# Multi-scale magnetic field structures in an expanding elongated plasma cloud
with hot electrons subject to an external magnetic field

M. A. Garasev1  A. A. Nechaev1  A. N. Stepanov1  V. V. Kocharovsky2  Vl. V.
Kocharovsky1<EMAIL_ADDRESS>1Institute of Applied Physics, Russian Academy
of Sciences, Nizhny Novgorod, 603950, Russia 2Department of Physics and
Astronomy, Texas A&M University, College Station, TX 77843, USA

###### Abstract

We carry out 3D and 2D PIC-simulations of the expansion of a magnetized plasma
that initially uniformly fills a half-space and contains a semi-cylindrical
region of heated electrons elongated along the surface of the plasma boundary.
This geometry is related, for instance, to ablation of a plane target by a
femtosecond laser beam under quasi-cylindrical focusing. We find that a decay
of the inhomogeneous plasma–vacuum discontinuity is strongly affected by an
external magnetic field parallel to its boundary.

We observe various transient phenomena, including an anisotropic scattering of
electrons and an accompanying Weibel instability, and reveal various spatial
structures of the arising magnetic field and current, including multiple
flying apart filaments of a z-pinch type and slowly evolving current sheets
with different orientations. The magnitude of the self-generated magnetic
field can be of the order of or significantly exceed that of the external one.
Such phenomena are expected in the laser and cosmic plasmas, including the
explosive processes in the planetary magnetospheres and stellar coronal
arches.

##  1 Introduction

The problem of generation and coexistence of large- and small-scale magnetic
fields during the expansion of a non-equilibrium collisionless plasma with hot
electrons into a cold background plasma or vacuum is typical for many
situations in the laboratory and cosmic plasma physics, including the
phenomena of laser ablation, dynamics of the planetary magnetosheath regions,
evolution of the solar (stellar) flares and wind formation (see, e.g., Huang
et al., 2019; Srivastava et al., 2019; Quinn et al., 2012; Romagnani et al.,
2008; Göde et al., 2017; Gruzinov, 2001; Lyubarsky & Eichler, 2006; Garasev &
Derishev, 2016; Medvedev & Loeb, 1999; Spitkovsky, 2008; Huntington et al.,
2015; Sakawa et al., 2016; Ruyer et al., 2020; Balogh et al., 2018). A plasma
discontinuity or a transition layer could be inhomogeneously heated and
exposed to an external magnetic field comparable in magnitude to the self-
generated one and oriented at various angles relative to the direction of the
plasma density gradient or the axis of a plasma anisotropy induced by the
expansion of a hot electron cloud.

We consider a particular, though quite typical case of a plasma uniformly
filling a half-space with a semi-cylindrical region of heated (Maxwellian)
electrons elongated along the surface of the plasma boundary. This case is
relevant, e.g., to the expansion of a laser plasma produced by a femtosecond
laser beam under quasi-cylindrical focusing on a plane target or the injection
of a current filament with hot electrons into the upper part of a stellar
coronal arch. By means of the 3D and 2D PIC-simulations, we describe the
inhomogeneous plasma flow and the evolution of the current structures of
different scales, paying particular attention to the anisotropic cooling of
electrons and the interplay between the self-generated and external magnetic
fields in the dynamics of particles.

We resolve the formation of the stratified plasma flows and current sheets or
filaments with dimensions of the order of or less than the particle
gyroradius, i.e., the structures of a kinetic origin and reduced
dimensionality, which are responsible for a small-scale stratification of the
plasma. These structures and an accompanying reconnection of the magnetic
field lines cannot be described within the MHD approximation (cf. Patel et
al., 2021; Srivastava et al., 2019; Priest, 2014; Plechaty et al., 2013;
Moritaka et al., 2016). In the laser and cosmic plasmas with weak, rare
collisions of particles, the leading mechanism of the formation of relatively
small-scale current structures is the Weibel instability (Weibel, 1959;
Davidson, 1989; Kocharovsky et al., 2016; Borodachev et al., 2017). It is
caused by the anisotropy of the charged particle velocity distribution and can
initiate or change various transient processes involving the generation of
strong small-scale magnetic fields.

