# Identification of active magnetic reconnection using magnetic flux transport
in plasma turbulence

Tak Chu Li  Yi-Hsin Liu  Department of Physics and Astronomy, Dartmouth
College, Hanover, NH, USA  Yi Qi  Department of Earth, Planetary, and Space
Sciences, University of California, Los Angeles, CA, USA

###### Abstract

Magnetic reconnection has been suggested to play an important role in the
dynamics and energetics of plasma turbulence by spacecraft observations,
simulations and theory over the past two decades, and recently, by
magnetosheath observations of MMS. A new method based on magnetic flux
transport (MFT) has been developed to identify reconnection activity in
turbulent plasmas. This method is applied to a gyrokinetic simulation of two-
dimensional (2D) plasma turbulence. Results on the identification of three
active reconnection X-points are reported. The first two X-points have
developed bi-directional electron outflow jets. Beyond the category of
electron-only reconnection, the third X-point does not have bi-directional
electron outflow jets because the flow is modified by turbulence. In all
cases, this method successfully identifies active reconnection through clear
inward and outward flux transport around the X-points. This transport pattern
defines reconnection and produces a new quadrupolar structure in the
divergence of MFT. This method is expected to be applicable to spacecraft
missions such as MMS, Parker Solar Probe, and Solar Orbiter.

magnetic reconnection — turbulence — plasmas

##  1 Introduction

Magnetic reconnection and plasma turbulence are both fundamental processes
ubiquitously operating throughout the universe. Reconnection has been
suggested to contribute to energy dissipation (Dmitruk et al., 2004; Sundkvist
et al., 2007; Osman et al., 2011, 2012; Markovskii & Vasquez, 2011; Perri et
al., 2012; Wan et al., 2012; Karimabadi et al., 2013; TenBarge & Howes, 2013;
Wu et al., 2013; Zhdankin et al., 2013; Shay et al., 2018) and potential
changes in the cascade (Loureiro & Boldyrev, 2017; Boldyrev & Loureiro, 2017;
Mallet et al., 2017a; Franci et al., 2017; Mallet et al., 2017b; Loureiro &
Boldyrev, 2017; Vech et al., 2018; Stawarz et al., 2019) of turbulence by in
situ observations, numerical simulations and theory. In heliospheric
turbulence, reconnection was first observed in situ in the terrestrial
magnetosheath by Cluster (Retinò et al., 2007). Recently, high resolution
measurements from MMS (Burch et al., 2016) have enabled the detection of
electron jets in small-scale current sheets in the turbulent magnetosheath
(Yordanova et al., 2016; Vörös et al., 2017; Phan et al., 2018; Wilder et al.,
2018), including notably, electron-only reconnection (Phan et al., 2018).

Reconnection occurs in a small-scale electron diffusion region (EDR) within a
thin current sheet. As upstream field lines flow into the EDR, they reconnect
at the X-point. The reconnected field possesses strong magnetic tension, which
drives the reconnected field away from the X-point, ejecting plasma that is
coupled to it as bi-directional outflow jets. The fundamental process of
reconnection can be described as inward and outward transport of magnetic flux
and associated plasmas at an X-point. The transport of magnetic flux and
plasma flows across a separatrix was used to to define reconnection
(Vasyliunas, 1975).

At the frontier of turbulence and reconnection research, important questions
include how reconnection occurs in a dynamical turbulent system and how the
rich dynamics of turbulence and reconnection, such as turbulent energy
dissipation and cascade, interplay. Nevertheless, there is still no clear,
reliable method to identify reconnection X-points in turbulent plasmas. In 2D
turbulence simulations, the method of saddle points that define an X-point
topology was applied (Servidio et al., 2009, 2010; Wan et al., 2013; Haggerty
et al., 2017). However, among a large number of identified X-points, only a
few displayed significant reconnection electric fields (Servidio et al.,
2009). It would be possible that many identified X-points are not actively
reconnecting.

In observations, a commonly used method to identify reconnection is the
detection of bi-directional Alfvénic ion outflow jets. In a turbulent system
such as the terrestrial magnetosheath, reconnection can happen at sub-ion or
electron scales (Wilder et al., 2018; Phan et al., 2018), and electron jets
becomes the conclusive signature of reconnection. However, fast turbulent
flows at sub-ion scales can make the detection challenging. In fact, only one
out of several tens of sub-ion-scale current sheets detected by Phan et al.
(2018) displayed clear bi-directional reconnection electron jets.

Recently, the transport of magnetic flux around an X-point was considered in
kinetic simulations of reconnection (Liu & Hesse, 2016; Liu et al., 2018). MFT
takes into account the decoupling of electron flow and magnetic flux
(slippage) arising from a non-ideal electric field, and thus correctly
captures the inward and outward transport of magnetic flux around a
reconnection X-point. In a symmetric reconnection simulation with shear flows,
the electron flow can be highly distorted (Liu et al., 2018). Under stronger
shear flows or asymmetry likely in turbulence, the electron flow may not show
typical reconnection outflows. In fact, in a highly asymmetric configuration,
active reconnection with only one electron jet is possible (Liu & Hesse,
2016).

##  2 Theory

The transport of magnetic flux inherent to reconnection represents an
innovative way for identifying active reconnecting X-points in turbulence. The
presence of inward flux transport also indicates reconnection is actively
taking place. The MFT velocity $\mathbf{U}_{\psi}$ was previously derived in
one and two dimensions (Liu & Hesse, 2016; Liu et al., 2018). The key steps
leading to the definition of $\mathbf{U}_{\psi}$ are summarized here. In 2D,
the magnetic field can be represented as an in-plane and out-of-plane (guide
field) component directed along $\hat{\mathbf{z}}$:
$\mathbf{B}=\mbox{$\hat{\mathbf{z}}$}\times\nabla\psi+B_{0}\mbox{$\hat{\mathbf{z}}$}$.
Curling the Faraday’s law:
$\mbox{$\hat{\mathbf{z}}$}\times[\,\partial_{t}\mathbf{B}+c\nabla\times\mathbf{E}=0\,]$
results in $\partial_{t}\psi=cE_{z}$. We then consider the electron momentum
equation:
$\mathbf{E}+\mathbf{v}_{e}\times\mathbf{B}/c=\mathbf{E}_{e}^{\prime}$, where
$\mathbf{E}_{e}^{\prime}$ is the non-ideal electric field in the electron
frame. Taking the $z$ component of this equation and casting it into the form
of the 2D advection equation of magnetic flux:
$\partial_{t}\psi+\mathbf{U_{\psi}}\cdot\nabla_{\perp}\psi=0$, the in-plane
MFT velocity is then given by:

