11institutetext: Departament de Física, Universitat de les Illes Balears,
E-07122, Palma de Mallorca, Spain 22institutetext: Institute of Applied
Computing & Community Code (IAC3), Universitat de les Illes Balears, E-07122,
Palma de Mallorca, Spain ; 22email<EMAIL_ADDRESS>
# Transition to turbulence in nonuniform coronal loops driven by torsional
Alfvén waves.

II. Extended analysis and effect of magnetic twist

Sergio Díaz-Suárez Transition to turbulence in nonuniform coronal loops driven
by torsional Alfvén waves.Transition to turbulence in nonuniform coronal loops
driven by torsional Alfvén waves.Transition to turbulence in nonuniform
coronal loops driven by torsional Alfvén waves.Transition to turbulence in
nonuniform coronal loops driven by torsional Alfvén waves. Roberto Soler
Transition to turbulence in nonuniform coronal loops driven by torsional
Alfvén waves.Transition to turbulence in nonuniform coronal loops driven by
torsional Alfvén waves.Transition to turbulence in nonuniform coronal loops
driven by torsional Alfvén waves.Transition to turbulence in nonuniform
coronal loops driven by torsional Alfvén waves.

(Received xxx /Accepted yyy)

It has been shown in a previous work that torsional Alfvén waves can drive
turbulence in nonuniform coronal loops with a purely axial magnetic field.
Here we explore the role of the magnetic twist. We model a coronal loop as a
transversely nonuniform straight flux tube, anchored in the photosphere, and
embedded in a uniform coronal environment. We consider that the magnetic field
is twisted and control the strength of magnetic twist by a free parameter of
the model. We excite the longitudinally fundamental mode of standing torsional
Alfvén waves, whose temporal evolution is obtained by means of high-resolution
three-dimensional ideal magnetohydrodynamic numerical simulations. We find
that phase mixing of torsional Alfvén waves creates velocity shear in the
direction perpendicular to the magnetic field lines. The velocity shear
eventually triggers the Kelvin-Helmholtz instability (KHi). In weakly twisted
magnetic tubes, the KHi is able to grow nonlinearly and, subsequently,
turbulence is driven in the coronal loop in a similar manner as in the
untwisted case. Provided that magnetic twist remains weak, the effect of
magnetic twist is to delay the onset of the KHi and to slow down the
development of turbulence. In contrast, magnetic tension can suppress the
nonlinear growth of the KHi when magnetic twist is strong enough, even if the
KHi has locally been excited by the phase-mixing shear. Thus, turbulence is
not generated in strongly twisted loops.

###### Key Words.:

Magnetohydrodynamics (MHD) – Sun: atmosphere – waves – Methods: numerical –
Sun: oscillations

##  1 Introduction

High-resolution and high-cadence observations have shown the ubiquity of the
magnetohydrodynamic (MHD) waves throughout the solar atmosphere (see, e.g.,
Aschwanden et al., 1999; Nakariakov et al., 1999; Tomczyk et al., 2007; De
Pontieu et al., 2007; McIntosh et al., 2011; Morton et al., 2015; Jafarzadeh
et al., 2017; Srivastava & Dwivedi, 2017, among many others). MHD wave
dissipation could play an important role in the heating of the solar corona
(see, e.g., Hollweg, 1978; Cranmer & van Ballegooijen, 2005; Cargill & de
Moortel, 2011; Mathioudakis et al., 2013; Soler et al., 2019; Van Doorsselaere
et al., 2020; Nakariakov & Kolotkov, 2020), and the acceleration of the solar
wind (see, e.g., Charbonneau & MacGregor, 1995; Cranmer, 2009; Matsumoto &
Suzuki, 2012; Shoda et al., 2018).

Torsional Alfvén waves are a sub-type of axisymmetric Alfvén waves in
cylindrical flux tubes. They are nearly incompressible, the restoring force is
the magnetic tension, and their direction of polarization is perpendicular to
the magnetic field lines. This kind of waves produce axisymmetric
perturbations in the perpendicular components of velocity and magnetic field.
Furthermore, unlike Alfvén waves with other azimuthal symmetries, torsional
Alfvén waves do not couple with magneto-acoustic waves when the magnetic field
is straight (Goossens et al., 2011). Torsional Alfvén waves have been reported
in bright points (Jess et al., 2009) and in a coronal active region structure
(Kohutova et al., 2020). Torsional motions found in the chromosphere and
transition region by De Pontieu et al. (2012, 2014) can also be interpreted as
torsional Alfvén waves. Aschwanden & Wang (2020) interpreted oscillations in
the magnetic energy during solar flares as torsional Alfvén waves. This kind
of waves has also been reported in the solar wind during the interaction
between coronal mass ejections (Raghav & Kule, 2018).

In coronal loops, closed magnetic structures with two footpoints anchored at
the solar photosphere, there is no report of such waves. Nonetheless,
torsional Alfvén waves can be excited at their footpoints through
vortex/twisting plasma motions in the solar photosphere (see, e.g., Fedun et
al., 2011; Shelyag et al., 2011, 2012; Wedemeyer-Böhm et al., 2012; Mumford et
al., 2015; Srivastava et al., 2017). Recently, Soler et al. (2021) have
investigated the propagation of torsional Alfvén waves from the photosphere to
a coronal loop. As the waves reach the coronal loop, they can resonate with
standing modes of the loop and drive global torsional oscillations (see also
Hollweg, 1984b, a). Soler et al. (2021) found that large amounts of energy can
be transmitted at coronal heights despite the chromospheric filtering.
Particularly, they find that the energy flux is channeled mostly through the
fundamental standing torsional mode of the loop.

Inspired by these results, in Díaz-Suárez & Soler (2021; hereafter Paper I),
we investigated the nonlinear evolution of torsional Alfvén waves in a
straight coronal flux tube with a constant axial magnetic field. In Paper I,
see also the torsional model in Guo et al. (2019), we showed that the phase
mixing of the torsional Alfvén waves generates azimuthal shear flows. These
flows eventually excite the Kelvin-Helmholtz instability (KHi), as Heyvaerts &
Priest (1983) and Browning & Priest (1984) analytically predicted. Phase
mixing is a linear phenomenon that occurs when waves in adjacent radial
positions oscillate with different frequencies. The cause of phase-mixing is
the presence of a frequency continuum, whose origin is in the inhomogeneities
in density and/or magnetic field (see, e.g., Heyvaerts & Priest, 1983; Nocera
et al., 1984; De Moortel et al., 2000; Smith et al., 2007; Prokopyszyn et al.,
2019). Phase-mixing increases the values of vorticity and current density and
generates small scales, although at a relatively slow pace.

The KHi triggered by phase mixing can evolve nonlinearly forming large eddies
that break into smaller eddies. This process initiates and drives turbulence.
Turbulence accelerates the energy cascade to small scales, which might enhance
the efficiency of wave energy dissipation (Hillier et al., 2020). There is
evidence that coronal loops may be in a turbulent state (De Moortel et al.,
2014; Hahn & Savin, 2014; Liu et al., 2014; Xie et al., 2017). The generation
of turbulence is also a result obtained by numerous studies of numerical
simulations of kink oscillations of coronal loops (see, e.g., Terradas et al.,
2008, 2018; Antolin et al., 2015; Magyar & Van Doorsselaere, 2016; Howson et
al., 2017a; Karampelas et al., 2019; Antolin & Van Doorsselaere, 2019, among
others).

In the present paper, we extend the investigation of Paper I by replacing the
constant axial magnetic field by a twisted magnetic field. The existence of
twisted coronal loops has been confirmed by observations (see, e.g., Kwon &
Chae, 2008; Aschwanden et al., 2012; Thalmann et al., 2014; Aschwanden, 2019).
The behavior of linear MHD waves in twisted flux tubes has extensively been
investigated analytically or semi-analytically (see, e.g., Bennett et al.,
1999; Erdélyi & Fedun, 2006, 2007, 2010; Ruderman, 2007; Karami & Barin, 2009;
Karami & Bahari, 2010; Terradas & Goossens, 2012; Ebrahimi et al., 2017;
Ebrahimi & Soler, 2022, among many others). The nonlinear evolution of kink
MHD waves in twisted flux tubes has also been investigated numerically (Howson
et al., 2017a; Terradas et al., 2018). Here, we explore how the presence of
magnetic twist affects the nonlinear evolution of torsional Alfvén waves.

##  2 Numerical setup

###  2.1 Initial Configuration

As in Paper I, we use the standard coronal loop model (see, e.g., Edwin &
Roberts, 1983). The model is made of an overdense cylindrical flux tube of
radius, $R$, and length, $L,$ embedded in a low-$\beta$ uniform coronal
environment. We set $L/R=10$. We used a shorter loop length than what is
reported from observations, typically $L/R\sim 100$. The main reason for doing
so is to speed up the simulation times. The periods of the standing torsional
oscillations are proportional to the loop length. Thus, if a longer loop were
considered, the periods and simulation times would be longer, but the dynamics
would be essentially the same. The effect of considering larger loop lengths
was explored in Paper I.

The loop footpoints are fixed at two rigid walls representing the solar
photosphere. We neglect the curvature of the coronal loop and the thin
chromospheric layer at the loop feet. In Paper I we assumed a uniform magnetic
field along the cylinder axis. Here, we improve the model by considering a
twisted magnetic field, which is a more realistic representation of the
magnetic field in coronal loops, namely

${\bf B}=B_{\varphi}(r)\hat{e}_{\varphi}+B_{z}(r)\hat{e}_{z},$ (1)

where $B_{\varphi}$ and $B_{z}$ are the azimuthal and longitudinal components
of the background magnetic field in a cylindrical coordinate system denoted by
$r$, $\varphi$, and $z$. In our model, the $z$-axis coincides with the flux
tube axis. We consider the same force-free twist model as that used in
Terradas et al. (2018), namely

$\displaystyle B_{\varphi}(r)$ $\displaystyle=$
$\displaystyle\left\\{\begin{matrix}cr/R,&\mbox{if}\;r<R,\\\
cR/r,&\mbox{if}\;r\geq R,\end{matrix}\right.$ (2) $\displaystyle B_{z}(r)$
$\displaystyle=$
$\displaystyle\left\\{\begin{matrix}\sqrt{B^{2}_{0}+2c^{2}\left(1-r^{2}/R^{2}\right)},&\mbox{if}\;r<R,\\\
B_{0},&\mbox{if}\;r\geq R,\end{matrix}\right.$ (3)

where $B_{0}$ is a constant that corresponds to the magnetic field strength at
the tube axis and $c$ is a dimensionless parameter that controls the amount of
magnetic twist.