In this work we continue our recent studies (Nechaev et al., 2020a, b) of the
Weibel instability during the decay of a strong discontinuity in the density
and temperature of a non-relativistic plasma with hot electrons by means of
numerical simulations using the code EPOCH (Arber et al., 2015). Investigating
the plasma expansion into vacuum (in this case the electrostatic shock wave
does not form), we focus mainly on the role of a uniform external magnetic
field that is parallel to the plane boundary of the plasma and oriented either
along or across the inhomogeneously heated region elongated along this
boundary. To our best knowledge, the decay of a magnetized plasma–vacuum
discontinuity in this geometry has not been studied in detail (cf. Moreno et
al., 2020; Dieckmann et al., 2018; Fox et al., 2018; Thaury et al., 2010;
Schoeffler & Silva, 2018), although under other conditions the Weibel-type
instabilities and, in particular, the filamentation instability have been
frequently investigated numerically, e.g., for colliding plasma flows,
including magnetized ones (see Spitkovsky, 2008; Chang et al., 2008; Sironi et
al., 2013; Sironi & Spitkovsky, 2009; Bret, 2009; Silva, 2006; Dieckmann,
2009; Ruyer et al., 2015).

The strongly non-equilibrium expansion of a magnetized plasma from an
initially heated region is possible only if the energy density of an external
magnetic field $B_{0}$ is less than or of the order of the kinetic energy
density of the hot electrons. Let $n_{0}$ and $T$ denote the initial number
density and isotropic temperature (in energy units) of the hot electrons,
respectively. In the limit $B_{0}^{2}/8\pi\ll n_{0}T$, the field weakly
affects the profile of the expanding plasma density, but can significantly
influence the arising anisotropy of the electron velocity distribution and the
accompanying Weibel-type instability. In the considered geometry, the latter
can develop even in the absence of an external field.

The contents of the paper is as follows. Section 2 describes the initial
geometry of the plasma–vacuum discontinuity in our PIC-simulations of its
decay in the presence of an external magnetic field of various orientations. A
typical example of the results of the 3D3V simulations of this plasma
discontinuity decay is presented in section 3. Section 4 describes the main
features of such a decay which are clarified by means of 2D3V simulations for
a representative set of the external magnetic fields and various plasma
densities. In section 5 we discuss the revealed qualitative patterns and some
open problems of the phenomenon in question. General conclusions are stated in
section 6.

##  2 An initial-value problem of the plasma–vacuum discontinuity decay in the
case of a semi-cylindrical region of heated electrons

Bearing in mind a typical laser-ablation experiment, at the initial moment of
time we choose the following density and temperature of ions (with a mass of
$M=100\,m$, where $m$ is the electron mass), placed below the plane $y=0$ in
Cartesian coordinates: $n(y>0)=0$, $n(y\leq 0)=$ $n_{0}=1.7\cdot 10^{22}$,
$10^{21}$, or $10^{20}$ cm-3 (in different simulations) and
$T_{\mathrm{i}}=10$ eV. We take the initial temperature of heated Maxwellian
electrons (see fig. 1) to be independent of the coordinate $z$ and vary with
the radius according to the Gaussian law with the maximum temperature $T=1$
keV at the point $x=0$, $y=0$ and the asymptotic value
$T_{\mathrm{e},\infty}=T_{\mathrm{i}}=10$ eV at a large distance
$r=(x^{2}+y^{2})^{1/2}$ from that point:
$T_{\mathrm{e}}=T_{\mathrm{e},\infty}+(T-T_{\mathrm{e},\infty})\exp\left(-r^{2}/r_{0}^{2}\right)$.
We assume that $r_{0}=25$ $\mu$m in the simulations with $n_{0}=10^{20}$,
$10^{21}$ cm-3 and $r_{0}=5$ $\mu$m in the simulations with $n_{0}=1.7\cdot
10^{22}$ cm-3. The geometry of this temperature distribution corresponds to a
long semi-cylinder that has the axial section lying on the surface of the
plasma, $y=0$, and the axis directed along the $z$-axis. The external magnetic
field $\vec{B}_{0}$ is directed along either the $y$\- or $z$-axis.

