[1]Grzegorz Kowal These authors contributed equally to this work.

These authors contributed equally to this work.

[1]Escola de Artes, Ciências e Humanidades, University of São Paulo, Av.
Arlindo Béttio, 1000 - Vila Guaraciaba, São Paulo, 03828-000, SP, Brazil

# Quenching of Tearing Mode Instability by Transverse Magnetic Fields in
Reconnection Current Sheets

<EMAIL_ADDRESS>Diego A. Falceta-Gonçalves<EMAIL_ADDRESS>*

###### Abstract

The tearing mode instability is a key process for magnetic energy conversion
in magnetohydrodynamics, once anti-parallel components are allowed to
reconnect, leading to the formation of magnetic islands. It has been employed
to explain phenomena at different scales in nature, from galactic nuclei, to
solar flares and laboratory fusion devices. In this study, we investigate the
dynamics of a current sheet in the presence of a transverse magnetic field
component, in the framework of viscoresistive, incompressible
magnetohydrodynamics (MHD), both analytically and by means of direct numerical
simulations. Firstly, we obtain analytical solution for the time-varying one-
dimensional profile of an initial Harris current sheet in the presence of a
transverse field. We find that the introduction of a transverse magnetic field
disrupts the system’s equilibrium, leading to the natural development of a
neutral layer with shear flows within the current sheet, one along the
antiparallel magnetic component and another along the guide field direction.
Secondly, through numerical analysis, we examine the dispersion relation of
the incompressible MHD equations in the context of a modified equilibrium
profile due to the transverse field. Our findings indicate a rapid suppression
of unstable modes of tearing instability with the width of the neutral layer,
confirming the analytical predictions. These results offer new insightful
understanding on the interplay between transverse magnetic fields, shear
flows, and tearing mode instabilities in current sheet environments.

###### keywords:

magnetohydrodynamics, plasma instabilities, tearing mode, magnetic
reconnection, numerical methods

Tearing instability is a fundamental process in magnetohydrodynamics (MHD)
that plays a critical role in the dynamics of astrophysical plasmas. First
analyzed in detail by Furth, Killeen, and Rosenbluth (1963) [1], the tearing
is a type of resistive instability that disrupts current sheets in plasmas,
leading to faster magnetic reconnection as compared to standard Sweet-Parker
model [2, 3], a process essential for converting magnetic energy into kinetic
and thermal energy, as well as accelerating particles, which is significant in
various astrophysical environments.

In astrophysical contexts, magnetic reconnection is crucial for understanding
phenomena such as solar flares, magnetic storms, and the heating of the solar
corona [4]. These events are driven by the release of stored magnetic energy
through reconnection, which is supposedly facilitated by the tearing
instability. This instability enables the breaking and reconnection of
magnetic field lines, leading to a rapid reconfiguration of the magnetic field
topology and the associated energy release [5].

Recent research on the interplay between turbulence and tearing mode
instability suggests a crucial role of the instability in shaping the energy
dissipation and magnetic field structure in magnetohydrodynamic (MHD)
turbulence [6, 7, 8]. Current sheets formed within turbulent eddies can become
unstable to tearing modes when their thickness lies below a critical scale,
leading to the formation of plasmoids and altering the energy cascade. This
tearing-mediated regime introduces a sub-inertial interval characterized by a
distinct scaling of the energy spectrum, different from the standard
Kolmogorov-like turbulence. For instance, the scale of eddies where
reconnection becomes significant is larger than the resistive diffusion scale,
resulting in a $k_{\perp}^{-5/2}$ spectrum in the reconnection interval. The
instability also affects the alignment of magnetic field lines, reducing
anisotropy at smaller scales and modifying the turbulence spectrum to
$k_{\perp}^{-11/5}$. Furthermore, in the context of the turbulent dynamo,
tearing instability disrupts folded magnetic structures at high magnetic
Reynolds numbers, enhancing viscous-to-resistive dissipation and steepening
the magnetic energy spectrum [9]. These findings highlight that tearing
instability not only competes with but also significantly influences the
nonlinear evolution of turbulent eddies, leading to a complex and dynamic
interplay that is essential for understanding energy dissipation and magnetic
field topology in various astrophysical and laboratory plasma environments.

Plasmoid instability, which has become synonymous of tearing mode instability
in a three-dimensional scenario, is a significant mechanism in magnetic
reconnection processes. This instability manifests when thin current sheets,
formed during magnetic reconnection events, become susceptible to high-
wavenumber perturbations, leading to the rapid formation of fluxtubes trapped
between reconnecting active regions, known as plasmoids or secondary islands.
Theoretical models and numerical simulations [10, 11, 12, 13], have shown that
the growth rate of plasmoid instability scales with the Lundquist number,
$S=Lv_{A}/\eta$, following the relation $\gamma\sim S^{1/4}$. This rapid
instability causes the current sheets to break into a chain of plasmoids,
accelerating the reconnection process beyond the rates predicted by the
classical Sweet-Parker model. In high-Lundquist-number plasmas, plasmoid
instability induces a regime of fast reconnection characterized by dynamic and
impulsive behaviors, leading to significantly higher reconnection rates than
previously understood. The ejection of the plasmoids, however, is necessary
for an efficient magnetic energy conversion - and particle acceleration -,
which are crucial in explaining the rapid energetic events observed in
astrophysical and laboratory plasmas.

