# Simulation of the spatial shift in detector response for polarized protons
within a calorimeter

A. Blitstein<EMAIL_ADDRESS>B. Wojtsekhowski  Department of Physics
and Astronomy, University of North Carolina, Chapel Hill, North Carolina,
27599, USA Thomas Jefferson National Accelerator Facility, Newport News,
Virginia 23606, USA

###### Abstract

Measurement of the helicity dependent elastic electron-proton scattering cross
section provides a key means of investigating parity violation within the
proton. However, such measurements exhibit potential instrumental effects
associated with the detection of polarized recoiled protons. In particular,
spin-orbit interactions within a massive detector induce a systematic spatial
shift in the detector signal. In this study, we determine the size of this
shift using the Geant4 simulation toolkit. For a typical hadron calorimeter,
we found a polarization dependent shift on the order of 0.01-0.1 mm, multiple
orders of magnitude smaller than the typical spatial resolution seen in
hadronic calorimeters. Additionally, we provide the custom modifications
required of the Geant4 source code to implement the quasi-elastic scattering
of polarized protons incident on nuclei in the detector. The modifications are
readily extendable to generic matter sources, and can be used for the study of
additional spin dependent observables in Geant4.

###### keywords:

Polarized Protons , Spin-Orbit Interaction , Geant4

###### PACS:

25.30.Bf , 13.40.Gp , 14.20.Dh

††journal: NIM A

##  1 Introduction

Much of what is currently known about the structure of the proton has been
ascertained from electron scattering experiments [1, 2, 3, 4]. In such
experiments, a beam of electrons is incident on a proton target. By measuring
the distribution of kinematic variables associated with the scattered
particles, such as their momentum and energy, one obtains data on various
proton form factors. These form factors are then used to infer the spatial
distribution of quarks and gluons within the proton. In this way, the use of
polarized beams and polarized targets allow one to probe the origin of the
spin of the nucleon.

The observation of the small value of the quark polarization in the nucleon
(“spin crisis”) [5] raises a question about the strange quark contribution to
the spin of the nucleon. The parity violation present in elastic electron-
proton scattering allows for the determination of the proton form factors
related to the strange quark [6]. In such experiments the cross section beam
helicity asymmetry is measured. The detected particles are also polarized,
which could require systematic corrections to the measurement results on
account of spin-dependent interactions within the detector. These instrumental
effects become significant for the detection of polarized recoiled protons.

Consequently, it is necessary to ensure that these spin dependent interactions
that occur within the detector do not collude to disrupt the determination of
the coordinate measurement of the polarized protons by the detector. It is
known that polarized protons are affected by a spin-orbit interaction when
scattering off nuclei in matter [7, 8]. This spin-dependent, quasi-elastic
scattering can be accounted for via the introduction of an empirically known
asymmetry in the azimuthal distribution of scattered protons. In many cases,
this data is already available. Here, we make use of data [9] for hydrocarbon
based targets obtained with a standard hadron sampling calorimeter.

The specific motivation for the current study is a possible experiment
proposed in Ref. [10]. In what follows, we present a polarization-dependent
prediction for the expected size of the spatial shift in detector signal
induced by spin-orbit interactions within the proposed calorimeter. To do so,
simulations were run in Geant4, a Monte-Carlo based toolkit for simulating
nuclear physics that is sourced in C++ [11]. Currently, generic Geant4 has no
default implementation for many of the spin-dependent interactions that are
necessary to accurately simulate parity violating effects in nuclear physics.
Consequently, it was necessary to introduce custom modifications to the source
code of Geant4 associated with the modeling of the elastic scattering of
protons incident on nuclei in matter. This implementation is readily
generalizable to account for alternative modifications to the azimuthal part
of the scattering statistics, and can be used in further Geant4 studies
involving spin dependent observables (such as those seen in Ref. [12]).

##  2 The Propagation of Polarized Protons in Matter

Due to their interaction with the incident beam of polarized particles, the
protons that undergo elastic scattering with said particles are polarized as
well. In particular, for elastic $e-p$ scattering in the single photon
exchange approximation, the transverse polarization $P_{t}$ of the recoiled
protons for a longitudinally polarized electron beam is:

$P_{t}=-2h\,P_{e}\sqrt{\tau(1+\tau)}\,\,\tan(\frac{\theta_{e}}{2})\frac{G_{{}_{E}}\cdot\,G_{{}_{M}}}{G_{{}_{E}}^{2}+(\tau/\epsilon)G_{{}_{M}}^{2}},$
(1)

where $P_{e}$ is the electron beam polarization, $h=\pm 1$ the beam helicity,
$\theta_{e}$ the electron scattering angle, $\tau=Q^{2}/4m_{p}^{2}$ ($Q^{2}$
the negative four-momentum transfer squared, $m_{p}$ the proton mass),
$\epsilon$ the virtual photon polarization parameter, and $G_{{}_{E}}$ and
$G_{{}_{M}}$ the electric and magnetic form factors [4].

To first order, when polarized protons elastically scatter off nuclei in
matter, they experience a spin-orbit interaction [7, 8]. Depending on the path
they take around a given nucleus, the direction of their angular momentum
relative to the nucleus differs, as seen in Fig. 1. If the spin orbit
interaction is such that it prefers anti-alignment of proton spin with its
angular momentum, it will preferentially scatter off the nucleus in the
direction that fulfills that requirement (the top/red trajectory in Fig. 1).
For polarized protons in matter, it is empirically known that protons do
preferentially scatter in the direction that promotes anti-alignment of their
spin and angular momentum, the same direction as their spin axis crossed with
their incident direction (see Ref. [7] for a summary of the spin-orbit
interaction for nucleon-nucleus scattering).

Figure 1: Diagram illustrating the spin-orbit interaction felt by polarized
protons elastically scattering off nuclei in matter. It is known empirically
that protons prefer to take the path with $\bm{L}\parallel-\bm{P}$, indicated
by the top/red trajectory. Color online.

The elastic scattering of polarized protons incident on spinless nuclei
depicted in Fig. 1 has a few key features which must be considered when
modeling the interaction. First, the spin orbit piece to the interaction
should not modify the distribution of the scattering angle, $\theta$, for
scattered protons. This piece of the scattering statistics reflects the
typical treatment of protons elastically scattering off nuclei, providing an
important benchmark that our modified interaction should reproduce.
Accordingly, all that will be changed is the distribution associated with the
azimuthal angle, $\phi$, which sets the scattering direction in the plane
normal to the incident direction.

For standard proton-nucleus elastic scattering, the azimuthal distribution is
taken to be uniform, with no preference in direction for scattering. To add in
such a preference, we will need to introduce an asymmetry in the azimuthal
distribution for scattered protons. Taking the z-axis to be along the incident
direction and proton spin to be measured along the y-axis, the scattering
angular distribution takes the form

$N(\theta,\phi)=N_{0}(\theta)\left(1+A_{y}P_{y}\cos\phi\right),$ (2)

where $N_{0}(\theta)$ is the angular distribution in the absence of proton
polarization [9]. The side (transverse) polarization, $P_{y}$, is defined as

$P_{y}=\frac{N_{y}^{\uparrow}-N_{y}^{\downarrow}}{N_{y}^{\uparrow}+N_{y}^{\downarrow}},$
(3)

where $N_{y}^{\uparrow/\downarrow}$ is the total number of spin-up/down
protons with respect to the y-axis. One can also view the side polarization as
twice the average y-component of incident proton spin. The other variable that
appears in Eq. 2, the analyzing power $A_{y}$, is there for the sole purpose
of accounting for any additional azimuthal dependence seen in
experiments/theory.

For our purposes, we are primarily interested in the elastic scattering of
protons incident on nuclei within a plastic scintillator. For protons incident
with lab frame momentum $\vec{p}_{\text{lab}}=p_{\text{lab}}\,\,\hat{z}$, it
is known from experiment that the analyzing power is well approximated by

$A_{y}(\theta)=\frac{\sum_{n=1}^{4}c_{n}\left(p_{\text{lab}}\sin\theta\right)^{n}}{p_{\text{lab}}},$
(4)

where $c_{1}=3.02\pm 0.13$, $c_{2}=-7.33\pm 0.66$, $c_{3}=6.17\pm 1.11$, and
$c_{4}=-1.74\pm 0.59$, with $p_{\text{lab}}$ in GeV/c (see Ref. [9]).

Looking at Eq. 2, we see that the effect of the added asymmetry in the
azimuthal distribution is to preferentially scatter protons in the direction
corresponding to $\phi=0$ (where the extra term has a maximum) while making it
less likely to scatter in the direction with $\phi=\pm\pi$ (where the extra
term has a minimum). Further, the additional factor averages over
$\phi\in(-\pi,\pi)$ to 1, thereby not changing the net scattering angle part
of the angular distribution.

##  3 Geant4 Custom Modifications

To model the elastic scattering of polarized protons in matter, simulations
were run in Geant4. Geant4 is a toolkit composed of C++ source files intended
for simulating nuclear physics via Monte Carlo techniques (see Ref. [11] for
more than what is presented here). Interactions between particles are handled
in a generic way by associating with each an interaction length that goes as
the inverse of the interaction cross section, thereby allowing for modular
adjustments to its code.