The largest value of the magnetic twist parameter used here is $c=0.4$, which
is chosen according to observational constraints. Aschwanden (2013) computed
the twist angle, $\theta$, in coronal loops as
$\theta=\arctan\sqrt{q_{\mathrm{free}}}$, where $q_{\mathrm{free}}$ is the
free energy ratio. The calculation of $q_{\mathrm{free}}$ can be achieved from
observations through line-of-sight magnetograms, the three-dimensional (3D)
position of the coronal loop, and a nonlinear force-free field code. Tables 3
in Aschwanden et al. (2014) and Aschwanden et al. (2016) show that the free
energy ratio ranges from 0 to $\sim 0.25$. Therefore, twist angles lower than
$\sim 25^{\circ}$ are expected. The twist angle of our model can be calculated
as

$\theta=\arcsin\left(\frac{B_{\varphi}(r)}{B(r)}\right),$ (4)

where $B(r)=\sqrt{B^{2}_{\varphi}\;(r)+B^{2}_{z}\;(r)}$ is the modulus of the
magnetic field. For $c=0.4$, the twist angle is $\theta=21.8^{\circ}$ at
$r=R$, which is slightly lower than the largest angle inferred by Aschwanden
et al. (2014) and Aschwanden et al. (2016). On the other hand, the number of
turns of the magnetic field lines over the cylinder length can be computed as,

$N\left(r\right)=\frac{1}{2\pi}\frac{L}{r}\frac{B_{\varphi}\left(r\right)}{B_{z}\left(r\right)},$
(5)

where the factor
$LB_{\varphi}\left(r\right)/rB_{z}\left(r\right)\equiv\Phi(r)$ is the absolute
twist. Kwon & Chae (2008) considered a sample of 14 loops from Transition
Region And Coronal Explorer (TRACE; Handy et al. 1999) observations in extreme
ultraviolet and determined that their absolute twist values ranged from
$0.22\pi$ to $1.73\pi$. This result implies that $0.11<N<0.865$. At $r=R$ in
our model, where $B_{\varphi}$ is maximum, we obtain $N=0.64$ for $c=0.4$ and
$L/R=10$, so that the maximum twist considered here is below the upper limit
of the interval inferred by Kwon & Chae (2008).

A twisted magnetic field is prone to develop the kink instability. In the case
of uniform twist, Hood & Priest (1979) found that the kink instability can
develop when the absolute twist is larger than $3.3\pi$. With an improved
analysis, Hood & Priest (1981) reduced the threshold value to $2.49\pi$. Using
the definition of $N$ in Eq. (5), we find that the kink instability may appear
in our model if $N\geq 1.65$ according to the critical twist of Hood & Priest
(1979) and if $N\geq 1.245$ according to the critical twist of Hood & Priest
(1981). Clearly, we are sufficiently far away from the critical twist for the
kink instability because our maximum twist ($N=0.64$) is still relatively
weak.

We note that we used a pre-existing twisted magnetic field in our model. A
different way to proceed would be to create twist in the model by dynamically
rotating the magnetic field lines anchored in the photosphere, as done in
Ofman (2009). This alternative approach imposes a more complex scenario and it
is not considered here for simplicity.

Since the magnetic field is force-free, we consider in the model a background
uniform gas pressure, $p_{0}$. In turn, the equilibrium density, $\rho_{0}$,
is the same as that used in Paper I. Thus, we used

$\rho_{0}(r)=\left\\{\begin{array}[]{lll}\rho_{\mathrm{i}},&\mbox{if}&r\leq
R-\frac{l}{2},\\\
\rho_{\mathrm{tr}}(r),&\mbox{if}&R-\frac{l}{2}<r<R+\frac{l}{2},\\\
\rho_{\mathrm{e}},&\mbox{if}&r\geq R+\frac{l}{2}.\end{array}\right.$ (6)

In Eq. (6), $\rho_{\rm i}$ is the density in the uniform inner core,
$\rho_{\rm e}$ is the uniform external density, and $\rho_{\rm tr}(r)$ is the
density in a nonuniform transitional layer of thickness $l$ that continuously
connects both uniform regions as

$\rho_{\mathrm{tr}}\left(r\right)=\frac{\rho_{\mathrm{i}}}{2}\left\\{\left[1+\frac{\rho_{\mathrm{e}}}{\rho_{\mathrm{i}}}\right]-\left[1-\frac{\rho_{\mathrm{e}}}{\rho_{\mathrm{i}}}\right]\sin\left[\frac{\pi}{l}\left(r-R\right)\right]\right\\},$
(7)

where $l$ may range from 0 (abrupt transition) to $2R$ (fully nonuniform
loop). We set $\rho_{\rm i}/\rho_{\rm e}=2$ and $l=0.4R$. A scheme of our
model is depicted in Fig. 1.

Figure 1: Sketch of the coronal flux tube model. The isosurface of density
with value equal to $(\rho_{\mathrm{i}}+\rho_{\mathrm{e}})/2$ is shown in
orange. Magnetic field lines located at $r=2R$, $r=R$ and $r=0$ are drawn in
green, blue, and red colors, respectively, for the model with $c=0.4$. The
bottom and top gray planes represent the solar photosphere.

The equilibrium Alfvén, $v_{A}(r)$, and sound, $v_{s}(r)$, speeds are,

$\displaystyle v_{\rm A}(r)$ $\displaystyle=$
$\displaystyle\frac{B(r)}{\sqrt{\mu\rho_{0}(r)}},$ (8) $\displaystyle
v_{s}(r)$ $\displaystyle=$ $\displaystyle\sqrt{\frac{\gamma
p_{0}}{\rho_{0}(r)}},$ (9)

where $\gamma$ is the adiabatic constant. Figure 2 shows the radial profiles
of both speeds for a particular set of parameters. Both Alfvén and sound
speeds vary inside the transition region because the density is nonuniform.
However, when $c\neq 0$ the Alfvén speed is also position-dependent in the
inner core and in the external medium, where the density is uniform. This
happens because the magnetic field is nonuniform everywhere when twist is
present.

Figure 2: The equilibrium radial profiles of the Alfvén, $v_{\rm A}(r)$, and
sound speeds, $v_{s}(r)$, for the twist parameter ranging from $c=0$ to
$c=0.4$. The sound speed is the same in all cases. The region between the
vertical dashed lines denotes the inhomogeneous region in density. The values
are normalized with respect to the Alfvén speed at $r\to\infty$. We used
$\rho_{\rm i}/\rho_{\rm e}=2$ and $l/R=0.4$.

According to Halberstadt & Goedbloed (1993), the line-tied Alfvén frequency
continuum is

$\omega_{\rm A}(r)=\frac{n\pi}{L}\frac{B_{z}(r)}{\sqrt{\mu_{0}\rho_{0}(r)}},$
(10)

where $n$ is the harmonic number. A continuous spectrum is present because of
the radial variation of the $z$-component of the magnetic field, $B_{z}(r)$,
and the density, $\rho_{0}(r)$. The density is only nonuniform in the
transition region. However, $B_{z}(r)$ is also nonuniform in the inner core of
the loop when twist is present. Because of twist, the Alfvén continuum also
extends into the uniform core of the loop, as it can be seen in Fig. 3.

Figure 3: The radial profile of the line-tied Alfvén frequency continuum for
the twist parameter ranging from $c=0$ to $c=0.4$. The region between the
green vertical dashed lines corresponds to the nonuniform region in density.
The values are normalized to the external Alfvén frequency. We used $\rho_{\rm
i}/\rho_{\rm e}=2$ and $l/R=0.4$.

Throughout the paper, length and density are normalized with respect to the
radius of the flux tube, $R$, and the external density, $\rho_{e}$,
respectively. Velocities are normalized with respect to the external Alfvén
speed at $r\to\infty$, namely $v_{A,\infty}=v_{A}\left(r\to\infty\right)$. The
normalized time is $\overline{t}=t/\tau_{\rm A}$, with $\tau_{\rm
A}=R/v_{A,\infty}$.

###  2.2 Numerical code

As in Paper I, the numerical simulations are performed with the PLUTO code
(Mignone et al., 2007). We solve the 3D ideal MHD equations with a finite-
volume, shock capturing spatial discretization on a structured mesh. The
equations are as follows:

$\displaystyle\frac{\partial\rho}{\partial t}$ $\displaystyle=$
$\displaystyle-\nabla\cdot\left(\rho\mathbf{v}\right),$ (11)
$\displaystyle\rho\frac{\mathrm{D}\mathbf{v}}{\mathrm{D}t}$ $\displaystyle=$
$\displaystyle-\nabla
p+\frac{1}{\mu_{0}}\left(\nabla\times\mathbf{B}\right)\times\mathbf{B},$ (12)
$\displaystyle\frac{\partial\mathbf{B}}{\partial t}$ $\displaystyle=$
$\displaystyle\nabla\times\left(\mathbf{v}\times\mathbf{B}\right),$ (13)
$\displaystyle\frac{\mathrm{D}p}{\mathrm{D}t}$ $\displaystyle=$
$\displaystyle\frac{\gamma p}{\rho}\frac{\mathrm{D}\rho}{\mathrm{D}t}.$ (14)

In Eqs. (11)-(14),
$\frac{\mathrm{D}}{\mathrm{D}t}\equiv\frac{\partial}{\partial
t}+\mathbf{v}\cdot\nabla$ denotes the total derivative, $\rho$ is the mass
density, $\mathbf{v}$ is the velocity, $\mathbf{B}$ is the magnetic field, $p$
is the gas pressure, and $\mu_{0}$ is the vacuum magnetic permeability.
Gravity and nonideal terms are neglected.

The PLUTO code solves Eqs. (11)-(14) in Cartesian coordinates. As in Paper I,
the code uses a Roe-Riemann solver (Roe, 1981) to compute the numerical
fluxes, a piecewise parabolic scheme for spatial reconstruction, and a second-
order Runge-Kutta algorithm for temporal evolution. In order to maintain the
solenoidal constraint, the Generalized Lagrange Multiplier (GLM) method
(Dedner et al., 2002) is used. The background magnetic field is not current-
free because of magnetic twist, so the background splitting technique cannot
be used in the present simulations, contrary to Paper I. For comparison
purposes, the background splitting technique is not used for the straight
magnetic field case either. So, here the code advances the full magnetic field
vector.

We used the same base computational box as in Paper I, namely a numerical
resolution of 100x100x100 cells distributed uniformly from $-3R$ to $3R$ in
the $x$\- and $y$-directions, and from $-L/2$ to $L/2$ in the $z$-direction.
The code implements the adaptive mesh refinement (AMR) strategy (Mignone et
al., 2012) by which the cells split in two, and so the spatial resolution
doubles, if a certain criterion is satisfied. The refinement criterion is
based on the second derivative error norm (Lohner, 1987) of a parameter
closely related with the kinetic energy density, since the goal of the AMR
strategy in the present simulations is to conserve wave energy as much as
possible. The use of AMR decreases the computational cost by keeping low
resolution where the dynamics is not relevant and generating high resolution
in the regions of interest. We included four levels of refinement, so that the
maximum effective resolution is 1600x1600x1600. For typical values of a
coronal loop radius, the effective transverse resolution is $\sim$ 10 km.

Although the AMR scheme allows us to deal with extremely fine scales, the
dynamics developed in the flux tube eventually cascades the energy to scales
below the maximum effective resolution. As in Paper I, to determine the
simulation time at which this unphysical energy loss starts to happen, we
monitored the kinetic, magnetic, and internal energies. In such a way, we
could determine the time at which numerical dissipation is beginning to play
an important role. This time also coincides with the time when the total
integrated vorticity saturates and decreases (see later).

Regarding the boundary conditions, we set the same set of boundary conditions
as in Paper I. We considered outflow conditions, that is, zero gradient for
pressure, density, and the $x$\- and $y$-components of the magnetic field at
all boundaries. We also set the outflow condition for the $z$-component of
magnetic field, $B_{z}$, at the lateral boundaries. To mimic the line-tying of
the magnetic field lines at the solar photosphere, $B_{z}$ is fixed to the
equilibrium value through Eq. (3) at the top and bottom boundaries. The three
components of velocity vanish at all boundaries to avoid energy injection from
outside the domain.

