also remains consistent across a broad
range of $P_{m}$ values and is considered to be independent of the helical
structure of the flow (See Fig. 27).

(a)

(b)

(c)

Figure 27:  Calculation of the kinetic energy spectral density ($E(k)$) for
helical ($\beta=1.0$) and non-helical ($\beta=0.0$) dynamos, where
$k=\sqrt{k_{x}^{2}+k_{y}^{2}+k_{z}^{2}}$ and $\int|E(k,t)|^{2}dk$ represents
the total energy at time t. We have computed it for $P_{m}$ values (a) $2.0$,
(b) $1.0$, and (c) $0.25$. For all the examples addressed here, we have found
that the kinetic energy spectrum is close to the Kolmogorov spectrum, that is,
$E(k)\propto k^{-\frac{5}{3}}$.

Figure 28 provides a comparison of the magnetic energy spectral density
($B(k)$) for various values of magnetic Prandtl numbers ($P_{m}$). It is
evident that as the value of $P_{m}$ decreases, the peak of $B(k)$ shifts
towards decreasing wave numbers. The rationale for this is that a decrease in
$P_{m}$ while holding the kinetic Reynolds number ($R_{e}$) constant signifies
an increase in the resistivity of the plasma, which causes the resistive scale
to kicks of earlier in modes.

(a)

(b)

Figure 28:  Estimation of the magnetic energy spectral density $B(k)$ at
$P_{m}$ values of $2.0$, $1.0$, and $0.25$ for (a) helical-drive and (b) non-
helical-drive, where $\int|B(k,t)|^{2}dk$ is the total energy at time t and
$k=\sqrt{k_{x}^{2}+k_{y}^{2}+k_{z}^{2}}$. It is evident that as the value of
$P_{m}$ is reduced, the peak of the spectral energy density ($B(k)$) shifts
towards a lower mode ($k$). Plots are represented on both a linear and log-log
scale (inset view).

##  VI Summary and Conclusion

Direct numerical simulations (DNS) of self-consistent dynamos have been
performed in this study utilizing a three-dimensional magnetohydrodynamic
model. We have explored several aspects of self-consistent dynamos based on
the type of driving force present in the system. The newly proposed Yoshida-
Morrison (YM) flow drive has been taken into consideration as a potential
driver in our simulation. One distinguishing feature of this drive is that its
helicity can be altered by changing the magnitude of a physically significant
parameter. When driven in a helical manner, we classify the dynamo as a
helical dynamo, and when driven non-helically, we refer to it as non-helical
dynamos.

Our major findings are:

$\bullet$ The differences between helical and non-helical dynamos has been
investigated. Our numerical simulation has shown that helical dynamos exhibit
a significantly greater growth rate of magnetic energy compared to non-helical
dynamos. The strong “back-reaction” of the magnetic field on the velocity
field resulting from this high growth rate causes non-linearity to manifest
much earlier in the helical case and causes the exponential growth rate to
decelerate earlier in comparison to the non-helical case. As a result, the
helical turbulent dynamos has been observed to reach saturation state prior to
the non-helical turbulent dynamos.

$\bullet$ It is observed that the initial seed magnetic field has no influence
on the time evolution of volume-averaged magnetic energy. It is observed that
the saturation value and the growth rate of magnetic energy in the kinematic
stage are not influenced by the seed field. This observation holds true in the
case of helical and non-helical dynamos alike. It appears that all memory of
the original seed field has been lost.

$\bullet$ It is observed that the Alfven speed has a substantial impact on
both helical and non-helical dynamos. It has been noted that the prominence of
the dynamo activity strengthens in the super Alfvenic regime. This observation
is consistent with helical and non-helical dynamos.

$\bullet$ For both helical and non-helical driven dynamos, the impact of the
magnetic Prandtl number ($P_{m}$) has been studied. Our numerical experiments
have revealed that dynamo action is suppressed at lower $P_{m}$ limits for
both helical and non-helical dynamos similarly.

$\bullet$ This investigation also encompasses the dynamics of the magnetic
field iso-surface (IsoB) at various $P_{m}$ limits. The findings indicate that
during the kinematic stage, small scale structures predominate, whereas during
the saturated stage, relatively large scale structures are observed. It
appears that this finding holds true for both helical and non-helical drives.