A generic Geant4 simulation is called a run, which is composed of a specified
number of events, or repetitions of the user defined initial conditions. Each
event is then split up into multiple tracks for each of the primary particles
and any secondaries that are generated. Then, each track is updated in small
increments called steps, implementing dynamic changes to each of the particles
with a mix of at rest (e.g. radioactive decay), continuous (e.g. ionization),
and discrete (e.g. pair production) processes.

The typical Geant4 user need only modify a few concrete instances of the core
base classes which guide the workflow of a typical simulation. Changes to
detector geometry and materials are handled in a concrete instance of the
G4VUserDetectorConstruction class. Particle definitions and their interactions
are set in an instance of the G4VUserPhysicsList class. Finally, event
initialization, such as setting the initial particles and their kinematic
properties, is specified in an instance of the G4VUserActionInitialization
class. Though the aforementioned classes are the only mandatory classes the
user should create, additional classes can be defined, which allow one to
interface with the particles at the beginning and end of each run, event,
track, and step. These are set in concrete instances of the G4UserRunAction,
G4UserEventAction, G4UserTrackingAction, and G4UserSteppingAction base classes
respectively.

While users usually only need to associate predefined physics processes with
the particles they involve, this fails when Geant4 does not have a default
implementation for the interaction of interest. As it turns out, this is the
case for the elastic scattering of polarized protons incident on nuclei. To
implement the interaction, it is necessary to directly modify the source code
containing the default implementation of hadron elastic scattering,
G4HadronElasticProcess.cc.

Within the default code, the azimuthal angle is chosen randomly from -$\pi$ to
$\pi$. In order to account for the asymmetry present in Eq. 2, we instead need
to sample $\phi$ from the following normalized probability density function

$F(\phi)=\frac{1}{2\pi}\left(1+A_{y}P_{y}\cos\phi\right).$ (5)

In the modified code, this is accomplished by first integrating $F(\phi)$ to
get the cumulative distribution function,

$Q(\phi)=\int_{-\pi}^{\phi}F(\phi^{\prime})\differential{\phi^{\prime}}=\frac{1}{2}+\frac{\phi}{2\pi}+A_{y}P_{y}\sin\phi.$
(6)

Then, a random sample $\phi_{s}$ from $F(\phi)$ can be obtained by first
taking a random sample $x_{s}$ from the uniform distribution on $[0,1]$,
$\mathcal{U}(0,1)$, followed by solving the implicit equation

$x_{s}=Q(\phi_{s})=\frac{1}{2}+\frac{\phi_{s}}{2\pi}+A_{y}P_{y}\sin\phi_{s},$
(7)

for $\phi_{s}$ such that $\phi_{s}=Q^{-1}(x_{s})$. In the modified code, Eq. 7
is solved using the bisection method with an accuracy of $\pi/2^{11}$. We find
the sampling of $\phi_{s}$ from $F(\phi)$ via the bisection method induces a
10% increase in computation time relative to the default code. As we only
require a few days of computation time to produce sufficiently small
systematic uncertainty, we choose not to employ a more efficient sampler such
as the standard rejection method.

Though we now have a procedure to sample $\phi$ from an asymmetrical
distribution, we need to be careful about what azimuth $\phi=0$ corresponds
to. In Sec. 2, this was the azimuth of the cross product of the incident
proton’s spin axis with its incident direction. Geant4, however, has a
different process for assigning this azimuth, one that rotates the coordinate
system that the scattering analysis is performed in differently each time.
This analysis frame, in which the scattering angles are set, is defined to be
the right handed coordinate system with the z-axis along the direction of
propagation of the proton immediately before the scattering event, and the
x-axis chosen such that the proton’s direction at the start of the simulation
is contained within the xz plane of this new coordinate system. This still
leaves us with two possible choices for the direction of the x-axis, so the
unique analysis frame is specified as the one in which the inner product of
the unit vector along the x-axis with the proton direction at the start of the
simulation is positive. We note that the proton direction at the start of the
simulation and the proton direction immediately before the scattering event
will generally differ due to continuous Geant4 processes. Refer to Fig. 2 for
an example of this choice of analysis frame.

Figure 2: Depiction of the analysis frame used to set scattering angles in
Geant4. Color online.

In Geant4, three vectors are defined as instances of the G4ThreeVector class.
To rotate vectors from the analysis frame into the default lab frame, the
public member function rotateUz(indir) is used, where indir is a normalized
G4ThreeVector pointing in the direction of the proton immediately before
elastic scattering. To determine the azimuth corresponding to $\phi=0$ in the
analysis frame, we will need to transform the direction along which the spin
of the proton is measured from the lab frame into the analysis frame. This is
encoded by the inverse of the rotation accomplished by rotateUz(indir), which
had to be added into the code as it was not one of the default G4ThreeVector
member functions. More specifically, if
$\bm{v}=(v_{1},v_{2},v_{3})^{\text{T}}$ is a vector and indir =
$(u_{1},u_{2},u_{3})^{\text{T}}$ a unit vector, we can represent the action of
rotateUz(indir) on $\bm{v}$, v.rotateUz(indir), as

$\matrixquantity(\frac{u_{1}u_{3}}{u_{1}^{2}+u_{2}^{2}}&-\frac{u_{2}}{u_{1}^{2}+u_{2}^{2}}&u_{1}\\\
\frac{u_{2}u_{3}}{u_{1}^{2}+u_{2}^{2}}&\frac{u_{1}}{u_{1}^{2}+u_{2}^{2}}&u_{2}\\\
-u_{1}^{2}-u_{2}^{2}&0&u_{3})\matrixquantity(v_{1}\\\ v_{2}\\\
v_{3})=\bm{R}\bm{v}$ (8)

upon which we added in the inverse rotation rotateUzInv(indir) as follows:

$\matrixquantity(\frac{u_{1}u_{3}}{u_{1}^{2}+u_{2}^{2}}&\frac{u_{2}u_{3}}{u_{1}^{2}+u_{2}^{2}}&-u_{1}^{2}-u_{2}^{2}\\\
-\frac{u_{2}}{u_{1}^{2}+u_{2}^{2}}&\frac{u_{1}}{u_{1}^{2}+u_{2}^{2}}&0\\\
u_{1}&u_{2}&u_{3})\matrixquantity(v_{1}\\\ v_{2}\\\ v_{3})=\bm{R}^{-1}\bm{v}.$
(9)

Since we take proton polarization to be measured along the y-axis of the lab
frame, we first get the components of the $\hat{y}$ unit vector in the
analysis frame using Eq. 9. Taking a cross product of this vector with a unit
vector along the z-axis in the analysis frame then gives a vector who’s
azimuth corresponds to $\phi=0$ when sampling from Eq. 5. To take this into
account, we find the current azimuthal angle of this new vector in the
analysis frame, add it to our sampled $\phi_{s}$ from before, and set that as
the new azimuthal angle in the analysis frame. After some (default) post
processing, the scattered proton momentum is rotated back into the lab frame
through Eq. 8 and changes to the kinematic properties of the particles
involved are set by the Geant4 code.

In order to actually carry out these changes, one must ensure they are
changing the G4ThreeVector outdir that is being passed to the
ProposeMomentumDirection() member function associated with the variable
theTotalResult, which gets returned by the
G4HadronElasticProcess::PostStepDoIt() process. Accordingly, the changes to
outdir can be made anywhere between its initial definition and setting of the
proposed momentum direction in the default G4HadronElasticProcess.cc source
code.

Besides the modified hadron elastic scattering code, the other standard
interactions and particles needed to simulate protons incident on calorimeters
are included. In particular, we use the physics lists defined within the
default FTFP_BERT.cc/.hh files. Doing so accounts for electromagnetic and
synchroton physics, particle decays, hadronic physics, stopping physics, ion
physics, and cuts for tracking neutrons.

##  4 Benchmarks with Constant Analyzing Power

With the modifications to the code for elastic proton scattering made, it is
important to benchmark our code with experimental results to show that
everything is working correctly and that the results properly reflect reality.
First, the scattering angle part of the differential scattering cross section
should be unchanged. To check this, we send in a beam of protons with initial
momentum 3.8 GeV/c and side polarization 1 onto a 51.6 g/cm2 block of
ethylene, a common plastic scintillator. We then record the distribution of
the component of momentum transverse to the incident proton direction,
$p_{t}=p_{\text{lab}}\sin\theta$, where $p_{\text{lab}}$ is the magnitude of
the lab frame proton momentum and $\theta$ the scattering angle. Carrying out
a run with one million events, we arrive at the following distribution in Fig.
3, which agrees qualitatively with empirical data recorded by Azhgirey et al.
in Ref. [9]. We omit a graphical comparison with Ref. [9] due to the inability
of Geant4 to reproduce the exact experimental conditions used in the
reference.