Viscosity plays a significant role in modulating the tearing mode instability,
particularly in high-Lundquist-number plasmas [14, 15, 16, 17]. The
introduction of viscosity alters the scaling of the growth rate and the
behavior of the unstable modes. For instance, when viscosity is considered
alongside small shear flow, the scaling of tearing growth rate changes to
$S^{-2/3}(S_{v}/S)^{1/6}$, where $S_{v}$ is the ratio of the viscous time to
the Alfvén time. This adjustment in scaling indicates that viscosity can
strongly reduce the instability growth rate compared to the inviscid case.
Additionally, at large magnetic Prandtl numbers ($P_{m}=\nu/\eta\gg 1$), the
growth rate and wavenumber of the most unstable mode scale differently,
leading to predictions such as $\gamma_{\mathrm{max}}\sim S^{1/4}P_{m}^{-5/8}$
and $k_{\mathrm{max}}L_{CS}\sim S^{3/8}P_{m}^{-3/16}$. This implies that
higher viscosity allows for larger aspect ratios of current sheets before they
become unstable. In the context of Sweet-Parker current sheets, the presence
of viscosity also affects the onset and growth rate of the plasmoid
instability and, in some cases, it enables the Kelvin-Helmholtz instability to
grow faster than the plasmoid instability. Overall, viscosity introduces
additional complexity to the tearing mode dynamics influencing the stability
and structure of current sheets in both laboratory and astrophysical plasmas.

The effects of compressibility and velocity shear significantly influence the
tearing mode instability in various ways. Compressibility enhances the
development of tearing instabilities, particularly in regions where
incompressible plasma would otherwise remain stable [18]. This destabilizing
effect is triggered by the longitudinal magnetic field, playing a crucial role
in dynamic solar phenomena such as flares. On the other hand, velocity shear,
particularly in the presence of equilibrium shear flow and viscosity,
introduces complex modifications to the tearing mode growth rate [14, 15, 19,
20]. Shear flow alters the eigenfunction profiles, creating new peaks outside
those found in flow-free scenarios, and modifies growth rate scalings to
$S^{-2/3}(S_{v}/S)^{1/6}$. High shear flow can induce Kelvin-Helmholtz
instabilities, further complicating the dynamics. Additionally, shear flow can
stabilize the current sheet by stretching and evacuating magnetic islands,
limiting the duration of instability growth and leading to non-exponential
behavior. The interplay between these factors—compressibility and shear
flow—demands detailed numerical and analytical studies to fully understand
their combined impact on magnetic reconnection processes in astrophysical and
laboratory plasmas.

In a three-dimensional (3D) tearing scenario, one expects a complex interplay
between multiple resonant surfaces and the influence of a guide field on the
instability dynamics [12, 21]. Unlike the two-dimensional (2D) case, where
instability occurs primarily at the null surface, 3D geometries with a guide
field exhibit a spectrum of unstable modes across various resonant surfaces
within the current sheet. These surfaces are characterized by unique angles of
obliquity, with the most unstable modes being oblique rather than parallel.
The presence of a guide field introduces a dispersive nature to the
propagation of these modes, altering their growth rates and structures. For
instance, the most unstable wavenumber is found at the intersection of
constant-$\psi$ and non-constant-$\psi$ regimes (where $\psi$ is the magnetic
flux function), with the growth rate scaling as $\gamma_{\mathrm{max}}\sim
S_{L}^{1/4}(1-\mu^{4})^{1/2}$. In the presence of the Hall effect [22, 21], a
strong guide field does not change the dominance of the fastest-growing
parallel mode but suppresses the wavelike structure of oblique modes, making
the eigenfunctions asymmetric. These findings underscore the significance of
including guide fields in tearing mode analyses, as they fundamentally alter
the instability characteristics and contribute to the formation and dynamics
of plasmoids in 3D MHD systems.

As discussed above, the tearing instability may be suppressed, or have its
scaling laws severely modified, under some circumstances, either by viscous
effects or the presence of velocity shears. Another stabilizing factor for
tearing is the presence of a transverse, i.e. normal to the current sheet
plane, magnetic field component. The stabilization effect of the transverse
field $B_{n}$ becomes essential for $\xi\equiv{B_{n}}/{B_{0}}\geq S^{-3/4}$
[23], decreasing the growth rate $\gamma$ with increasing the value of $\xi$.
Once $\xi\gg S^{-3/4}$, the instability is completely stabilized. This
conclusion was, however, drew based on the stability of linearized
incompressible MHD equations under the assumption of the same equilibrium as
in the original work by Furth, Killeen, and Rosenbluth (1963).

In this work we demonstrate that the presence of transverse field modifies the
initial magnetic shear profile. We provide the analytical solution of an
initial Harris current sheet profile in the presence of transverse field, and
perform the stability analysis of the incompressible MHD equations for the
modified profile.

## Harris Current Sheet in the Presence of a Transverse Field

First, we analyze the effect of a transverse magnetic field on the evolution
of the initial configuration of a one-dimensional (1D) Harris current sheet
with thickness $a$. The initial magnetic field configuration is given by
$B_{x}(t=0,z)=B_{0}\tanh\left(\frac{z}{a}\right)$ for the field parallel to
the current sheet, and
$B_{y}(t=0,z)=B_{0}\mathrm{sech}\left(\frac{z}{a}\right)$ for the guide field,
with no initial flow ($U_{x}(t=0,z)=U_{y}(t=0,z)=0$), and a non-zero, uniform
transverse field ($B_{z}=\text{const}$). The evolution of the system along the
$z$-direction is governed by the incompressible magnetohydrodynamic (MHD)
equations:

$\frac{\partial U_{x}}{\partial t}=B_{z}\frac{\partial B_{x}}{\partial
z}+\nu\frac{d^{2}U_{x}}{dz^{2}}$ (1)

$\frac{\partial U_{y}}{\partial t}=B_{z}\frac{\partial B_{y}}{\partial
z}+\nu\frac{\partial^{2}U_{y}}{\partial z^{2}}$ (2)

$\frac{\partial U_{z}}{\partial t}=-\frac{\partial}{\partial
z}\left[p+\frac{1}{2}\left(B_{x}^{2}+B_{y}^{2}\right)\right]$ (3)

$\frac{\partial B_{x}}{\partial t}=B_{z}\frac{\partial U_{x}}{\partial
z}+\eta\frac{\partial^{2}B_{x}}{\partial z^{2}}$ (4)

$\frac{\partial B_{y}}{\partial t}=B_{z}\frac{\partial U_{y}}{\partial
z}+\eta\frac{\partial^{2}B_{y}}{\partial z^{2}}$ (5)

$\frac{\partial B_{z}}{\partial t}=0$ (6)

with the magnetic field expressed in terms of the Alfvén speed ($V_{A}\equiv
B_{0}$), under the normalization used.