Below we describe the results of typical simulations and the revealed physical
phenomena for the initial density of heated electrons $n_{0}$ (equal to the
density of ions) within the $10^{21}$ – $1.7\cdot 10^{22}$ cm-3 range for
strong external magnetic fields in the range of 13–2500 T and a lower density,
$n_{0}=10^{20}$ cm-3, for milder fields in the range of 0.5–13 T. A typical
field value of 13 T is chosen based on the parameters of the experiment
planned in the Institute of Applied Physics of the Russian Academy of Sciences
where an observation of the multi-scale magnetic field structures is possible.
For the indicated densities, according to estimates (see also section 4), the
external fields less than 1 T and 0.1 T, respectively, have little effect on
the energy and structure of the magnetic fields generated in the plasma. The
external fields above 2500 T and 200 T, respectively, practically exclude the
explosive decay of a hot plasma discontinuity with the parameters mentioned
above.

PIC-simulations using the EPOCH code (Arber et al., 2015) are carried out
either as completely three-dimensional 3D3V with periodic boundary conditions
at $z=\pm L_{z}/2$ (with $L_{z}=40$ $\mu$m) or simplified 2D3V, omitting all
dependencies on the coordinate $z$, but still taking into account all three
components of all vectors (including the particle velocity vectors). In the
latter case, the $z$-axis is directed across the computational plane $xy$ and
the development of the Weibel instability or the plasma-boundary currents, as
a rule, leads to the formation of filaments or current sheets elongated mainly
along this $z$-axis. On the side boundaries of the computational domain
($x=\pm L_{x}/2$), parallel to the $y$-axis, the periodic boundary conditions
for particles and fields are used. The lower boundary ($y=-L_{y}/4$) reflects
particles but allows the fields to escape (be absorbed). The upper boundary
($y=3L_{y}/4$) is open for both particles and fields. The dimensions are
$L_{x}\times L_{y}=240\times 240$ $\mu$m2 for the density $n_{0}=10^{20}$,
$10^{21}$ cm-3 and $L_{x}\times L_{y}=36\times 36$ $\mu$m2 for the density
$n_{0}=1.7\cdot 10^{22}$ cm-3.

The plasma comprises $2.5\cdot 10^{9}$ ($2\cdot 10^{8}$) macroparticles of
each fraction, electrons and ions, in 3D3V (2D3V) calculations and the
computational domain consists of $400\times 400\times 400$ ($1200\times 1200$)
cells. The simulation duration is mainly limited by the moment of time
$\tau_{\mathrm{R}}=6\cdot 10^{4}\,\omega_{\mathrm{pe}}^{-1}$ when the
transient phenomena have been already fully manifested but the qualitative
differences between the 3D3V and simplified 2D3V calculations have not yet
usually arisen. Here $\omega_{\mathrm{pe}}=(4\pi e^{2}n_{0}/m)^{1/2}$ is the
plasma frequency and $e$ is the elementary charge. We have
$\tau_{\mathrm{R}}\approx$ 8 ps and 100 ps for $n_{0}=1.7\cdot 10^{22}$ and
$10^{20}$ cm-3, respectively. The skin depths are
$c/\omega_{\mathrm{pe}}\approx$ 0.04 $\mu$m and 0.5 $\mu$m, respectively, and
do not appear in our simulations, so that the observed phenomena are not
related to this scale.

For the chosen parameters, particle collisions are weak in the rarefied plasma
regions of interest where the expansion into the vacuum (towards $y>0$) and
the formation of currents take place. Indeed, already for the densities of
$0.1\,n_{0}=10^{19},1.7\cdot 10^{21}$ cm-3 the mean free path of hot
electrons, $L_{\mathrm{f}}$, is within the range 5000–50 $\mu$m, thus greater
than $r_{0}$, $L_{x,y}$ and a typical size of the region where the small-scale
magnetic fields are formed (see, e.g., figs. 2–7). Particle collisions are
significant in the dense plasma below the target surface where the
collisionless PIC-simulations may not be valid.

Figure 1: Geometry of a semi-cylindrical cloud (below the target surface
$y=0$) of heated electrons and their initial temperature distribution in the
PIC-simulations.