Figure 3: Histograms depicting the number of outgoing protons observed with
transverse momentum $p_{t}$. Both were normalized by the total number of
particles and bin size to produce the differential efficiency on the y-axis.
These histograms are the result of simulating one million protons with initial
momentum 3.8 GeV/c and side polarization 1 scattering off of a 51.6 g/cm2
block of ethylene, which agree qualitatively with empirical results in Ref.
[9].

In Fig. 3, it should be noted that the scattering peak at low $p_{t}$ is
primarily due to multiple-scattering within the block of ethylene, whereas the
gradual drop off afterwards is due to elastic scattering [9]. To analyze only
the elastic scattering statistics, we need to enforce a low $p_{t}$ cut in the
data. For protons incident with a few GeV of energy, it suffices to cut out
data with $p_{t}<0.1$ GeV/c.

To check that our code produces an asymmetry of the correct size in the
azimuthal distribution of protons that are elastically scattered off nuclei in
the block of ethylene, we test the code with three representative values of
the analyzing power. In particular, we generate histograms for the azimuthal
angle of scattered protons with $A_{y}$ set to 0.5, 0.15, and 0.05, which are
shown in Fig. 4. To determine the best fit analyzing power, we perform a least
squares fit to the data by choosing $C$ and $A_{y}$ such that

$\sum_{i=1}^{\text{\\# of
Bins}}\left[N(\phi_{i})-C(1+A_{y}P_{y}\cos\phi_{i})\right]^{2}=\text{minimum}.$
(10)

The resulting fits (shown in Fig. 4) yielded analyzing powers of $0.54\pm
0.02$, $0.16\pm 0.02$, and $0.06\pm 0.02$ respectively for the simulations run
with $A_{y}=$ 0.5, 0.15, 0.05. Since all of the fits agree within 1-2 standard
deviations, we conclude that the azimuthal asymmetry has been properly
implemented.

Figure 4: Histograms depicting the number of outgoing protons observed with
azimuthal angle $\phi$ overlayed with least squares fits, in red, to determine
the analyzing power. For each plot, $10^{5}$ protons were simulated with
initial momentum 3.8 GeV/c and side polarization 1 with constant analyzing
power set to 0.5, 0.15, and 0.05 respectively from left to right. Color
online.

As these checks were made for analyzing powers over the full range of values
predicted by the empirical parameterization in Eq. 4, we proceed with a direct
substitution of $A_{y}(\theta)$ where before we had $A_{y}$ equal to a
constant.

##  5 Results

With the code properly checked and the analyzing power parameterized by Eq. 4,
the shift in detector signal can now be quantified. To mimic as close as
possible the calorimeter that would be used for the proposed measurement of
the strange form factor of the proton in Ref. [10], we model the longitudinal
structure of our test detector off of the HCAL-J detector at JLab [13], which
is similar to the one developed in Ref. [14]. In particular, the test detector
consists of a 15 by 15 array of $5\times 5\times 100$ cm3 modules. Each module
is a sampling calorimeter with 40 layers composed of 1.5 cm of Fe and 1 cm
plastic scintillator (vinyltoluene). Energy deposits are collected only within
the plastic scintillator layers, and position is resolved with a spatial
segmentation of 5 cm (see Fig. 5).

Figure 5: Schematic of the detector used for simulations with a single
detection event depicted. The plastic scintillator layers are shaded orange,
and iron layers shaded silver. The incident proton is colored in blue, and
generated photons are colored in green. Color online.

To estimate the shift in detector signal, 10 million 2.5 GeV protons with side
polarization 1 were incident on the center of the detector. For each event,
the first moment, or center, of energy deposit was computed, with the results
accumulated in histograms shown in Fig. 6.

Figure 6: Histograms depicting the center of detector signal for 10 million
2.5 GeV protons with side polarization 1 incident on the described sampling
calorimeter. Both distributions were fit to Gaussians shown in red. Color
online.

The histograms in Fig. 6 were fit to Gaussians, yielding a mean center of
detector signal of $\bar{x}=0.021\pm 0.001$ cm and $\bar{y}=0.000\pm 0.001$
cm. The reported uncertainties only account for statistical uncertainty,
neglecting all systematic effects. As expected, the shift is in the direction
of the polarization axis crossed with the incident proton direction, or the
positive x-axis.

When $A_{y}\,P_{y}<<1$, the shift in detector signal is approximately
proportional to the side polarization. Thus, for an arbitrary side
polarization $P_{y}$, the expected shift is approximately

$\Delta x\approx P_{y}\times(0.21\pm 0.01)\text{ mm}.$ (11)

##  6 Discussion and Conclusions

Ultimately, we found a polarization dependent shift in detector signal on the
order of 0.01-0.1 mm given by Eq. 11 for polarized protons incident on a
calorimeter. This result is three orders of magnitude smaller than the typical
spatial resolution of 2-5 cm in hadronic calorimeters. This small value of the
polarization dependent shift is vital for parity violation scattering
measurements, as it implies that cuts made in data analysis neglecting this
shift in detector signal are still consistent at the level of precision
required, for instance, in the proposed experiment in Ref. [10].

Among other things, these results allow us to begin work on a proposal to the
JLab PAC and work on experimental design for a measurement of the strange form
factor of the proton at a momentum transfer $Q^{2}$ of 3 (GeV/c)2. If shown to
be large, we will be provided with yet another clue about the internal
dynamics of the quark-gluon sea residing within the proton. Such information
is useful for ab initio treatments of the proton and could pave the way for
future advances in proton physics.

Additionally, the process by which changes were made to the Geant4 source code
in this study are readily generalizable to similar changes that would be
needed for empirical corrections to other processes. This would allow for
similar spin dependent effects with known azimuthal dependence to be added to
Geant4. Such changes are especially useful for the simulation of other parity
violating effects, a growing sub-branch of nuclear physics studies [15].

##  7 Acknowledgments

This work was made possible by the US DOE SULI program at Thomas Jefferson
National Accelerator Facility. We would like to extend gratitude towards Dr.
John Annand, Dr. Xinzhan Bai, and Dr. Vardan Khachatryan for assistance with
Geant4. This work was supported in part by the US DOE Office of Science and
Office of Nuclear Physics under the contracts DE-AC02-05CH11231, DE-
AC02-06CH11357, and DE-SC0016577, as well as DOE contract DE-AC05-06OR23177,
under which JSA, LLC operates JLab.

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# Laminar and Turbulent Plasmoid Ejection in a Laboratory Parker Spiral
Current Sheet

Ethan E. Peterson1<EMAIL_ADDRESS>Douglass A. Endrizzi2  Michael
Clark2  Jan Egedal2  Kenneth Flanagan2  Nuno F. Loureiro1  Jason Milhone2
Joseph Olson2  Carl R. Sovinec3  John Wallace2  and Cary B. Forest2  1Plasma
Science and Fusion Center, MIT, Cambridge, MA 02139, USA 2Department of
Physics, University of Wisconsin–Madison, Madison, WI 53706, USA 3Engineering
Physics Department, University of Wisconsin–Madison, Madison, WI 53706, USA

###### Abstract

Quasi-periodic plasmoid formation at the tip of magnetic streamer structures
is observed to occur in experiments on the Big Red Ball as well as in
simulations of these experiments performed with the extended-MHD code, NIMROD.
This plasmoid formation is found to occur on a characteristic timescale
dependent on pressure gradients and magnetic curvature in both experiment and
simulation. Single mode, or laminar, plasmoids exist when the pressure
gradient is modest, but give way to turbulent plasmoid ejection when the
system drive is higher, producing plasmoids of many sizes. However, a critical
pressure gradient is also observed, below which plasmoids are never formed. A
simple heuristic model of this plasmoid formation process is presented and
suggested to be a consequence of a dynamic loss of equilibrium in the
high-$\beta$ region of the helmet streamer. This model is capable of
explaining the periodicity of plasmoids observed in the experiment and
simulations and produces plasmoid periods of 90 minutes when applied to 2D
models of solar streamers with a height of $3R_{\odot}$. This is consistent
with the location and frequency at which periodic plasma blobs have been
observed to form by LASCO and SECCHI instruments.

##  1 Introduction

Over the past few decades, in-situ measurements of the solar wind have
produced an enormous amount of data that can be used to characterize
properties of the solar wind and to discover its source regions on the Sun.
One of the earliest characterizations was the observation of “fast” and “slow”
streams of wind as measured by Mariner II (Neugebauer & Snyder, 1962).
However, later on it was discovered that the slow wind was better
characterized by the charge state ratios of oxygen — indicating a much higher
electron temperature in the source region (Neugebauer et al., 2016; Fu et al.,
2017; Cranmer et al., 2017), consistent with the temperatures and charge
states in well confined coronal loops. This led to the understanding that the
slow solar wind likely originates from transport between closed flux and open
flux in the equatorial streamer belt and that it can only occur via magnetic
reconnection. This formation process is drastically more complex than the fast
wind acceleration in coronal holes, which agrees with the original Parker
solution (Parker, 1958) and produces a relatively quiescent, supersonic flow
with photospheric abundances. Invoking magnetic reconnection in the formation
mechanism of the slow wind inherently leads to a dynamic process capable of
explaining its high degree of variability. However, spontaneous magnetic
reconnection is difficult to achieve in high Lundquist number plasmas and so a
mechanism with enough free energy for facilitating or driving the reconnection
must be included in the theory. To this end, a number of theories have been
postulated including “interchange reconnection” (Crooker et al., 2000; Fisk et
al., 1998; Fisk, 2003), the S-Web theory (Antiochos et al., 2011; Higginson &
Lynch, 2018; Antiochos et al., 2007), and streamer top reconnection (Einaudi
et al., 2001, 1999; Lapenta & Knoll, 2005; Endeve et al., 2004; Wu et al.,
2000).