$\displaystyle\mathbf{U}_{\psi}\equiv\mathbf{v}_{ep}-(\mathbf{v}_{ep}\cdot\hat{b}_{p})\hat{b}_{p}+\frac{cE_{ez}^{\prime}}{B_{p}}(\mbox{$\hat{\mathbf{z}}$}\times\hat{b}_{p}),$
(1)

where $\hat{b}_{p}\equiv\mathbf{B}_{p}/B_{p}$ is the unit vector of the in-
plane magnetic field $\mathbf{B}_{p}$ and $\mathbf{v}_{ep}$ the in-plane
electron flow. The first two terms represent the in-plane electron flow
perpendicular to $\mathbf{B}_{p}$. They come from the
$\mathbf{v_{e}}\times\mathbf{B}$ term in the electron momentum equation. For
$\mathbf{E}_{e}^{\prime}$=0, the electron flow is frozen-in to the magnetic
field and they move together. When $\mathbf{E}_{e}^{\prime}$$\not=$0, slippage
between magnetic flux and electron flow arises as the last term. Without
separating the perpendicular electron flow and slippage terms, which provide a
relation between the transport of magnetic flux and electron flow, Equation
(1) can be simplified to:

$\displaystyle\mathbf{U}_{\psi}=\frac{cE_{z}}{B_{p}}(\mbox{$\hat{\mathbf{z}}$}\times\hat{b}_{p}).$
(2)

To the first order in gyrokinetics, $\mathbf{U}_{\psi}$ is given by Equation
(1) or (2) with $\mathbf{v}_{ep}$, $\mathbf{B}_{p}$ and $E_{ez}^{\prime}$
replaced by $\delta\mathbf{u}_{ep}$, $\delta\mathbf{B}_{p}$ and $\delta
E_{ez}^{\prime}=\delta
E_{z}+(\delta\mathbf{u}_{ep}\times\delta\mathbf{B}_{p}/c)_{z}$, where
fluctuating quantities in turbulence are the in-plane electron bulk flow
$\delta\mathbf{u}_{ep}$ and so on. Note that Equation (1) is not applicable at
the X-point because a source or sink term, representing flux generation or
annihilation at the X-point, is not included in this advection equation.

A new quantity, the divergence of MFT, $\nabla\cdot\mathbf{U}_{\psi}$, is
considered here. $\nabla\cdot\mathbf{U}_{\psi}<0$ and $>0$ can capture the
converging inflows and diverging outflows of magnetic flux, respectively.
These bi-directional inflows and outflows of magnetic flux at an X-point
signifies active reconnection. $\nabla\cdot\mathbf{U}_{\psi}$ also informs
about the time scale of diverging magnetic flux from the X-point. Having the
dimension of inverse time, $\nabla\cdot\mathbf{U}_{\psi}$ is frame-independent
in 2D in the non-relativistic limit. Therefore, one can compute
$\nabla\cdot\mathbf{U}_{\psi}$ for moving X-points without changing frames.

##  3 Code

The 2D gyrokinetic turbulence simulation has been previously performed (Li et
al., 2016) using the the Astrophysical Gyrokinetics Code, or AstroGK,
described in details in (Numata et al., 2010). AstroGK has been extensively
used to investigate turbulence in weakly collisional plasmas (Howes et al.,
2008; Tatsuno et al., 2009; Howes et al., 2011; TenBarge & Howes, 2012;
Nielson et al., 2013; TenBarge & Howes, 2013; Howes, 2016; Li et al., 2016;
Howes et al., 2018; Li et al., 2019) and collisionless strong-guide-field
reconnection (Numata et al., 2011; TenBarge et al., 2014; Kobayashi et al.,
2014; Numata & Loureiro, 2015). AstroGK is an Eulerian continuum code with
triply periodic boundary conditions. It has a slab geometry elongated along
the straight, uniform background magnetic field,
$\mathbf{B}_{0}=B_{0}\mbox{$\hat{\mathbf{z}}$}$. The code evolves the
perturbed gyroaveraged Vlasov-Maxwell equations in five-dimensional phase
space (three-dimensional-two-velocity) (Frieman & Chen, 1982; Howes et al.,
2006). The evolved quantities are the electromagnetic gyroaveraged
complementary distribution function for each species $s$, the scalar potential
$\varphi$, parallel vector potential $A_{\parallel}$ and parallel magnetic
field perturbation $\delta B_{\parallel}$, where $\parallel$ is along the
total local magnetic field
$\mathbf{B}=B_{0}\mbox{$\hat{\mathbf{z}}$}+\delta\mathbf{B}$. The total and
background magnetic fields are the same to first-order accuracy retained for
perturbed fields in gyrokinetics. The velocity grid is specified by pitch
angle $\lambda=v_{\perp}^{2}/v^{2}$ and energy $\varepsilon=v^{2}/2$. The
background distribution functions for both species are stationary uniform
Maxwellians. Collisions are incorporated using a fully conservative,
linearized gyro-averaged Landau collision operator (Abel et al., 2008; Barnes
et al., 2009).