###  2.3 Initial perturbation

In the solar atmosphere, torsional Alfvén waves can be driven by photospheric
plasma motions (see, e.g., Fedun et al., 2011; Shelyag et al., 2011, 2012;
Wedemeyer-Böhm et al., 2012; Mumford et al., 2015; Srivastava et al., 2017).
The waves propagate to the corona through the chrosmosphere, where reflection
and dissipation heavily reduces the upward wave transmission (see Soler et
al., 2019). When a broadband spectrum of waves driven at the photosphere are
transmitted into a closed coronal structure, like a coronal loop, Soler et al.
(2021) showed that most of the energy is deposited in the longitudinally
fundamental mode. In view of this result and as in Paper I, we excited at
$t=0$ the longitudinally fundamental mode of standing torsional Alfvén waves.
We did so by perturbing the components of velocity. To avoid that the
perturbation also excites slow MHD waves initially, we impose an initial
velocity field in the same manner as in Díaz-Suárez & Soler (2021), so that
the velocity is purely perpendicular to the magnetic field lines, namely

$\displaystyle v_{r}(r,z)$ $\displaystyle=$ $\displaystyle 0,$ (15)
$\displaystyle v_{\varphi}(r,z)$ $\displaystyle=$ $\displaystyle
v_{\perp}(r,z)\frac{B_{z}(r)}{B(r)},$ (16) $\displaystyle v_{z}(r,z)$
$\displaystyle=$ $\displaystyle-v_{\perp}(r,z)\frac{B_{\varphi}(r)}{B(r)},$
(17)

where $v_{\perp}$ is the perpendicular velocity to the magnetic field lines.
We consider the same form as that used in Paper I, namely

$v_{\perp}(r,z)=v_{0}A\left(r\right)\cos\left(\frac{\pi}{L}z\right),$ (18)

where $v_{0}$ is the maximum velocity amplitude and $A\left(r\right)$ contains
the radial profile (see Paper I). We set $v_{0}=0.1v_{A,\infty}$. The radial
profile of $v_{\varphi}$ and $v_{z}$ at $z=0$ is shown in Fig. 4 for different
values of the twist parameter, $c$.

Figure 4: Radial profiles of the azimuthal (top) and longitudinal (bottom)
components of velocity at $t=0$ for the twist parameter ranging from $c=0$ to
$c=0.4$. The inset in the top panel shows a close-up view of the azimuthal
velocity around its maximum. Both velocity components are normalized to the
maximum amplitude of the perpendicular velocity perturbation, $v_{0}$.

We recover the same initial condition as in Paper I for the untwisted case,
$c=0$. When we increase the magnetic twist, the maximum of $v_{\varphi}$
slightly decreases, but its profile remains mostly unaltered. The longitudinal
component of velocity, $v_{z}$, displays a significant increase when the twist
grows. This is a direct effect of the relative variation of the azimuthal and
longitudinal components of the magnetic field.

Although the initial perturbation will mostly excite torsional Alfvén waves,
other types of MHD can also appear in the flux tube. The inclusion of a
twisted magnetic field introduces linear coupling between torsional Alfvén
waves and fast magneto-acoustic sausage waves (Sakurai et al., 1991; Goossens
et al., 1992, 2011; Giagkiozis et al., 2015). The waves excited by the initial
perturbation when $c\neq 0$ will develop mixed properties of both torsional
Alfvén waves and fast magneto-acoustic sausage waves. Nevertheless, since the
twist is weak, the torsional Alfvén wave character will be dominant. In
addition, later in the evolution the ponderomotive force will nonlinearly
drive slow MHD waves (Hollweg, 1971; Rankin et al., 1994; Tikhonchuk et al.,
1995; Terradas & Ofman, 2004).

##  3 Revisiting the straight magnetic field case

The main goal of this paper is to illustrate the effects of a twisted magnetic
field in the nonlinear evolution of the standing torsional Alfvén waves. In
particular, we investigate its influence on the triggering of the KHi and the
associated turbulence generation. However, before tackling the investigation
of twist, we seize the opportunity to revisit the straight magnetic field case
of Paper I and to perform an extended study.

We rerun the simulation corresponding to the thin-layer case of Paper I but
for larger times and deeper into the turbulent phase. So, the simulation is
continued beyond the capacity of the AMR scheme to correctly describe the
smallest spatial scales developed in the computational box. Although the
dynamics of the smallest scales will then be affected by numerical
dissipation, we can assume that the evolution at spatial scales larger than
the maximum effective resolution will remain valid. Having this in mind, we
can study how turbulence evolves globally at larger times despite the non-
reliable information at the smallest scales.

We refer readers to Paper I for detailed explanations of the simulation
evolution. We give a brief summary. Standing torsional Alfvén waves excited in
the flux tube develop the process of phase mixing (see, e.g., Heyvaerts &
Priest, 1983; Nocera et al., 1984; De Moortel et al., 2000; Smith et al.,
2007; Prokopyszyn et al., 2019; Ebrahimi et al., 2020). In the transition
region, where density is nonuniform, waves in adjacent magnetic surfaces
oscillate with a slightly different frequency (see Eq. (10)) and become out of
phase as time increases. The effect of phase mixing is to generate an
azimuthal velocity shear, which eventually triggers the KHi (see, e.g.,
Heyvaerts & Priest, 1983; Soler et al., 2010; Zaqarashvili et al., 2015;
Barbulescu et al., 2019). Initially, the KHi develops locally in the
transition region, so it does not globally perturb the flux tube. Nonetheless,
the KHi evolves nonlinearly as time increases. The nonlinear evolution of the
KHi breaks large eddies into smaller eddies. The instability naturally drives
turbulence in the flux tube that develops perpendicularly to the background
magnetic field. Turbulence extends beyond the boundaries of the initial
nonuniform transition, and further accelerates the energy cascade towards
small scales. Because of turbulence, mixing of the internal and external
plasmas occurs. Thus, the dynamics of torsional Alfvén oscillations is similar
to that of linearly polarized kink (see, e.g., Terradas et al., 2008; Antolin
et al., 2015; Magyar & Van Doorsselaere, 2016; Karampelas et al., 2019;
Antolin & Van Doorsselaere, 2019; Pascoe et al., 2020; Martínez-Gómez et al.,
2022) or circularly polarized kink oscillations (Magyar et al., 2022).
Nonetheless, the flux tube is not displaced laterally in the case of torsional
oscillations, as occurs with the excitation of kink oscillations, and the
resonant absorption process is absent for torsional Alfvén waves (see, e.g.,
Goossens et al., 2011).

As shown in Paper I, a parameter that helps us understand the different phases
of the evolution is the vorticity. We investigated the vorticity squared,
$\omega^{2}(\mathbf{r},t)=\left|\nabla\times\mathbf{v}(\mathbf{r},t)\right|^{2}$,
and its value integrated over the whole computational domain, $\Omega^{2}$,
namely

$\Omega^{2}\left(t\right)=\int\omega^{2}\left(\mathbf{r},t\right)\mathrm{d}^{3}\mathbf{r}.$
(19)

Figure 5 shows a cross-sectional cut at the tube center, $z=0$, of
$\omega^{2}\left(\mathbf{r},t\right)$ in logarithmic scale. Only a subdomain
of the complete numerical domain is shown, where the relevant dynamics occurs.
The inset to the right shows the integrated vorticity squared,
$\Omega^{2}\left(t\right)$, as a function of the computational time. The blue
dot tracks the position on the curve of $\Omega^{2}$ as time varies. The
complete temporal evolution is available as a movie.

In the small panel of Fig. 5, we see that the temporal evolution of
$\Omega^{2}$ undergoes three phases (see Paper I). In the first phase for
$\bar{t}\lesssim 65$, $\Omega^{2}$ oscillates and slightly increases owing to
the linear development of phase-mixing. In this phase, the spatial
distribution of $\omega^{2}\left(\mathbf{r},t\right)$ is mostly in the form of
concentric rings. The concentric rings deform after the KHi is locally
excited. In the second phase for $65\lesssim\bar{t}\lesssim 87$, the KHi has
grown enough to behave nonlinearly and induces turbulence. The KHi vortices
break into smaller and smaller vortices. As a result, there is a dramatic
increase in vorticity. The spatial distribution of
$\omega^{2}\left(\mathbf{r},t\right)$ clearly reveals the formation of very
fine scales owing to turbulence. Finally, in the third phase for
$\bar{t}\gtrsim 87$, the AMR scheme can no longer fully describe the smallest
scales that appear in the evolution. Thus, vorticity decreases unphysically
because of numerical dissipation. As a reference, the periods of the internal
and external Alfvén oscillations are $20\sqrt{2}$ and $20$ normalized time
units, respectively.

The particular snapshot displayed in the large panel of Fig. 5 shows the
spatial distribution of $\omega^{2}\left(\mathbf{r},t\right)$ at the time for
which $\Omega^{2}$ is maximum. We can clearly see that the loop boundary is
fully turbulent and extremely fine structures in vorticity have already been
generated. Several authors have studied the evolution of vorticity in
numerical simulations where the kink mode is excited (see, e.g., Terradas et
al., 2008; Antolin et al., 2015; Howson et al., 2017b; Karampelas et al.,
2017; Karampelas & Van Doorsselaere, 2018; Guo et al., 2019; Howson et al.,
2020, 2021) and some of their results are comparable with the present ones.
Similar vorticity structures as those displayed in Fig. 5 can also be seen in
previous works of nonlinear kink oscillations (see, e.g., Antolin et al.,
2014; Howson et al., 2017a; Antolin & Van Doorsselaere, 2019). Nonetheless,
the spatial distribution is different due to the different azimuthal symmetry
of torsional and kink oscillations.

Figure 5: Cross-sectional cut at $z=0$ of the logarithm with base 10 of the
vorticity squared, $\omega^{2}\left(\mathbf{r},t\right)$, in arbitrary units
when $\overline{t}=87$. The small panel to the right shows the temporal
evolution of $\omega^{2}\left(\mathbf{r},t\right)$ integrated over the whole
computational domain, $\Omega^{2}(t)$, normalized with respect to the value at
$\overline{t}=0$. The blue dot marks the position in the curve for the
considered simulation time. The complete temporal evolution is available as a
movie.

The turbulent evolution of the flux tube is clearly highlighted by the
formation of extremely fine vorticity structures captured by the high
resolution of the AMR scheme. A consequence of turbulence is the mixing of the
internal and external plasmas, which heavily modifies the transverse density
profile adopted initially. To illustrate this effect, Fig. 6 displays an
azimuthal average of the transverse density profile at $z=0$ for various
times. Figure 6 is similar to density panels from Fig. 5 of Karampelas & Van
Doorsselaere (2018) for nonlinear kink waves. On average, we find that the
width of the nonuniform region increases with time. For large times in our
simulation, part of the initially uniform core becomes turbulent, and so
nonuniform. Consequently, at sufficiently large times beyond those simulated
here, the whole flux tube should become turbulent. It is likely that for kink
waves, which are global oscillations of the whole tube, the fully turbulent
regime is reached faster than for torsional Alfvén waves, which are local
waves in nature. This would require further analysis. We note that, because of
the azimuthal average, the density variations are smoothed. Along specific
directions, we can find density variations that are much more important than
those displayed in Fig. 6.In addition, the full temporal evolution of the
results displayed in Fig. 6 also shows periodic density increases in the loop
core caused by the ponderomotive force (see, e.g., Hollweg, 1971).

Figure 6: Azimuthally averaged density at the tube center, $z=0$, for
different simulation times. The vertical dashed purple lines show the limits
of the nonuniform region at $\overline{t}=0$. Density is normalized to the
external density.