The chosen geometry and the condition of incompressibility enforce that the
partial derivatives of the velocity component $U_{z}$ and the magnetic field
component $B_{z}$ with respect to $z$ are zero ($\partial_{z}U_{z}=0$ and
$\partial_{z}B_{z}=0$). Without loss of generality, $U_{z}=0$ is initially
assumed. The total pressure within the system is also found to remain uniform.
From Equation (6), we infer that the transverse component of the magnetic
field, $B_{z}$, likewise, remains constant over time allowing us to treat
$B_{z}$ as a fixed parameter. Moreover, one can notice that Equations (1) and
(4) are separable from Equations (2) and (5).

The influence of the transverse field $B_{z}$ is manifested in Equations (1),
(2), (4), and (5). Specifically, the term $B_{z}\partial_{z}B_{x}$ in Equation
(1) induces a constant acceleration, resulting in a velocity shear in the
X-direction, initially proportional to $B_{z}\text{sech}^{2}(z)$, provided
$B_{z}$ is nonzero. In Equation (4), the shear in $U_{x}$ alters the profile
of the current sheet through the term $B_{z}\partial_{z}U_{x}$. Furthermore, a
non-zero $B_{z}$ impacts the Equations (2) and (5), where the non-uniform
guide field $B_{y}$ generates another shear in the plane of the current sheet
in the Y-direction via $B_{z}\partial_{z}B_{y}$, affecting the guide field
$B_{y}$ profile due to $B_{z}\partial_{z}U_{y}$. The remaining non-ideal terms
represent viscosity ($\nu$) in Equations (1) and (2), and resistivity ($\eta$)
in Equations (4) and (5).

By neglecting the effects of viscosity and resistivity, it becomes possible to
analytically solve the system. Taking the time derivative of Equation (4) and
substituting the time derivative of $U_{x}$ from Equation (1) leads us to a
second-order differential equation for $B_{x}$:

$\frac{\partial^{2}B_{x}}{\partial
t^{2}}=B_{z}^{2}\frac{\partial^{2}B_{x}}{\partial z^{2}}$ (7)

Given the initial Harris current sheet, described by a hyperbolic tangent, the
solution to this second-order differential equation is:

$B_{x}(t,z)=\frac{B_{0}}{2}\left[\tanh{\left(\frac{z+B_{z}t}{a}\right)}+\tanh{\left(\frac{z-B_{z}t}{a}\right)}\right]$
(8)

It is easily verified that, at $t=0$, this profile corresponds to
$B_{0}\tanh{\left(\frac{z}{a}\right)}$. By introducing a relative measure of
the transverse field with respect to the upstream field, i.e. $\xi\equiv
B_{z}/B_{0}$, we have $B_{z}=(B_{z}/B_{0})B_{0}=\xi V_{A}$.

The solution describes a current sheet of the thickness $a$ that develops a
neutral layer of vanishing $B_{x}$ expanding around the midplane with the
half-width increasing with time as $w=({\xi}/{a})V_{A}t$. The solution can be
rewritten as:

$B_{x}(t,z)=\frac{B_{0}}{2}\left[\tanh{\left(\frac{z+\xi
V_{A}t}{a}\right)}+\tanh{\left(\frac{z-\xi V_{A}t}{a}\right)}\right]$ (9)

From the magnetic field profile, we can deduce the solution for the shear
$U_{x}$ by integrating the $z$-derivative of $B_{x}$ over time, as described
by Equation 1. The resultant expression for the $U_{x}$ shear is:

$U_{x}(t,z)=\frac{V_{0}}{2}\left[\tanh{\left(\frac{z+\xi
V_{A}t}{a}\right)}-\tanh{\left(\frac{z-\xi V_{A}t}{a}\right)}\right]$ (10)

It can be easily verified that at $t=0$, $U_{x}=0$. Upon examining this
profile, it becomes apparent that its maximum occurs at $z=0$, which reveals
the temporal increase in the amplitude of the $U_{x}$ shear as follows:

$U_{x}(t,z=0)=V_{0}\tanh{\left(\frac{\xi}{a}V_{A}t\right)}$ (11)

This equation implies that the amplitude of the $U_{x}$ shear initially
increases from zero, growing linearly with time at a rate proportional to $\xi
V_{A}/a$, and eventually saturates at the Alfvén speed beyond the saturation
time $t_{\mathrm{sat}}\approx a/\xi V_{A}$.

Applying similar reasoning, we can determine the analytical solutions for
$B_{y}$ and $U_{y}$, assuming negligible viscosity and resistivity. Taking the
time derivative of Equation 5 and inserting Equation 2 leads to a second-order
differential equation:

$\frac{\partial^{2}B_{y}}{\partial
t^{2}}=B_{z}^{2}\frac{\partial^{2}B_{y}}{\partial z^{2}}$ (12)

For the Harris current sheet model under consideration, the solution to this
equation is:

$B_{y}(t,z)=\frac{B_{0}}{2}\left[\mathrm{sech}{\left(\frac{z+\xi
V_{A}t}{a}\right)}+\mathrm{sech}{\left(\frac{z-\xi
V_{A}t}{a}\right)}\right]+B_{g},$ (13)

where $B_{g}$ is the uniform guide field component. Similarly, the solution
for the $U_{y}$ shear profile:

$U_{y}(t,z)=\frac{B_{0}}{2}\left[\mathrm{sech}{\left(\frac{z+\xi
V_{A}t}{a}\right)}-\mathrm{sech}{\left(\frac{z-\xi V_{A}t}{a}\right)}\right].$
(14)

The solutions for $B_{y}$ and $U_{y}$ hold under the condition that the guide
field $B_{g}$ vanishes. Initially, the amplitude of this shear increases
linearly from zero, reaching a saturation value of $V_{A}/2$. However, as the
uniform component of the guide field $B_{g}$ increases, the amplitude of
$U_{y}$ correspondingly decreases. Importantly, the evolution of $B_{x}$ and
$U_{x}$ does not depend on the $y$-components of these fields, highlighting a
distinct separation in the dynamics of the field components.