##  4 Setup

The 2D Orszag-Tang Vortex (OTV) problem has been widely used to study plasma
turbulence (Politano et al., 1989; Dahlburg & Picone, 1989; Picone & Dahlburg,
1991; Politano et al., 1995; Grauer & Marliani, 2000; Mininni et al., 2006;
Parashar et al., 2009, 2014). It is given by

$\displaystyle\delta\mathbf{u}$ $\displaystyle=\delta
u[-\sin(k_{\perp}y)\mbox{$\hat{\mathbf{x}}$}+\sin(k_{\perp}x)\mbox{$\hat{\mathbf{y}}$}]$
$\displaystyle\delta\mathbf{B}$ $\displaystyle=\delta
B[-\sin(k_{\perp}y)\mbox{$\hat{\mathbf{x}}$}+\sin(2k_{\perp}x)\mbox{$\hat{\mathbf{y}}$}],$

where $\delta u=\delta B/\sqrt{4\pi\rho_{0}}$, $\delta\mathbf{u}$ and
$\delta\mathbf{B}$ are perturbations in the ion and electron bulk flow and the
magnetic field, and $k_{\perp}=2\pi/L_{\perp}$ are positive constants.

To follow the turbulent cascade from the inertial range ($k_{\perp}\rho_{i}\ll
1$) to below electron scales ($k_{\perp}\rho_{e}>1$) (TenBarge & Howes, 2013;
TenBarge et al., 2013, 2014), we specify a reduced mass ratio,
$m_{i}/m_{e}=25$, which, in a simulation domain of $L_{\perp}=8\pi\rho_{i}$
and dimensions
$(n_{x},n_{y},n_{z},n_{\lambda},n_{\varepsilon},n_{s})=(128,128,2,64,32,2)$,
enables us to resolve a dynamic range of $0.25\leq k_{\perp}\rho_{i}\leq
10.5$, or $0.05\leq k_{\perp}\rho_{e}\leq 2.1$. Plasma parameters are ion
plasma $\beta_{i}=8\pi n_{i}T_{0i}/B_{0}^{2}=0.01$ and $T_{0i}/T_{0e}=1$.
Collision frequencies of $\nu_{i}$ = 10${}^{-5}\omega_{A0}$ and $\nu_{e}$ =
0.05 $\omega_{A0}$ (where $\omega_{A0}\equiv k_{\parallel}v_{A}$ is a
characteristic Alfvén wave frequency in 3D) are sufficient to keep velocity
space well resolved (Howes et al., 2008, 2011). Length, time and velocity are
normalized to the ion gyroradius $\rho_{i}\equiv v_{ti}/\Omega_{ci}$, where
$\Omega_{ci}\equiv eB_{0}/m_{i}c$, domain turnaround time $\tau_{0}\equiv
L_{\perp}/\delta u$ and electron thermal speed
$v_{te}\equiv\sqrt{2T_{0e}/m_{e}}$. $\tau_{0}$ can be converted to the inverse
ion gyro-frequency, a relevant time scale for reconnection, by $\tau_{0}$ = 25
$\Omega_{ci}^{-1}$. The divergence of velocity is normalized to
$v_{te}/\rho_{e}=\Omega_{ce}$.

Figure 1:  (a) The out-of-plane current density $J_{z}$ (color) overlaid with
contours of $A_{\parallel}$ of the OTV configuration, and (b) the
$x$-component of $\mathbf{U}_{\psi}$ at $t/\tau_{0}$=0.12, showing X1 and X2
and their mirrors (labeled). At $t/\tau_{0}$=1.48, (c) $J_{z}$ showing
developed turbulence and (d) $U_{\psi x}$ revealing X3, X4 (labeled) and X3
mirror. The bi-directional outflows of magnetic flux at X1 and X4, and inflows
at X2 and X3 are observed. $\delta B_{p}$ is offsetted by adding a 1% of its
maximum value in the domain such that $\mathbf{U}_{\psi}$ remains finite at
the X- and O-points that have vanishing $\delta B_{p}$. This does not
qualitatively affect the profile of $\mathbf{U}_{\psi}$. Dashed boxes indicate
regions in zoomed-in figures. See an animation of $J_{z}$ online. The
animation lasts for $t/\tau_{0}$=0–2.01. It shows the evolution of
reconnection driven by turbulent flows.

##  5 Results

Figure 1(a) shows the out-of-plane current density $J_{z}$ (color) and
contours of the parallel vector potential $A_{\parallel}$ representing
magnetic field lines of the OTV at an early time of $t/\tau_{0}$=0.12. The OTV
has an initial flow configuration that rotates the two vortices near the
center of the domain, forming a current sheet in between. The symmetry of the
two vortices allows symmetric reconnection to take place at the current sheet.
The flows also drive two asymmetric vortices at the top right and bottom left,
resulting in two mirroring asymmetric reconnection X-points by symmetry of the
system. A fourth reconnection X-point, which is a mirror of the central
symmetric reconnection X-point, is located at $(x,y)\simeq$(0,12.6). The
central symmetric (X1) and top-right asymmetric (X2) X-points are two of the
cases we will discuss in details.

As the total turbulence energy dissipates over time (Li et al., 2016), the
driving of reconnection weakens and reconnection at later times is generally
weaker than early-time events. Figure 1(c) shows $J_{z}$ at late time
$t/\tau_{0}$=1.48 when multiscale features, including small-scale current
sheets, have developed. A turbulent cascade in the dissipation range (see
Figure A1 for the magnetic energy spectrum) is also developed. At this time,
an asymmetric reconnection X-point forms at the bottom left. This X-point (X3)
does not develop bi-directional electron outflow jets and therefore cannot be
identified through electron flows. Below we discuss the application of MFT and
the identification of each reconnection X-point.

Figure 1(b) shows the $x$-component of the MFT velocity, $U_{\psi x}$, of the
whole domain at $t/\tau_{0}$=0.12, showing X1 and X2 as well as their mirrors,
and (d) at $t/\tau_{0}$=1.48, showing X3, its mirror and a reconnection
X-point (X4) formed at the center of an evolved, elongated vortex (flux tube).
The factor of $\delta B_{p}^{-1}$ in the definition of $\mathbf{U}_{\psi}$
could tend to infinity at the X- and O-points where $\delta B_{p}$ vanishes.
As a practical step, we add a 1% offset to $\delta B_{p}$ everywhere so that
$\mathbf{U}_{\psi}$ remains finite at the X- and O-points that have vanishing
$\delta B_{p}$. For the range of 0.01–4% offsets, the amplitudes of
$\mathbf{U}_{\psi}$ and $\nabla\cdot\mathbf{U}_{\psi}$ only vary by a factor
of 2. Note that masking the X-points by a grid point of size $\rho_{e}$ yields
similar amplitudes to applying a 1% $\delta B_{p}$ offset. Below we zoom in to
X1–X3 to investigate the X-points more thoroughly.