##  4 Exploring the effect of magnetic twist

Here we turn to investigate the influence of magnetic twist. Besides the
simulation without magnetic twist ($c=0$) discussed before, we performed four
additional simulations including magnetic twist with $c=$ 0.1, 0.2, 0.3, and
0.4.

Before the onset of the KHi, when the dynamics is governed by the development
of phase-mixing, the results of all simulations are quite similar
independently of the strength of magnetic twist. The reason is that the shape
of the Alfvén frequency continuum in the nonuniform layer is very similar in
all cases (see Fig. 3). Therefore, we focus on later times. In Fig. 7 we show
cross-sectional cuts of density at the tube center, $z=0$, at three different
simulation times corresponding to already advanced stages of the simulations.
The results for $c=0$ are shown in the first row. The twist parameter
increases from $c=0.1$ to $c=0.4$ as we move from top to bottom in the
remaining rows. The results for $c=0.3$ are not shown as they are visually
indistinguishable from those for $c=0.4$. Figure 7 suggests that the inclusion
of the magnetic twist in our model delays the onset of KHi. For the same time,
the KHi vortices appear less and less developed when the twist increases.

Figure 7: Cross-sectional cuts of density at the tube centre, $z=0$, for
different simulation times: $\overline{t}=76$ (left column), $\overline{t}=82$
(mid column), and $\overline{t}=88$ (right column). The first row corresponds
to the simulation with no magnetic twist, $c=0$. In descending order, the
remaining rows correspond to the simulations with magnetic twist parameter of
$c=$ 0.1, 0.2, and 0.4. Density is normalized to the external density.

On the other hand, when magnetic twist is included the KHi vortices are not
strictly perpendicular to the flux tube axis and can also develop in the
$z$-direction (Terradas et al., 2018). To illustrate this fact, we show in
Fig. 8 a longitudinal cut of density of the simulation with $c=0.2$. This cut
is done for the same time as that in the third column of Fig. 7.

Figure 8: Longitudinal cut of density at $y=0$ for the simulation with
magnetic twist parameter, $c=0.2$ when $\overline{t}=88$, i.e., the same time
as in the right column of Fig. 7. Density is normalized to the external
density.

###  4.1 Delay or suppression of the KHi

We aim to quantify how magnetic twist postpone the KHi onset time.
Theoretically, a component of the magnetic field along the direction of the
shear flow has a stabilizing effect on the KHi (see, e.g., Chandrasekhar,
1961). Significant efforts have been made to understand this problem for the
KHi driven by transverse MHD waves. However, the analytical obtainment of the
onset time of the KHi in a cylindrical tube with time-varying flows under the
presence of a twisted magnetic field remains a challenge. Browning & Priest
(1984) analytically obtained an expression for the onset time of the KHi for
time-varying flows in Cartesian geometry and with a straight field. Soler et
al. (2010) considered the KHi at the boundary of a cylindrical tube, but with
constant azimuthal flows. Soler et al. (2010) studied the effect of magnetic
twist by considering a Cartesian analog with an inclined magnetic field and
found that twist can decresase the growth rates and even suppress the
instability. Zaqarashvili et al. (2015) studied the instability of a
cylindrical, twisted, and rotating jet. They found that jets are unstable to
the KHi only when the kinetic energy of rotation is larger than the magnetic
energy of the twist. Hillier et al. (2019) studied the linear stability of a
discontinuous oscillatory shear flow in Cartesian geometry in the presence of
a uniform magnetic field. They show that parametric instabilities can also
grow in addition to the KHi. Barbulescu et al. (2019) also considered a
Cartesian model of time-varying flows, but including an inclined magnetic
field on one side of the interface to mimic the effect of twist, as was
previously done by Soler et al. (2010) for constant flows. Barbulescu et al.
(2019) conclude that the magnetic shear may reduce the instability growth
rate, but it cannot completely stabilize the interface, contrary to the
constant flow case of Soler et al. (2010).

To understand the effect of twist on the triggering of the KHi, here we follow
the method introduced by Terradas et al. (2018). Their method is based on
studying the excitation of different azimuthal wave numbers. This approach was
also used in Antolin & Van Doorsselaere (2019) and in Paper I. We proceed as
follows. In a cross-sectional cut at the tube center, $z=0$, we calculated the
perpendicular component of velocity to the magnetic field lines, $v_{\perp}$,
at the middle of the transition region, $r=R$, as a function of the azimuthal
angle, $\varphi$, from $\varphi=0$ to $\varphi=2\pi$. After that, we computed
the discrete Fourier transform to the data using the fast Fourier transform
(FFT) algorithm with the Scipy module (Virtanen et al., 2020). Following the
notation of Terradas et al. (2018), the discrete Fourier transform is

$G\left(p\right)=\sum^{N-1}_{k=0}g\left(k\right)\exp{\left(\frac{-2\pi
ipk}{N}\right)},$ (20)

where $N$ is the number of sample points, $g\left(k\right)$ is the angular
sampling of $v_{\perp}$, and $p$ is an integer that plays the role of the
azimuthal wavenumber. Unlike in Paper I, here, we consider positive and
negative values of $p$, namely $p=0,\pm 1,\pm 2,...,\pm(N-1)$, since twist
breaks the degeneracy in the sign of the azimuthal wavenumber. $p=0$ is the
torsional plus the sausage mode. $p=\pm 1$ are the kink modes, and $|p|\geq 2$
are the fluting modes (see, e.g., Roberts, 2019). In the presence of magnetic
twist, the torsional mode and the fast sausage mode are coupled linearly (see,
e.g., Goossens et al., 2011). However, the contribution of the fast sausage
mode is weak because the magnetic twist considered here is relatively weak,
even for $c=0.4$. Thus, for $p=0$, the torsional mode dominates over the fast
sausage mode.

Figure 9: Top: temporal evolution of the Fourier coefficient associated with
the torsional mode, $p=0$, at $z=0$ and $r=R$ for the twist parameter ranging
from $c=0$ to $c=0.4$. Bottom: same as the top panel but for the sum of the
first twenty positive and the first twenty negative Fourier coefficients
excluding $p=0$. Arbitrary units are used.

Figure 9 shows the results of the azimuthal Fourier analysis. To better
visualize the results, in the top panel, we plot the temporal evolution of the
$p=0$ mode alone. In the bottom panel, we plot the temporal evolution of the
sum of the first twenty positive modes and the first twenty negative modes,
excluding the $p=0$ mode. We checked that adding more modes to the sum does
not affect the results.

As expected, we obtain that the $p=0$ mode is dominant during the linear
evolution of phase-mixing independently of the magnetic twist. The dominance
of the $p=0$ mode lasts until the onset of KHi, which excites higher azimuthal
wave numbers. At that moment, the sum of the $p\neq 0$ Fourier coefficients
begins to display significant variations. These results are consistent with
those of Terradas et al. (2018) and Antolin & Van Doorsselaere (2019) for the
standing kink mode and also agree with those of Paper I. During the
development of KHi and the subsequent turbulence the initially low-amplitudes
of the sum of the $p\neq 0$ Fourier coefficients become comparable with or
even larger than the amplitude of the $p=0$ mode. Moreover, we find that the
$p=0$ mode loses its periodicity.

Similarly to the results of Paper I, independently on the presence or absence
of magnetic twist, we verified that the excitation of high-order azimuthal
wave numbers initially occurs in the nonuniform transition region. This
analysis is not shown here for simplicity. We also verified the persistence of
the $p=0$ mode in the core where the KHi cannot develop. However, once the
turbulence expands into the core, high-order azimuthal wave numbers are also
excited in the core. This is consistent with the results from Fig. 6 for the
untwisted case.

Therefore, the azimuthal Fourier analysis confirms what was already visualized
in Fig. 7: the inclusion of the magnetic twist delays the development of the
KHi. Indeed, if magnetic twist is sufficiently weak, the KHi is just delayed
but the dynamics is similar to that in the absence of twist, although evolving
at a slower pace. In a sense, this conclusion is not completely new, although
this is the first time that the role of twist in the nonlinear evolution of
standing torsional Alfvén waves has been investigated. Similar results can be
found in simulations of nonlinear kink waves in twisted flux tubes (see, e.g.,
Howson et al., 2017a; Terradas et al., 2018). For the different twist profiles
considered by Howson et al. (2017a), they find that the KHi vortices grow less
as the magnetic twist is increased. Howson et al. (2017a) also find that the
location of the maximum twist may affect at the development of KHi vortices.
Terradas et al. (2018) also studied nonlinear kink waves considering the same
twist profile used here. Similarly to Howson et al. (2017a), Terradas et al.
(2018) find that the KHi vortices grow less when the magnetic twist increases.

When the magnetic twist is strong enough, however, the physical scenario can
be different. The simulations with strong magnetic twist suggest that the KHi
can be suppressed altogether even if the phase-mixing generated shear flows
should remain unstable according to linear theory (see Barbulescu et al.,
2019). Although the KHi itself can linearly be triggered in a local fashion,
its perturbations are not allowed to fully grow into the nonlinear regime when
twist provides a strong enough tension force (Díaz-Suárez & Soler, 2021). The
important role of magnetic tension in nonlinearly preventing the growth of the
KHi vortices was already discussed by, e.g., Galinsky & Sonnerup (1994), Ryu
et al. (2000), and Hillier (2019). In our model, the nonlinear suppression of
the KHi seems to occur for $c\gtrsim 0.3$.

Since the KHi suppression for strong twist is a nonlinear phenomenon, we
cannot exclude that the initial amplitude of the excited torsional wave may
affect the subsequent stability. Indeed, one should expect that for
sufficiently large amplitudes, the KHi may break the stabilizing effect of
magnetic tension even if twist is strong. So, we speculate that the ability of
the KHi to grow might depend on the relation between the kinetic energy of the
shear flow that drives the KHi and the magnetic energy of the twisted magnetic
field, in a similar fashion as in the model by Zaqarashvili et al. (2015). The
investigation of the effect of the initial amplitude is not done here and is
left for forthcoming works.

###  4.2 Turbulence and vorticity generation

The delay or suppression of the KHi has important consequences in the dynamics
of the flux tube. In our model, the generation of turbulence, which results
from the KHi evolution, can just be delayed or be absent depending on the
strength of magnetic twist. In addition, magnetic twist may affect the
subsequent evolution of turbulence. To explore that, we resort again to the
vorticity to study how magnetic twist affects the evolution of turbulence.

Figure 10 shows cross-sectional cuts of $\omega^{2}\left(\mathbf{r},t\right)$
at the tube center, $z=0$, for the same three times as in Fig. 7. In each row,
the strength of magnetic twist is different and sorted in increasing order
from top to bottom. For comparison purposes, we also included in the first row
the results from the simulation without magnetic twist. Unlike in Fig. 7, we
find slightly different vorticity patterns for the cases with a magnetic twist
of $c=0.3$ and $c=0.4$. This will be highlighted later. Figure 10 evidences
that magnetic twist not only plays a role in delaying or suppressing the KHi,
but also it heavily affects the value and spatial distribution of vorticity
when turbulence is present. Another plot that helps us in the present
discussion is Fig. 11. There, we show the temporal evolution of the vorticity
squared integrated in the whole computational domain, $\Omega^{2}$, as a
function of the computational time for all simulations. The curve for $c=0$
plotted in Fig. 11 is the same as that already shown in the small panel of
Fig. 5.