## Numerical solutions of the current sheet evolution in the presence of a
transverse field

To validate our analytical predictions, we conducted numerical simulations of
Equations (1)-(6) using the Dedalus framework (Dedalus Project:
https://dedalus-project.org/) on a setup consisting of two adjacent Harris
current sheets within a periodic box. The initial current sheet thickness $a$
was set to unity. Our exploration included various values of the transverse
magnetic field, $\xi$.

Figure 1 illustrates the growth in the shear amplitudes of $U_{x}$ and $U_{y}$
within the current sheet setup across different $\xi$ values. Initially, both
shear amplitudes demonstrate linear growth, progressing to a saturation level,
in agreement with the analytical predictions. The time it takes to reach this
saturation point scales with the magnitude of $\xi$. Notably, the $U_{x}$
shear reaches saturation at the Alfvén velocity ($V_{A}$), whereas the $U_{y}$
shear saturates at half the Alfvén velocity ($V_{A}/2$).

Beyond the saturation phase, the current sheet undergoes expansion, indicated
by the widening of the $B_{x}$ profile depicted in Figure 2. This expansion
affects not only the $U_{x}$ shear region but also leads to the broadening of
the profiles for $U_{y}$ and $B_{y}$, again, in agreement with the analytical
predictions.

The analysis of the one-dimensional Harris current sheet in the presence of a
transverse magnetic field reveals significant dynamics that are governed by
the incompressible MHD equations. The study highlights how a constant
transverse field $B_{z}$ influences the evolution of magnetic and velocity
field profiles, inducing shears in both the $x$\- and $y$-directions. These
shears evolve due to the coupling between velocity and magnetic field
components, as mediated by $B_{z}$. The analytical solutions derived
demonstrate that the magnetic field profile $B_{x}$ and velocity profile
$U_{x}$ both split and propagate away from the midplane, leading to the
development of dynamic shears that increase linearly in amplitude and
eventually saturate. This saturation occurs at the Alfvén speed for $U_{x}$
and at half the Alfvén speed for $U_{y}$, dependent on the magnitude of
$B_{z}$. Furthermore, the evolution exhibits a distinct separation in the
dynamics of the $x$\- and $y$-components, underscoring the robust separability
in their behaviors. Numerical simulations confirm these analytical findings,
illustrating the key role of the transverse field in shaping the temporal and
spatial dynamics of the Harris current sheet.

Figure 1: The growth of the shear amplitude $U_{x}$ (solid) and $U_{y}$
(dashed) as functions of the transverse field strength, $\xi$. The initial
current sheet thickness was assumed to unity, $a=1$ and the null uniform guide
field, $B_{g}=0$. The horizontal gray lines indicate the saturation levels of
$V_{A}$ (solid) and $V_{A}/2$ (dashed) for the shear velocities. The
saturation time, $\tau_{\mathrm{sat}}$, scales proportionally with the
strength of $\xi V_{A}$ ($t_{\mathrm{sat}}\sim a/\xi V_{A}$). These
simulations were conducted with $S=10^{6}$ and $Pr=0$ ($\eta=10^{-6}$ and
$\nu=0$, respectively). Figure 2: The field profiles at selected times
($t=0,100,200,$ and $300$) for a fixed transversal field, $B_{z}=0.01$. The
panels from top to bottom depict $U_{x}$, $U_{y}$, $B_{x}$, and $B_{y}$,
respectively.

## Tearing Instability Analysis in the Quasi-Steady Expanding Current Sheet
Limit

In the previous section, we established that the presence of a transverse
field disrupts the equilibrium of the 1D Harris current sheet, altering its
profile and creating an expanding neutral layer. If $B_{z}$ is sufficiently
small, this expansion may be slow enough to allow the development of tearing
instability, which can be studied in a quasi-steady current sheet scenario.

In this section, we will perform a stability analysis using the current sheet
profile with a neutral layer as defined by the solution found in the previous
section. We perform such analysis using different profiles, that correspond to
different evolutionary stages of the current sheet. Apart from the $B_{x}$
profile, we will also consider a corresponding solution for $U_{x}$. We
parameterize the strength of the transverse field by $\xi$ as defined earlier,
and the half-width of the expanding neutral layer by $w=\xi V_{A}t$. Assuming,
without loss of generality, that $B_{0}=1$ and $V_{A}=1$.