Figure 2:  Application of the MFT method to X1, a symmetric reconnection
X-point. Plotted quantities are (a) vectors of $\mathbf{U}_{\psi}$, (b) the
divergence of $\mathbf{U}_{\psi}$, (c) vectors of the fluctuating in-plane
electron flow velocity $\delta\mathbf{u}_{e}$, and (d) the divergence of
$\delta\mathbf{u}_{e}$, overlaid with $A_{\parallel}$ contours. The amplitudes
of vectors are denoted by the color and relative length of the arrows. The
divergence of velocity is normalized to $v_{te}/\rho_{e}$.

Figure 3:  Same quantities as Figure 2 plotted for X2, an asymmetric
reconnection X-point.

###  5.1 X1: Symmetric reconnection X-point

Figure 2 shows (a) vectors of $\mathbf{U}_{\psi}$, (b)
$\nabla\cdot\mathbf{U}_{\psi}$, and for comparison, (c) vectors and (d) the
divergence of $\delta\mathbf{u_{e}}$ in a zoomed-in region around X1. Clear
bi-directional outflows and converging inflows of magnetic flux around X1 are
captured in $\mathbf{U}_{\psi}$. (b) $\nabla\cdot\mathbf{U}_{\psi}$ reveals
negative (blue) and positive (red) amplitudes highly localized to X1,
representing converging inward and diverging outward MFT at the X-point. This
is the inherent flux transport pattern of reconnection. It results in a new
quadrupolar structure in $\nabla\cdot\mathbf{U}_{\psi}$. The quadrupolar
structure reflects the bi-directional flux transport at the two sides upstream
and downstream of the X-point. Both quantities are highly localized to the
X-point, and can serve as local signatures of reconnection.

Bi-directional electron outflow jets in the outflow region can be seen in (c)
$\delta\mathbf{u_{e}}$. (d) $\nabla\cdot\delta\mathbf{u}_{e}$ reveals positive
amplitude, representing the diverging outflows. In comparison to
$\mathbf{U}_{\psi}$, the electron outflow develops further from the X-point
and over a much broader region.

###  5.2 X2: Asymmetric reconnection X-point

The same quantities as Figure 2 are plotted around X2 in Figure 3. Similarly,
clear bi-directional inflows and asymmetric bi-directional outflows of
magnetic flux are captured in (a) $\mathbf{U}_{\psi}$, with the downward
transport being stronger. (b) $\nabla\cdot\mathbf{U}_{\psi}$ reveals the
presence of converging inward and diverging outward flux transport as
$\nabla\cdot\mathbf{U}_{\psi}<0$ and $>0$, respectively, at X2. Both signify
active reconnection.

In (c) $\delta\mathbf{u_{e}}$, asymmetric electron outflow jets are seen, with
a stronger downward jet from X2. (d) The divergence of the electron flow
reveals negative and positive amplitudes located broadly around and downstream
from the X-point, representing converging inflows and diverging outflows of
electrons at this X-point.

Figure 4:  Same format as Figure 2 plotted for X3, a reconnection X-point with
only one outflow jet in electrons (and ions), at $t/\tau_{0}=1.48$. Velocity
vectors are measured at the X-point frame.

###  5.3 X3: Reconnection X-point Without bi-directional plasma jets

As the turbulent flows that drive reconnection are significantly dissipated at
late times (Li et al., 2016), reconnection activity becomes weaker than early-
time reconnection. Nevertheless, converging inflows and bi-directional
outflows of magnetic flux are captured in Figure 4(a) $\mathbf{U}_{\psi}$ at
X3. (b) $\nabla\cdot\mathbf{U}_{\psi}$ also reveals positive and negative
amplitudes highly localized to the X-point, sharing a similar quadrupolar
structure to X2 and X1.

On the other hand, the electron flow is highly modified by turbulence. No
clear bi-directional outflow jets are seen in the electrons (or in the ions
(Appendix Figure A2)) at X3. Only one electron and one ion outflow jet are
present. (d) $\nabla\cdot\delta\mathbf{u}_{e}$ also does not show clear
evidence of reconnection. Plasma flows cannot be used for identifying
reconnection at this X-point. However, the MFT method is able to identify
reconnection through its clear inward and outward flux transport at this
X-point, demonstrating the sensitivity of MFT in identifying reconnection
activity in turbulence.

###  5.4 Super-Alfvénic $\mathbf{U}_{\psi}$

While $\mathbf{U}_{\psi}$ is normalized to $v_{te}$, it is meaningful to
compare it with the upstream Alfvén speed. Using the electron plasma
$\beta_{e}\equiv(v_{te}/c_{Ae})^{2}$=0.01, where $c_{Ae}=\sqrt{B_{0}/4\pi
n_{0}m_{e}}$, in the simulation, and estimates of the upstream $\delta
B_{p}/B_{0}\sim$ 0.1 and density $n/n_{0}\sim$ 0.7–1.1 for the three X-points,
we can relate the upstream electron Alfvén speed (Cassak & Shay, 2007) to
$v_{te}$ as $c_{Ae,p}/v_{te}\sim$ 1\. Therefore, at X1 and X2, $U_{\psi}$ is
of order $c_{Ae,p}$. The flux transport velocity is electron Alfvénic.
Similarly, at X3, $U_{\psi}\sim$ 1.2 $c_{A,p}$ is super-Alfvénic. The higher
velocity at early-time reconnection is associated with strong driving by
initial turbulent flows. The Alfvénic velocity at late times is consistent
with undriven reconnection simulations (Liu & Hesse, 2016).
$\mathbf{U}_{\psi}$ is between orders $c_{A,p}$ and $c_{Ae,p}$ based on the
simulation.

