# Laser-Driven, Ion-Scale Magnetospheres in Laboratory Plasmas. II. Particle-
in-cell Simulations

Filipe D. Cruz<EMAIL_ADDRESS>GoLP/Instituto de Plasmas e
Fusão Nuclear,
Instituto Superior Técnico, Universidade de Lisboa, 1049-001 Lisboa, Portugal
Derek B. Schaeffer  Department of Astrophysical Sciences, Princeton
University, Princeton, NJ 08540, USA  Fábio Cruz  Luis O. Silva
<EMAIL_ADDRESS>GoLP/Instituto de Plasmas e Fusão Nuclear,
Instituto Superior Técnico, Universidade de Lisboa, 1049-001 Lisboa, Portugal

###### Abstract

Ion-scale magnetospheres have been observed around comets, weakly-magnetized
asteroids, and localized regions on the Moon, and provide a unique environment
to study kinetic-scale plasma physics, in particular in the collisionless
regime. In this work, we present the results of particle-in-cell simulations
that replicate recent experiments on the Large Plasma Device at the University
of California, Los Angeles. Using high-repetition rate lasers, ion-scale
magnetospheres were created to drive a plasma flow into a dipolar magnetic
field embedded in a uniform background magnetic field. The simulations are
employed to evolve idealized 2D configurations of the experiments, study
highly-resolved, volumetric datasets and determine the magnetospheric
structure, magnetopause location and kinetic-scale structures of the plasma
current distribution. We show the formation of a magnetic cavity and a
magnetic compression in the magnetospheric region, and two main current
structures in the dayside of the magnetic obstacle: the diamagnetic current,
supported by the driver plasma flow, and the current associated to the
magnetopause, supported by both the background and driver plasmas with some
time-dependence. From multiple parameter scans, we show a reflection of the
magnetic compression, bounded by the length of the driver plasma, and a higher
separation of the main current structures for lower dipolar magnetic moments.

††preprint: APS/123-QED

##  I Introduction

A vast range of space and astrophysical scenarios are driven by the rapid
expansion of plasmas through space. Such examples include interplanetary
coronal fast ejecta [1], the expansion of the stellar material from supernova
remnants [2], and artificial magnetospheric releases of tracer ions [3]. When
these expanding plasmas encounter obstacles of magnetic nature, the resultant
interaction leads to highly nonlinear and complex dynamics. In the solar
system, the interaction between the plasma flow (i.e. the solar wind) and
planetary-sized magnetic obstacles leads to the formation of magnetospheres
[4].

The effective size of the magnetic obstacles is determined by the equilibrium
position between the kinetic pressure of the solar wind and the magnetic
pressure exerted by the planetary magnetic fields [5]. The region of
equilibrium, called the magnetopause, can be described using the pressure
balance derived from magnetohydrodynamics (MHD)

$n_{d}m_{i,d}v_{0}^{2}=\frac{B^{2}}{8\pi}\ ,$ (1)

where $n_{d}$ is the density of the solar wind, $v_{0}$ is its flow velocity,
$m_{i,d}$ is the mass of its ions, and $B$ is the total magnetic field at the
magnetopause. The total magnetic field can be written as
$B=B_{0}+B_{\textrm{dip}}$, where $B_{0}$ is the collective magnetic field and
$B_{\textrm{dip}}=M/L_{0}^{3}$ is the magnetic field of the obstacle, often
well described by a dipolar profile of magnetic moment $M$. The distance
$L_{0}$ between the center of the dipole and the magnetopause, often referred
to as the plasma standoff distance, measures the effective size of the
magnetic obstacle.

For planetary-sized magnetospheres, the obstacle size is typically tens of
thousands of kilometers. However, magnetospheres with a few hundreds of
kilometers are also observed in space environments such as the lunar surface.
When the magnetic obstacle size is smaller or of the order of the ion kinetic
scales of the plasma, i.e. the ion skin depth or the ion gyroradius, the
interaction with the solar wind results in ion-scale magnetospheres, or mini-
magnetospheres.

The study of mini-magnetospheres in past years was mainly motivated by the
observation of crustal magnetic anomalies on the lunar surface [6, 7, 8, 9,
10]. Although the Moon does not have a global magnetic field like Earth, it
does have small localized regions of crustal magnetic field, of 10-100 nT over
distances of 100-1000 km [6], which are of the same order as the gyroradius of
solar wind ions near the Moon’s surface. As a result, when these regions of
the lunar surface are exposed to the solar wind, mini-magnetospheres are
formed. The deflection of charged particles off of lunar mini-magnetospheres
commonly leads to the formation of “lunar swirl” structures [11]. Similar
interactions between the solar wind and small-sized patches of magnetic field
also occur in other planets and natural satellites without a planetary
magnetosphere, such as Mars [12], Mercury [13], Ganymede [14], and comets and
asteroids [15].

Multiple experiments have been performed in laboratory environments that
replicate the interaction between plasma flows and magnetic obstacles. With a
proper re-scaling of parameters [16], these experiments represent highly
controlled configurations where a large variety of diagnostics can be used to
obtain more accurate measurements than those obtained from the direct probing
of astrophysical events. In experimental studies, fast-moving plasma flows are
usually driven resorting to high-intensity lasers focused onto solid targets
of plastic or metal composition [17, 18]. These laser-ablated plasmas can be
mildly collisional or collisionless, replicating astrophysical conditions [19,
20]. By adding dipole field sources against the plasma flow, previous
experiments of mini magnetospheres studied possible applications for
spacecrafts [21, 22, 23], the formation of lunar swirls [11], and the
conditions for the formation of magnetosphere features [24, 25, 26, 27].
Although these experiments achieved important breakthroughs in the study of
ion-scale magnetospheric physics, they were limited to i) 1D measurements of
the magnetic field and plasma density profiles and ii) fixed properties of the
obstacle and plasma flow.

Numerical simulations play a key role in interpreting and designing
experiments. Early MHD simulations attempted to explain the formation and
characteristics of lunar mini-magnetospheres and validate experimental and
analytical models [28, 29, 25]. Hybrid simulations were used to study the role
of ion kinetic effects, and obtain conditions for the formation of
magnetospheres [30] and replicate previous experimental results [31]. However,
these simulations do not resolve the electron scales and do not capture
important kinetic effects on the magnetosphere’s boundary, e.g. charge
separation effects and nonthermal particle distributions. Particle-in-cell
(PIC) simulations were used to fully resolve the micro-physics of these
systems and study its role in the formation of lunar mini-magnetospheres [32,
33, 34, 35, 36], the scaling of their properties with solar wind speed and
magnetic field orientation [37] and the conditions for the formation of
collisionless shocks [38].

In this work, we use PIC simulations of ion-scale magnetospheres to interpret
the results of recent experiments [39] performed at the LArge Plasma Device
(LAPD), University of California, Los Angeles. In these experiments, fast
collisionless plasma flows generated by high-repetition-rate lasers were
collided with the magnetized ambient plasma provided by the LAPD and with a
dipolar magnetic field obstacle, leading to the formation of ion-scale
magnetospheres. Using motorized probes, high spatial and temporal resolution
measurements of the magnetic field allowed characterization of 2D magnetic
field and current density structures. Apart from validating the experimental
results, the simulations presented in this work explore a set of upstream and
magnetic parameter scans and configurations not accessible in the laboratory
to determine the importance of each system parameter on the magnetospheric
properties. The simulations show that the background ions, and then the driver
ions, are responsible for the formation of the magnetopause observed in the
experiments. They also show that a reflection of the downstream magnetic
compression is observed for certain parameters of the driver plasma, and that
the distance between the main current features is dependent on the dipolar and
driver plasma parameters.