Figure 10: Cross-sectional cuts at $z=0$ of the logarithm with base 10 of the
vorticity squared, $\omega^{2}\left(\mathbf{r},t\right)$, for different
simulation times: $\overline{t}=76$ (left column), $\overline{t}=82$ (mid
column), and $\overline{t}=88$ (right column). The first row corresponds to
the simulation with no magnetic twist, $c=0$. In descending order, the
remaining rows correspond to the simulations with magnetic twist parameter of
$c=$ 0.1, 0.2, 0,3, and 0.4. Arbitrary units are used.

Figure 11: Temporal evolution of the vorticity squared integrated in the whole
computational domain, $\Omega^{2}$, for the twist parameter ranging from $c=0$
to $c=0.4$. The curve for $c=0$ (red dot-dashed line) is the same as in the
small panel of Figure 5. All curves are normalized to their values at $t=0$.

To start with, we focus on comparing the simulations with $c=0$ (no twist) and
$c=0.1$ (weakest magnetic twist considered). Figure 11 indicates that the
onset of the KHi occurs practically at the same time in both simulations,
since the sharp increase in vorticity in the two cases is almost simultaneous.
Thus, the delaying effect of twist when $c=0.1$ is still not very relevant.
However, slightly smaller values of integrated vorticity are found when
$c=0.1$ compared with those of $c=0$. Indeed, the first two rows in Fig. 10
show that the spatial distribution of vorticity once turbulence sets in is
very similar in the two cases. Nevertheless, smaller vorticity structures
appear when magnetic twist is absent. This effect is clearly visible when one
compares the two simulations when $\overline{t}=88$, a time for which
turbulence is already well established.

The effect of magnetic twist becomes more pronounced when we consider the
simulation with $c=0.2$. Figure 11 shows that the KHi onset is significantly
delayed with respect to the cases with $c=0$ and $c=0.1$ discussed before.
Figure 10 plainly evidences that the dynamics advances at a slower pace when
$c=0.2$. For instance, the spatial distribution of
$\omega^{2}\left(\mathbf{r},t\right)$ at $\overline{t}=88$ is similar to that
found at $\overline{t}=76$ in the simulation with $c=0.1$. The case with
$c=0.2$ corresponds to an intermediate situation in the sense that twist
already has an important impact in delaying the KHi onset and in the
appearance and evolution of turbulence. However, magnetic twist is still not
strong enough to avoid the KHi growth and so prevent the generation of
turbulence.

The simulations with stronger magnetic twist, $c=0.3$ and $c=0.4$, represent a
completely different scenario in which turbulence is not driven because the
KHi is not allowed to grow. Even for the largest considered simulation times,
the shape of $\omega^{2}\left(\mathbf{r},t\right)$ in the two simulations is
essentially in the form of concentric rings, which is the typical structure
caused by phase-mixing. However, when $\overline{t}=88$ a close look at the
shape of $\omega^{2}\left(\mathbf{r},t\right)$ in the case with $c=0.3$
reveals that some of the phase-mixing-driven concentric rings are slightly
deformed in a wavy fashion. Such a deformation is not seen in the simulation
with $c=0.4$ at that time. This deformation in the vorticity rings when
$c=0.3$ is caused by the very local triggering of the KHi within the
nonuniform transition layer. Such a local inception of the KHi is not
appreciable as discernible deformations in density. Nonetheless, vorticity,
which is computed from the velocity field, is much more sensitive to the very
initial appearance of the KHi.

We have run the simulation with $c=0.3$ up to larger times than those
displayed in Fig. 10 and have confirmed that the KHi does not grow and
vortices do not form. As explained before, the reason is the nonlinear
suppresion of the KHi by magnetic tension (see, e.g., Galinsky & Sonnerup,
1994; Ryu et al., 2000; Hillier, 2019; Díaz-Suárez & Soler, 2021). Thus,
turbulence is not driven. In the simulation with $c=0.4$, the local inception
of the KHi is eventually observed at later times, but the perturbations are
neither allowed to grow and turbulence is equally absent. However, Fig. 11
shows that vorticity keeps increasing for both $c=0.3$ and $c=0.4$ at large
times. The reason is that the linear phase mixing remains at work for large
times and it is responsible for the increase of vorticity. Nevertheless, the
increase of vorticity owing to phase mixing alone occurs at a much slower pace
than when the KHi is present.

Again, our results about the effect of magnetic twist on the vorticity are
comparable with those obtained from nonlinear kink wave simulations (Howson et
al., 2017a; Terradas et al., 2018). Howson et al. (2017a) showed cross-
sectional cuts of the magnitude of vorticity at the loop apex for a twisted
and an untwisted case at two different simulation times. In both cases, they
find big and small vortices due to KHi in their untwisted case, but their
twisted case does not show small vortices. Thus, the magnetic twist is
suppressing the small vortices associated with KHi. Howson et al. (2017a) also
find that the increase in vorticity due to KHi happens earlier when twist is
absent.

###  4.3 Kinetic energy cascade to small scales

In Paper I we showed that the onset of turbulence in the flux tube accelerates
the cascade of energy from large to small spatial scales. Such a process is
slower before the appearance of turbulence, when phase mixing is the only
working mechanism. The delay or even suppression of the turbulent regime when
magnetic twist is included should affect the rate at which small scales are
generated.

We investigate how kinetic energy is transported to small scales depending on
the strength of twist. For this purpose, we only consider the kinetic energy
associated with motions perpendicular to the magnetic field lines, namely
$E=\frac{1}{2}\rho v^{2}_{\perp}$. We exclude the contribution from the other
components of velocity because we are mainly interested in the Alfvénic part
of the energy. The presence of twist linearly couples torsional Alfvén waves
and fast magneto-acoustic sausage waves (see, e.g., Sakurai et al., 1991).
Fast magneto-acoustic sausage waves would predominantly perturb the radial
component of velocity. In turn, Alfvén waves and slow magneto-acoustic waves
are nonlinearly coupled through the ponderomotive force (see, e.g., Hollweg,
1971). Slow waves would mainly affect the longitudinal component of velocity.

We consider a cross-sectional cut at the tube centre, $z=0$, and average the
Alfvénic kinetic energy in the azimuthal direction using 512 radial cuts.
Thus, the angular resolution is approximately $\sim 0.7^{\circ}$. We denote
the averaged energy by $\overline{E}$, which is a function of $r$ and $t$.
Then, we apply the continuous Fourier transform in the radial direction,
discretized due to the limited numerical resolution. Following Paper I, it can
be defined as

$\displaystyle\overline{E}_{F}(k,t)\approx\frac{\Delta
r}{\sqrt{2\pi}}\exp{\left(-ikr_{0}\right)}\sum^{N-1}_{m=0}\overline{E}(r,t)\exp{\left(-\frac{2\pi
imk}{N}\right)},$ (21)

where $N$ is the number of samples, $k$ is the radial wave number, $\Delta r$
is the spatial resolution, and $r_{0}$ is the upper limit of the radial
domain. The summatory in Eq. (21) is exactly the 1D discrete Fourier
transform, which we compute using the fft module of Scipy (Virtanen et al.,
2020) based on the FFT algorithm (Cooley & Tukey, 1965). Finally, we calculate
the modulus of $\overline{E}_{F}(k,t)$ and normalize it with respect to the
maximum value in the spectrum at each time.

Figure 12 displays the results of the Fourier transform for all simulations at
the last common simulation time for which the vorticity still increases in all
simulations, $\overline{t}=87$. The reason for choosing this time is that, for
larger times, the simulations with $c=0$ and $c=0.1$ are already significantly
affected by numerical dissipation (see the decrease of the integrated
vorticity in Fig. 11), which may also affect the results of the Fourier
transform at the small scales.

Figure 12: Azimuthally-averaged amplitude spectrum of the Alfvénic
(perpendicular to the magnetic field) part of the kinetic energy for the twist
parameter ranging from $c=0$ to $c=0.4$. We considered the last common time
for which vorticity increases in all simulations, $\bar{t}=87$. Arbitrary
units are used.

At large scales, $kR<10^{0}$, the amplitude spectrum is independent on the
wave number. For intermediate scales, $10^{0}<kR<10^{2}$, we do not find that
the values of the amplitude spectrum are ordered in any particular way
regarding the twist parameter, $c$. The temporal evolution of the amplitude
spectrum (not shown here) shows oscillations by which the various curves with
different $c$ exchange their relative positions depending on the considered
time. The reason for such a behavior is unclear to us. The understanding of
the behavior of the amplitude spectrum in these intermediate scales would
require additional research.

We are more interested in the spectrum at small scales, $10^{2}<kR<10^{3}$,
where the amplitude spectrum decreases as the wavenumber increases. In the
absence of twist, the values of the amplitude spectrum are larger than those
obtained when twist is considered. In addition, we find a decreasing trend
with respect to the twist parameter, $c$, so that the stronger the magnetic
twist, the lower the values of the amplitude spectrum. These results just
confirm again the relevant role of magnetic twist in suppressing the
generation of small scales and in the rate at which Alfvénic energy is
deposited into those small scales. We intentionally avoided $kR>10^{3}$ in our
analysis owing to the possible role of numerical dissipation at the smallest
scales.

##  5 Concluding remarks

We conclude the investigation started in Paper I about the ability of
torsional Alfvén waves to drive turbulence in coronal loops. Because of plasma
and/or magnetic nonuniformity, standing torsional Alfvén waves undergo the
process of phase mixing, generating shear flows perpendicular to the magnetic
field direction in adjacent radial positions as time increases. Eventually,
the shear flows trigger the KHi, as Heyvaerts & Priest (1983) and Browning &
Priest (1984) first predicted.

In the absence of magnetic twist (the case studied in Paper I), the KHi can
grow nonlinearly without opposition. The nonlinear evolution of the KHi
naturally induces turbulence. First, large KHi eddies are formed, which later
break into smaller eddies in a cascade-like dynamics. Mixing of plasma takes
place. An important increase in vorticity is found and the generation of small
scales speeds up compared with the initial phase dominated by phase mixing
alone. Turbulence evolves perpendicularly to the magnetic field, so that for a
straight magnetic field turbulence is pseudo-2D.

In the present paper, we have explored the role of magnetic twist. While the
linear evolution of phase mixing is similar in the presence or in the absence
of twist, the dynamics of the subsequent KHi growth and turbulence generation
depends strongly on the strength of the twist.

If magnetic twist is sufficiently weak, the onset and growth of the KHi is
just delayed, but the dynamics is similar to that in the case of a straight
magnetic field. In this regime, the stronger the magnetic twist, the longer
the delay. Although the vorticity still increases dramatically during the
development of turbulence, the increase is smaller compared with that in
absence of magnetic twist. Small scales are still quickly generated by
turbulence, although at a slower pace than when twist is absent. Turbulence
still evolves perpendicularly to the magnetic field lines, but since the field
is twisted, turbulence is no longer confined to perpendicular planes to the
tube axis.

Conversely, under the presence of a strong enough magnetic twist, the scenario
is completely different. If the strength of magnetic twist surpasses a
critical value, or a critical twist angle, the KHi vortices cannot grow. The
KHi itself is still locally excited by the phase-mixing generated shear flows,
but the tension of the twisted magnetic field prevents now the nonlinear
development of the instability (see, e.g., Galinsky & Sonnerup, 1994; Ryu et
al., 2000; Hillier, 2019; Díaz-Suárez & Soler, 2021). As a consequence, there
is no enhancement of vorticity, neither turbulence is driven. Thus, the
generation of small scales is not accelerated and continues slowly evolving at
the rate dictated by phase-mixing.