Figure 5:  (a) The divergence of MFT of the whole domain at (a)
$t/\tau_{0}$=0.12, showing X1 and X2 (labeled) and their mirrors, and (c)
$t/\tau_{0}$=1.48, showing X3 and X4 (labeled) among the turbulence; (b) and
(d) $\mathbf{J}\cdot\delta\mathbf{E}_{e}^{\prime}$, the non-ideal energy
conversion in the electron frame, at the two times.

###  5.5 Divergence of MFT

Plotted in Figure 5 is the divergence of MFT of the whole domain at (a)
$t/\tau_{0}$=0.12, showing X1 and X2 and their mirrors, and at (c)
$t/\tau_{0}$=1.48, when turbulence is developed, revealing X3 and X4.
$\nabla\cdot\mathbf{U}_{\psi}$ shows significant amplitudes only at the active
reconnection X-points, even among the turbulence. It remains small throughout
the domain, and is thus suitable for the identification of reconnecting
X-points in turbulence. $\nabla\cdot\delta\mathbf{u}_{e}$ is much more
structured throughout the system, and at late times, becomes highly turbulent
(not shown). For comparison, (b)
$\mathbf{J}\cdot\delta\mathbf{E}_{e}^{\prime}$, energy conversion (Zenitani et
al., 2011) in the electron frame, is much more broadly distributed over the
current sheets and throughout the system. (d) At late times, it is dominated
by turbulent flows far away from the reconnection X-points, and thus may not
help in locating reconnection in turbulence. The amplitude of
$\nabla\cdot\mathbf{U}_{\psi}$ is of order 0.1–1 $\Omega_{ce}$ at the three
reconnection X-points.

##  6 Discussion

The flux transport velocity has been generally considered as the E$\times$B
drift velocity. In Equation (1), the slippage between magnetic flux and
electron flow arising from an non-ideal electric field
$\mathbf{E}_{e}^{\prime}$ is included. For the three reconnection X-points,
the slippage provides the major contribution to the inflows and outflows of
magnetic flux near the X-point, being $\sim$2–3 times larger than the
perpendicular electron flow. Further away from the X-point where the the
slippage becomes small, $\mathbf{U}_{\psi}$ follows the perpendicular electron
flow, which is mainly the E$\times$B drift.

$\nabla\cdot\mathbf{U}_{\psi}$ consistently shows a quadrupolar structure at
all reconnection X-points in turbulence. However, a signal is possible at
O-points, where magnetic flux annihilation could happen. This process is
recently explored by MMS (Hasegawa et al., 2020). MFT activity at O-points
deserves future investigation.

A new category of reconnection in turbulence beyond electron-only reconnection
(Phan et al., 2018) is revealed by X3. Only a single electron Alfvénic
electron jet and Alfvénic ion jet are observed at X3. This category has
$\mathbf{U}_{\psi}$ reversals, but no plasma outflow jet reversal. Electron-
only reconnection with only one jet is also reported in simulations of shock-
driven turbulence (Bessho et al., 2020).

##  7 Application to heliospheric plasmas

Application of the MFT method to heliospheric plasmas requires the following
conditions: (i) $k_{\parallel}\ll k_{\perp}$, where ”$\parallel$” is along the
background magnetic field, and (ii) the reconnection magnetic fields primarily
reside on a local reconnection plane. $k_{\parallel}\ll k_{\perp}$ is based on
$k_{\parallel}/k_{\perp}\ll\delta E_{\parallel}/\delta E_{\perp}$ for deriving
$\partial_{t}\psi$ (§2), a condition well satisfied in the simulation.
Equation (1) is then a good approximation of $\mathbf{U}_{\psi}$ even in 3D
systems. Physically, this represents quasi-planar reconnection with parallel
length scales much longer than perpendicular. $k_{\parallel}\ll k_{\perp}$ is
well satisfied in the cascade of kinetic Alfvén wave turbulence (Cho &
Lazarian, 2004; Schekochihin et al., 2009), which is consistent with solar
wind and magnetosheath observations (Alexandrova et al., 2008, 2009; Sahraoui
et al., 2013; Chen, 2016; Chen & Boldyrev, 2017). The model of planar
reconnection is adopted by the local current sheet (LMN) coordinate (Sonnerup
& Cahill Jr., 1967), commonly used in space reconnection observations.
Observations of reconnection in small-scale current sheets in the turbulent
magnetosheath are consistent with this model (e.g. (Phan et al., 2018; Wilder
et al., 2018)). Thus, the conditions for applying MFT is expected to be
realistic for reconnection in heliospheric turbulence. Recent 3D PIC
simulations further show that a long extended X-line, satisfying
$k_{\parallel}\ll k_{\perp}$, easily arising in sub-ion-scale current sheets
in 3D (Li et al., 2020), also favors reconnection activity (Liu et al., 2019;
Huang et al., 2020).

##  8 Conclusion

The MFT method is a new way of identifying reconnection X-points in turbulent
plasmas. It captures bi-directional inflows and outflows of magnetic flux at
the X-points to signify reconnection, even without bi-directional plasma
outflow jets. $\nabla\cdot\mathbf{U}_{\psi}$ is suitable for use in multi-
spacecraft missions such as MMS. The first application to a 2D gyrokinetic
turbulence simulation demonstrates the capability of this method in clearly
capturing active reconnection signatures, as an inflow-outflow pattern or a
quadrupolar structure in $\nabla\cdot\mathbf{U}_{\psi}$. It also reveals a new
category of reconnection in turbulence beyond electron-only reconnection. This
method has the potential to replace the plasma outflow jet reversal signature
for reconnection. Applications to 3D simulations and heliospheric observations
from spacecraft missions will present new opportunities to study the role of
reconnection and identify new types of reconnection in turbulence.

The authors thank Tai Phan, Prayash Sharma Pyakurel and Daniel Verscharen for
fruitful discussions. This work is supported by NSF award AGS-2000222 and NASA
grants 80NSSC18K0754 and MMS mission 80NSSC18K0289.