Specifically with regards to streamer top reconnection, a number of
computational studies have attempted to recreate these periodic density
structures (PDSs). This process can be driven by instabilities in the coronal
loop or by converging flows at the streamer cusp (Einaudi et al., 2001, 1999;
Lapenta & Knoll, 2005; Wu et al., 2000) and has also revealed that two-fluid
effects can alter plasmoid characteristics emanating from helmet streamers
(Endeve et al., 2003, 2004). These multi-fluid simulations prescribe a fixed
amount of coronal heating at the base of the helmet streamer, which results in
a dynamic system with no equilibrium that oscillates periodically. However,
the periodicity of the plasmoids in these simulations is longer than that
observed, $\sim$15-20 hours.

With the improvements to imaging diagnostics on many of the current satellite
missions (SOHO, STEREO, Parker Solar Probe), as well as novel data analysis
techniques, increasingly smaller and more dynamic features are constantly
being revealed in the solar wind (DeForest et al., 2018; Bale et al., 2019).
One example of this pertains specifically to the slow solar wind and the
observation of PDSs, also known as plasma blobs or plasmoids, that are
released into the solar wind at the tips of helmet streamers (Wang et al.,
1997; Sheeley, Jr. et al., 1997; Lavraud et al., 2020). Running difference
calculations of white light images produced by SOHO’s Large Angle and
Spectrometric Coronograph (LASCO) (Brueckner et al., 1995) reveal bipolar
signatures indicative of small scale structures propagating outwards into the
solar wind (Wang et al., 1998). Recently these PDSs have also been identified
by the SECCHI instrument suite onboard STEREO (Viall & Vourlidas, 2015), in
old Helios data (Di Matteo et al., 2019), and during Parker Solar Probe’s
first orbit (Lavraud et al., 2020). They have also been observed to have
magnetic signatures (Rouillard et al., 2011) and be the product of magnetic
reconnection at the open-closed flux boundary of helmet streamers (Kepko et
al., 2016; Sanchez-Diaz et al., 2019).

While the work of Viall & Vourlidas (2015) shows that blobs form at or below
$2.5R_{\odot}$ and have a typical period of 90 minutes with a range of 65-100
minutes, other work suggests that blobs can also form at larger radii
($2-6R_{\odot}$) and have longer periods of a few hours (Wang & Hess, 2018;
Wang et al., 1998). The implied correlation between these observations is that
when helmet streamer tips are closer to the Sun they release higher frequency
PDSs, and lower frequency PDSs when they are further away. This hypothesis is
consistent with the observations of a wide number of variable discrete
frequencies that are observed in the slow solar wind at 1AU over the course of
the solar cycle (Viall et al., 2008).

Aside from the presence of multiple coherent frequencies of observed PDSs, it
is also well understood that the heliospheric current sheet (HCS) — and the
solar wind as a whole — is extremely turbulent (Bavassano et al., 1997;
Coleman et al., 1968; Bavassano & Bruno, 1989a, b; Luttrell & Richter, 1987;
Belcher & Davis, 1971; Marsch & Tu, 1990a, b; Bale et al., 2019) and evolves
as a function of distance from the Sun (Bavassano et al., 1982). Even though
fully developed turbulence is typically observed to exist by $0.3R_{\odot}$ in
the slow wind near the HCS, it is often not enough to completely decorrelate
the coherent PDS fluctuations generated in the corona as they are routinely
observed to drive magnetospheric fluctuations at 1AU (Viall et al., 2008;
Kepko et al., 2002; Stephenson & Walker, 2002; Kepko & Spence, 2003).

The conclusion from this plethora of observational insight is that any
mesoscale model that wishes to accurately describe the acceleration of the
solar wind near the magnetic equator must be able to produce these coherent
fluctuations embedded in a turbulent background that evolves as it travels
away from the source. It is also necessary that the frequencies be on the
proper time scales and that the fluctuations are consistent with density and
magnetic signatures indicative of plasmoids.

In this article we report on experimental observations of axisymmetric
plasmoid ejection via helmet streamer tip reconnection in a laboratory Parker
Spiral. We discuss a potential plasmoid formation mechanism and observations
of plasmoid scaling properties by comparisons between experimental
measurements and extended-MHD simulations performed with the two-fluid
modeling capabilities of the NIMROD code (Sovinec et al., 2004; Sovinec &
King, 2010). These scalings indicate higher frequency plasmoids for more
elongated streamers with thinner current sheets, as well as an evolution
toward turbulence further downstream as plasmoids of many sizes interact.
Lastly, a simple heuristic model is presented suggesting that the periodicity
of plasmoids in the experiments and simulations and those produced in the
solar wind is set by a dynamic transition from a state of quasi-equilibrium to
one where no such equilibrium exists — a process we will refer to as
equilibrium loss. It is important to note that the helmet streamers produced
in the lab and those in the solar wind represent drastically different
systems; the former is governed by Hall-MHD and exists on sub-ion scales,
whereas the latter is much better described by ideal MHD and exists on scales
much, much larger than any kinetic scale. This article does not make any
claims about the relationship of the dominant transport mechanisms between the
two systems, nor does it claim that the underlying physics of the subsequent
reconnection events are the same. The simple association is made that both
systems exhibit sonic outflows that result in periodic plasmoid ejection from
the tip of the helmet streamers and that the loss of equilibrium responsible
for this phenomenon can be driven by pressure gradients and magnetic curvature
both in the experiments and in the solar wind.

##  2 Experimental and Simulation Methodologies

The Big Red Ball (BRB) at the Wisconsin Plasma Physics Laboratory is a
versatile plasma confinement device well-suited to the study of high-$\beta$
and flow-dominated systems. The capabilities of the BRB are discussed in more
detail in other works (Forest et al., 2015; Olson et al., 2016; Flanagan et
al., 2020; Endrizzi et al., 2021) and the experimental setup of the Parker
Spiral solar wind experiments and initial findings are detailed in Peterson et
al. (2019). A depiction of the experimental configuration of the BRB for these
experiments as well as the computational model of the experiment are shown in
Fig. 1. The summary of the experimental methodology for generating the Parker
Spiral in the BRB is as follows: Thermally emissive Lanthanum-Hexaboride
cathodes are used to generate a warm, dense, unmagnetized plasma atmosphere
($T_{e}\sim 7$ eV, $n_{e}\sim 4\times 10^{17}$ m-3). A permanent dipole magnet
is placed inside this background plasma with two ring electrodes located near
its north and south poles. Current is driven from molybdenum anodes in the
plasma atmosphere into the dipole magnetosphere and collected on the polar
electrodes. This current path is visualized in Fig. 1(b) in the context of the
NIMROD simulation domain. These cross field currents generate a torque on the
plasma which causes the magnetosphere to rotate, thereby twisting the magnetic
field into a Parker Spiral.

Figure 1: Experimental (a) and computational (b, c, d) configurations. The Big
Red Ball is shown in (a) along with the local cylindrical coordinate system,
dipole magnet, and diagnostics used for mapping the magnetosphere. The 2D
finite element mesh with current injection boundary conditions and initial
vacuum field configuration are shown in (b). Panels (c) and (d) show the
location of probes within the NIMROD simulation for outputting high time
resolution field measurements. Both the initial magnetic field configuration
(c) and time-averaged magnetic field configuration after the current injection
has reached steady state (d) are shown to demonstrate the probe positions
relative to where the plasmoid formation process occurs.

At the interface between the closed field lines of the magnetosphere and the
open field lines of the Parker Spiral, periodic reconnection occurs, ejecting
axisymmetric plasmoids, much like the density structures observed in the
heliospheric current sheet which emanate from streamer top reconnection. A
number of diagnostics were employed to map the 2D time dynamics of the Parker
Spiral including two arrays of 3 axis Hall sensors and an array of triple
probes and 2D Mach probes for measuring density, temperature, and flows in the
plasma. One of the Hall sensor arrays was kept stationary and displaced
azimuthally from the 2D scanning plane to provide phase reference measurements
critical for the reconstruction of the plasmoid dynamics.