Figure 2:  (a) The current density component $j_{z}$ representing the self-
generated filamentary structure that is mostly aligned with the $z$-axis and
(b) the magnitude of the transverse magnetic field,
$\left(B_{x}^{2}+B_{y}^{2}\right)^{1/2}$, from the 3D PIC-simulation of the
expansion of the plasma cloud with hot electrons into vacuum at the moment of
time $t=3.0$ ps $\approx 3\cdot 10^{4}\omega_{\mathrm{pe}}^{-1}$. The initial
plasma density is $n_{0}=1.7\cdot 10^{22}$ cm-3. The external magnetic field
is oriented along the $z$-axis, $\vec{B}_{0}\|Oz$, $B_{0z}=250$ T.

##  3 3D simulation for a typical set of plasma parameters with the external
magnetic field parallel to the elongated heated region

Due to a lack of room, we illustrate a full 3D calculation by just one fig. 2.
Fig. 2b shows a typical tufted structure of the transverse component of the
self-generated magnetic field $B_{\bot}=(B_{x}^{2}+B_{y}^{2})^{1/2}$ at a high
initial density $n_{0}=1.7~{}\cdot~{}10^{22}$ cm-3 of hot electrons with an
average temperature $T\sim 0.5$ keV and a strong external magnetic field
$B_{0z}=250$ T directed along the axis $z$ of the heated semi-cylinder. This
distribution of the field is created by the electric current filaments, on
average parallel to $z$ and shown in fig. 2a. At the short time $t=3.0$ ps
after the expansion begins, there is only a small semi-cylinder with a radius
less than 3 $\mu$m where the longitudinal self-generated field $B_{z}-B_{0z}$
is of the order of (and directed opposite to) the external field. At this time
and during further expansion at the picosecond timescale, when the plasma
pressure dominates over the magnetic one, the displacement of the external
field by the large-scale transverse electron currents (as in a solenoid)
occurs approximately with an ion-acoustic speed $\propto(T/M)^{1/2}\sim
10^{6}$ m/s, i.e., the deceleration of the plasma flow in the $xy$-plane only
slightly exceeds its deceleration in the absence of the external field. (In
actual experiments with heavier ions of $M\sim 50000\,m$ and higher electron
temperatures of $T\sim 5$ keV, the plasma will expand with a smaller ion-
acoustic speed $\sim 10^{5}$ m/s.)

Hereafter attention should be paid first of all to the rather universal effect
of a small-scale structuring, in particular pinching, of electric currents and
the magnetic fields generated by them due to the Weibel-type instability,
which itself results from the growth of the electron velocity distribution
anisotropy in the nonequilibrium plasma expanding into vacuum. Namely, the
temperature of electrons along the $z$-axis practically does not change for a
long time, while their temperature in the transverse $xy$-plane decreases
rather rapidly and significantly (this is a 3D version of the mechanism
originally proposed in Thaury et al. (2010) and further studied in Nechaev et
al. (2020a)). The instability has the maximum growth rate for the perturbation
wave vectors orthogonal to the direction of the maximum electron temperature
(the $z$-axis here), and the growth rate decreases (Vagin & Uryupin, 2014)
with a decrease of the angle between a wave vector and the $z$-axis. So, at
later times when the growth of the $z$-oriented filaments is saturated due to
the self-generated magnetic field, the oblique filaments become quite
pronounced and the entire current structure becomes more irregular and
strongly dependent on the $z$-coordinate. It is shown already in fig. 2a where
there are some oblique filaments of finite lengths, especially in the center
region of the expanding plasma.