The linearized equations are derived from the incompressible MHD equations in
a traditional manner, assuming a 2D setup in X and Z coordinates. To eliminate
the dependency on pressure, we take a curl of the momentum equation. We assume
that the perturbations take the form $\hat{q}(t,x,z)=q(z)\exp{\left(-\gamma
t+ikx\right)}$. Furthermore, we substitute the x components of velocity and
magnetic field Fourier amplitudes using the incompressibility condition, i.e.,
$b_{x}=i\partial_{z}b_{z}/k$ and $u_{x}=i\partial_{z}u_{z}/k$. This procedure
leads to the following linearized independent equations:

$\displaystyle\gamma\left(u_{z}^{\prime\prime}-k^{2}u_{z}\right)=ik\left[F\left(b_{z}^{\prime\prime}-k^{2}b_{z}\right)-F^{\prime\prime}b_{z}-G\left(u_{z}^{\prime\prime}-k^{2}u_{z}\right)+G^{\prime\prime}u_{z}\right]$
(15)
$\displaystyle{\color[rgb]{0,0,1}+\xi\left(b_{z}^{\prime\prime\prime}-k^{2}b_{z}^{\prime}\right)}+P_{r}S^{-1}\left(u_{z}^{\prime\prime\prime\prime}-2k^{2}u_{z}^{\prime\prime}+k^{4}u_{z}\right),$

$\displaystyle\gamma b_{z}=ik\left(Fu_{z}-Gb_{z}\right){\color[rgb]{0,0,1}+\xi
u_{z}^{\prime}}+S^{-1}\left(b_{z}^{\prime\prime}-k^{2}b_{z}\right),$ (16)

where $\xi\equiv B_{t}/B_{0}$ is the relative strength of the transverse
field, and ′, ′′, ′′′, and ′′′′ are the first, second, third, and fourth-order
derivatives with respect to $z$, respectively. The functions
$F(z)=\frac{1}{2}\left[\tanh\left(\frac{z+w}{a}\right)+\tanh\left(\frac{z-w}{a}\right)\right]$
and
$G(z)=\frac{1}{2}\left[\tanh\left(\frac{z+w}{a}\right)-\tanh\left(\frac{z-w}{a}\right)\right]$
represent the analytical solutions for $B_{x}$ and $U_{x}$, respectively, at
time $t$, corresponding to $w=\xi V_{A}t$. Here, $L$ represents the assumed
unit of length. Furthermore, we define two Alfvén time scales:
$\tau_{A}=a/V_{A}$, related to the thickness of the current sheet, and
$t_{A}=L/V_{A}$, related to the characteristic scale $L$. For most cases
presented here, we assume $L=a=1$, resulting in $\tau_{A}$ and $t_{A}$ being
equal. The linearized equations (15)-(16) resemble those derived in previous
studies on tearing instability [see, e.g., 24, 20]. The distinction arises
from the presence of terms related to the shear, represented by $G$ and
$G^{\prime\prime}$, and terms containing the constant transverse field, $\xi$.

We perform the stability analysis of the Equations (15)-(16) using the Psecas
framework [25], which offers a robust tool for studying the stability of
various physical systems by facilitating the solution of linear stability
problems. It is particularly adapted for handling differential equations that
describe perturbations in continuous media, such as those found in fluid
dynamics and plasma physics. The framework employs spectral methods, which are
advantageous for their accuracy and efficiency in resolving complex waveforms
in stability analyses. By decomposing the problem space into basis functions,
Psecas efficiently transforms the differential equations into a matrix
eigenvalue problem. This allows for the computation of growth rates and
eigenmodes, essential for understanding the stability characteristics of the
system. The flexibility of Psecas in selecting appropriate basis functions and
boundary conditions makes it a versatile tool in the theoretical investigation
of instabilities, enhancing both the depth and scope of analysis possible
within computational research environments.

Figure 3: The dispersion relations for the case of Lundquist number $S=10^{4}$
are shown for several values of $w=0,0.5,1,1.5,$ and $2$ (colors), and
strengths of the transverse field $\xi=0,10^{-4},10^{-3},$ and $10^{-2}$
(solid, dashed, dotted, and dashdot line styles, respectively). The upper and
lower plots correspond to cases with Prandtl numbers 0 and 1, respectively.
For this analysis we assumed here $L=a=1$.

Figure 3 shows dispersion relations for the equilibrium profiles of $B_{x}$
and $U_{x}$, described by functions $F$ and $G$, respectively, with several
values of $w$. The solid lines correspond to $\xi=0$ and are derived by
neglecting the terms related to $\xi$ in Equations (15)-(16). The system’s
response is solely due to the equilibrium profiles.

Two regimes of tearing instability can be recognized: the constant-$\psi$
regime at large $\alpha$, where the growth rate $\gamma$ decreases with the
wavenumber, and the non-constant-$\psi$ regime at small $\alpha$, with
$\gamma$ increasing with $\alpha$. The existence of these two regimes results
in a maximum growth rate $\gamma_{\mathrm{max}}$ and corresponding wavenumber
$\alpha_{\mathrm{max}}$. The figure shows that the dispersion relations
decrease in terms of the growth rate $\gamma$ and shift their maximum to
longer wavelengths with increasing $w$. For $w=2$, the dispersion relation is
characterized by growth rates smaller than $5\times 10^{-4}$ with wavelengths
$\alpha=ka$ smaller than 0.2. Including the viscous terms in Equation (15)
results in a further decrease of the growth rates $\gamma$ (compare upper and
lower panels in Fig. 3).

The dispersion relations calculated by taking into account the terms with
$\xi$ in Equations (15)-(16) are plotted with dashed, dotted, and dash-dotted
lines for $\xi=10^{-4}$, $10^{-3}$, and $10^{-2}$, respectively. Small values
of $\xi\ll 10^{-2}$, result in a gradual decrease of the growth rates at all
wavelengths. For $w=0$ and $\xi=10^{-2}$, we observe a significant increase of
the growth rate, especially in the region of $\gamma_{\mathrm{max}}$. However,
as the system evolves and the neutral layer width ($w$) increases, the growth
rate in the non-constant-$\psi$ regime drops, and most of the wavelengths in
the constant-$\psi$ regime are stabilized.

These results demonstrate the significant impact of the $\xi$-related terms on
tearing mode instability, enhancing the growth rates for the classical profile
of the Harris current sheet ($w=0$) with a sufficiently large transverse field
$\xi=10^{-2}$, but quickly stabilizing most of the tearing instability modes
once the equilibrium profiles expand due to the action of a non-zero $\xi$. It
is important to note that the value of $w=1$ corresponds to the half-width of
the neutral layer being equal to the initial Harris current sheet thickness
$a=1$. Therefore, once the action of the transverse field expands the neutral
layer to more than twice the thickness of the initial current sheet, the
expected result is the stabilization of essentially all tearing instability
modes.