The computational domain and vacuum magnetic field for the accompanying NIMROD
simulations are shown in Fig. 1. Figure 1(b) represents the hemispherical mesh
in cylindrical coordinates, which extends down to $R=5$ cm rather than the
experiment’s support rod at $R=2$ cm. This is to allow for a small dipole
magnet to be placed outside the computational domain and to facilitate the
boundary condition manipulation to model current injection/extraction in a
manner representative of the experiment. By prescribing $B_{\varphi}$ along
the boundary as a function of time, we can set the normal component of $J$, or
the current into and out of the vessel. In all the presented simulation work
that follows, the current injection linearly ramps from zero up to a
prescribed steady state value. The ramp duration in some of the initial
simulations was 1 ms, but was shortened to 100 $\mu$s to reduce the required
simulation time for some of the higher current injection cases that have
smaller time steps. For all simulations discussed in this work, there are zero
heat flux and zero particle flux conditions applied to the boundary as well as
a no-slip boundary condition on the velocity. All simulations were performed
with experimental parameters of $n_{\mathrm{e}}=4\times 10^{17}$ m-3,
$T_{\mathrm{e}}=7$ eV, $T_{\mathrm{i}}=0.5$ eV, which give viscous and
resistive diffusivities of $\nu=50$ m${}^{2}/$s and $\eta=35$ m${}^{2}/$s
respectively. We also note that in the experiments and simulations the ion
skin depth is set to $d_{i}=70$ cm. In terms of resolution, the simulations
were performed with 2400 bicubic poloidal elements which provides centimeter-
scale resolution in the current sheet.

In order to better compare results from simulation to the experimental
measurements, the NIMROD code was modified in order to take a list of R, Z
coordinates for placing history nodes, or probes. For the simulations in this
study, four probes were placed in the current sheet at $R=25,40,55,70$ cm and
$Z=0$ cm as shown in Fig. 1. The probe locations relative to the vacuum
magnetic flux configuration and the flux distribution near the end of a
simulation can be seen in Fig. 1(c) and Fig. 1(d) respectively. These probes
provide outputs for the solution fields after every time step to allow for
higher frequency phenomena to still be captured at a few select locations in
the simulation. All NIMROD simulations presented in this work are
axisymmetric.

##  3 Observation of Streamer Top Reconnection and Plasmoid Formation in
Experiment and Two-Fluid Simulations

While the basic operating parameters, mean magnetic field, and plasma flows
that develop in both the experiment and simulations are presented in Peterson
et al. (2019), this article sheds more light on the details of the fluctuation
measurements as well as their scaling properties. Previous work demonstrated
that, during the generation of the Parker Spiral, two fluid effects are
critical as the system size is small compared to the ion skin depth, the
electrons are the only magnetized species, and $T_{e}\gg T_{i}$. The
consequence of this is a radially outward flow of electrons that advects the
dipolar magnetic field into a Parker Spiral and simultaneously generates an
inward electric field via the Hall effect. This Hall electric field drives
accretion of ions into the magnetosphere where a density and pressure gradient
begin to build until a loss of equilibrium occurs. In both experiment and
simulation, this loss of equilibrium occurs in the current sheet associated
with the Parker Spiral where $\beta>10$ (Peterson et al., 2019) and increases
up to $\beta\sim 50$ further downstream.

Figure 2: Time histories and frequency content of magnetic signals measured in
the Parker Spiral current sheet by 3-axis Hall sensors in the experiment. The
difference between the turbulent, non-axisymmetric Phase I and axisymmetric,
single mode Phase II can be seen in (a) where two probes at the same radius,
but different azimuthal angles have very different mean field values as well
as high frequency characteristics in (a.i.), but are nearly identical in Phase
II (a.ii). A spectrogram of one of these time histories (b) shows broadband
fluctuations in Phase I, followed by a coherent downward chirping spectrum of
laminar plasmoid ejection.

Measurements of the magnetic field as well as the plasma density show
interesting fluctuations whose frequency depends on the amount of current
injected. As shown in Fig. 2(a), the vertical component of the magnetic field,
$B_{Z}$ (the poloidal component perpendicular to the current sheet) is non-
axisymmetric and highly uncorrelated in the initial, high current phase of the
experiment (Phase I). However, as shown in Fig. 2(a.ii), these magnetic
fluctuations become coherent as the current injection falls with the
characteristic timescale set by the RC circuit of the current injection system
($\sim 100$ms). In addition, the $B_{Z}$ fluctuations are bi-polar and thus
suggestive of closed loops of magnetic flux disconnected from the inner
magnetosphere of the rotating plasma. A spectrogram typical of these magnetic
fluctuations in the current sheet can be seen in Fig. 2(b) which shows the
turbulent nature of the high current Phase I and coherent mode Phase II with a
high degree of correlation between the plasmoid frequency and current
injection of the system. This scaling relationship will be discussed in
subsequent sections.

Figure 3: Plasmoids are ejected from the helmet streamer cusp at a frequency
of 20 kHz when the current injection is 150 A in the experiment. Panel (a)
shows the magnetic field ($B_{Z}$) and density fluctuation ($\tilde{n}_{i}$)
signals as measured by the probe in the current sheet at $R=42$ cm and denoted
by the teal dot in panels (b), (c), and (d). Panels (b), (c), and (d) show the
flux map and density perturbation map at three successive 10 $\mu$s time steps
corresponding to the magenta, cyan, and yellow vertical lines in (a). These
panels show how the magnetic field at the streamer cusp expands outwards with
higher density plasma until reconnection occurs, releasing a plasmoid into the
current sheet.

Performing phase correlation measurements between the stationary Hall array
and the scanning array over many discharges allows us to reconstruct the
plasmoid dynamics — both magnetic field and density — through conditional
averaging. This reconstruction method first identifies 500 $\mu$s time windows
for every shot that exhibit the highest degree of shot-to-shot similarity as
measured by a single stationary hall probe. A center frequency of 20 kHz was
used for this process as it showed the best correlation shot-to-shot between
1.1 and 1.15 seconds as shown in Fig. 2(b). Once the best time window from
each shot is selected, a phase shift is calculated for each shot using the
stationary reference probe which allows us to align in time the many different
shots at different locations and build up a poloidal map of the 2D plasma
dynamics. This process reveals periodic reconnection occurring near the top of
the streamer structures resulting in plasmoids being ejected into the Parker
spiral current sheet as shown in Fig. 3. Figure 3(a) shows the $B_{Z}$
measurement as well as ion density fluctuations $\tilde{n}_{i}$ as measured by
the probe located at ($R=42$cm, $Z=-8$cm) and shown as a cyan dot in Fig.
3(b-d). We believe the up-down asymmetry of the current sheet, or “droop”, is
likely due to preferential current draw to an anode in the southern hemisphere
that was larger than the others, which is not captured in simulations since
the current injection is uniform over the range of $\pm 15^{\circ}$ latitude.
Figure 3(b-d) show subsequent time steps of 10$\mu$s through one half period
of the reconnection process, which exhibits a periodic build up of plasma
density (and pressure) inside the streamer leading to field line stretching,
reconnection, and plasmoid ejection.

In addition to plasmoids observed experimentally, they also manifest in
extended MHD simulations when the Hall term and electron pressure gradient
term are used in Ohm’s law. 2D cross sections of the magnetic flux and density
fluctuations from the NIMROD simulations as well as the time resolved signals
in the current sheet as would be measured by probes in the experiment or
satellites in space are shown in Fig. 4 and Fig. 5. The time histories of both
experimental and laminar plasmoids can be seen in movies published in previous
work (Peterson et al., 2019), whereas the evolution of the turbulent plasmoids
in Fig. 5 can be viewed in the movie turbulent_plasmoids.mp4. Figure 4
represents a moderate current injection level of 400 A in the simulation which
is slightly above the threshold necessary to observe plasmoids. As a result,
we observe relatively large single plasmoids emitted with a very regular
frequency. We refer to these plasmoids as laminar since the magnetic flux
evolves very smoothly with regular ejection of similarly sized plasmoids.

Figure 4: Laminar plasmoid formation in an axisymmetric NIMROD simulation with
400A of injected current. The time history of $B_{Z}$ and ion density
fluctuations as measured by a probe at $R=70$cm (top panel) shows the periodic
ejection of high density plasmoids that are roughly similar size over time.

On the other hand, Fig. 5 represents a high current drive case of 1000 A where
we see plasmoids of many sizes interacting in a thinner current sheet
resulting in a more turbulent medium downstream.

Figure 5: Turbulent plasmoid formation in an axisymmetric NIMROD simulation
with 1000A of injected current. The time history of $B_{Z}$ and ion density
fluctuations as measured by a probe at $R=70$cm (top panel) shows the high
variability of plasmoids in both frequency and size.

The observation of plasmoids in the experiment as well as in the NIMROD
simulations begs the question of whether this occurrence is coincidence or if
the same physical processes are driving plasmoid formation in both cases.
Importantly, theoretical and computational studies (Bhat & Loureiro, 2018), as
well as experiments (Hare et al., 2017) have demonstrated that plasmoid
formation can occur at values of the Lundquist number much below the usual
$10^{4}$ required in resistive MHD when ion-scale kinetic effects are not
negligible, as is the case here. We now turn our attention to a discussion of
the properties and scaling relationships for these observed plasmoids.