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Two supplementary figures are available. Figure A1 shows the turbulent cascade
in the dissipation range in the magnetic energy spectrum. Figure A2 shows the
fluctuating in-plane ion flow velocity $\delta\mathbf{u}_{i}$ for X3.

Figure A1:  Evolution of the magnetic energy spectrum as a function of scale,
showing a developed turbulent cascade in the dissipation range.

Figure A2: Ion flow velocity $\delta\mathbf{u}_{i}$ for X3, showing one
outflow jet in ions. Compare with the electron flow velocity in Figure 4(c).
$\delta\mathbf{u}_{i}$ is normalized to the ion thermal speed $v_{ti}$.


# Self-similar shock dynamics satisfying the inviscid Burgers equation in
planar, cylindrical and spherical problems

M. I. Radulescu<EMAIL_ADDRESS>Department of Mechanical Engineering,
University of Ottawa, Ottawa (ON) K1N 6N5 Canada

###### Abstract

The solution of self-similar shock dynamics satisfying the inviscid Burgers
equation are provided in closed form for planar, cylindrical and spherical
problems. The approach follows Lee’s method for obtaining self-similar
solutions for the Euler equations describing compressible fluid dynamics.
Closed form solutions are provided for the two types of self-similar
solutions: of the first kind, where the shock dynamics are constrained by
integral relations, and the second kind, constrained by the internal
requirement of the regularity of solutions along limiting characteristics. The
solutions obtained illustrate simply the theoretical underpinning of Taylor-
Sedov blast waves (self-similarity of the first kind) and Guderley implosion
problems (self-similarity of the second kind).

inviscid Burgers’ eauation, similarity solutions, shock waves

###### pacs:

boh

##  I Introduction

The inviscid Burgers’ equation Burgers (1948) for a scalar $c$, namely,

$\displaystyle\frac{\partial c}{\partial t}+c\frac{\partial c}{\partial x}=0$
(1)

is a single-sided non-linear wave equation which serves as a prototype for
studying problems of compressible flows and fluid dynamics alike, since it
contains the convective non-linearity of the Euler equations of inviscid flow.
The convection speed $c$ in front of the space derivative being the same as
the transported variable inside the time and space partial derivatives, gives
rise to wave steepening in compressive regions leading to shock formation.

There are a wide range of problems where the Euler equations degenerate into
Burgers equation describing their dynamics. In gas dynamics, weak shocks are
described by the inviscid Burgers equation Chandrasekhar (1943); Whitham
(1974). Reactive compressible flow problems also often degenerate into
reactive forms of the inviscid Burgers equation Bdzil and Stewart (1986);
Clavin and Williams (2002); Faria, Kasimov, and Rosales (2015). Ad hoc
extensions of the inviscid Burgers equation have also been proposed as toy
mathematical models to explore certain non-linear aspects of the dynamics of
compressible reactive flows Fickett (1979); Radulescu and Tang (2011);
Kasimov, Faria, and Rosales (2013); Mi and Higgins (2015).

In the present communication, we wish to derive the similarity solutions for
shock waves satisfying the inviscid Burgers equation. They will be the
analogue of the well known Taylor-Sedov Taylor (1950); Sedov (1959) point
blast explosion problems of compressible flows and the Guderley solution for
self-similar shock implosion Guderley (1942). We follow the systematic method
used by Lee to unify such problems of self-similar shock propagation in
perfect gases Lee (2016). To the best of our knowledge, only explosion
solutions have been proposed by Mi et al. for the planar problem modelled by
the inviscid Burgers equation, using the method of characteristics Mi and
Higgins (2015).

##  II The Burgers-Fickett equation

The general continuity equation is

$\displaystyle\frac{\partial\rho}{\partial t}+\frac{\partial q}{\partial x}$
$\displaystyle=-q\frac{j}{x}$ (2)

with $j$=0, 1 or 2 for planar, cylindrical and spherical geometries
respectively. The density of the quantity being conserved is $\rho$ and $q$ is
its flux (transfer rate of the quantity conserved per unit time per unit
area).

Following Fickett Fickett (1979, 1985), the simplest model for describing gas
dynamics in a toy model is to model the flux $q$ as a function of $\rho$
alone. The simplest model for the flux is the Burgers flux given by
$\frac{\rho^{2}}{2}$. The resulting non-linear wave equation becomes the
Burgers-like equation

$\frac{\partial\rho}{\partial t}+\rho\frac{\partial\rho}{\partial
x}=-\frac{\rho^{2}}{2}\frac{j}{x}$ (3)

We present similarity solutions to this model equation. This also serves to
illustrate the solution methodology that can be used for extensions to this
model for other flux functions in kinematic waves of material flow (e.g.,
traffic flow, pedestrian flow, flood waves, etc….) as pioneered by Lighthill
and Whitham Lighthill and Whitham (1955a, b).

##  III Dynamics of Burgers’ shocks in similarity variables

We are concerned with problems of shock dynamics. We will consider the problem
of strong shocks, where $\rho=0$ ahead of the shock. Let $X(t)$ denote the
shock’s trajectory and $\dot{X}(t)$ its speed. We introduce the similarity
variables

$\displaystyle\zeta=\frac{x}{X(t)}$ (4)
$\displaystyle\phi=\frac{\rho}{\dot{X}(t)}$ (5)

We change variables from $(x,t)$ to
$(\zeta(x,t)=\frac{x}{X(t)},t^{\prime}(t)=t)$. With this change of variables,
partial derivatives become:

$\displaystyle\left(\frac{\partial}{\partial t}\right)_{x}$
$\displaystyle=\left(\frac{\partial}{\partial\zeta}\right)_{t^{\prime}}\left(\frac{\partial\zeta}{\partial
t}\right)_{x}+\left(\frac{\partial}{\partial
t^{\prime}}\right)_{\zeta}\left(\frac{\partial t^{\prime}}{\partial
t}\right)_{x}$ (6)
$\displaystyle=-\zeta\frac{\dot{X}}{X}\left(\frac{\partial}{\partial\zeta}\right)_{t^{\prime}}+\left(\frac{\partial}{\partial
t^{\prime}}\right)_{\zeta}$ (7)

and

$\displaystyle\left(\frac{\partial}{\partial
x}\right)_{t}=\left(\frac{\partial}{\partial\zeta}\right)_{t^{\prime}}\left(\frac{\partial\zeta}{\partial
x}\right)_{t}=\frac{1}{X}\left(\frac{\partial}{\partial\zeta}\right)_{t^{\prime}}$
(8)