This paper is organized as follows. In Sec. II, we briefly review the LAPD
experiments and their main results. In Sec. III, we present PIC simulations of
ion-scale magnetospheres. In Sec. III.1, we outline the standard configuration
and parameters used for the simulations. In Sec. III.2, we provide an overview
of the temporal evolution of these systems and show that the simulations agree
with the results of the LAPD experiments. We discuss the origin of the
structures observed in current density and magnetic field synthetic
diagnostics and use particle phase spaces to interpret them. In Sec. III.3, we
present the results for different lengths of the plasma flow and define the
conditions required to reproduce the features observed experimentally. The
coupling between the laser-ablated driver and background plasmas is
characterized in Sec. III.4 with simulations with different driver densities.
In Sec. III.5, different magnetic moments are considered, and we show that the
main current density features are highlighted and more easily visible for
weaker magnetic obstacles. In Secs. III.6 and III.7, we discuss and illustrate
the validity of the key simplifications and approximations used for the
parameter scans presented in Secs. III.2-III.5. Finally, we outline the
conclusions of this work in Sec. IV.

This paper is the second part of a two part series. Detailed experimental
results are presented in Part I [39].

##  II LAPD Experiment

A new experimental platform has been developed on the LAPD to study mini-
magnetospheres. The platform combines the large-scale magnetized ambient
plasma generated by the LAPD, a fast laser-driven plasma, and a pulsed dipole
magnet, all operating at high-repetition-rate ($\sim 1$ Hz). In the
experiments, a supersonic plasma is ablated from a plastic target and then
expands into the dipole magnetic field embedded in the ambient magnetized
plasma. By measuring 2D planes of the magnetic field over thousands of shots,
detailed maps of the magnetic field evolution are constructed. Additional
details on the platform and results can be found in Part I.

Example results are shown in Fig. 1 for the measured change in magnetic field
$\Delta B_{z}=B_{z}-B_{z,\textrm{initial}}$ and the current density
$J_{x}=\partial\Delta B_{z}/\partial y$. Here, $B_{z}$ is the total magnetic
field, $B_{z,\textrm{initial}}=B_{0}+B_{\textrm{dip}}$ is the total initial
magnetic field, $B_{0}$ is the background LAPD field, and $B_{\textrm{dip}}$
is the dipole magnetic field. These results are taken along $y$ at $x=0$ from
the $z=0$ plane probed experimentally. In the experiments, the dipole is
centered at $(x,y,z)=(0,0,0)$ and has a magnetic moment $M=475$ Am2.

As seen in Fig. 1(a), the expanding laser-driven plasma creates a leading
magnetic field compression followed by a magnetic cavity. The cavity reaches a
peak position of $y\approx-13$ cm, while the compression propagates closer to
the dipole before being reflected back towards the target. The current density
in Fig. 1(b) shows two prominent structures. Following the expansion of the
magnetic cavity is a diamagnetic current, which reaches a peak position of
$y\approx-15$ cm before stagnating for approximately 1 $\mu$s and then
dissipating. Ahead of the diamagnetic current is the magnetopause current near
$y\approx-13.5$ cm, which lasts for about 0.5 $\mu$s.

Here, we aim to qualitatively model these experiments in order to address key
questions that aid in the interpretation of the experimental results. In
particular, simulations can explain the role of each system component in the
features observed, address which plasma component (ambient or laser-driven) is
responsible for the features observed and which pressure balances are most
relevant.

For convenience, the notations used in this paper are different for the ones
used in Part I. Here, we use the CGS system, and the axis system is rotated
from the used in Part I.

Figure 1:  LAPD experimental results for the temporal evolution of a) the
variation of the magnetic field $\Delta B_{z}$ and b) the current density
$J_{x}$ at $x=z=0$. The experimental results are discussed with more detail in
Part I.

##  III PIC simulations

###  III.1 Configuration of the simulations

Motivated by the results of experiments described in Sec. II, we performed 2D
simulations with OSIRIS, a massively parallel and fully relativistic PIC code
[40, 41]. With PIC simulations, we can accurately resolve the plasma kinetic
scales characteristic of mini-magnetospheres dynamics.

The numerical simulations presented in this work stem from a simplified
description of the LAPD experimental setup, represented in Fig. 2. In these
simulations, a driver plasma moves against a background plasma permeated by a
uniform magnetic field $\mathbf{B_{0}}$ and a dipolar magnetic field
$\mathbf{B_{dip}}$. $\mathbf{B_{0}}$ and $\mathbf{B_{dip}}$ are oriented along
the $z$ direction and are transverse to the driver plasma flow. Since the most
relevant dynamics of the simulations occurs at the ion kinetic scales, all the
spatial scales are normalized to the ion skin depth of the background plasma
$d_{i}=c/\omega_{pi}=\sqrt{m_{i,0}c^{2}/4\pi n_{0}e^{2}}$, where $c$ is the
speed of light in vacuum, $\omega_{pi}$ is the ion plasma frequency, $m_{i,0}$
is the mass of the background plasma ions, $n_{0}$ is the background density,
and $e$ is the electron charge. In turn, the temporal scales are normalized to
$1/\omega_{ci}$, where $\omega_{ci}=eB_{0}/m_{i,0}c$ is the ion cyclotron
frequency of the background. The simulation box is a 12 $d_{i}$ $\times$ 12
$d_{i}$ area with open and periodic boundary conditions in the $x$ and $y$
directions, respectively. The flow is in the $x$ direction and the size of the
simulation domain in the $y$ direction is large enough to avoid re-circulation
of the particles through the whole interaction. The simulations considered 25
particles per cell per species. To resolve the dynamics of the electron
kinetic scales, we used 10 grid cells per electron skin depth
$d_{e}=d_{i}\sqrt{m_{e}/m_{i,0}}$ in both $x$ and $y$ directions, where
$m_{e}$ is the electron mass.

Figure 2:  Schematic illustration of the initial setup of the 2D PIC
simulations performed. The system considers a vacuum region at the left, a
driver plasma (I) of density $n_{d}$ and length $L_{x}$, travelling to the
right with flow velocity $v_{0}$, and a background plasma (II) with constant
density $n_{0}$ and with an internal magnetic field $B_{0}$. A dipole is
included at the center of the background region. Both the uniform and the
dipolar magnetic fields are oriented in the $z$ direction. An illustration of
the effective magnetic obstacle created by the dipole and of the magnetic
field profile at $y=0$ are also shown in a dashed circumference and in a solid
black line, respectively.

The driver plasma, shown in region I in Fig. 2, represents ideally the plasma
ablated from the plastic target in the experiments. We assume that this driver
has a length $L_{x}$ that is typically 2 $d_{i}$, and a width $L_{y}$ that is
typically infinite. It has a constant density $n_{d}$, and it is initialized
moving to the right side with initial flow velocity $v_{0}$. The driver is
composed of an electron species and a single ion species, with ion mass
$m_{i,d}$. Because the driver plasma is reflected during the interaction with
the background, an empty region at the left of the driver was added to
accommodate the reflecting particles.

The background plasma is represented in region II. It is an 8 $d_{i}$ length
and infinite width plasma and it has uniform density $n_{0}$. The initial
interface between the driver and background plasma is located at $x_{B}=-4\
d_{i}$. Like the driver plasma, it has an electron species and a single ion
species, of mass $m_{i,0}$. The background plasma is magnetized with an
internal uniform magnetic field $\mathbf{B_{0}}=B_{0}\mathbf{\hat{z}}$, and
its magnitude is defined such that the Alfvénic Mach number of the flow,
$M_{A}\equiv v_{0}/v_{A}=v_{0}\sqrt{4\pi n_{0}m_{i,0}}/B_{0}$ matches the peak
experimental value $M_{A}=1.5$, where $v_{A}$ is the Alfvén velocity.