Although the effect of magnetic twist may oppose the process, the nonlinear
evolution of torsional Alfvén waves remains as a viable mechanism to induce
turbulence as long as coronal loops are weakly twisted. There are other
effects that should be explored in the future. For instance, the tension of
the background field in curved coronal loops may also affect the triggering
and evolution of the KHi and associated turbulence. It could be also
interesting to investigate turbulence generation in multithreaded loops (see,
e.g., Ofman, 2009).

###### Acknowledgements.

This publication is part of the R+D+i project PID2020-112791GB-I00, financed
by MCIN/AEI/10.13039/501100011033. S.D. acknowledges the financial support
from MCIN/AEI/10.13039/501100011033 and European Social Funds for the
predoctoral FPI fellowship PRE2018-084223. We also thank the UIB for the use
of the Foner cluster. For the simulation data analysis we have used VisIT
(Childs et al., 2012) and Python 3.6. In particular, we have used Matplotlib
(Hunter, 2007), Scipy (Virtanen et al., 2020), and Numpy (Harris et al.,
2020). We are thankful to B. Vaidya and his contributors for the tool pyPLUTO.

## References

  * Antolin et al. (2015) Antolin, P., Okamoto, T. J., De Pontieu, B., et al. 2015, ApJ, 809, 72
  * Antolin & Van Doorsselaere (2019) Antolin, P. & Van Doorsselaere, T. 2019, Frontiers in Physics, 7, 85
  * Antolin et al. (2014) Antolin, P., Yokoyama, T., & Van Doorsselaere, T. 2014, ApJ, 787, L22
  * Aschwanden (2013) Aschwanden, M. J. 2013, Sol. Phys., 287, 369
  * Aschwanden (2019) Aschwanden, M. J. 2019, ApJ, 874, 131
  * Aschwanden et al. (1999) Aschwanden, M. J., Fletcher, L., Schrijver, C. J., & Alexander, D. 1999, ApJ, 520, 880
  * Aschwanden et al. (2016) Aschwanden, M. J., Reardon, K., & Jess, D. B. 2016, ApJ, 826, 61
  * Aschwanden & Wang (2020) Aschwanden, M. J. & Wang, T. 2020, ApJ, 891, 99
  * Aschwanden et al. (2012) Aschwanden, M. J., Wuelser, J.-P., Nitta, N. V., et al. 2012, ApJ, 756, 124
  * Aschwanden et al. (2014) Aschwanden, M. J., Xu, Y., & Jing, J. 2014, ApJ, 797, 50
  * Barbulescu et al. (2019) Barbulescu, M., Ruderman, M. S., Van Doorsselaere, T., & Erdélyi, R. 2019, ApJ, 870, 108
  * Bennett et al. (1999) Bennett, K., Roberts, B., & Narain, U. 1999, Sol. Phys., 185, 41
  * Browning & Priest (1984) Browning, P. K. & Priest, E. R. 1984, A&A, 131, 283
  * Cargill & de Moortel (2011) Cargill, P. & de Moortel, I. 2011, Nature, 475, 463
  * Chandrasekhar (1961) Chandrasekhar, S. 1961, Hydrodynamic and hydromagnetic stability
  * Charbonneau & MacGregor (1995) Charbonneau, P. & MacGregor, K. B. 1995, ApJ, 454, 901
  * Childs et al. (2012) Childs, H., Brugger, E., Whitlock, B., et al. 2012, in High Performance Visualization–Enabling Extreme-Scale Scientific Insight, 357–372
  * Cooley & Tukey (1965) Cooley, J. W. & Tukey, J. W. 1965, Mathematics of computation, 19, 297
  * Cranmer (2009) Cranmer, S. R. 2009, Living Reviews in Solar Physics, 6, 3
  * Cranmer & van Ballegooijen (2005) Cranmer, S. R. & van Ballegooijen, A. A. 2005, ApJS, 156, 265
  * De Moortel et al. (2000) De Moortel, I., Hood, A. W., & Arber, T. D. 2000, A&A, 354, 334
  * De Moortel et al. (2014) De Moortel, I., McIntosh, S. W., Threlfall, J., Bethge, C., & Liu, J. 2014, ApJ, 782, L34
  * De Pontieu et al. (2012) De Pontieu, B., Carlsson, M., Rouppe van der Voort, L. H. M., et al. 2012, ApJ, 752, L12
  * De Pontieu et al. (2007) De Pontieu, B., McIntosh, S. W., Carlsson, M., et al. 2007, Science, 318, 1574
  * De Pontieu et al. (2014) De Pontieu, B., Rouppe van der Voort, L., McIntosh, S. W., et al. 2014, Science, 346, 1255732
  * Dedner et al. (2002) Dedner, A., Kemm, F., Kröner, D., et al. 2002, Journal of Computational Physics, 175, 645
  * Díaz-Suárez & Soler (2021) Díaz-Suárez, S. & Soler, R. 2021, ApJL, 922, L26
  * Díaz-Suárez & Soler (2021) Díaz-Suárez, S. & Soler, R. 2021, A&A, 648, A22
  * Ebrahimi et al. (2017) Ebrahimi, Z., Karami, K., & Soler, R. 2017, ApJ, 845, 86
  * Ebrahimi & Soler (2022) Ebrahimi, Z. & Soler, R. 2022, MNRAS, 511, 3477
  * Ebrahimi et al. (2020) Ebrahimi, Z., Soler, R., & Karami, K. 2020, ApJ, 893, 157
  * Edwin & Roberts (1983) Edwin, P. M. & Roberts, B. 1983, Sol. Phys., 88, 179
  * Erdélyi & Fedun (2006) Erdélyi, R. & Fedun, V. 2006, Sol. Phys., 238, 41
  * Erdélyi & Fedun (2007) Erdélyi, R. & Fedun, V. 2007, Sol. Phys., 246, 101
  * Erdélyi & Fedun (2010) Erdélyi, R. & Fedun, V. 2010, Sol. Phys., 263, 63
  * Fedun et al. (2011) Fedun, V., Shelyag, S., Verth, G., Mathioudakis, M., & Erdélyi, R. 2011, Annales Geophysicae, 29, 1029
  * Galinsky & Sonnerup (1994) Galinsky, V. L. & Sonnerup, B. U. Ö. 1994, Geochim. Res. Lett., 21, 2247
  * Giagkiozis et al. (2015) Giagkiozis, I., Fedun, V., Erdélyi, R., & Verth, G. 2015, ApJ, 810, 53
  * Goossens et al. (2011) Goossens, M., Erdélyi, R., & Ruderman, M. S. 2011, Space Sci. Rev., 158, 289
  * Goossens et al. (1992) Goossens, M., Hollweg, J. V., & Sakurai, T. 1992, Sol. Phys., 138, 233
  * Guo et al. (2019) Guo, M., Van Doorsselaere, T., Karampelas, K., et al. 2019, ApJ, 870, 55
  * Hahn & Savin (2014) Hahn, M. & Savin, D. W. 2014, ApJ, 795, 111
  * Halberstadt & Goedbloed (1993) Halberstadt, G. & Goedbloed, J. P. 1993, A&A, 280, 647
  * Handy et al. (1999) Handy, B. N., Acton, L. W., Kankelborg, C. C., et al. 1999, Sol. Phys., 187, 229
  * Harris et al. (2020) Harris, C. R., Millman, K. J., van der Walt, S. J., et al. 2020, Nature, 585, 357
  * Heyvaerts & Priest (1983) Heyvaerts, J. & Priest, E. R. 1983, A&A, 117, 220
  * Hillier (2019) Hillier, A. 2019, PhPl, 26, 16
  * Hillier et al. (2019) Hillier, A., Barker, A., Arregui, I., & Latter, H. 2019, MNRAS, 482, 1143
  * Hillier et al. (2020) Hillier, A., Van Doorsselaere, T., & Karampelas, K. 2020, ApJ, 897, L13
  * Hollweg (1971) Hollweg, J. V. 1971, J. Geophys. Res., 76, 5155
  * Hollweg (1978) Hollweg, J. V. 1978, Sol. Phys., 56, 305
  * Hollweg (1984a) Hollweg, J. V. 1984a, Sol. Phys., 91, 269
  * Hollweg (1984b) Hollweg, J. V. 1984b, ApJ, 277, 392
  * Hood & Priest (1979) Hood, A. W. & Priest, E. R. 1979, Sol. Phys., 64, 303
  * Hood & Priest (1981) Hood, A. W. & Priest, E. R. 1981, Geophysical and Astrophysical Fluid Dynamics, 17, 297
  * Howson et al. (2017a) Howson, T. A., De Moortel, I., & Antolin, P. 2017a, A&A, 607, A77
  * Howson et al. (2017b) Howson, T. A., De Moortel, I., & Antolin, P. 2017b, A&A, 602, A74
  * Howson et al. (2021) Howson, T. A., De Moortel, I., & Pontin, D. I. 2021, A&A, 656, A112
  * Howson et al. (2020) Howson, T. A., De Moortel, I., & Reid, J. 2020, A&A, 636, A40
  * Hunter (2007) Hunter, J. D. 2007, Computing in Science & Engineering, 9, 90
  * Jafarzadeh et al. (2017) Jafarzadeh, S., Solanki, S. K., Gafeira, R., et al. 2017, ApJS, 229, 9
  * Jess et al. (2009) Jess, D. B., Mathioudakis, M., Erdélyi, R., et al. 2009, Science, 323, 1582
  * Karami & Bahari (2010) Karami, K. & Bahari, K. 2010, Sol. Phys., 263, 87
  * Karami & Barin (2009) Karami, K. & Barin, M. 2009, MNRAS, 394, 521
  * Karampelas & Van Doorsselaere (2018) Karampelas, K. & Van Doorsselaere, T. 2018, A&A, 610, L9
  * Karampelas et al. (2017) Karampelas, K., Van Doorsselaere, T., & Antolin, P. 2017, A&A, 604, A130
  * Karampelas et al. (2019) Karampelas, K., Van Doorsselaere, T., & Guo, M. 2019, A&A, 623, A53
  * Kohutova et al. (2020) Kohutova, P., Verwichte, E., & Froment, C. 2020, A&A, 633, L6
  * Kwon & Chae (2008) Kwon, R. Y. & Chae, J. 2008, ApJ, 677, L141
  * Liu et al. (2014) Liu, J., McIntosh, S. W., De Moortel, I., Threlfall, J., & Bethge, C. 2014, ApJ, 797, 7
  * Lohner (1987) Lohner, R. 1987, Computer Methods in Applied Mechanics and Engineering, 61, 323
  * Magyar et al. (2022) Magyar, N., Duckenfield, T., Van Doorsselaere, T., & Nakariakov, V. M. 2022, A&A, 659, A73
  * Magyar & Van Doorsselaere (2016) Magyar, N. & Van Doorsselaere, T. 2016, A&A, 595, A81
  * Martínez-Gómez et al. (2022) Martínez-Gómez, D., Soler, R., Terradas, J., & Khomenko, E. 2022, A&A, 658, A106
  * Mathioudakis et al. (2013) Mathioudakis, M., Jess, D. B., & Erdélyi, R. 2013, Space Sci. Rev., 175, 1
  * Matsumoto & Suzuki (2012) Matsumoto, T. & Suzuki, T. K. 2012, ApJ, 749, 8
  * McIntosh et al. (2011) McIntosh, S. W., de Pontieu, B., Carlsson, M., et al. 2011, Nature, 475, 477
  * Mignone et al. (2007) Mignone, A., Bodo, G., Massaglia, S., et al. 2007, ApJS, 170, 228
  * Mignone et al. (2012) Mignone, A., Zanni, C., Tzeferacos, P., et al. 2012, ApJS, 198, 7
  * Morton et al. (2015) Morton, R. J., Tomczyk, S., & Pinto, R. 2015, Nature Communications, 6, 7813
  * Mumford et al. (2015) Mumford, S. J., Fedun, V., & Erdélyi, R. 2015, ApJ, 799, 6
  * Nakariakov & Kolotkov (2020) Nakariakov, V. M. & Kolotkov, D. Y. 2020, ARA&A, 58, 441
  * Nakariakov et al. (1999) Nakariakov, V. M., Ofman, L., Deluca, E. E., Roberts, B., & Davila, J. M. 1999, Science, 285, 862
  * Nocera et al. (1984) Nocera, L., Priest, E. R., & Leroy, B. 1984, A&A, 133, 387
  * Ofman (2009) Ofman, L. 2009, ApJ, 694, 502
  * Pascoe et al. (2020) Pascoe, D. J., Goddard, C. R., & Van Doorsselaere, T. 2020, Frontiers in Astronomy and Space Sciences, 7, 61
  * Prokopyszyn et al. (2019) Prokopyszyn, A. P. K., Hood, A. W., & De Moortel, I. 2019, A&A, 624, A90
  * Raghav & Kule (2018) Raghav, A. N. & Kule, A. 2018, MNRAS, 476, L6
  * Rankin et al. (1994) Rankin, R., Frycz, P., Tikhonchuk, V. T., & Samson, J. C. 1994, J. Geophys. Res., 99, 21291
  * Roberts (2019) Roberts, B. 2019, MHD Waves in the Solar Atmosphere (Cambridge University Press)
  * Roe (1981) Roe, P. L. 1981, Journal of Computational Physics, 43, 357
  * Ruderman (2007) Ruderman, M. S. 2007, Sol. Phys., 246, 119
  * Ryu et al. (2000) Ryu, D., Jones, T. W., & Frank, A. 2000, ApJ, 545, 475
  * Sakurai et al. (1991) Sakurai, T., Goossens, M., & Hollweg, J. V. 1991, Sol. Phys., 133, 227
  * Shelyag et al. (2011) Shelyag, S., Fedun, V., Keenan, F. P., Erdélyi, R., & Mathioudakis, M. 2011, Annales Geophysicae, 29, 883
  * Shelyag et al. (2012) Shelyag, S., Mathioudakis, M., & Keenan, F. P. 2012, ApJ, 753, L22
  * Shoda et al. (2018) Shoda, M., Yokoyama, T., & Suzuki, T. K. 2018, ApJ, 853, 190
  * Smith et al. (2007) Smith, P. D., Tsiklauri, D., & Ruderman, M. S. 2007, A&A, 475, 1111
  * Soler et al. (2019) Soler, R., Terradas, J., Oliver, R., & Ballester, J. L. 2019, ApJ, 871, 3
  * Soler et al. (2021) Soler, R., Terradas, J., Oliver, R., & Ballester, J. L. 2021, ApJ, 909, 190
  * Soler et al. (2010) Soler, R., Terradas, J., Oliver, R., Ballester, J. L., & Goossens, M. 2010, ApJ, 712, 875
  * Srivastava & Dwivedi (2017) Srivastava, A. K. & Dwivedi, B. N. 2017, Journal of Astrophysics and Astronomy, 38, 61
  * Srivastava et al. (2017) Srivastava, A. K., Shetye, J., Murawski, K., et al. 2017, Scientific Reports, 7, 43147
  * Terradas et al. (2008) Terradas, J., Andries, J., Goossens, M., et al. 2008, ApJ, 687, L115
  * Terradas & Goossens (2012) Terradas, J. & Goossens, M. 2012, A&A, 548, A112
  * Terradas et al. (2018) Terradas, J., Magyar, N., & Van Doorsselaere, T. 2018, ApJ, 853, 35
  * Terradas & Ofman (2004) Terradas, J. & Ofman, L. 2004, ApJ, 610, 523
  * Thalmann et al. (2014) Thalmann, J. K., Tiwari, S. K., & Wiegelmann, T. 2014, ApJ, 780, 102
  * Tikhonchuk et al. (1995) Tikhonchuk, V. T., Rankin, R., Frycz, P., & Samson, J. C. 1995, Physics of Plasmas, 2, 501
  * Tomczyk et al. (2007) Tomczyk, S., McIntosh, S. W., Keil, S. L., et al. 2007, Science, 317, 1192
  * Van Doorsselaere et al. (2020) Van Doorsselaere, T., Srivastava, A. K., Antolin, P., et al. 2020, Space Sci. Rev., 216, 140
  * Virtanen et al. (2020) Virtanen, P., Gommers, R., Oliphant, T. E., et al. 2020, Nature Methods, 17, 261
  * Wedemeyer-Böhm et al. (2012) Wedemeyer-Böhm, S., Scullion, E., Steiner, O., et al. 2012, Nature, 486, 505
  * Xie et al. (2017) Xie, H., Madjarska, M. S., Li, B., et al. 2017, ApJ, 842, 38
  * Zaqarashvili et al. (2015) Zaqarashvili, T. V., Zhelyazkov, I., & Ofman, L. 2015, ApJ, 813, 123