With these transformations and seeking solutions where $\phi=\phi(\zeta)$
only, (3) can be re-written as an ordinary differential equation:

$\dot{\phi}=-\frac{\frac{j}{2}\frac{\phi^{2}}{\zeta}+\theta\phi}{\phi-\zeta}$
(9)

provided

$\theta\equiv\frac{\ddot{X}(t)X(t)}{\dot{X}(t)^{2}}$ (10)

is not a function of $t$. Since it is not a function of $\zeta$ by
construction, it can only be a constant. We have thus reduced the problem to
determining $\theta$ and the distribution $\phi(\zeta)$ given by (9).

Since we can re-write (10) as:

$\displaystyle\theta=\frac{\ddot{X}X}{\dot{X}^{2}}=\frac{\ddot{X}/\dot{X}}{\dot{X}/X}=\frac{\mathrm{\ln\dot{X}}}{\mathrm{\ln
X}}=\mathit{constant}$ (11)

we obtain immediately that the shock speed depends on distance as a power law:

$\displaystyle\dot{X}=BX^{\theta}$ (12)

where $B$ is an integration constant.

The shock decay coefficient $\theta$ is problem dependent. Below, we
illustrate how to obtain $\theta$ in two types of self-similar problems of the
Burgers equation: for similarity solutions of the first and second kind. We
draw from blast wave theory of gas dynamics Lee (2016).

##  IV Similarity solutions of the first kind: shock motion given release of
material at a point source

The first problem we wish to solve is the analogous problem to the Taylor-
Sedov problem of blast waves, where the energy addition in a gas at a point,
along a line or along a plane gives rise to a decaying strong shock followed
by a self-similar profile of the field variables (speed, density and
pressure). For the Burgers equation, the analogous problem is thus the sudden
release of fixed material $m_{j}$ in a medium initially at $\rho=0$. $m_{1}$
is the amount of material per unit area in the planar problem, $m_{2}$ the
material per unit length for the cylindrical problem and $m_{3}$ the material
for the spherical problem. We wish to determine the law of shock decay and the
self-similar distribution of density behind the shock $\rho(x,t)$ satisfying
(3) for the planar, cylindrical and spherical problems.

The point, line or plane release problems posed requires the material engulfed
by the decaying shock to remain constant, since $\rho=0$ ahead of the shock,
i.e.,

$\displaystyle\int_{0}^{X(t)}\rho k_{j}x^{j}\mathrm{d}x=m_{j}$ (13)

where $m_{j}$ is a constant. The geometrical factor $k_{j}$ takes values
$4\pi$, $2\pi$, and $1$ for spherical, cylindrical and planar waves. In terms
of the similarity variables, and using (12), we obtain

$\displaystyle m_{j}=BX(t)^{\theta+j+1}I_{j}$ (14)

where

$\displaystyle I_{j}=\int_{0}^{1}\phi k_{j}\zeta^{j}\mathrm{d}\zeta$ (15)

Since $m_{j}$ is not a function of $t$, the exponent of $X(t)$ needs to vanish
for a self-similar solution. This requires

$\displaystyle\theta=-(1+j)$ (16)

The blast decay coefficient $\theta$ takes values of -1, -2 and -3 for planar,
cylindrical and spherical problems respectively. We also now relate the mass
deposition to the integral $I_{j}$:

$\displaystyle m_{j}=BI_{j}$ (17)

Note that the mass integral permitted us to determine the shock decay profile
for the kinematic wave considered. In the Taylor-Sedov solution, the integral
determining the blast solution is the energy integral of the total energy
engulfed by the shock, made equal to the energy deposition.

For decaying shock waves from $X(t=0)=0$, (12) can be further integrated to
obtain

$X=Ct^{N}$ (18)

with the constants $C=\left(B(1-\theta)\right)^{N}$ and
$N=\frac{1}{1-\theta}$. Spherical blast wave radius $X$ grows like $t^{1/4}$,
cylindrical waves grow like $t^{1/3}$ and planar waves grow like $t^{1/2}$.

The speed of the shock satisfies the shock jump condition:

$\dot{X}=\frac{[\frac{1}{2}\rho^{2}]_{0}^{s}}{[\rho]_{0}^{s}}=\frac{1}{2}\rho$
(19)

where $[\alpha]_{0}^{s}=\alpha_{s}-\alpha_{0}$, such that

$\phi(\zeta=1)=2$ (20)

With this boundary condition and the blast decay coefficient $\theta$ known,
the self-similar profile $\phi(\zeta)$ satisfying (9) is readily found. By
inspection, the solution is

$\phi=2\zeta$ (21)

for the planar, cylindrical and spherical geometries alike, as it satisfies
(9) identically.

With $\phi(\zeta)$ known, the constant $I_{j}$ given by (15) can readily be
found

$\displaystyle I_{j}=\frac{2k_{j}}{j+2}$ (22)

It takes values of 1, $4/3\pi$ and $2\pi$ for planar, cylindrical and
spherical waves respectively. The constant $B$ in the self-similar solution
becomes $m_{1}$, $\frac{3m_{2}}{4\pi}$ and $\frac{m_{3}}{2\pi}$ for planar,
cylindrical and spherical waves respectively. The self-similar shock motion
given by (12) becomes

$\displaystyle\mathrm{planar:}\;$ $\displaystyle\dot{X}=m_{1}X^{-1}$ (23)
$\displaystyle\mathrm{cylindrical:}\>$
$\displaystyle\dot{X}=\left(\frac{3m_{2}}{4\pi}\right)X^{-2}$ (24)
$\displaystyle\mathrm{spherical:}\;$
$\displaystyle\dot{X}=\left(\frac{m_{3}}{2\pi}\right)X^{-3}$ (25)

Figure 1: Space-time diagram of the spherical explosion problem, with $m=1$;
blue line is the shock trajectory, black solid lines are contours of $\rho$,
black dashed lines are contours of the similarity variable $\zeta$ and red
lines are characteristics.