A dipolar magnetic field is externally imposed in our simulations (i.e., it is
added to the plasma self-consistent electromagnetic fields to advance particle
momenta but is not included in Maxwell’s equations to advance the fields). The
dipole is centered at $(x,y)=(0,0)$ and its associated magnetic field is
$\mathbf{B_{dip}}=B_{\mathrm{dip}}\mathbf{\hat{z}}$, with
$B_{\mathrm{dip}}=M/r^{3}$, where $M$ is the dipolar magnetic moment,
$r=\sqrt{x^{2}+y^{2}+\delta^{2}}$ is the distance to the origin of the dipole
and $\delta=0.25\ d_{i}$ is a regularization parameter. For most simulations,
the magnetic moment $M$ was chosen such that the expected standoff, obtained
from Eq. (1), is similar to the experimental value $L_{0}=1.8\ d_{i}$. For
this particular magnetic moment, the total initial magnetic field
$B_{0}+B_{\mathrm{dip}}$ is $\approx 3.0~{}B_{0}$ at the standoff distance.
Near the interface between the driver and background plasmas, the magnetic
field of the dipole is relatively small and the initial magnetic field is
$\approx 1.2~{}B_{0}$.

In this work, we present simulations with different drivers and magnetic
dipole moments. All the simulations presented here, and their respective
parameter sets, are listed in Table 1. Simulations B-G are discussed through
Sec. III on equally labeled subsections. Simulation B is used to discuss the
overall dynamics of the system, while simulations C, D, and E illustrate the
role of the driver length, the density ratio, and the magnetic moment,
respectively. Simulations F show the results for more realistic choices of
parameters and simulation G for a more realistic driver shape. The physical
parameters of the simulations (e.g. $M_{A}$, $L_{0}/d_{i}$) were adjusted to
be similar to the LAPD experiments, whereas other parameters (e.g.
$m_{i}/m_{e}$, $v_{0}$, $v_{the}$) were chosen to make simulations
computationally feasible. The experimental and numerical parameters are
presented in Table 2 and compared with lunar mini-magnetospheres.

Table 1:  List of simulations performed and their parameters. $v_{the,x}$ and $v_{thi,x}$ represent the $x$ component of the electron and ion thermal velocities, respectively. All the runs considered $v_{th,x}=v_{th,y}=v_{th,z}$ for the electrons and ions. Name | $v_{the,x}/v_{0}$ | $v_{thi,x}/v_{0}$ | $n_{d}/n_{0}$ | $m_{i}/m_{e}$ | $m_{i,0}/m_{e}$ | $L_{x}/d_{i}$ | $L_{y}/d_{i}$ | $L_{0}/d_{i}$
---|---|---|---|---|---|---|---|---
B/D2/E2 | 0.1 | 0.01 | 2 | 100 | 100 | 2 | $+\infty$ | 1.8
C1 | 0.1 | 0.01 | 2 | 100 | 100 | 1 | $+\infty$ | 1.8
C2 | 0.1 | 0.01 | 2 | 100 | 100 | 4 | $+\infty$ | 1.8
C3 | 0.1 | 0.01 | 2 | 100 | 100 | $+\infty$ | $+\infty$ | 1.8
D1 | 0.1 | 0.01 | 1 | 100 | 100 | 2 | $+\infty$ | 1.8
D3 | 0.1 | 0.01 | 4 | 100 | 100 | 2 | $+\infty$ | 1.8
E1 | 0.1 | 0.01 | 2 | 100 | 100 | 2 | $+\infty$ | 2.3
E3 | 0.1 | 0.01 | 2 | 100 | 100 | 2 | $+\infty$ | 1.4
F1 | 0.1 | 0.002 | 2 | 1836 | 1836 | 2 | $+\infty$ | 1.8
F2 | 2.5 | 0.033 | 2 | 1836 | 1836 | 2 | $+\infty$ | 1.8
F3 | 2.5 | 0.033 | 2 | 100 | 100 | 2 | $+\infty$ | 1.8
G | 0.1 | 0.01 | 2 | 100 | 100 | 2 | 6 | 1.8

In most simulations, we considered a reduced mass ratio $m_{i}/m_{e}=100$, a
flow velocity $v_{0}/c=0.1$, and cold plasmas to reduce the required
computational resources, allow extended scans over the different parameters of
the system, and simplify our analysis. The thermal effects are negligible for
the main results, and the chosen ion-to-electron mass ratio is high enough to
ensure sufficient separation between electron and ion spatial and temporal
scales. We confirm the validity of our assumptions in Sec.III.6.

In most of the simulations presented in this work, we have assumed that ions
and electrons are initially in thermal equilibrium, and thus used the electron
thermal velocities $v_{the}$ shown in Table 1, to compute the ion thermal
velocities $v_{thi}$. Because we aim to study the role of the hydrogen ions of
the experimental driver in the interaction with the background plasma, these
simulations considered equal ion masses for the driver and background plasmas,
i.e. $m_{i,d}=m_{i,0}$.

Table 2:  Typical parameters associated with lunar mini-magnetospheres [4, 11, 6], the range of parameters of LAPD [39] and the canonical simulation B. The parameters are written in both physical and normalized units to facilitate the comparison between the space, the laboratory environments and the PIC simulations. The experimental parameters are presented in ranges of values computed with the possible LAPD values for the flow velocity $v_{0}$, the density $n_{0}$ and the temperature $T$. The plasma parameters shown for lunar mini-magnetospheres are relative to the solar wind, while for the experiments and the simulations, they are relative to the background plasma. The ion data shown corresponds to Hydrogen ions. The magnetic field $B_{\textrm{std}}$ is calculated at the standoff position, i.e., at a distance $L_{0}$ from the center of the obstacle. Parameters | Lunar mini-magnetospheres | LAPD experiments | PIC simulations
---|---|---|---
Physical units | Normalized units | Physical units | Normalized units | Normalized units
Flow velocity, $v_{0}$ |  $400$ km/s |  $10^{-3}$ $c$ | 200-300 km/s | 0.7-1.0$\times 10^{-3}$ $c$ | 0.1 $c$
Density, $n_{0}$ | 5 cm-3 | — |  $10^{12}$-$10^{13}$ cm-3 | — | —
Mass ratio, $m_{i}/m_{e}$ | — | 1836 | — | 1836 | 100
Ion skin depth, $d_{i}$ | 100 km | — | 7-23 cm | — | —
Electron skin depth, $d_{e}$ | 2 km | 2$\times 10^{-2}$ $d_{i}$ | 0.2-0.5 cm | 0.7-7.0$\times 10^{-2}$ $d_{i}$ | 0.1 $d_{i}$
Magnetic obstacle size, $L_{0}$ | 300 km | 3 $d_{i}$ | 14-18 cm | 0.6-2.5 $d_{i}$ | 1.8 $d_{i}$
Internal magnetic field, $B_{0}$ |  $10^{-4}$ G |  $10^{-2}$ $m_{e}c^{2}/ed_{e}$ | 300 G | 3-9$\times 10^{-2}$ $m_{e}c^{2}/ed_{e}$ | 0.67 $m_{e}c^{2}/ed_{e}$
Ion gyroradius, $\rho_{i}$ | 500 km | 5 $d_{i}$ | 7-10 cm | 0.3-1.5 $d_{i}$ | 1.5 $d_{i}$
Electron gyroradius, $\rho_{e}$ | 800 m |  $8\times 10^{-3}$ $d_{i}$ | 4-6$\times 10^{-3}$ cm | 2-8$\times 10^{-4}$ $d_{i}$ | 0.15 $d_{i}$
Ion gyroperiod, $\omega_{ci}^{-1}$ | 1 s | — | 230-520 ns | — | —
Alfvén velocity, $v_{A}$ | 80 km/s |  $3\times 10^{-4}$ $c$ | 140-980 km/s | 0.5-3.3$\times 10^{-3}$ $c$ | 0.067 $c$
Alfvénic Mach number, $M_{A}$ | — | 5 | — | 0.3-1.5 | 1.5
Temperature, $T$ | 5 eV | — | 1-10 eV | — | —
Electron thermal velocity, $v_{the}$ | 1500 km/s | 4 $v_{0}$ | 730-2300 km/s | 2.4-11.5 $v_{0}$ | 0.1 $v_{0}$
Ram pressure, $n_{0}m_{i}v_{0}^{2}$ | 1 nPa | — | 70-1500 Pa | — | —
Standoff magnetic field, $B_{\textrm{std}}$ |  $5\times 10^{-4}$ G | 0.07 $m_{e}c^{2}/ed_{e}$ | 100-600 G | 0.02-0.2 $m_{e}c^{2}/ed_{e}$ | 2.0 $m_{e}c^{2}/ed_{e}$

###  III.2  Evolution and main features of the system

To identify the main magnetospheric and kinetic-scale structures that arise
from the initial configuration, simulation B was performed. It considered a
driver with length $L_{x}=2\ d_{i}$ and density $n_{d}=2\ n_{0}$ (twice the
background density). Figs. 3 a1-3) represent the total ion density
$n_{i}=n_{i,d}+n_{i,0}$, for three different times, and Figs. 3 b1-3) show the
variation of the $z$ component of the magnetic field, from its initial value,
$\Delta B_{z}=B_{z}-B_{z,\textrm{initial}}$.