##  4 Discussion of Laminar and Turbulent Plasmoid Formation, Properties, and
Scalings

We begin by discussing the characteristics of these plasmoids, particularly
with respect to the amount of current injected into the experiment (or
simulation). Shown in Fig. 6 are the time-averaged magnetic field
configurations in the simulations (a-c) and experiment (d-f) as a function of
increasing current injection. As more flux is advected outwards with the
electron flow, it results in a current sheet thinning effect as shown in Fig.
6, where the thinner current sheets display larger magnetic curvature near the
streamer cusp. As shown in both Fig. 7 and Fig. 8, these elongated streamers
associated with higher current injection values result in higher frequency
plasmoid ejection. However, plasmoids are not present for every current
injection value in the simulation and experiment. This is shown in Fig. 7
which shows $B_{Z}$ power spectral densities from four different simulations
with increasing amounts of current injection: 200A, 400A, 600A, and 1000A for
panels (a), (b), (c), and (d), respectively. We can see that no plasmoids are
present in the 200A simulation, likely because the accretion caused by the
Hall effect is not strong enough to build up pressure above the critical
gradient necessary for loss of equilibrium.

Figure 6: Mean-field evolution of the magnetic field as a function of current
injection is shown for axisymmetric NIMROD simulations (a-c) as well as the
experiment (d-f). The top row depicts both the time-averaged poloidal and
toroidal magnetic field at three different current injection values: 250 A,
600 A, and 1000 A, from left to right, respectively. In this progression, it
is clear that the current sheet becomes thinner and the toroidal magnetic
field increases as the poloidally injected current is increased. The same is
true in the experiment as shown in the bottom row, where the current injection
values are 150 A, 250 A and 350 A, respectively. For currents larger than 350
A in the experiment, the magnetic field is highly non-axisymmetric; as a
result, axisymmetric flux surface reconstruction is not possible.

Another characteristic that trends with increased system drive (or current
injection) is a decrease in coherence of the fluctuations and increase in
turbulence. In Fig. 7(b) we can see a well defined fundamental frequency at 15
kHz as well as multiple resolved harmonics. However, as the current is
increased in panels (c) and (d), the spectra become more broadband and the
fundamental mode increases in frequency, which is consistent with the
experimental observations reported in Peterson et al. (2019).

Figure 7: $B_{Z}$ fluctuation power spectra from four probes in four different
two-fluid NIMROD simulations with current values of 200A, 400A, 600A, and
1000A shown in panels (a), (b), (c), and (d), respectively. Probes 1-4 are
located at increasing radii in the current sheet as according to Fig. 1.
Increasing the injected current increases both the fundamental plasmoid
frequency as well as the amplitudes of higher frequency components.
Fluctuations are increasingly more broadband at larger radial distances as
well.

One more important observation from Fig. 7 is that field lines at inner radii
(closer to probe 1) are more coherent — that is, the dominant mode is much
stronger relative to the other higher frequencies. The physical interpretation
of this phenomenon is that the pressure inside the magnetosphere drives a
periodic loss of equilibrium that manifests as magnetospheric oscillations
that drive larger fluctuations further out in the current sheet where the
field is weaker, ultimately resulting in a turbulent current sheet that still
has a quasi-periodic nature.

A comparison of the plasmoid frequencies in simulation to those observed in
the experiment is shown in Fig. 8(a) as a function of current (orange and
green triangles). In Fig. 8(a) each experimental data point represents the
peak value in the $B_{Z}$ frequency spectrum from a single Hall probe in 10 ms
windows between 0.9 and 1.4 seconds. This generates 50 data points per shot
and is plotted for roughly 100 shots. The simulation results (blue circles)
show the dominant frequency for the duration of the simulation once the
current injection has reached steady state. Each data point for the NIMROD
simulations therefore represents a single simulation at a single current
injection value. It is important to note the strong linear scaling at modest
current injection up to $\sim 400$ A, as well as the abrupt disappearance of
plasmoids below 100 A and $\sim 10$ kHz. The high density of data points at
very low frequency for currents above $\sim 300$ A, are not particularly
germane to this discussion and just indicate that the frequency range with the
highest power spectral density was found to be at low frequencies during the
non-axisymmetric phase of the experiment and can be seen as well in the Phase
II portion of the spectrogram in Fig. 2(b). Also plotted is the fundamental
plasmoid frequency from the Hall-MHD NIMROD simulations as a function of
current. Both experimental and simulation plasmoid frequencies scale linearly
with the current, but with different slopes.

Figure 8: $B_{z}$ fluctuation peak frequencies in the current sheet at $R=30$
cm for Helium and Argon discharges as well as the frequencies of plasmoids
present in the NIMROD simulations (a). Plasmoids in both experiment and
simulation scale linearly with the injected current and exhibit some
stabilizing effect such that no plasmoids exists below roughly 10 kHz in
either the experiment or simulations. When the current injection value is
translated into a characteristic timescale dependent on the magnetic curvature
and pressure gradient at that current value, the plasmoid frequencies in both
experiment and simulation are found to scale similarly (b).

Since the plasmoid frequencies scale more strongly with current injection in
the experiment than in simulation, the drive mechanism is likely correlated
with some quantity other than the current, but likely influenced by it. One
possible explanation is that these plasmoids are pressure driven and that the
current drive in the experiment produces larger densities in the magnetosphere
as a result of ionization, which is not modeled in the simulations. Therefore,
calculating a pressure-curvature driven time scale for the loss of equilibrium
in experiment and simulations may provide a unifying scaling, as evidenced by
Fig. 8(b). Each data point for the experiment in Fig. 8(b) is derived from a
2D flux surface reconstruction and maps of pressure, and temperature averaged
over 10 ms windows. Where the flux surface reconstruction is possible (roughly
from $t=1.0$ s to $t=1.2$ s), we obtain an estimate of the characteristic
frequency $f=\frac{1}{2\pi}\sqrt{c_{s}^{2}{\bf\it\kappa}\cdot\gradient{p}/p}$
calculated near the helmet streamer tip at $R\sim 30$ cm. This results in
$\sim 20$ data points whose error in the x-direction is calculated from the
error in the magnetic field, temperature, and pressure measurements and whose
error in the y-direction represents the full width at half maximum of the peak
in the $B_{z}$ power spectral density. The theoretical model and justification
for this characteristic time scale are explained in the following section.

##  5 Heuristic Model of Plasmoid Evolution and Extrapolation to Solar
Streamers

A relatively simple heuristic model can be constructed for the plasma
expansion in the high-$\beta$ transition region between the hydrostatic
equilibrium of the closed flux corona and the hydrodynamic equilibrium of open
field lines outside the heliospheric current sheet (HCS). The geometry of such
a system is shown in Fig. 9. Writing the momentum equation in terms of the
magnetic curvature vector, ${\bf\it\kappa}$, we obtain the dynamic equation
for the plasma expansion:

$\rho\derivative{\mathbf{v_{\perp}}}{t}=-\gradient_{\perp}{p}+\frac{B^{2}}{\mu_{0}}{\bf\it\kappa}-\gradient_{\perp}\frac{B^{2}}{2\mu_{0}}.$
(1)

Figure 9: A cartoon drawing of helmet streamer topology with indications of
the zones where equilibrium exists; namely, deep in the streamer where there
is a hydrostatic solution, and outside the current sheet where Parker’s
original hydrodynamic solution is valid. The interface between the two
exhibits no equilibrium due to the particle and heat sources present in the
corona. This loss of equilibrium is caused by the extremely high values of
$\beta$ in this region and result in quasi-periodic plasma blobs which are
driven by strong pressure gradients. The length scale $L_{c}$ shown above is
the critical length scale of the current sheet that develops from field line
stretching before reconnection occurs.

Using the equation of state $p=\rho c_{s}^{2}$, rewriting the right hand side
of Eq. 1 in terms of the plasma $\beta=2\mu_{0}p/B^{2}$, and dotting both
sides with the curvature vector, ${\bf\it\kappa}$, gives:

${\bf\it\kappa}\cdot\derivative{\mathbf{v_{\perp}}}{t}=-\frac{c_{s}^{2}}{p}{\bf\it\kappa}\cdot\gradient_{\perp}\left[p\left(1+1/\beta\right)\right]+\frac{2c_{s}^{2}}{\beta}\kappa^{2}.$
(2)

The left-hand side of this equation can be taken to define a characteristic
timescale for the driven loss of equilibrium:
${\bf\it\kappa}\cdot\derivative{\mathbf{v_{\perp}}}{t}\sim-\gamma_{dr}^{2}$.
This can be understood as the time required to form a plasmoid of radius
$R_{c}=\kappa^{-1}$ due to the acceleration provided by the net force on the
right-hand side of Eq. 2. As discussed before, the values of $\beta$ in the
current sheet are very large for both the experiment and simulations and range
from $10<\beta<50$ throughout the region of interest, denoted as the
high-$\beta$ dynamic region in Fig. 9. Therefore, ignoring terms of order
$\beta^{-1}$ results in

$\gamma_{dr}^{2}\sim\frac{c_{s}^{2}}{p}{\bf\it\kappa}\cdot\gradient{p}.$ (3)

This characteristic time scale can be understood as the time for a sound wave
to traverse a distance equivalent to the geometric mean of the pressure
gradient scale length and radius of curvature, or equivalently the free-fall
time of a plasma parcel under the action of a pressure gradient and adverse
magnetic field curvature.