Figure 4: The dependence of maximum growth rate $\gamma_{\mathrm{max}}$ (left)
and corresponding wavenumber $\alpha_{\mathrm{max}}$ (right) on the neutral
layer width $w$ for different strength of transverse field $\xi$ ($0$,
$10^{-5}$, $10^{-4}$, $10^{-3}$, and $10^{-2}$). The case of $S=10^{4}$ for
$P_{r}=0$ and $P_{r}=1$ are shown in the upper and lower panels, respectively.
For this analysis we assumed here $L=a=1$.

Now, focusing on the fastest growing mode, Figure 4 displays the maximum
growth rate, $\gamma_{\mathrm{max}}$, and the corresponding wavenumber,
$\alpha_{\mathrm{max}}$, in the left and right panels, respectively,
dependence on the half-width of the neutral layer $w$ for cases with varying
magnitudes of the transverse field, $\xi$. It is noticeable that for very weak
transverse fields, $\xi<10^{-4}$, the maximum growth rate decays nearly
exponentially with increasing $w$. Once the transverse field strengthens,
i.e., $\xi\geq 10^{-4}$, the maximum growth rate drops faster than the
exponential rate, eventually stabilizing completely at a certain thickness of
the neutral layer.

In the inviscid case shown in the upper panel, the estimated stable half-
widths are approximately $w\sim 2.6$, $\sim 1.7$, and $\sim 1.2$ for
$\xi=10^{-4}$, $10^{-3}$, and $10^{-2}$, respectively. A similar behavior is
observed in the viscous case (lower panel), with stable half-widths slightly
larger. We anticipate that the action of the viscous terms in Equation (15)
decreases the gradients in the $u_{z}$ profiles, which somewhat diminishes the
impact of the $\xi$-related term in Equation (16).

Figure 5: The dependence of maximum growth rate $\gamma_{\mathrm{max}}$ (left)
and corresponding wavenumber $\alpha_{\mathrm{max}}$ (right) on the initial
current sheet thickness $a$ for different neutral layer widths $w$ ($0$,
$0.1$, $0.2$, and $0.3$) and Lundquist numbers ($10^{4}$, $10^{5}$, and
$10^{6}$). The case of $Pr=0$ and $Pr=1$ are shown in the upper and lower
panels, respectively.

Finally, we study the dependence on the initial current sheet thickness, $a$.
In the classical tearing mode analysis the maximum growth rate increases with
decreasing current sheet thickness as
$\gamma_{\mathrm{max}}\propto(a/L)^{-3/2}$. As shown in the left panels of
Figure 5, this behaviour is recovered only for $w=0$. Once the neutral layer
increases, the behaviour of $\gamma_{\mathrm{max}}$ with $a$ changes
drastically. We show that in the presence of the expanding neutral layer this
relation does not hold anymore, mostly due to the changing profile of the
equilibrium field, and not the action of $\xi$ related terms. The saturation
time $t_{\mathrm{sat}}\propto a$, therefore, for smaller $a$ the tearing is
stabilized faster. Indeed, as seen in Figure 5, already for 10 times thinner
current sheet the growth rate decays with decreasing $a$ for $w>0.1$. At the
same time, the corresponding wavenumber $k_{\mathrm{max}}$ tends to larger
scales, as shown in the right panels of Figure 5.

## Direct Numerical Simulations of Tearing Mode in an Expanding Current Sheet

The stability analysis of the previous section, performed for fixed
equilibrium profiles, indicates that the tearing mode instability should be
suppressed for a sufficiently thick neutral layer. To verify if this
conclusion holds for time-varying profiles of magnetic and velocity shears, we
conducted direct numerical simulations of the initial Harris current sheet,
represented by a hyperbolic tangent profile of the $x$-component of the
magnetic field and varying transverse field strength. The numerical solution
of the incompressible MHD equations in 2D was obtained using the Dedalus
framework (Dedalus Project: https://dedalus-project.org/).

(a)

(b)

(c)

(d)

Figure 6: Field profiles in simulations of the initial Harris current sheet
equilibrium with two values of transverse field, $\xi=0$ (subfigures a and c)
and $\xi=10^{-3}$ (subfigures b and d) at two different moments, $t=1000$
(subfigures a and b) and $t=1500$ (subfigures c and d). Each subfigure
contains colormaps of $v_{x}$, $v_{z}$, $\omega_{y}$ (left panels from top to
bottom, respectively), and $B_{x}$, $B_{z}$, $j_{y}$ (right panels from top to
bottom, respectively). The simulations were conducted with a Lundquist number
$S=10^{4}$, magnetic Prandtl number $Pr_{m}=0$, and initial current sheet
thickness $a=1$.

Figure 6 shows examples of simulations performed with a Lundquist number
$S=10^{4}$ and a magnetic Prandtl number $Pr_{m}=0$, for the initial Harris
current sheet of thickness $a=1$ and different strengths of the initial
transverse field, $B_{z}=0$ (subfigures a and c) and $B_{z}=10^{-3}$
(subfigures b and d). The amplitude of the initial magnetic field perturbation
was set to $10^{-4}$. The velocity components and vorticity are shown in the
left panels of each subfigure, while the magnetic field components and current
density are shown in the right panels. The growth rate of the most unstable
tearing mode for this case is $\gamma_{\mathrm{max}}\approx 0.006$, resulting
in an instability time scale of $t_{\mathrm{tearing}}\approx 165\tau_{a}$,
where $\tau_{a}$ is the current sheet crossing time. Two moments are depicted:
$t=1000$ (subfigures a and b) and $t=1500$ (subfigures c and d). At these
times, the tearing mode should grow by factors of approximately 420 and 8,600,
respectively. For comparison, the estimated stabilization time
$t_{\mathrm{sat}}$ due to the presence of the transverse field for the case of
$\xi=10^{-3}$ is around 1000, several times longer than the instability growth
time.