Figure 3:  Spatiotemporal evolution of a) the total ion density and b) the
variation of the $z$ component of the magnetic field in simulation B (see
Table 1 for a list of parameters). Columns 1-3 correspond to three different
times in the simulation. The vertical and circular dashed lines mark the
initial border between the driver and background plasma and the dipolar
magnetic obstacle with radius $L_{0}$, respectively.

In Fig. 3 a1), we see the total ion density for an early time
($t\omega_{ci}=1.5$). Given the small distance propagated by the driver plasma
at this time, the dipolar magnetic field does not significantly affect the
interaction between the plasmas. For this reason, we can express the early
system as a driver flowing against a uniform magnetized background plasma. In
Fig. 3 b1), we observe that this interaction creates a region of compressed
magnetic field in the downstream region, where the background plasma is
located, and expels the magnetic field in the region of the driver, leading to
a magnetic cavity in the upstream region with approximately null magnetic
field [20].

In Figs. 3 a2) and b2), we start to observe the effects of the dipolar
magnetic field for a later time ($t\omega_{ci}=3.0$). As the magnetic pressure
exerted against the plasmas increases, a region of compressed background
plasma forms in front of the dipole, as Fig. 3 a2) shows. After the
interaction between the background and the dipole, the magnetic field pressure
becomes large enough to counterbalance the kinetic pressure of the driver,
reflecting it upstream. This can be seen seen in Fig. 3 a3) for a subsequent
time ($t\omega_{ci}=4.5$). After the reflection, there is no longer a plasma
flow pushing the magnetic compression forward or holding the decompression by
the left side of the background region, and as a result, the region near the
dipole quickly decompresses — see Fig. 3 b3).

To compare the numerical results with the experimental data shown in Fig. 1,
synthetic diagnostics were obtained from the simulations. In Fig. 4, the
variation of the magnetic field $\Delta B_{z}$ and the density current $J_{y}$
measured at the axis of symmetry $y=0$ and as a function of time are plotted
for simulation B. These diagnostics are important to comprehend the system
dynamics, due to the importance of the $z$ direction of the magnetic field in
the motion of the particles.

Figure 4:  Temporal evolution of a) the variation of the magnetic field
$B_{z}$ and b) current density $J_{y}$ at $y=0$ for the simulation B. The
driver has a 2 $d_{i}$ length and a density $n_{d}=2\ n_{0}$. The dashed lines
have slopes that match the flow velocity $v_{0}$, the coupling velocity
$v_{c}$ and the reflection velocity $v_{r}$.

The main features of Fig. 4 are consistent with the experimental results. In
the magnetic field plot of Fig. 4 a), both the upstream magnetic cavity and
the downstream magnetic compression are present. Between $t\omega_{ci}=0$ and
$t\omega_{ci}\approx 1.5$, the system behaves approximately as a driver piston
moving against a uniform magnetized plasma. As the driver pushes the
background plasma and magnetic field, the discontinuity that separates these
two media travels at constant coupling velocity $v_{c}<v_{0}$, measured as
$v_{c}\approx 0.49\ v_{0}$ for this simulation. The leading edge of the
compression of the magnetic field travels with a velocity close to $v_{0}$ for
the runs considered.

The driver experiences increasingly higher magnetic fields until the magnetic
pressure is enough to reflect the driver near the expected standoff
$x_{0}=-L_{0}$, at $t\omega_{ci}\approx 3$. The magnetic cavity and magnetic
compression are also reflected, and the boundary between these two regions
travels with a velocity $v_{r}$ after reflection. The background magnetic
decompression is seen after $t\omega_{ci}=5$.

In the current density plot of Fig. 4 b), we can observe the diamagnetic
current that supports the magnetic field gradient between the driver and
background plasmas and that identifies the leading edge of the magnetic
cavity. During the driver reflection, this current branches into multiple
components due to the multi-stream velocity distributions developed in the
driver and background plasmas. We can also verify that this structure is
reflected near the expected standoff $x_{0}=-L_{0}$. Between
$t\omega_{ci}\approx 2$ and $t\omega_{ci}\approx 3$, a second current
structure is present in the background region. It is associated with the
magnetopause of the system and the small decompressed field region that we see
in Fig. 4 a), and it arises from the interaction of the accelerated background
ions with the dipole, as we show in Sec. III.5. The presence of these two
current structures is consistent with the experimental results.

In Fig. 4 b) we can also see the formation of waves in the background plasma,
near the dipole region. These waves are excited in regions of highly non-
uniform density and magnetic field, and have periods and wavelengths between
the ion and electron kinetic scales. We have verified that their properties
change significantly for different ion thermal velocities. In particular, we
have found these waves to be more clearly excited for lower ion temperatures,
which may explain why these waves have not been observed in the experiments
performed at the LAPD. A detailed characterization of these waves and the
conditions for their formation is out of the scope of this paper, and shall be
addressed in a future work.

To better understand the particle motion during the events described, we show
in Fig. 5 the phase spaces of ions and electrons located near $y=0$. For the
ions, the $x$ component of the velocity of the particles is presented, to
illustrate their reflection and accumulation, while for the electrons, the $y$
component is shown instead, to show the formation of the currents. The
magnetic field $B_{z}$ and the current density $J_{y}$ profiles for $y=0$ are
also represented. Once again, we used the parameter set B of Table 1.

Figure 5:  Ion (a) and electron (b) phase spaces, and magnetic field $B_{z}$
and current density $J_{y}$ profiles at $y=0$, for simulation B and for three
different times (1-3). The particles shown were randomly selected in the
region $-0.2\ d_{i}<y<0.2\ d_{i}$. The frames labeled a1) to a3) show the
$v_{x}$ velocity of the ions, while the frames labeled b1) to b3) show the
$v_{y}$ velocity of the electrons. Blue/orange markers correspond to
background/driver plasma particles. The green and purple lines correspond to
the magnetic field $B_{z}$ and current density $J_{y}$, respectively. The left
dashed line marks the initial border between the driver and the background
plasmas, and the right dashed line marks the expected standoff $x_{0}=-L_{0}$.

Fig. 5 a1) shows the $v_{x}$ velocity of the ions when the dipole field is
still negligible. The ions initially move upstream with velocity $v_{0}$ until
they interact with the background field. After reaching the background, they
are mostly decelerated and reflected by electric field in the interface
between the plasmas [20], and end up with a flow velocity that is close to
zero for the simulation considered. The reflection occurs near the boundary of
the magnetic cavity, which moves with velocity $v_{c}$ through the background,
as mentioned above. During this stage, the background ions accelerate from
rest to velocities of average close to $v_{c}$.

The driver and the accelerated background ions continue to approach the dipole
until they are reflected. This can be seen in Fig. 5 a2). During this
interaction, two main current structures are visible in the $J_{y}$ profile.
The first one (from the left) corresponds to the typical diamagnetic current,
while the second one corresponds to the magnetopause. To the right of these
two main current structures, we can see the background waves observed in Fig.
4 b). In Fig. 5 a3), the driver ions are totally reflected. The ions reflected
by the dipole obtain a velocity close to $-v_{0}$, while the magnetic cavity
moves back with velocity $v_{r}$.