# A compact cryogenic configurable slit unit for a multi-object infrared
spectrograph:
Design and Development of a prototype at TIFR

P. P. Madhwani  A. P. K. Kutty  B. Mookerjeaa J. V. Parmar  V. N. Kurhade  S.
L. D’Costa  P. Manoj  A. Surya  Department of Astronomy & Astrophysics, Tata
Institute of Fundamental Research,
Homi Bhabha Road, Mumbai 400005, India

((to be inserted by publisher); (to be inserted by publisher); (to be inserted
by publisher))

###### Abstract

We present a cryogenic configurable slit unit (CSU) for a multi object
infrared spectrograph with an effective field of view of
9$.\mkern-4.0mu^{\prime}$1$\times$9$.\mkern-4.0mu^{\prime}$1 that was
completely conceived and designed in the laboratory at TIFR. Several
components of the CSU including the controller for the commercially procured
piezo-walkers, controlled loop position sensing mechanism using digital slide
callipers and a cryogenic test facility for the assembled prototype were also
developed in-house. The principle of the CSU involves division of the field of
view of the spectrometer into contiguous and parallel spatial bands, each one
associated with two opposite sliding metal bars that can be positioned to
create a slit needed to make spectroscopic observations of one astronomical
object. A three-slit prototype of the newly designed CSU was built and tested
extensively at ambient and cryogenic temperatures. The performance of the CSU
was found to be as per specifications.

###### keywords:

Infrared; Multi-Object Spectrograph; Configurable Slit Unit; Piezowalker-based
positioning

aCorresponding author: B. Mookerjea<EMAIL_ADDRESS>
; ; ;

##  1 Introduction

Near-infrared (NIR) spectroscopy is particularly well-suited for observation
of vibrational and rotational molecular lines that are indicative of the
chemo-dynamical structure of stars of the Milky Way and can be used to study
stellar radial velocities and abundances [Wallace & Hinkle, 1997]. The NIR
spectra of young stellar objects and stars provide important constraints on
the accretion of material, the structure of protostellar disks that eventually
form planets [Scoville et al., 1983, Alcalá et al., 2014]. Spectroscopic
observations in the NIR can be used to access the high redshift galaxies whose
important rest frame optical diagnostic emission lines are redshifted to the
NIR [Petitjean, 2011].

Multi Object Spectroscopy (MOS)is an efficient way to observe the spectra of
an aggregate of sources that are within the field of view of an instrument at
a single pointing. With the advent of technologies enabling fabrication of
large format detector arrays, the demand for capabilities to perform
simultaneous spectroscopy of multiple objects within the fields of view (FOV)
covering several square arcminutes up to several square degrees of the large
telescopes has been on the rise [Ellis & Parry, 1988, Allington-Smith, 2006].

In the instruments capable of MOS, the light from the source is focused by the
telescope onto the focal plane, where narrow slits or fiber optics are used to
isolate the source light and is finally dispersed using a spectrograph with a
dispersing element like a prism or a diffraction grating. The spectra are then
imaged onto a detector. The data undergo detector level processing and
extraction, as well as wavelength and flux calibration, typically using
software designed specifically to handle the multiplexed spectra. Presently,
most large 8–10 meter class ground-based observatories offer at least one MOS
instrument option. In ground-based instruments, apertures are either created
using slit masks or using fibers arranged into patterns on the sources in the
field. For optical telescopes the field-specific slit masks are milled out on-
site prior to the observations. Such pre-fabricated slit masks are
inconvenient in the near-infrared (1–2 $\mu$m) where there is a strong
contribution from background thermal emission that necessitate cooling the
slits to temperatures of 120 K or below along with the entire spectrograph. A
plausible solution for such instruments is the use of a configurable slit unit
(CSU) to precisely position the slit mask that can be configured in-situ in
the cryogenically cooled environment shortly before observations begin.
Configurable slit spectrographs have a number of benefits over slit mask-based
spectrographs, that include flexibility, efficiency (easy to customize the
slit width based on the properties of the target) and reduced overhead (in-
situ configuration in cryogenic environment). Such configurable slit units are
currently installed on the MOSFIRE instrument on the Keck telescope [McLean et
al., 2010] and also on Espectrógrafo Multiobjeto Infra-Rojo (EMIR) instrument
on the Gran Telescopio de Canarias (GTC). MOSFIRE can observe up to 46 objects
in a 6$.\mkern-4.0mu^{\prime}$12$\times$6$.\mkern-4.0mu^{\prime}$12 FOV and
EMIR on GTC can observe up to 53 objects in a FOV of
6$.\mkern-4.0mu^{\prime}$64$\times$4′ [Garzón et al., 2022]. The physical size
of the EMIR FOV in the focal plane is 307 mm$\times 307$ mm and for the
MOSFIRE the FOV is 267 mm$\times 267$ mm.

We present here the design of a configurable slit unit operational at
cryogenic temperatures (120 K) meant for a Multi Object Infrared Spectrograph
(MOIS) [Surya et al., 2023] operational between 0.8–2.5 $\mu$m, to be
installed on an Indian optical/near-infrared observing facility. The present
prototype for the CSU has 5 configurable slits designed as per the optics of
the 3.6 m Devasthal Optical Telescope (DOT). However the modular and scalable
nature of the design will enable installation of the CSU on future 10 m class
Indian telescopes that are being planned. The entire design has been developed
in-house and a prototype consisting of three slits has already been fabricated
and tested at cryogenic temperatures.

##  2 Design of the CSU

The design goals for the CSU was to create 5 slits using 10 bars each of which
can have a maximum displacement of up to 100 mm for a field of view of 85.77
mm$\times 85.77$ mm. The position of the slit in the FOV and its width are
decided upon based on the location and size of the astronomical source. The
CSU is required to generate a multi-slit configuration, a long slit, or an
imaging aperture while being at cryogenic temperatures, necessary for its
operation at the infrared wavelengths.

Figure 1: Single slit pair assembly along with the drive electronics. The
functional components of the bar assembly are labeled.

Each slit bar assembly for the CSU consists of i) Vernier scale, ii) Piezo-
walker,iii) Piezo-walker mounting base, iv) Titanium link rods, v) Slit bar
and Slit edge and the vi) Controller electronics (Figure 1). In the following
text we present details of the crucial components of this assembly.