We justify this conclusion on the Weibel-type mechanism of the growth and
saturation of the small-scale current filaments by comparing the gyroradius
and gyrofrequency of hot electrons in the self-generated magnetic field with
the scale of inhomogeneity and the growth rate of the magnetic field
characteristic of the Weibel instability (Vagin & Uryupin, 2014; Kocharovsky
et al., 2016; Nechaev et al., 2020a). As the simulations show, in the region
of interest, $0<y<5$ $\mu$m (see fig. 2), the plasma density and the
longitudinal electron temperature are $n_{\mathrm{e}}\sim 0.1n_{0}\approx
1.7\cdot 10^{21}$ cm-3 and $T_{z}\sim 0.5$ keV, respectively. Adopting also
the observed values of the self-generated transverse field $\sim 50$ T and the
temperature anisotropy $A=T_{z}/T_{x}-1\sim 1$, we find that the gyroradius of
a hot electron, $r_{B}=(2T_{z}/m)^{1/2}\omega_{B}^{-1}\approx 1.5$ $\mu$m, is
approximately equal to the optimal scale of the electron Weibel instability,
$\lambda\sim
10\,c/\omega_{\mathrm{pe}}\,(n_{0}/n_{\mathrm{e}})^{1/2}A^{-1/2}\sim 1.4$
$\mu$m, and its gyrofrequency $\omega_{B}\approx 9\cdot 10^{12}$ rad/s is
close to the maximum Weibel growth rate $\Gamma\sim
3\,(T_{x}/m)^{1/2}\lambda^{-1}(1+A^{-1})^{-1}\sim\omega_{\mathrm{pe}}/900\sim
8\cdot 10^{12}$ s-1. Hence, the magnetic field value satisfies the well-known
saturation condition for the Weibel instability (see, e.g., Kocharovsky et
al., 2016). Its characteristic growth time $\tau_{B}$ in simulations is also
consistent with the Weibel mechanism: $\tau_{B}\sim 10\,\Gamma^{-1}$.
According to the next section, similar qualitative conclusions ($\lambda\sim
r_{B}$ and $\tau_{B}\sim 10\Gamma^{-1}$) follow from the 2D simulations (fig.
3), which have been compared with the 3D simulations and shown to be correct
with respect to the main features of the small- and large-scale structures of
the generated magnetic fields.

Also noticeable is the cumulation (focusing) of the plasma flow, formed
collectively by the hot electrons and the cold ions, in the direction of the
maximum deformation of the magnetic field lines in the region of the greatest
pressure of escaping electrons. The cumulation takes place when this field is
oriented along the long heated region on the target surface as is discussed in
the next section (see fig. 3). Note that in all of these simulations the
dynamics of electric currents is quasi-static, since the evolution time of the
characteristic structures exceeds the scale of their inhomogeneity divided by
the velocity of typical particles, and the role of the induction electric
field is inessential.

##  4 2D simulations for a wide range of the plasma density and external
magnetic field values. A comparative analysis

The established validity of the simplified 2D3V calculations for a qualitative
analysis of physical phenomena at a considerably long period of the decay of
the plasma discontinuity with an elongated (semi-cylindrical) region of heated
electrons, allows us to describe in more detail a number of features of this
process at various plasma densities as well as various values and orientations
of an external magnetic field. It is convenient to illustrate these features
by comparing figs. 3, 4, 5, 6, 7.

Figure 3:  Magnetic field structures generated in the 2D simulation of the
expansion of the plasma cloud with hot electrons into vacuum at $t=8$ ps
$\approx 6\cdot 10^{4}\omega_{\mathrm{pe}}^{-1}$ after the beginning of the
expansion. The initial plasma density is $n_{0}=1.7\cdot 10^{22}$ cm-3. The
external magnetic field is $B_{0}=250$ T and orthogonal to the simulation
plane, $\vec{B}_{0}\|Oz$. (a) $B_{z}$ component of the total magnetic field
minus the external magnetic field. (b) $B_{x}$ component of the magnetic
field; black contours show the isolines of the normalized density for
$n/n_{0}=0.01,0.1,1$.

First of all, it is clear from figs. 2, 3, 4, and 7 that, in the region of the
expanding collisionless plasma cloud, the magnetic field could substantially
weaken or significantly change its direction and even become directed opposite
to the external one, keeping almost the same order of magnitude ($\sim
B_{0}/3$). Moreover, under certain conditions, according to the laws of
magnetostatics, in the outside boundary region next to the expanding cloud
(practically in vacuum) the magnetic field can also change its direction and
even increase, differently in different regions and depending on the strength
of the local quasi-surface currents and the orientation of the external
magnetic field. At the same time, for both orthogonal orientations of the
external field ($B_{0z}$ and $B_{0x}$), there is an axial asymmetry, i.e., a
difference in directions of the generated magnetic fields, especially near the
initial plasma discontinuity to the left and to the right with respect to the
center of the heated section. This asymmetry arises due to the vertical,
parallel to the $y$-axis, component of the electric current created by the
central ”fountain” of the escaping electrons and the currents flowing in the
dense plasma and compensating the charge of these escaped electrons, as can be
seen from the magnetic field distributions in figs. 3b and 4a (on the
”fountain” mechanism in the absence of an external magnetic field, as in fig.
6, see, e.g., Sakagami et al. (1979); Kolodner & Yablonovitch (1979);
Albertazzi et al. (2015)).