$\bullet$ It has been determined through magnetic field iso-surface (IsoB)
visualization that the magnetic field structure for helical dynamos becomes
more pronounced at lower $P_{m}$ values. The emergence of such enormous
structures is a consequence of the system’s increasing magnetic resistivity,
which subsequently eliminates fluctuations.

$\bullet$ It is observed that the variation of magnetic field structures for
non-helical dynamos with respect to $P_{m}$ follows the same pattern that has
been observed for helical dynamos, but only during the kinematic stage. During
the saturation stage, the observation is diametrically opposite; as $P_{m}$
increases, the magnetic field structures are observed to reveal large-scale
structures. The velocity field has also been visualized, and it has been
observed that as $P_{m}$ increases, the structures of the velocity field
become more pronounced; this is consistent with the dynamics of the magnetic
field. This observation is noteworthy in comparison to previous works.

$\bullet$ For both helical and non-helical dynamos, the spectral distribution
of the magnetic field has also been presented. It has been observed that the
majority of power is concentrated in higher mode numbers for each scenario
examined in this study; therefore, the dynamos are small-scale dynamos (SSD).
The magnetic field exhibits the established Kazantsev scaling ($B(k)\propto
k^{\frac{3}{2}}$), which remains consistent for both helical and non-helical
dynamos. It is also observed that the variation of $P_{m}$ has no effect on
the Kazantsev scaling. This is another significant finding of the
investigation.

$\bullet$ Controlled helicity injection has been observed to impact the
spectral structure of small scale dynamo (SSD) in a variety of $P_{m}$ values
in our numerical experiments. Without magnetic back-reaction, the same
observation has been reported. The presence of magnetic back-reaction does not
impact the nature. Likewise, this is a noteworthy finding.

$\bullet$ According to our numerical experiments, as $P_{m}$ decreases, the
peak of the magnetic energy spectral density ($B(k)$ migrates toward smaller
modes.

$\bullet$ The kinetic energy spectral density ($E(k)$) has been calculated for
both helical and non-helical dynamos. Our spectral analysis shows that the
kinetic energy spectrum closely adheres to Kolmogorov scaling ($E(k)\propto
k^{-\frac{5}{3}}$). This Kolmogorov scaling justifies the turbulent nature of
the velocity field.

$\bullet$ We have also developed various diagnostic tools such as those
utilized to determine the coherence length scale [$l_{u}(t)$ & $l_{B}(t)$],
examine the probability density function (PDF) of velocity and magnetic field,
evaluate the PDF of alignment angle between different fields, calculate the
Skewness and Kurtosis [$S(f)$ & $K(f)$], determine the structure functions of
different orders [$S^{p}(l)$], and evaluate the hyper-flatness
[$\mathcal{F}^{6}(l)$]. The PDF calculation reveals that the magnetic field
structure displays higher levels of intermittency during the kinematic stage
in comparison to the self-consistent stage. This increased intermittency is
also indicated by the higher values of Kurtosis and Skewness. Through
structure function calculation, it is observed that Kolmogorov’s hypothesis
has been satisfied for both helical and non-helical dynamos

In conclusion, we have studied the impact of helical and non-helical drive on
the development of fully self-consistent dynamos through numerical simulation.
The findings of our numerical analysis indicate that small scale dynamos (SSD)
are generated by both helical and non-helical drives. Kazantsev’s
$k^{\frac{3}{2}}$ scaling is observed to persist for both helical and non-
helical dynamos, and this scaling remains consistent regardless of the
magnetic Prandtl number ($P_{m}$). In the nonlinear limit, our investigation
also demonstrates that the helical nature of the drive does influence the
dynamics of small-scale dynamos which are significant in various astrophysical
objects, particularly in the interstellar and intergalactic medium, stars,
planets, and liquid metal experiments.