The next step in the derivation of the plasmoid frequency is to show that it
is essentially this drive frequency which sets the frequency of reconnection
in the current sheet. It will be shown that in this situation the reconnection
rate is relatively insensitive to the details of the tearing mode growth rate.
We can assume that the current sheet is lengthening at the rate given by the
drive timescale $\gamma_{dr}$ because the plasma is frozen to the magnetic
field. We can also assume that the lengthening of the current sheet is
exponential in time: a result of a transition from quasi-equilibrium to a
dynamic system. Incompressibility then requires that the forming sheet is
likewise thinning exponentially, such that its thickness $a(t)$ can be
described by

$a(t)=a_{0}e^{-\gamma_{dr}t},$ (4)

where $a_{0}$ is the thickness at the beginning of the expansion.

As the aspect ratio $L/a$ of this forming current sheet increases, it becomes
unstable to a progressively broader spectrum of tearing modes. As argued by
Uzdensky & Loureiro (2016), one of those modes — the one whose growth rate,
$\gamma_{tear}(t)$, first satisfies $\gamma_{tear}(t_{crit})\sim\gamma_{dr}$ —
will eventually grow to become as wide as the forming sheet, thereby
disrupting it and leading to plasmoid ejection. This happens at a so-called
critical time, $t_{crit}$, whereupon the sheet thickness is

$a_{crit}\equiv a(t_{crit})=a_{0}e^{-\gamma_{dr}t_{crit}}.$ (5)

This can be inverted to yield

$t_{crit}=\gamma_{dr}^{-1}\ln{\left(\frac{a_{0}}{a_{crit}}\right)}.$ (6)

As we can see, the critical time for reconnection to occur is essentially the
same as the drive timescale, $\gamma_{dr}^{-1}$, since the ratio of the
initial to the critical current sheet thickness represents only a logarithmic
correction. That is, while reconnection is essential to the formation and
ejection of plasmoids, the physics of the reconnection onset are such that
details of the tearing instability that underlies it (such as the functional
form of $\gamma_{tear}(t)$) are not essential to the prediction of the
timescale associated with the plasmoid ejection.

Casting the drive timescale for $\gamma_{dr}$ into a characteristic frequency
in Hz we obtain

$f=\frac{1}{2\pi}\sqrt{c_{s}^{2}{\bf\it\kappa}\cdot\gradient{p}/p}.$ (7)

This time scale is associated with a high-$\beta$ pressure-curvature driven
loss of equilibrium. The characteristics of these oscillations — namely that
they are electromagnetic, axisymmetric perturbations localized to the region
of bad magnetic curvature with a frequency dependent on the pressure gradient
— support the notion that these plasmoids are driven by a mechanism similar in
nature to flute modes, but likely represent a loss of equilibrium due to
particle and heat transport into the streamer rather than a linear
instability. It is the frequency in Eq. 7 that empirically provides the
unifying scaling between the experimental results and the extended-MHD
simulations presented in Fig. 8. Computing this frequency along field lines
from the density, temperature and magnetic flux measured by diagnostics in the
experiment as well as the simulation shows a local maximum located at the
outboard midplane where the curvature is largest. Plotting the measured
plasmoid frequencies against this calculated pressure-curvature frequency
gives the results in Fig. 8(b) which shows much better agreement between the
experiment and simulations and is consistent with the idea that these
plasmoids are pressure driven.

If we normalize both the pressure gradient scale length and the magnetic
curvature by the critical current sheet length, $L_{c}$, we obtain two
dimensionless quantities: $\ell_{b}^{-1}=L_{c}\kappa$ and
$\ell_{p}^{-1}=L_{c}\gradient{p}/p$. We can assume, in the laminar plasmoid
case, that this critical length, $L_{c}$, is simply the plasmoid length just
as reconnection occurs. Substituting these parameters into Eq. 7 provides us
with the dimensional scaling below:

$f=\frac{1}{2\pi}\frac{c_{s}}{L_{c}\sqrt{\ell_{b}\ell_{p}}}.$ (8)

Given similar normalized length scales, $\ell_{b}$ and $\ell_{p}$, the scaling
between experimental and solar wind frequencies is simply the ratio of
critical length scales, or plasmoid size, and sound speeds; i.e.:

$f_{sw}\sim\frac{L_{c,exp}}{L_{c,sw}}\frac{c_{s,sw}}{c_{s,exp}}f_{exp}.$ (9)

As mentioned before, we will use the plasmoid length as the proxy for $L_{c}$
as it is reasonable to assume for single plasmoids that the associated
plasmoid is roughly the size of the current sheet just before it reconnects.
For plasmoids in the simulations and experiment, we will take this length
scale to be $L_{c,exp}=0.25$ m and in the solar wind we will take this scale
to be $L_{c,sw}=1R_{\odot}=7\times 10^{8}$ m which is consistent with
observations of plasmoids appearing around $2-4R_{\odot}$ and having a length
of $1R_{\odot}$ and width of $0.1R_{\odot}$(Sheeley et al., 2009). Combining
these plasmoid length scales with the sound speed typical of the experiment
($c_{s,exp}\sim 13$ km/s) and solar corona ($c_{s,sw}\sim 200$ km/s) gives us
a simple scaling relationship between frequencies observed in the lab and in
the solar wind as shown in Eq. 10

$f_{sw}\sim 5.5\times 10^{-9}f_{exp}.$ (10)

Therefore, the $20-40$ kHz plasmoids observed in both the experiment and
simulations correspond to a plasmoid frequency in the solar wind of
$110-220~{}\mu$Hz, or periods of $75-150$ minutes — in remarkable agreement
with observations (Viall & Vourlidas, 2015).

We note that this model also offers a natural explanation for the existence of
laminar and turbulent plasmoid regimes. Increasing the drive (i.e., increasing
the injected current in experiments and simulations) leads to a larger
$\gamma_{dr}$. When the drive is strong enough such that $\gamma_{dr}\gtrsim
v_{A}/L_{c}$, an ejected plasmoid has insufficient time to advect downstream a
distance comparable to its length before the subsequent plasmoid is ejected.
This leads to plasmoids of different sizes interacting downstream of the
reconnection region and more stochastic behavior. For the high drive cases in
the simulation, many of the plasmoids are small ($L_{c}\sim 5$ cm), and
$v_{A}\sim 2$ km/s, which results in the condition $\gamma_{dr}\gtrsim 40$ kHz
which is in good agreement with the onset of the turbulent plasmoids shown in
Fig. 5 and Fig. 7(d).

Another mechanism that may be contributing to the turbulent dynamics and to
the non-axisymmetric nature of Phase I in the experiment is the transition
from single plasmoid formation to multiple simultaneous plasmoid formation. To
support this hypothesis, the calculation in Appendix A computes the threshold
drive frequency necessary to destabilize shorter wavelength tearing modes in
the current sheet. If this threshold is reached we may conclude the subsequent
stochastic dynamics are those of a plasmoid chain (Uzdensky et al., 2010;
Loureiro et al., 2012) and are responsible for the turbulence in the high-
drive cases. Remarkably, as outlined in Appendix A, this threshold frequency
is computed to be $f=\gamma_{dr}/(2\pi)\gtrsim 60\,$kHz and is precisely in
the range we would expect based on experimental evidence ($\sim 60-100$ kHz).

These two mechanisms allow us to infer a hierarchy of stochasticity associated
with the drive strength and therefore the frequency of plasmoid formation. As
the drive strength increases, the system experiences a loss of equilibrium and
laminar ‘single’ plasmoid ejection. This phase is followed by plasmoids of
different ‘generations’ catching up with each other to generate a turbulent
medium downstream, but still in the ‘single’ plasmoid regime. This phase
applies particularly to the NIMROD simulations which are constrained to be
axisymmetric and are below the threshold for multiple plasmoids, yet still
exhibit increased turbulence at higher drive. Finally, at the highest drives
obtained in the experiment, the threshold to multiple simultaneous plasmoid
formation is reached as discussed in Appendix A.

This analysis can be made more quantitative by formally taking into account
the geometric effects of the magnetic field and pressure gradient for helmet
streamer structures in the solar wind. This process likewise results in blob
periodicities in the 1-2 hour range consistent with observations (Viall &
Vourlidas, 2015). This model is constructed by taking the streamer-like
poloidal field geometry generated during NIMROD simulations and scaling the
radius where reconnection occurs to coincide with $3R_{\odot}$ as a
characteristic location for PDS formation (Viall & Vourlidas, 2015; Wang et
al., 1998; Wang & Hess, 2018). Scaling the magnetic field strength using fits
to solar wind data in the ecliptic plane according to Köhnlein (1996),
produces a plausible helmet streamer geometry at the proper scale with
realistic magnetic field strength. Using fits for plasma temperature and
density in the ecliptic (likewise from Köhnlein (1996)) and mapping them to
the respective flux surfaces produces a mock helmet streamer with plausible
temperature and density profiles.