Because the simulation considers a cold plasma approximation, the ion thermal
velocities remain small most of the time, except for the boundary between the
two plasmas, where the velocity of the ions changes abruptly. The same does
not occur for the electrons. We can see in the $v_{y}$ velocity of the
electrons, represented in Figs. 5 b1) to b3) that, although the electron
thermal velocities are initially small, they rapidly increase considerably. At
the boundary, the electrons can reach thermal velocities of $6\ v_{0}$, much
higher than the ion velocities. Because the electron and ion density profiles
are very similar during the entire evolution of the system, the current
density $J_{y}=e(n_{i}v_{iy}-n_{e}v_{ey})$ is then mainly transported by the
electrons, where $n_{j}$ is the density and $v_{jy}$ the $y$ component of the
velocity of the ions and electrons ($j=i,e$, respectively). This is also
consistent with the observed spatial distribution of electrons during the
reflection, which shows an excess of fast electrons around the standoff
position.

###  III.3 Driver length

To choose a driver length that best reproduces the experimental results shown
in Fig. 1 and to understand its role on the magnetic field and current density
structures, we performed simulations C1 to C3 (see Table 1) with varying
driver length $L_{x}$. In Fig. 6, we show $\Delta B_{z}$ and $J_{y}$ at $y=0$
for $L_{x}=1\ d_{i}$ (C1), $L_{x}=4\ d_{i}$ (C2) and for an infinite driver
(C3). For these simulations, the properties of the background plasma and the
width of the driver $L_{y}$ were kept unchanged. The density of the driver was
$n_{d}=2\ n_{0}$.

In Figs. 6 a1) and b1), we see the magnetic field and current density plots
for the short driver length $L_{x}=1\ d_{i}$. We observe most of the features
of Fig. 4, namely the reflection of the compressed magnetic field in a1) and
the diamagnetic and magnetopause currents in b1). For this length, however,
the driver never fully interacts with the dipole. The closest that the
diamagnetic current structure gets to the dipole is $x_{r}\approx-3.0\ d_{i}$,
i.e., much farther than the expected standoff $x_{0}=-L_{0}=-1.8\ d_{i}$. To
replicate the experimental results and ensure that the driver can reach the
dipole, we should thus use a sufficiently long driver such that $x_{r}>x_{0}$.
Additionally, short drivers risk entering in a decoupling regime between the
two plasmas [42], which can compromise the observation of a magnetopause. The
coupling effects on the results are discussed in detail in Sec. III.4.

The position where the driver is fully reflected by the background can be
estimated as $x_{r}\approx x_{B}+L_{x}v_{c}/(v_{0}-v_{c})$, where $x_{B}$ is
the initial boundary position between the two plasmas. This estimate is
obtained by computing the volume of the background plasma required for the
driver plasma to deposit its kinetic energy, i.e. $x_{r}-x_{B}$ corresponds to
the magnetic stopping radius of the system [43].

In the simulation with $L_{x}=4\ d_{i}$, represented in Figs. 6 a2) and b2),
we observe once more the main features identified in Fig. 4, but unlike the
$L_{x}=1\ d_{i}$ case, the driver is long enough and ends up reflected by the
dipole. We observe that the diamagnetic current reaches the expected standoff
and has enough plasma to maintain it near the dipole for a time period
($t\omega_{ci}\approx 3$ to $t\omega_{ci}\approx 5$) longer than the $2\
d_{i}$ case shown in Fig. 4. As a result, the magnetic decompression in the
background region is delayed for longer drivers. However, because the full
driver reflection also occurs later, longer drivers will result in short-lived
reflections of the compression of the magnetic field.

In Figs. 6 a3) and b3), we show the results for a driver with infinite length
($L_{x}=+\infty$). In this simulation, the driver plasma is only partially
initialized inside the simulation domain, and a flow is continuously injected
from the lower $x$ boundary. An infinite driver configuration allows us to
understand the dynamics of the system in an asymptotic regime in which the
driver plasma stays close to the dipole. As expected, until $t\omega_{ci}=3$,
the features observed are very similar to $L_{x}=2\ d_{i}$ and $L_{x}=4\
d_{i}$. After this time, the magnetic and the driver kinetic pressures balance
each other near $x_{0}$, so the diamagnetic current remains stationary.
Because the driver can hold for longer near the dipole, the decompression in
the background region is much slower and is not visible for the time range of
the plot. We can also observe that the background waves are only visible
during a transient.

In all the three simulations, the coupling velocity measured was always
$v_{c}\approx 0.49\ v_{0}$. Given the results shown in Fig. 6, we chose a
driver length of $2\ d_{i}$ to reproduce the experimental results. This driven
length is large enough to ensure that the driver arrives at the dipole and
small enough to observe a significant reflection of the compression of the
magnetic field as we see in the experiments.

Figure 6:  Temporal evolution of the variations of the magnetic field $\Delta
B_{z}$ and current density $J_{y}$ at $y=0$, for driver lengths of a) 1
$d_{i}$, b) 4 $d_{i}$ and for c) an infinite driver length (see Table 1 for a
full list of the parameters). The dashed lines represent the slopes of the
flow velocity $v_{0}$, the coupling velocity $v_{c}$, and the reflection
velocity $v_{r}$.

###  III.4 Plasma coupling with density ratio

As expected from previous works, increasing the ratio between the driver and
background plasma densities should improve the coupling between the two
plasmas [20, 42], meaning that, for denser drivers, the transfer of momentum
and energy from the driver to the background plasma is more efficient. To
better understand the role of the coupling mechanism, we performed simulations
with different values of the driver density, namely $n_{d}=n_{0}$ (D1),
$n_{d}=2\ n_{0}$ (D2) and $n_{d}=4\ n_{0}$ (D3), while keeping a constant
background density $n_{0}$ and a driver length $L_{x}=2\ d_{i}$. For each run,
the magnetic moment was chosen such that the expected standoff obtained from
Eq. (1) was always $L_{0}=1.8\ d_{i}$. The synthetic magnetic field and
current density diagnostics were obtained for these simulations and are shown
in Fig. 7.

Figure 7:  Temporal evolution of the variations of the magnetic field $\Delta
B_{z}$ and current density $J_{y}$ at $y=0$, for different ratios between the
driver and background densities $n_{d}/n_{0}$. The magnetic moment was chosen
so that the expected standoff distance $L_{0}$, calculated from Eq. (1), was
kept as 1.8 $d_{i}$ for all the simulations. Panels a-c) show results for
$n_{d}=n_{0}$, $n_{d}=2\ n_{0}$ and $n_{d}=4\ n_{0}$, respectively.

In Figs. 7 a1) and b1) we can see $\Delta B_{z}$ and $J_{y}$ for the lowest
driver density considered, $n_{d}=n_{0}$ (i.e., background and driver with the
same initial density). In this regime, the coupling is less efficient and, as
a result, the coupling velocity $v_{c}\approx 0.38\ v_{0}$ is lower than
obtained in the higher densities cases represented in Figs. 7 b) and c). Due
to the low coupling velocity, the driver plasma is reflected more quickly by
the background than for denser drivers, and the expected position $x_{r}$ for
the total reflection on the background is farther from the dipole than the
expected standoff $x_{0}$, meaning $x_{r}<x_{0}$. As a result, Fig. 7 a) shows
similarities with the short driver length represented in Fig. 6 a), because,
in both simulations, the driver parameters do not ensure that the driver
arrives near the dipole.

In Figs. 7 a2) and b2), we show the results for $n_{d}=2\ n_{0}$, which is the
same run represented in Fig. 4. For this density, the coupling velocity,
measured as $v_{c}\approx 0.49\ v_{0}$, is high enough to secure a reflection
of the driver by the dipole, as we observe at $t\omega_{ci}\approx 3$. In
Figs. 7 a3) and b3) we show the case with the highest driver density $n_{d}=4\
n_{0}$, which is similar to the $n_{d}=2\ n_{0}$ case, because the measured
coupling velocity for Fig. 7 c) was $v_{c}\approx 0.56\ v_{0}$, i.e., only
slightly larger than the $v_{c}$ measured for Fig. 7 b). In the high density
case, we also see that the current density structures during the plasma
reflection are filamented, due to analogous multi-stream velocity
distributions discussed for Fig. 6 b).