In subfigures a and c of Figure 6, one can recognize the typical evolution of
tearing mode instability in the case of a null transverse field: thickening of
the current sheet in the center, accompanied by outflows from the center in
both directions, resulting in the creation of two plasmoids on both sides. In
the case of a non-zero but weak transverse field, shown in subfigures b and d,
we observe the formation of an expanding neutral layer with a strong velocity
shear, as predicted by the analytical solution of the 1D Harris current sheet
evolution. Additionally, we see a much weaker current density with no
characteristic plasmoid formation, separated by a thin current sheet.

In conclusion, the 2D numerical simulations of the initial Harris current
sheet with a non-zero transverse field confirm the prediction from the linear
stability analysis regarding the stabilization effect of the transverse field
on tearing mode instability. The presence of a transverse field indeed
suppresses the tearing mode instability, leading to the formation of a stable,
expanding neutral layer as opposed to the plasmoid-dominated structure seen in
the absence of a transverse field.

## Conclusions

In this work, the impact of a transverse magnetic field on the evolution of a
Harris current sheet for magnetic reconnection is thoroughly investigated,
including its impact on the tearing/plasmoid instability. The showed that the
presence of the transverse field $B_{z}$ modifies the initial shear profile of
the magnetic field, as demonstrated through both analytical and numerical
methods. The derived analytical solutions for the magnetic and velocity fields
show that $B_{z}$ induces, and enhances through time, a velocity shear in the
$x$-direction, which eventually saturates at the Alfvén speed. This transverse
field also affects the $y$-direction shear, although it saturates at half the
Alfvén speed. In a time evolution analysis, we showed that this increased
shear induces the broadening of the current sheet, resulting in the formation
of a neutral layer whose width increases over time. This effect becomes more
important for stronger transverse fields, Numerical simulations validate these
analytical predictions, indicating a clear alignment between theoretical and
computational results.

Major consequences of the time-evolution described above is seen in the
tearing/plasmoid instability. The stability analysis of the current sheet,
taking into account the transverse field, indicates significant modifications
on the modes of tearing instability. The analysis reveals that for small
values of $B_{z}$ the growth rates of tearing instabilities are reduced,
leading to stabilization as the neutral layer expands. For strong transverse
fields the growth rates initially increase but are quickly reduced as the
neutral layer broadens. This demonstrates that, despite what has been
previously thought, the combined effect of a strong transverse field in
tearing, and the time evolution of the current sheet, actually stabilizes the
tearing modes by expanding the neutral layer beyond a critical thickness.

Direct numerical simulations of the tearing instability with varying
transverse field strengths further support the theoretical predictions,
demonstrating the stabilizing effect of the transverse field. The direct
numerical simulations show that, in the absence of a transverse field, the
tearing mode instability leads to the formation of plasmoids and a thickened
current sheet. However, with a non-zero transverse field, an expanding neutral
layer with strong velocity shear develops, and the characteristic plasmoid
formation is suppressed. The stabilization effect is clearly observed, as the
transverse field prevents the typical tearing mode evolution and results in a
stable, expanding neutral layer. These results validate the theoretical
stability analysis presented in this work, confirming that a sufficiently
strong transverse field can effectively quench the tearing mode instability.

Our study emphasizes the role of the transverse magnetic field in shaping the
evolution and stability of current sheets, highlighting the intricate
interplay between magnetic field configurations and plasma instabilities.
These findings may change considerable the understanding of magnetic
reconnection and turbulence in plasma physics, with potential implications for
space and astrophysical phenomena, given that one of the fast mechanisms of
magnetic reconnection has been shown here to be quickly quenched in real
systems.

Stabilization effects of transverse fields has been already pointed by Somov
and Verneta [26, 23, 4]. In their work, Somov and Verneta demonstrated a
significant stabilizing influence of a transverse magnetic field within the
magnetohydrodynamics (MHD) framework, highlighting how a small transverse
field can suppress the tearing mode instability in current layers. They also
explained the discrepancies found in earlier studies, which had yielded
negative results regarding the stabilizing effects. The current study advances
this understanding by providing detailed analytical solutions for the
evolution of magnetic and velocity shears in the presence of a transverse
field and confirming these theoretical predictions through direct numerical
simulations using the Dedalus framework. The simulations show that the
transverse field indeed leads to the formation of an expanding neutral layer
and suppresses the characteristic plasmoid formation associated with tearing
modes, thus corroborating the theoretical insights provided by Somov and
Verneta and extending them to time-varying profiles in a dynamically evolving
current sheet. This alignment between analytical and numerical results offers
a robust confirmation of the stabilizing effects of transverse fields on
tearing mode instabilities in reconnecting current layers.