Figure 10: The work of Köhnlein (1996) provides doubly logarithmic fits to
Helios data for density, temperature, and magnetic field as a function of
heliocentric distance in the ecliptic plane. Mapping these quantities onto the
magnetic streamer structure shown in (a) allows us to compute
$f=1/2\pi\sqrt{c_{s}^{2}{\bf\it\kappa}\cdot\gradient{p}/p}$ along field lines
just inside and just outside the reconnection radius in the same fashion as
shown in Fig. 8(b). This results in plasmoid frequencies that are peaked at
the streamer cusp as expected and produce periodicities in close agreement
with in situ observations for plausible solar wind parameters.

A diagram of this 2D magnetic geometry for this heuristic model is shown in
Fig. 10(a). This model enables us to calculate the same characteristic
frequency used to unify the scaling between experiment and simulation (Eq. 7)
along the field lines in the vicinity of the reconnection site shown as the
cyan dashed lines in Fig. 10(a). The result of calculating this frequency
along flux surfaces between the cyan dashed flux surfaces in Fig. 10(a) is
shown in Fig. 10(b) and plotted as a function of field line distance away from
the outboard midplane. In this context, a field line distance of 0 corresponds
to the streamer top or the point of highest magnetic curvature where we might
expect the highest growth rate of any pressure-curvature driven loss of
equilibrium. We can see from this model that the same characteristic timescale
used to unify experimental and simulation plasmoid frequencies results in a
blob periodicity in the solar wind of $\sim 90$ minutes if formed at
$3R_{\odot}$ (Fig. 10b). If the PDS origin radius is increased or decreased,
the PDS periodicity likewise increases or decreases respectively in accordance
with observations (Wang & Hess, 2018).

##  6 Conclusions

To accompany the experimental measurements of streamer top reconnection and
plasmoid formation in the Parker Spiral current sheet, a wide range of
extended-MHD simulations were performed with the NIMROD code. Through
measurements and comparisons between experiment and simulation, we showed that
this high-$\beta$ loss of equilibrium is related to the pressure gradient and
magnetic curvature in the streamer. Specifically, the frequency of expelled
plasmoids scales with the pressure gradient and becomes more turbulent as the
pressure gradient increases and as one moves further downstream in the current
sheet.

A heuristic model for this loss of equilibrium is presented and demonstrates
that pressure-curvature driven outflows in the high-$\beta$ transition region
of the streamer belt may be responsible for the streamer top reconnection that
fuels a portion of the slow solar wind near the HCS. Although the parameters
and scale lengths in the experiment are considerably different from those in
the solar wind, the pressure driven loss of equilibrium allows both systems to
expand outwards at their respective sound speed, advecting the magnetic flux
with the ions in the case of the solar wind and with the electrons in the
experiment and simulations. While the dynamics of magnetic reconnection are
likewise vastly different between the two systems — occurring on sub-ion
scales in the experiment and macroscopic (MHD) scales in the solar wind — it
is likely that the plasmoid formation rate is governed more by the drive
timescale, $\gamma_{dr}$, than by the specifics of magnetic reconnection. As a
result, the streamer top may be reconnecting at a rate governed by the
particle and heat sourcing on these outer field lines which results in loss of
equilibrium rather than a linear instability. This allows for a unified theory
to connect observations in drastically different regimes of plasma physics
based on empirical evidence. While the underlying reconnection dynamics which
set the critical length scale of the current sheet are certainly different,
the resulting phenomenon was found to be remarkably similar between experiment
and simulation and was also reminiscent of observations of the solar corona
performed by the LASCO and SECCHI instrument suites as well as recently by
Parker Solar Probe (Lavraud et al., 2020).

The present work was supported by the NASA Earth and Space Sciences -
Heliophysics Division Fellowship award no. NNX14AO16H. The BRB facility was
constructed with support from the National Science Foundation and is now
operated as a Department of Energy National User Facility under DOE fund DE-
SC0018266. In addition, this work was supported by the NSF-DOE Partnership in
Basic Plasma Science and Engineering award no. PHY-2010136.

##  Appendix A Reconnection onset in a forming current sheet in the
collisional Hall-MHD regime

In Section 5 we alluded to the onset of the tearing mode in the forming
current sheet driven by the equilibrium loss. In this Appendix, we present a
quantitative, though simplified, derivation aimed at capturing what we think
are the key features of this process in our experiments and simulations.

The plasma regime of relevance here can be described by the resistive Hall-MHD
framework; namely, we take the ions to be cold, the reconnection dynamics to
be happening at sub-ion-skin-depth scales, and the frozen-flux condition to be
broken by resistivity. Under these constraints, the expressions for the growth
rate of the tearing instability for small and large values of the tearing
instability parameter $\Delta^{\prime}$ can be obtained from Attico et al.
(2000)111Attico et al. (2000) consider the case where resistivity is
negligible and the frozen flux constraint is instead broken by electron
inertia. The resistive scalings that we use here are directly retrievable from
theirs upon the substitution $d_{e}^{2}\rightarrow\eta/\gamma$, where $d_{e}$
is the electron skin depth.. They are

$\gamma\tau_{w}=0.47\,\Delta^{\prime}(\tau_{w}\eta)^{1/2},$ (11)

in the low $\Delta^{\prime}$ case; and

$\gamma\tau_{w}=0.69\,(ka)^{3/4}a^{-1/2}(\tau_{w}\eta)^{1/4},$ (12)

in the large $\Delta^{\prime}$ case. The normalizing timescale that appears in
these expressions is sometimes called the whistler time,
$\tau_{w}=a^{2}/(d_{i}v_{A})$, with $v_{A}$ the Alfvén speed based on the
upstream (reconnecting) magnetic field.

Given a spectrum of unstable wavenumbers, the fastest growing tearing mode is
given by the intersection of these two scalings:

$k_{max}a\approx 1.2\,(\tau_{w}\eta)^{1/7}a^{-2/7},$ (13)

where we have assumed that the upstream magnetic field is well represented by
a $\tanh{x/a}$ profile, whose instability parameter is
$\Delta^{\prime}a\approx 2/(ka)$ for $ka\ll 1$. The corresponding growth rate
is

$\gamma_{max}\tau_{w}\approx 0.8(\tau_{w}\eta)^{5/14}a^{-5/7}.$ (14)

This mode is the fastest-growing mode if the current sheet is long enough that
it fits inside the layer; i.e., if $k_{max}L\geq 1$.

From here, the calculation proceeds exactly as prescribed in Uzdensky &
Loureiro (2016). We assume, as in Section 5, that the length and the thickness
of the forming current sheet expand, or contract, exponentially, with the
drive rate $\gamma_{dr}$. Then we find that the $N=1$ mode transitions from
the low to the large $\Delta^{\prime}$ regime at the time

$t_{tr}=\frac{1}{2\gamma_{dr}}\ln\left(\frac{2\pi}{1.2}\frac{a_{0}}{L_{0}}S_{H}^{1/7}\right),$
(15)

where $S_{H}\equiv d_{i}v_{A}/\eta$ is the Hall Lundquist number, and $a_{0}$
and $L_{0}$ are the initial thickness and length of the current sheet.

On the other hand, one can compute the time $t_{cr}$ at which the growth rate
of the $N=1$ mode matches the current sheet formation rate; i.e., solve
$\gamma(t)=\gamma_{dr}$ for the $N=1$ mode. This yields

$t_{cr}\approx\frac{1}{4\gamma_{dr}}\ln\left(\frac{\pi}{0.47}\frac{a_{0}}{L_{0}}\frac{a_{0}^{2}\gamma_{dr}}{d_{i}v_{A}}S_{H}^{1/2}\right).$
(16)

Finally, one can ask if the $N=1$ mode has the time to transition from the low
to the large $\Delta^{\prime}$ regime before reaching its critical time. This
occurs when

$\gamma_{dr}>4.1\frac{a_{0}}{L_{0}}\frac{d_{i}v_{A}}{a_{0}^{2}}S_{H}^{-3/14}.$
(17)

In other words, if this condition is satisfied, one would expect the forming
current sheet to be disrupted by multiple plasmoids (large tearing mode
number, $N$); if it is not, then it is the $N=1$ tearing mode that disrupts
the forming sheet.

Inserting the values measured or inferred from the experiments into this
expression (namely, $a_{0}=0.05\,$m, $L_{0}=0.25\,$m, $d_{i}=0.7\,$m,
$c_{s}=13000\,$m s-1 and $\beta_{e}=20$) yields $f=\gamma_{dr}/(2\pi)\gtrsim
60\,$kHz. This frequency is in remarkable agreement with our experimental
results which indicate a transition to the multiple plasmoid regime in the
$60-100$ kHz range.

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