To guarantee that the driver reaches the expected standoff, we thus require
that $x_{r}>x_{0}$. In fact, the position where the driver is reflected
$x_{r}$, for no dipole cases, increases with the driver length $L_{x}$ and the
velocity ratio $v_{c}/v_{0}$, and thus, both quantities must be large enough
to guarantee that $x_{r}>x_{0}$. In turn, the ratio $v_{c}/v_{0}$ increases
with increasing driver density ratio $n_{d}/n_{0}$, and so, the driver should
be sufficiently long and dense to effectively couple to the background plasma.
Our results (in particular Sec. III.2) show that a driver with $L_{x}=2\
d_{i}$ and $n_{d}=2\ n_{0}$ qualitatively reproduces the experimental results.

A separate study was also performed to analytically determine the properties
of the driver-background plasma coupling. The results of this study will be
presented in a future paper.

###  III.5 Dependency of the magnetopause position with the magnetic moment

To confirm that the features previously associated with the magnetopause
location change according with its expected position, we performed simulations
with a 2 $d_{i}$ long driver with density $n_{d}=2\ n_{0}$ for three different
magnetic moments. Considering the magnetic moment that results in the expected
standoff $L_{0}=1.8\ d_{i}$ as $M_{0}$ (simulation B/E2 on Table 1),
simulations with the magnetic moments $2\ M_{0}$ (E1) and $M_{0}/2$ (E3) were
also performed, corresponding respectively to the expected standoffs
$L_{0}\approx 2.3\ d_{i}$ and $L_{0}\approx 1.4\ d_{i}$. Fig. 8 shows the
$\Delta B_{z}$ and $J_{y}$ synthetic diagnostics at $y=0$ for the three
simulations.

Figure 8:  Temporal evolution of the variation of the magnetic field $\Delta
B_{z}$ and current density $J_{y}$ at $y=0$, for three different magnetic
moments. The magnetic moments $M$ considered were a) $M=2\ M_{0}$, b)
$M=M_{0}$ and c) $M=M_{0}/2$, where $M_{0}$ represents the magnetic moment
that corresponds to a standoff $L_{0}=1.8\ d_{i}$ for a driver density
$n_{d}=2\ n_{0}$. The corresponding standoffs for these simulations are a)
$L_{0}\approx 2.3\ d_{i}$, b) $L_{0}=1.8\ d_{i}$ and c) $L_{0}\approx 1.4\
d_{i}$.

Figs. 8 a1) and b1) show the results for the highest magnetic moment $M=2\
M_{0}$. We see that the current structures associated with the magnetopause
and the background waves are less evident than for the lower magnetic moments,
as they are formed farther from the dipole. Figs. 8 a2) and b2) correspond to
the magnetic moment $M_{0}$ that leads to $L_{0}=1.8\ d_{i}$ and are the same
results shown in Fig. 4. As previously mentioned, there are two main
observable current structure standoffs. The first one is associated to the
diamagnetic current, which is reflected around $t\omega_{ci}\approx 3$ near
the expected value $x_{0}=-L_{0}=-1.8\ d_{i}$. This standoff is related to the
interaction between the driver ions and the dipole. The second standoff occurs
between $t\omega_{ci}\approx 2$ and $t\omega_{ci}\approx 3$ and it is located
in the background plasma region. This standoff also occurs near $x=-1.8\
d_{i}$.

In Figs. 8 a3) and b3), we show the results obtained for the half magnetic
moment $M=M_{0}/2$. In this case, the magnetic pressure exerted by the dipole
is lower, leading to a smaller $L_{0}$, and consequently, the diamagnetic
current feature visible in b3) is closer to the dipole than in Figs. 8 b1) and
b2). The main changes, however, occur in the magnetopause current. Unlike what
we observe for the other magnetic moments, the magnetopause current,
pinpointed in the current density plot, lasts for a longer time (until
$t\omega_{ci}\approx 4$). This current is also more separated from the
diamagnetic current standoff and is easier to identify. This is consistent
with the experimental observations.

To identify the pressure balances associated with the two observed standoffs,
and because the magnetic and kinetic pressures vary over time, we studied the
temporal evolution of the different plasma and magnetic pressure components of
the system. In particular, we calculated the spatial profiles of the magnetic
pressure $B^{2}/8\pi$, the ram pressure $n_{j}m_{j}v_{flj}^{2}$ and the
thermal pressure $n_{j}m_{j}v_{thj}^{2}$ as a function of time for $y=0$. In
these expressions, $n_{j}$, $m_{j}$, $v_{flj}$ and $v_{thj}$ refer to the
density, mass and flow and thermal velocities, respectively, of the ions
($j=i$) and electrons ($j=e$). The magnetic pressure was calculated from the
magnetic field measured in each PIC grid cell located at $y=0$. The flow and
thermal pressures, were calculated from averaged particle data. To ensure that
the calculation of each kinetic pressure considered a sufficiently large
number of particles, all the particles between $-0.1\ d_{i}<y<0.1\ d_{i}$ were
binned into equal-sized bins of width $0.05\ d_{i}$ over the $x$ direction.
For each bin we computed: i) the average density of each species, ii) the flow
velocity, corresponding to the average of the velocity of the particles, and
iii) the thermal velocity, corresponding to the standard deviation of the
velocity of the particles [44]. With these averaged quantities, the ram and
thermal pressures were calculated in each bin for each species of ions and
electrons and each component of the velocity $x$, $y$, and $z$. The $y$ and
$z$ components of the pressures, however, are negligible.

These pressure profiles were obtained for simulation E3 with magnetic moment
$M=M_{0}/2$ and are plotted in Fig. 9 for times where a) the magnetopause and
b) the diamagnetic current standoff can be observed. The kinetic pressures
represented were calculated by adding all the components of the ram and
thermal pressures of the ions and electrons for the background ($P_{0}$) and
the driver ($P_{\mathrm{d}}$) plasmas. The magnetic pressures represented were
calculated by considering the total and the relative magnetic field pressures
($P_{\mathrm{mag}}=B_{z}^{2}/8\pi$ and
$P_{\mathrm{rel}}=P_{\mathrm{mag}}-B_{0}^{2}/8\pi$, respectively). The
pressures were normalized to the initial ram pressure of the driver ions.

Figure 9:  Pressures profiles calculated for simulation E3 with a magnetic
moment $M=M_{0}/2$ (shown in Fig. 8 c)), during the occurrence of a) the
magnetopause and b) the standoff of the diamagnetic current. The magnetic
pressures are $P_{\mathrm{mag}}=B_{z}^{2}/8\pi$ and
$P_{\mathrm{rel}}=P_{\mathrm{mag}}-B_{0}^{2}/8\pi$. The kinetic pressures
$P_{\mathrm{d}}$ and $P_{0}$, corresponding to the driver and background
plasmas, respectively, consider both the ions and electrons and the flow and
thermal components of the velocity. c) Temporal evolution of the variation of
the total kinetic energies of the driver $\Delta W_{\mathrm{kin,d}}$ and
background $\Delta W_{\mathrm{kin,0}}$ plasmas, the magnetic energy $\Delta
W_{\mathrm{mag}}$, and the total energy of the simulation box $\Delta
W_{\mathrm{tot}}$. The total energy is calculated by adding all the kinetic
energies and the electric and magnetic energies. Since the background plasma
is magnetized, the electric energy term is many orders of magnitude smaller
than the magnetic energy term. The energies were normalized to the initial
total energy of the driver ions $W_{\textrm{d,ini}}$. The loss of energy
conservation near $t\omega_{ci}\approx 4$ is caused by the escape of
background plasma particles and magnetic field through the right hand side of
the simulation box.