###  2.1 Piezo-walker based drive mechanism

At the operating wavelengths of the MOIS it is necessary to suppress the
background emission from the components of the spectrometer, hence the slit
bars need to be in vacuum and in a cryogenic environment. This requirement of
low temperatures impose major constraints on the drive mechanisms that can be
used for the motion of the slit bars. Additionally, it is preferable that the
power dissipation from the bars and the driving motors is negligible when the
bars are not in motion. Based on these considerations we have identified
piezo-walkers as the drivers for the slit bars, which further have the
advantage of negligible backlash error as compared to regular rotary stepper
motors that are also used for such purposes. The PI NEXACT-310 piezo-walkers
identified for this purpose provide a travel range 10 to 125 mm and are
capable of generating a force of up to 10 N. The piezo-walker drives a ceramic
runner which was mechanically interfaced with the slit bar through a titanium
link rod and the motion is controlled by a controller designed in-house. The
NEXACT-310 piezo-walkers are functional only down to a temperature of 273 K,
hence the piezo-walkers were isolated from thermal radiation using enclosures
made out of space-grade multi-layer insulating (MLI) sheet. Only the slit bars
were cooled using copper braids connected to a cold plate at liquid N2
temperature of 77 K (Fig.1).

###  2.2 Closed-loop Vernier-scale based accurate Positioning of the Slit Bars

Accurate positioning of the slit bars was achieved using Grid-Capacitance
Linear Synchronous Sensors which consist of a set of grid-capacitances and
signal processing circuit similar to those used in commercially available
digital vernier calipers. The position information was tapped from the readout
to enable closed-loop correction to achieve an absolute positioning accuracy
of 10 $\mu$m for each slit bar. The positioning mechanism was also designed
and developed in-house. The slit length is fixed (17.15 mm) and the width can
range from 0 (fully shut) to 90 mm (fully open). The slit positioning accuracy
is $\pm 10$ $\mu$m and the slit width accuracy is $\pm 14$ $\mu$m. The design
goal is a slit width of 120 $\mu$m. The motion of the slit-bars is restricted
within the desired range by using limit switches, which when depressed stop
the movement in a particular direction and subsequently only motion in the
reverse direction is possible. The positioning accuracy of the CSU was tested
by shinning light through the assembly and the slits of different widths that
were formed at different positions of the FOV were imaged using a high
resolution camera fitted to a microscope. The attained positions and widths of
the slits were accurate to within 15 $\mu$m and were found to be consistently
reproducible.

###  2.3 Mechanical Design of the Slit Bars

Figure 2: Cross-section of the single slit bar designed to allow smooth motion
without light leak. The copper braids used to cool the slit bars are also
shown.

The mechanical design of the slit bars was challenging since the movement of
the slit bars had to be smooth while ensuring that there is no light leak
through them. In order to meet both design goals, a special design for the
cross-section of the slit bars was devised (Fig. 2) which ensured complete
overlap of the bars at their edges as well as free movement. Each slit bar
consists of two parts the main body or the bar and the edge. The bar, made of
stainless steel has poor thermal conductivity. The edge of the slit bar is
made from a hard aluminium alloy plate with a thickness of 800 $\mu$m attached
to an overlapping machined edge having a thickness of 300 $\mu$m. The choice
of aluminium alloy for the slit edge is to ensure efficient cooling to
cryogenic temperatures using copper braids connected to the cold plate of the
liquid N2 cylinder. The titanium link rods which were machined as per the
precision requirement play an important role in the assembly because these
isolate the slit bars from the rest of the assembly thus minimizing the
thermal load on the copper braids. The titanium rod in the slit bar assembly
works as a link between the Vernier scale, the piezo-walker and the slit bar.

####  2.3.1 Controller Electronics

The proto type CSU model developed comprises of 3 pairs of slit bar assemblies
and each slit bar assembly has a left arm and a right arm. The runner of the
piezo walker is coupled to the slit bar as well as a electronic Vernier scale
which provides the position information with an accuracy of 10 $\mu$m. Each of
the six piezo walkers is driven individually by a drive controller consisting
of closed loop circuits developed in-house. The working principle of the
control circuit can be understood from the functional block diagram of 1 slit
assembly (Fig. 3). In the following although the description is for the three
slit assembly for which the mechanical parts have been fabricated, the
electronics has already been designed and built and tested for the five-slit
final CSU configuration.

Figure 3: Schematic to show the working principle of the functional blocks of
the controller electronics for the piezowalkers.

The piezo walker has four legs which when excited with a 50 V trapezoidal wave
in a particular sequence produces a linear motion in the runner. The
controller circuit is built around a microcontroller which generates the
square waves of desired frequency, which are then converted to sine waves by
passing through low pass filters. The signals are further amplified, processed
and level shifted to achieve 4 trapezoidal waveforms of 50 V as required to
drive the four legs of the piezo walker. The functional block represents a
single slit assembly. Two more such assemblies together complete the three
slit assembly. These three slit assemblies termed as slaves are controlled
through a master which is commanded through a user interface.

Operationally, when powered up all the 6 slit bars are moved to the retracted
position so that the FOV of the detector is exposed and when fully open, the
limit switch is depressed to set the Vernier readings to a reference value. At
the engineering model level, using the user interface it is possible to
provide the location and size of all 3 slits, by providing 6 parameters i.e
the 3 positions corresponding to the location of the left arm of each slit and
3 slit widths. The parameters thus supplied are used by the master controller
to compute the value of distances to be traversed by the 6 arms of the CSU.
The computed values are then sent to the corresponding slaves, which enable
the movement of the slit bars by the piezos till the desired position and
width of all slits are achieved. The configuration of slits on the FOV so
achieved remain unchanged till a new set of parameters is transmitted by the
user to the master controller. The master controller stores the current
location of the slits, so that when a new set of parameters is entered it
calculates the range of motion required for each slit bar relative to that
position. The controller is programmed to enable motion of the slit bars with
a coarse and a fine step depending on the distance between the actual and the
desired positions of the bars. The controller is also designed and programmed
to avoid accidental collisions of the slit bars.

Figure 4: Completed CSU prototype creating 3-slits at the FOV in the
laboratory. The slit bar assembly on the top will be inside the cryogenic
dewar while the drive and monitoring electronics will be outside.

Figure 4 shows the 3-slit CSU prototype being tested in the laboratory. In the
final configuration the drive electronics consisting of ten cards will be kept
outside of the cryostat and will be used to control the movement of the slits,
monitor the positions and temperatures of the different parts of the CSU
through hermetically sealed connectors.

##  3 Cryogenic Functionality Testing of the CSU in the laboratory

Figure 5: (Left) The CSU connected to the cold plate (at the bottom of the
liquid N2 tank) inside the dewar prior to the cryogenic testing of the unit in
the laboratory. The shinning surface around the cold plate is a radiation
shield created to ensure efficient cooling by reflecting the heat due to the
uncooled parts of the dewar. (Right) Typical temperature profiles of the slit
bar, the body of the piezo-walker and the coldplate measured during one of the
cryogenic tests in the laboratory.

A dedicated aluminium dewar (supporting a vacuum of up to 10-6 mbar) was
designed and fabricated at TIFR for the cryogenic testing of the CSU. The
dewar consists of a top plate with a flange with four hermetically sealed
connectors each with 19 pins as well as an inlet for the 3 litre liquid N2
tank inside. The inlet pipe to the liquid nitrogen tank was fitted with a
solenoid valve based system for automated round-the-clock filling (without
spillage) of N2 that was developed in-house. The base of the slit bar assembly
was mounted on the bottom plate of the dewar with teflon rings to isolate the
two plates thermally. For ease of mounting the dewar was suspended upside-down
and the base plate with the CSU attached to it was attached to the dewar (Fig.
5). Multiple tests of the performance of the cooling scheme as well as copper
braids of different sizes (and hence thermal efficiencies) were carried out in
the laboratory using this set-up. No major issues with the motion of the CSU
in the final design presented here was noticed during the cryogenic tests. We
present the results of one of the tests which was continued uninterrupted for
over 72 hours and the motion of the slit bars measured at regular intervals
was found to be as per specifications for the entire duration of the test.
Figure 5 shows the results of the temperature profiles of the slit, the piezo-
walker and the coldplate measured using thermistors affixed at different parts
of the CSU. In order to maintain the piezo-walkers at a temperature above 273
K it was locally heated in an automated manner. The temperatures measured in
the rest of the CSU do not reflect any changes due to this heating, which
suggests that the MLI enclosure of the piezo-walkers may be adequate. A more
detailed analysis of the thermal background is currently in progress.

##  4 Summary and Future Directions

We presented here the new design of a configurable slit unit for a multi-
object infrared spectromgraph. The current technical specifications are based
on the optics of the DOT and for a five-slit unit. However, the modular nature
of the CSU design makes it easily scalable for much larger FOV spectrometers
meant for bigger (10 m or larger) telescopes and larger number of slits. The
mechanical and electronic design of the CSU presented here were developed
completely indigenously. Except for the commercially procured piezo-walkers
all mechanical and electronic parts were fabricated/built in-house for the
three-slit prototype of the CSU presented here. Based on a series of tests
performed in the laboratory we conclude that CSU performs as per specification
both at room and cryogenic temperatures. Keeping in mind the constraint
imposed by the piezo-walkers being non-functional at temperatures below 273 K,
we are working on a design using cryogenic piezo-drives. A definite conclusion
on whether a cryogenic piezo-drive is really needed will become clearer only
after the opto-mechanical design of the spectrometer including the thermal
modelling is completed. In this context we point out that the commercially
available cryo piezodrives typically have travel ranges of only up to 50 mm,
whereas for our design a movement range up to 100 mm is needed. Thus specially
customized commercially available piezo-drives are currently being explored.

## Acknowledgments

The project was entirely funded by Department of Atomic Energy, Government of
India, under Project Identification No. RTI 4002. The authors acknowledge the
contributions of G. S. Meshram towards the initial design of the slit bar
edges and also the support and help received from H. Shah, R. Jadhav, S.
Poojary, S. B. Bhagat and S. Gharat of the IR Astronomy group, TIFR.

## References

  * Alcalá et al. [2014] Alcalá, J. M., Natta, A., Manara, C. F., et al. 2014, A&A, 561, A2. doi:10.1051/0004-6361/201322254
  * Allington-Smith [2006] Allington-Smith, J. 2006, New Astronomy Reviews, 50, 244. doi:10.1016/j.newar.2006.02.024
  * Ellis & Parry [1988] Ellis, R. S. & Parry, I. R. 1988, Instrumentation for Ground-Based Optical Astronomy, 192
  * Garzón et al. [2022] Garzón, F., Balcells, M., Gallego, J., et al. 2022, A&A, 667, A107. doi:10.1051/0004-6361/202244729
  * McLean et al. [2010] McLean, I. S., Steidel, C. C., Epps, H., et al. 2010, Proceedings of the SPIE, 7735, 77351E. doi:10.1117/12.856715
  * Petitjean [2011] Petitjean, P. 2011, Astronomische Nachrichten, 332, 309. doi:10.1002/asna.201111544
  * Scoville et al. [1983] Scoville, N., Kleinmann, S. G., Hall, D. N. B., et al. 1983, ApJ, 275, 201. doi:10.1086/161526
  * Surya et al. [2023] Surya, A., Manoj, P., Mookerjea, B., accepted for Proc. SPIE ”Astronomical Optics: Design, Manufacture, and Test of Space and Ground Systems IV”, August 2023.
  * Wallace & Hinkle [1997] Wallace, L. & Hinkle, K. 1997, ApJS, 111, 445. doi:10.1086/313020