Fig. 1 illustrates the solution for the spherical explosion problem.
Interestingly, the interior solution is not a function of the initial mass
addition, and is given by $\rho=\frac{x}{2t}$; only the shock trajectory is.
The trajectory of characteristics is also shown; these are simple power laws
by integrating $\mathrm{d}x/\mathrm{d}t=\rho$. The region bounded by the shock
is the domain of dependence of the lead shock motion, with characteristics
catching up to the lead shock. The planar and cylindrical problems behave
similarly. The solution obtained for the planar problem agrees with the one
provided by Mi et al. Mi and Higgins (2015).

##  V Similarity solutions of the second kind: self-similar shock implosion

The second problem we will solve is one in which the shock decay coefficient
$\theta$ cannot be obtained from an integral relation of conservation of mass,
momentum or energy. In blast wave theory Lee (2016), these are said to be
similarity solutions of the second kind. We now solve the problem of shock
implosion for the cylindrical and spherical problems, i.e., the Guderley
problem of shock dynamics Guderley (1942).

Self-similarity still requires power law solutions for the shock dynamics
given by (12) with the distribution behind the shock given by (9). Inspection
of (9) suggests that the profiles will be singular when the denominator
vanishes, i.e, when $\zeta=\phi$. A regular solution without blow-up
corresponds to requiring that the numerator of (9) vanishes simultaneously at
this point. Denoting the internal point when this is satisfied by a *, we have
$\zeta^{*}=\phi^{*}$. Setting the numerator to zero yields immediately

$\theta=-j/2$ (26)

The profiles can now be obtained by direct integration of the ODE’s (9). With
the appropriate boundary condition at the shock $\phi(1)=2$, one obtains the
wave structure solution:

$\displaystyle\phi=\frac{2}{\zeta}\;\mathrm{for}\>j=2\mathrm{(spherical)}$
(27)
$\displaystyle\phi=\frac{2}{\sqrt{\zeta}}\;\mathrm{for}\;j=1\mathrm{(cylindrical)}$
(28)

The saddle point in the solution lies at $\zeta^{*}=2^{1/2}$ for spherically
imploding waves and at $\zeta^{*}=2^{3/2}$ for cylindrically imploding waves.

While the solution is complete, it is worth further elucidating the meaning of
the regularization criterion used to obtain the solution and its link to the
underlying wave motion dynamics of Burgers’ equation. Burgers equation (3) is
a hyperbolic equation with a single family of characteristics curves given

$\displaystyle\frac{\mathrm{d}x}{\mathrm{d}t}=\rho$ (29)

which can be re-written as:

$\displaystyle\frac{\mathrm{d}x}{\mathrm{d}t}=\phi\dot{X}$ (30)

A line of $\zeta=\mathit{constant}$ corresponds to

$\mathrm{d}\zeta(x,t)=\left(\frac{\partial\zeta}{\partial
x}\right)_{t}\mathrm{d}x+\left(\frac{\partial\zeta}{\partial
t}\right)_{x}\mathrm{d}t=0$ (31)

i.e.,

$\frac{\mathrm{d}x}{\mathrm{d}t}=-\frac{\left(\frac{\partial\zeta}{\partial
t}\right)_{x}}{\left(\frac{\partial\zeta}{\partial x}\right)_{t}}$ (32)

Evaluating the partial derivatives from (5), the last equation can be re-
written as:

$\frac{\mathrm{d}x}{\mathrm{d}t}=\zeta\dot{X}$ (33)

By comparing (30) and (33), this signifies that the locus of $\zeta=\phi$
corresponds to the location where a characteristic surface coincides with a
constant $\zeta$ line. This is the limiting characteristic, serving as an
event horizon for the dynamics of the imploding shock wave. Along the limiting
characteristic, $\phi$ remains constant by virtue of the right hand side of
(9) vanishing, this requires the nominator to vanish. Within the structure of
the imploding shock wave, non-linearity leads to wave compression waves
according to Burgers eauation, amplified by geometrical terms involving $j$ in
(9). The situation is thus perfectly analogous to the gas-dynamics problem of
Guderley.

Interestingly, the solutions for the implosion problem leads to interior
solutions independent of time, bounded by time-dependent shock motion and by
the time-dependent limiting characteristic. Fig. 2 shows the solution for the
spherical implosion problem. The density field is given by $\rho=2B/x$. The
trajectory of characteristics are given by simple power laws by integrating
$\mathrm{d}x/\mathrm{d}t=\rho$. The limiting characteristic is the one
coincident with the $\zeta=2^{3/2}$ curve. This separates the characteristics
affecting the implosion dynamics from the ones not affecting it. The behaviour
is thus perfectly analogous to the Guderley solution of gas dynamics.

The cylindrical implosion problem is qualitatively similar. The density field
is given by $\rho=2B/\sqrt{x}$.

Figure 2: Space-time diagram of the spherical implosion problem, with $B=-1$;
blue line is the shock trajectory, black solid lines are contours of $\rho$,
black dashed lines are contours of the similarity variable $\zeta$ and red
lines are characteristics.

##  VI Conclusions

We have derived closed form solutions form self-similar shock dynamics
satisfying the inviscid Burgers equation. Self-similar shock dynamics of the
first kind can be obtained by integral relations, while the second kind
requires the requirement of analyticity on the limiting characteristic. We
recover perfectly analogous results to those of the inviscid Euler equations,
generalizing the Taylor-Sedov blast wave problem (first kind) and Guderley
problem of shock implosion (second kind) in a transparent manner. The method
can be extended to other one-way non-linear wave equations modelling kinematic
waves in a straightforward way.

## Acknowledgements

This work was supported by AFOSR grant FA9550-23-1-0214, with Dr. Chiping Li
as program monitor. The author also acknowledges financial support from NSERC
Discovery Grant ”Predictability of detonation wave dynamics in gases:
experiment and model development”.

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