We have carried out the analysis for $P_{m}<1$ regime, which is significant
for stars, planets, and liquid metal experiments, as well as $P_{m}>1$ regime,
which is relevant to fluctuation dynamo in the interstellar and intergalactic
medium. The study can be further extended through various means. An immediate
extension would involve conducting the same analysis for dynamos utilizing a
Hall MHD model 17. In the nonlinear limit, the effect of flow shear on dynamo
activity has also not been investigated. Therefore, an investigation of the
impact of flow shear 89 on the activity of the self-consistent dynamo would be
of considerable interest. We have implemented the MHD approximation, although
plasma effects could also be significant. Comparing our results with those of
the plasma dynamo would be intriguing to observe how the relationship between
velocity and magnetic fields, as well as the magnetic field structure, change
when plasma effects are taken into account. Plasma effects are significant for
gases with weak collisions and partially ionized mediums such as the
interstellar medium (ISM). The MHD model is inadequate for describing these
plasmas due to its inability to differentiate the relative motion among
various species. Therefore, the incorporation of two fluid effects 90; 91 is
necessary. We intend to address these issues in our upcoming work.

##  VII SUPPLEMENTARY MATERIAL

The movie descriptions can be found in the supplementary material.

##  VIII 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 authors, S.B, acknowledges Prof. A. Alexakis of ENS
in Paris, France, for a valuable discussion during his visit to the ENS lab in
Paris. S.B is thankful to HPC support team of IPR for extending their help
related to ANTYA cluster.

##  IX DATA AVAILABILITY

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

##  X Conflict of Interest

The authors have no conflicts to disclose.

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##  Appendix A Initial condition invariance test for non-helical case

We have verified the independency of the initial condition for the non-
helically diven case. It is noted that the growth rate and saturated value of
the magnetic field are similar for both the uniform and random seed fields in
the non-helical driven case, as shown in Figure 29. Therefore, it can be
inferred from Figure 29 that the memory of the initial seed field is lost,
similar to the helical case discussed previously.

Figure 29: Time evolution of kinetic and magnetic energy for two distinct
initial conditions: random seed (red) & uniform (green) in a non-helical
dynamo. The evolution of magnetic and kinetic energy is observed to be
identical in both cases.

##  Appendix B Effect of $P_{m}$ on growth rate & saturation state of magnetic
energy

The magnetic energy growth rate ($\gamma=\frac{d}{dt}(\ln E_{B}(t))$) and
saturation value of magnetic energy ($E_{B}^{Sat}$) for both helical and non-
helical dynamos have been computed and plotted as functions of $P_{m}$ [See
Fig. 30]. It is evident from Fig. 30 that dynamos are more difficult to excite
at low $P_{m}$ limits. We have calculated the values of $\gamma$ and
$E_{B}^{Sat}$ based on the data presented in figures 6(b) & 15(b).

(a)

(b)

Figure 30:  (a) The growth rate of magnetic energy ($\gamma=\frac{d}{dt}(\ln
E_{B}(t))$) has been computed for various values of $R_{m}$. In the case of
both helical and non-helical drives, the growth rate ($\gamma$) is determined
to be proportional to the square root of $R_{m}$ (i.e,
$\gamma\propto\sqrt{R_{m}}$). (b) The relationship between the saturation
value of the magnetic energy ($E_{B}^{Sat}$) and $P_{m}^{-1}$ for helical and
non-helical dynamos. It is evident from both figures that dynamo action is
more difficult to achieve at lower values of the magnetic Prandtl number
($P_{m}$). We have tweaked the variable $R_{m}$ by adjusting $P_{m}$ while
maintaining $R_{e}$ constant. Thus, we can also view these two graphs as
illustrating the magnetic Prandtl number’s influence on $\gamma$ and
$E_{B}^{Sat}$.

##  Appendix C Effect of helicity injection on magnetic field dynamics

The magnetic field structures for the YM flow drive classes have been
visualized. Through the implementation of controlled helicity injection, more
finer magnetic field structures are generated, as demonstrated by our
numerical experiments (See Fig. 31). This observation suggests that the
helical structure of the flow drive influences the SSD structure.

(a)

(b)

(c)

Figure 31:  Visualization of the magnetic field in the kinematic stage at (a)
$\beta=0.0$, (b) $\beta=0.5$, and (c) $\beta=1.0$. The helical configuration
of the YM flow drive is regulated by the parameter $\beta$. With increasing
$\beta$, the magnetic field structures become more smaller in size.
Visualization is conducted using a logarithmic scale.