## References

  *   * [1] Furth, H. P., Killeen, J. & Rosenbluth, M. N.  Finite-Resistivity Instabilities of a Sheet Pinch.  _Physics of Fluids_ 6, 459–484 (1963).
  * [2] Parker, E. N.  Sweet’s Mechanism for Merging Magnetic Fields in Conducting Fluids.  _J. Geophys. Res._ 62, 509–520 (1957).
  * [3] Sweet, P. A.  Lehnert, B. (ed.) _The Neutral Point Theory of Solar Flares_.  (ed.Lehnert, B.) _Electromagnetic Phenomena in Cosmical Physics_ , Vol. 6, 123 (1958).
  * [4] Somov, B. V. & Verneta, A. I.  Tearing Instability of Reconnecting current Sheets in Space Plasmas.  _Space Sci. Rev._ 65, 253–288 (1993).
  * [5] Birn, J., Forbes, T. G. & Hesse, M.  Stability and Dynamic Evolution of Three-dimensional Flux Ropes.  _ApJ_ 645, 732–741 (2006).
  * [6] Loureiro, N. F. & Boldyrev, S.  Role of Magnetic Reconnection in Magnetohydrodynamic Turbulence.  _Phys. Rev. Lett._ 118, 245101 (2017).
  * [7] Boldyrev, S. & Loureiro, N. F.  Magnetohydrodynamic Turbulence Mediated by Reconnection.  _ApJ_ 844, 125 (2017).
  * [8] Walker, J., Boldyrev, S. & Loureiro, N. F.  Influence of tearing instability on magnetohydrodynamic turbulence.  _Phys. Rev. E_ 98, 033209 (2018).
  * [9] Galishnikova, A. K., Kunz, M. W. & Schekochihin, A. A.  Tearing Instability and Current-Sheet Disruption in the Turbulent Dynamo.  _Physical Review X_ 12, 041027 (2022).
  * [10] Loureiro, N. F., Schekochihin, A. A. & Cowley, S. C.  Instability of current sheets and formation of plasmoid chains.  _Physics of Plasmas_ 14, 100703–100703 (2007).
  * [11] Bhattacharjee, A., Huang, Y.-M., Yang, H. & Rogers, B.  Fast reconnection in high-Lundquist-number plasmas due to the plasmoid Instability.  _Physics of Plasmas_ 16, 112102 (2009).
  * [12] Baalrud, S. D., Bhattacharjee, A. & Huang, Y. M.  Reduced magnetohydrodynamic theory of oblique plasmoid instabilities.  _Physics of Plasmas_ 19, 022101 (2012).
  * [13] Comisso, L., Lingam, M., Huang, Y. M. & Bhattacharjee, A.  General theory of the plasmoid instability.  _Physics of Plasmas_ 23, 100702 (2016).
  * [14] Chen, X. L. & Morrison, P. J.  The effect of viscosity on the resistive tearing mode with the presence of shear flow.  _Physics of Fluids B_ 2, 2575–2580 (1990).
  * [15] Ofman, L., Chen, X. L., Morrison, P. J. & Steinolfson, R. S.  Resistive tearing mode instability with shear flow and viscosity.  _Physics of Fluids B_ 3, 1364–1373 (1991).
  * [16] Loureiro, N. F., Schekochihin, A. A. & Uzdensky, D. A.  Plasmoid and Kelvin-Helmholtz instabilities in Sweet-Parker current sheets.  _Phys. Rev. E_ 87, 013102 (2013).
  * [17] Tenerani, A., Rappazzo, A. F., Velli, M. & Pucci, F.  The Tearing Mode Instability of Thin Current Sheets: the Transition to Fast Reconnection in the Presence of Viscosity.  _ApJ_ 801, 145 (2015).
  * [18] Verneta, A. I. & Somov, B. V.  Effect of compressibility on the development of tearing instability in a non-neutral current sheet in the solar atmosphere.  _Astronomy Reports_ 37, 282–285 (1993).
  * [19] Tolman, E. A., Loureiro, N. F. & Uzdensky, D. A.  Development of tearing instability in a current sheet forming by sheared incompressible flow.  _Journal of Plasma Physics_ 84, 905840115 (2018).
  * [20] Shi, C., Velli, M. & Tenerani, A.  Marginal Stability of Sweet-Parker Type Current Sheets at Low Lundquist Numbers.  _ApJ_ 859, 83 (2018).
  * [21] Shi, C., Velli, M., Pucci, F., Tenerani, A. & Innocenti, M. E.  Oblique Tearing Mode Instability: Guide Field and Hall Effect.  _ApJ_ 902, 142 (2020).
  * [22] Pucci, F., Velli, M. & Tenerani, A.  Fast Magnetic Reconnection: “Ideal” Tearing and the Hall Effect.  _ApJ_ 845, 25 (2017).
  * [23] Somov, B. V. & Verneta, A. I.  Magnetic Reconnection in a High-Temperature Plasma of Solar Flares - Part Four.  _Sol. Phys._ 120, 93–115 (1989).
  * [24] Landi, S., Del Zanna, L., Papini, E., Pucci, F. & Velli, M.  Resistive Magnetohydrodynamics Simulations of the Ideal Tearing Mode.  _ApJ_ 806, 131 (2015).
  * [25] Berlok, T. & Pfrommer, C.  On the Kelvin-Helmholtz instability with smooth initial conditions - linear theory and simulations.  _MNRAS_ 485, 908–923 (2019).
  * [26] Somov, B. V. & Verneta, A. I.  Magnetic Reconnection in High-Temperature Plasma of Solar Flares - Part Three.  _Sol. Phys._ 117, 89–95 (1988).

## Acknowledgments

G.K. and D.F-G acknowledges support from FAPESP (grants 2013/10559-5,
2021/02120-0, 2021/06502-4, and 2022/03972-2). The simulations presented in
this work were performed using the clusters of the Group of Theoretical
Astrophysics at EACH-USP (Hydra HPC), which was acquired with support from
FAPESP (grants 2013/04073-2 and 2022/03972-2).

## Author contribution

G.K. derived the analytical solution of 1D MHD for initial Harris current
sheet and testes this solution numerically. D.F.G. performed the linearization
of incompressible MHD equations taking into account the viscosity, resistivity
and transverse field related terms. G.K. implemented these equations in the
Psecas framework and performed the calculations. G.K. performed 1D and 2D
simulations of current sheet in the presence of transverse field using the
Dedalus framework. Both authors analyzed and interpreted the obtained
dispersion relations for different profiles of the equilibrium, and Lundquist
and Prandtl numbers, as well as, the direct numerical simulations.

## Competing interests

The authors declare no competing financial interest.