Fig. 9 a) shows the magnetic and kinetic pressures at time
$t\omega_{ci}\approx 2.33$ where we observed the magnetopause in Fig. 8 b3).
When the driver starts pushing the background, the pressure of the driver at
the interface between the plasmas increases because the driver density and
thermal velocities also increase. During the flow, the driver transfers energy
and momentum to the background plasma, and as a result, the background
develops a strong kinetic pressure. At the time represented in Fig. 9 a), the
background plasma pressure equals the total and the dipolar magnetic pressures
in $x\approx-1.4\ d_{i}$, near the location of the magnetopause current of
Fig. 8 b3). This observation supports the hypothesis that this current emerges
from the standoff between the background and magnetic pressures. Fig. 9 b)
shows the pressures for $t\omega_{ci}=3$ where we see the beginning of the
reflection of the driver. The driver pressure equals the magnetic and dipolar
pressures near $x_{0}=-L_{0}\approx-1.4\ d_{i}$. After this time, the driver
is incapable of moving any further into the background because the magnetic
pressure exceeds its kinetic pressure.

The energy variations integrated over the entire simulation domain can also
help us understand the system dynamics. Fig. 9 c) shows the variation of the
total driver and background kinetic energies, $\Delta W_{\mathrm{kin,d}}$ and
$\Delta W_{\mathrm{kin,0}}$, respectively, as well as the variation of the
magnetic energy $\Delta W_{\mathrm{mag}}$, and of the total energy $\Delta
W_{\mathrm{tot}}$. The kinetic energies of the background and driver plasmas
consider all ions and electrons. In early times $t\omega_{ci}<3$, as the
driver and background plasmas interact, the driver transfers its energy to the
background plasma and the magnetic field. The total energy, given by the sum
of the electromagnetic energy and the kinetic energies, remains constant
during this period. After the driver is fully reflected by the dipole for
$t\omega_{ci}>3$, the magnetic field loses most of its energy to the
background and driver plasmas leading to a drop of the magnetic energy. After
$t\omega_{ci}\approx 4$, the background ions start to leave the simulation
box, and the total energy is no longer conserved. The background kinetic
energy remains approximately constant because the background plasma loses
energy to the sink at the right boundary of the simulation but gains energy
from the magnetic field. For both driver and background plasma, the ions carry
most of the energy.

From Fig. 9, we can identify the positions where multiple pressure balances
occur, and therefore, develop an insight into the pressure equilibria that are
behind the structures of the current density synthetic diagnostics. Using the
previously calculated pressures, we obtained the equilibrium positions where
certain pressure balances manifested and plotted them in Fig. 10 alongside
$J_{y}$.

Figure 10:  Temporal evolution of the current density $J_{y}$ at $y=0$, with
the closest locations to the dipole of different pressure balances for
multiple times. The represented locations of pressure balances are the
equilibria between the driver kinetic pressure $P_{\mathrm{d}}$ with the total
magnetic field pressure $P_{\mathrm{mag}}=B_{z}^{2}/8\pi$, represented by the
solid line; the background kinetic pressure $P_{0}$ with the pressure exerted
by the relative magnetic field
$P_{\mathrm{rel}}=P_{\mathrm{mag}}-B_{0}^{2}/8\pi$, by the dotted line, and
$P_{\mathrm{d}}=P_{\mathrm{rel}}$, by the dashed line. The results correspond
to simulation E3 (see Table 1).

This analysis shows that the system has, in general, two magnetopause
structures: one driven by the background, and one by the driver plasma. The
former structure is defined by the balance $P_{\mathrm{0}}=P_{\mathrm{rel}}$.
For the latter structure to form, the driver needs to have almost enough
energy to push the diamagnetic current up to the magnetopause, defined by Eq.
1. This is illustrated in Fig. 10, where we show the location of the pressure
equilibrium between the driver kinetic pressure and the total magnetic
pressure, $P_{\mathrm{d}}=P_{\mathrm{mag}}$.

As shown in Fig. 9, the current associated with the background magnetopause
seems to overlap with the region of background and magnetic pressure balance.
Unlike the driver, the background plasma is magnetized. If we neglect the
compression of the magnetic field in the downstream region, the pressure
balance that describes this magnetopause can then be estimated by the
equilibrium of the kinetic pressure of the background plasma with the relative
magnetic pressure, $P_{0}=P_{\mathrm{rel}}$. In Fig. 10, we show that this
pressure balance, represented by the dotted line, describes well the position
of the current feature identified as the magnetopause between times
$t\omega_{ci}\approx 2$ and $t\omega_{ci}\approx 3$.

After $t\omega_{ci}\approx 3$, the magnetopause current is well described by
the pressure balance $P_{\mathrm{d}}=P_{\mathrm{rel}}$, as illustrated by the
dashed line in Fig. 10. In fact, after inspecting the phase spaces in Figs. 5
a3) and b3), we can observe that a combination of driver plasma particles
(separated from the bulk distribution) and background ions pushes the dipolar
field and sets the position of the magnetopause.

We stress that, because we are determining equilibria via MHD pressure
balances but are checking the intersection between pressure curves with
kinetic resolution, some caution must be made to ensure that we are observing
the equilibrium between pressures and not merely the interface between the
different regions of interest. To ensure that the pressure equilibria were
correctly obtained, the corresponding pressure profiles were always carefully
inspected with additional diagnostics.

###  III.6 Realistic parameters

Due to the need for more extensive scans (and thus using physically equivalent
but computationally feasible parameters), the simulations shown so far
considered reduced ion mass ratios, cold plasmas, and higher velocities than
the ones used in the LAPD experiments - see Table 2. To ensure that the main
results presented in the previous sections are also valid with realistic
parameters, we have performed a set of simulations with parameters similar to
those expected experimentally.

Three simulations were performed, labeled as runs F1 to F3. Run F1 employs
realistic mass ratios $m_{i,d}/{m_{e}}=m_{i,0}/{m_{e}}=1836$. Additionally,
run F2 also considers a ratio between the electron thermal and flow velocities
close to the ones expected for the LAPD experiments, namely
$v_{the,x}/v_{0}=2.5$ and $v_{thi,x}/v_{0}=0.033$, leading to higher
temperatures than in the previous simulations, and thus allowing possible
thermal effects on the system. Finally, run F3 considers the same electron
thermal velocity ratios of F2 but the standard reduced mass ratios.

The $\Delta B_{z}$ and $J_{y}$ plots for these simulations are shown in Fig.
11. Note that, due to changes in $m_{i}/m_{e}$, the spatial and temporal
scales were recalculated for the new parameters. Once again, the magnetic
dipole moment for the three simulations was adjusted to ensure that
$L_{0}=1.8\ d_{i}$.

Figure 11:  Temporal evolution of a) the variation of the magnetic field
$\Delta B_{z}$ and b) the current density $J_{y}$ at $y=0$, for the
simulations with similar parameters to the experiments. Run F1 considers
realistic mass ratios for the driver and background plasmas and low ratios
between the thermal and flow velocities; run F2 uses realistic mass ratios and
thermal velocity ratios close to the ones expected in the experiments; run F3
uses the realistic thermal velocity ratios but reduced mass ratios.

As expected, these simulations show the same main structures discussed in the
previous sections. We observe the typical reflection of the compression of the
magnetic field and the current structures of the magnetopause and diamagnetic
cavity. However, some differences are also visible. In Figs. 11 a1) and b1),
i.e. for the realistic mass ratios but cold plasmas simulation, we observe a
stronger filamentation of the plasma flow reflected off the dipole and a
thinner diamagnetic current. This is because $d_{e}$ is the characteristic
length scale of the current layer and we have lower $d_{e}/d_{i}$ values for
larger $m_{i}/m_{e}$. Figs. 11 a2) and b2), for the simulation with higher
temperatures, show no major differences with Figs. 11 a1) and b1), even though
there is a significant increase in the thermal velocities.

In Figs. 11 a3) and b3), however, we observe significant differences for
reduced mass ratios with realistic thermal velocity ratios. In particular, we
observe in the current density plot smoother magnetic and current structures
and less defined background waves between the magnetopause and the dipole. We
also observed for increased ion thermal velocities, for example,
$v_{thi}/v_{0}\approx 0.25$, that the background waves are no longer visible.

Additionally, other simulations were performed to look for possible changes
with realistic parameters. A simulation with a lower flow velocity
$v_{0}=0.01\ c$ and realistic thermal velocity ratios lead to no significant
features observed, and the obtained synthetic diagnostics were very similar to
the ones in Figs. 11 a3) and b3), meaning that the system scales well with
$v_{0}$. Another simulation was performed to observe if the shape of the
initial density profiles of the plasmas would affect the main results. Namely,
the constant density profiles used on both the driver and background plasmas
were replaced by Gaussian density profiles with a typical gradient scale
$\sigma=1\ d_{i}$ on the edges of the plasmas. This simulation did not show
meaningful differences, in agreement with previous plasma coupling works,
which observed that the leading edge of the plasmas evolves similarly for
different initial density profiles [45].

###  III.7 Finite transverse size

For simplicity, and because we were more interested in studying the system
along the axis of symmetry $y=0$, the previous simulations only considered a
driver with infinite width $L_{y}$ and a length of $L_{x}=2\ d_{i}$. In the
experiments, however, the drivers had a width comparable to their lengths and
did not have the sharp boundaries used in the simulations. To investigate if
and how our results are modified with a more complex-shaped driver, we
performed a simulation with a finite width, semi-circular-shaped driver
plasma. This driver is initially defined with the conditions $(x+7.25\
d_{i})^{2}+y^{2}<(3.25\ d_{i})^{2}$ and $x>-6\ d_{i}$ and has length $L_{x}=2\
d_{i}$ and width $L_{y}=6\ d_{i}$. Fig. 12 shows the results of this
simulation and includes the initial shape of the driver in Fig. 12 a).

Figure 12:  a) Total ion density at time $t\omega_{ci}=3.0$, and temporal
evolution of b) the variation of the magnetic field $\Delta B_{z}$ and c) the
current density $J_{y}$ at $y=0$, for simulation G with a finite width driver
with a circular segment shape. The dashed lines at a) represent the initial
position of the driver and the left border of the background plasma.

Due to the finite width of the new driver and its particular shape, we should
expect to see significant differences in the regions of the simulation plane
far from $y=0$. In the total ion density plot of Fig. 12 a) for a time
$t\omega_{ci}=3$, when there is a strong interaction of the driver with the
dipolar magnetic field, we observe the propagation of waves at the lower and
upper sides of the dipole caused by the finite width of the driver, that was
not present for infinite width drivers.

In Figs. 12 b) and c), we see the usual magnetic and current density plots at
$y=0$ for this simulation. By shortening the driver plasma width, the
background particles escape from the bottom and top regions of the simulation
box, and the driver has more difficulty holding the magnetic decompression in
the background region. The decompression, therefore, occurs quicker for finite
drivers, as seen in Fig. 12 b), leading to short reflections of the magnetic
compression.

Although this complex-shaped driver gets us closer to the experimental
configuration, the simulations did not include all the properties of the
experimental driver, as for example, the non-uniform density, velocity
profiles of the plasmas and the flow divergence. Additionally, 3D effects
should also be considered. Future simulations are planned to study the effect
of these properties in the results. However, we expect that these features
will not change the main results of the simulations.

##  IV Conclusions

In this work, we have performed PIC simulations of mini-magnetospheres in the
interaction between a plasma flow and a magnetized background plasma. In
particular, we have successfully reproduced results from recent experiments
performed at the LAPD, validating the experimental platform to study mini-
magnetospheres in the laboratory. We have also explored an extensive parameter
space defining the interaction, allowing us to i) determine how the main
properties of the system change with the parameters and ii) identify the
required conditions for the creation of a mini-magnetosphere.

Our simulations have shown that some system features are present across
multiple regimes. The initial flow of the driver expels the magnetic field in
the upstream region, leading to a magnetic cavity, and compresses the
downstream magnetic field. The driver travels through the background until the
magnetic field pressure is large enough to counterbalance the driver plasma
pressure. A fast decompression of the background magnetic field then follows.
If the background decompression occurs after the total reflection of the
driver plasma, then we can observe the reflection of the compression of the
magnetic field. To see this feature, the driver needs to be short enough to
anticipate the driver reflection relative to the decompression but
sufficiently long to ensure that it can get close to the dipole.

For the super-Alfvénic flows considered, the driver particles are reflected
upstream during the interaction with the background plasma and the magnetic
field. The coupling velocity (i.e., the velocity at which the leading end of
the driver travels through the background) is lower than the flow velocity and
increases with the increase of the ratio between the driver and background
densities. The coupling velocity and the length of the driver determine how
far the driver can go through the background region without a dipole, for a
uniform driver plasma.

The interaction of the plasmas with the dipole results in two magnetopauses.
The first describes the balance between the kinetic pressure of the propelled
background plasma plus the pressure of the plasma internal magnetic field and
the total magnetic pressure. The seconds describes approximately the balance
between the kinetic pressure of the driver plasma separated from the bulk
distribution and the relative magnetic pressure. Using simulations with
different dipole moments, we have shown that, for lower magnetic moments, the
driver and background standoffs are closer to the center of the dipole, and
the magnetopause current is more clearly identified than for higher magnetic
moments. Furthermore, it is also easier to separate the magnetopause and
diamagnetic currents for lower magnetic moments, consistent with experimental
observations.

In the simulations performed, we also observed the formation of waves in the
background plasma region, between the magnetopause and the center of the
dipole, where the magnetic field gradient was significant. These waves result
from the excitation that always followed the formation of the magnetopause and
were only observed for background plasmas with relative low ion thermal
velocities. This condition may explain the absence of these waves in the
experimental plots.

Most of the simulations presented in this work were performed in idealized
configurations. In particular, we used reduced ion-to-electron mass ratios,
unrealistically high flow velocities, a simple flat-top driver density
profile, and neglected thermal effects. In Sec. III.6 and III.7, we presented
simulations that drop some of these simplifications. Replacing reduced ion
mass ratios with realistic ones and considering high thermal velocities ratios
close to the obtained in the experiments did not lead to significant changes
in the results. The same occurred when considering smoothed density profiles.
It was also possible to conclude that the main features of the system scaled
as expected with the absolute value of the driver flow velocity. We also
presented a simulation to study possible effects associated with the
complexity of the experimental laser-ablated driver. A simple circular
segment-shaped driver was considered and led to similar results in the axis of
symmetry as the infinite width driver simulations. However, wave-like
structures were observed on both the bottom and upper sides of the dipole. For
future studies on the regions outside the axis of symmetry, the driver shape
and complexity must be considered.

Additionally, we also performed other parameter scans related to the
complexity of the driver. For instance, we performed simulations where the
driver ions were heavier than the background ions to simulate the small role
of the carbon ions in the experimental driver. These studies showed no
significant differences to the lighter ions simulations.

In conclusion, the simulations were consistent with the LAPD experimental
results, and the multiple parameter scans performed dictated the formation
conditions of the main features of mini-magnetospheres. For future works, we
intend to exploit the features present in the sides of the dipole, exploit
anti-parallel magnetic field configurations, perform 3D simulations, and
consider even more realistic properties of the driver.

###### Acknowledgements.

We acknowledge the support of the European Research Council (InPairs
ERC-2015-AdG 695088), FCT (PD/BD/114307/2016 and APPLAuSE PD/00505/2012), the
NSF/DOE Partnership in Basic Plasma Science and Engineering (Award Number
PHY-2010248), and PRACE for awarding access to MareNostrum (Barcelona
Supercomputing Center, Spain). The simulations presented in this work were
performed at the IST cluster (Lisbon, Portugal) and at MareNostrum.

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