Figure 4:  Magnetic field structures generated in the 2D simulation of the
expansion of the plasma cloud with hot electrons into vacuum at $t=8$ ps after
the beginning of the expansion. The initial plasma density is $n_{0}=1.7\cdot
10^{22}$ cm-3. The external magnetic field is $B_{0}=250$ T and lies in the
simulation plane, $\vec{B}_{0}\|Ox$. (a) $B_{z}$ component of the magnetic
field. (b) $B_{x}$ component of the total magnetic field minus the external
magnetic field; black contours show the isolines of the normalized density for
$n/n_{0}=0.01,0.1,1$.

Figure 5:  The expansion of the plasma cloud with hot electrons (with the
initial density of $n_{0}=10^{20}$ cm-3) into vacuum with the external
magnetic field $B_{0}=13$ T, $\vec{B}_{0}\|Oz$, at $t=30$ ps $\approx 2\cdot
10^{4}\omega_{\mathrm{pe}}^{-1}$ after the beginning of the expansion. (a) The
magnitude of the total magnetic field. (b) The effective temperature $T_{z}$
along the $z$-axis; black contours show the isolines of the normalized density
for $n/n_{0}=0.01,0.1,1$.

Figure 6:  The expansion of the plasma cloud with hot electrons (with the
initial density of $n_{0}=10^{20}$ cm-3) into vacuum without an external
magnetic field at $t=30$ ps after the beginning of the expansion. (a) The
magnitude of the total magnetic field. (b) The effective temperature $T_{z}$
along the $z$-axis; black contours show the isolines of the normalized density
for $n/n_{0}=0.01,0.1,1$.

Figure 7:  The expansion of the plasma cloud with hot electrons (with the
initial density of $n_{0}=10^{20}$ cm-3) into vacuum with the external
magnetic field $B_{0}=13$ T, $\vec{B}_{0}\|Ox$, at $t=30$ ps after the
beginning of the expansion. (a) The magnitude of the total in-plane magnetic
field; arrows show the direction of the in-plane field component. (b) $B_{x}$
component of the total magnetic field minus the external magnetic field; black
contours show the isolines of the normalized density for $n/n_{0}=0.01,0.1,1$.

Figure 8:  The temperature anisotropy degree, $A=T_{z}/T_{x}-1$, induced by
the expansion of the plasma cloud with hot electrons (with the initial density
of $n_{0}=10^{20}$ cm-3) into vacuum (a) with the external magnetic field
$B_{0}=13$ T, $\vec{B}_{0}\|Oz$, (b) without an external magnetic field, both
at the moment of time $t=30$ ps $\approx 2\cdot
10^{4}\omega_{\mathrm{pe}}^{-1}$ after the beginning of the expansion.

According to the simulations, the resulting structure of the electric currents
and quasi-magnetostatic fields is practically independent of an external
magnetic field only if the latter is very weak ($B_{0}<0.1$ T and $B_{0}<1$ T
for the chosen plasma parameters with the initial hot electron density of
$n_{0}=10^{20}$ cm-3 and $1.7\cdot 10^{22}$ cm-3, respectively). Such a weak
field is easily displaced by the plasma and does not essemtially change an
electron distribution function. If the external magnetic field is stronger,
though still satisfies the inequality $B_{0}^{2}/8\pi\ll n_{0}T$ and is
displaced by the plasma, then the electron distribution function in a
considerable volume of a plasma cloud is significantly changed during the
plasma expansion. Nevertheless, the Weibel instability will proceed
qualitatively similar to that in the absence of the external field, if both
the electron gyrofrequency $eB_{\mathrm{r}}/(mc)$ for the remaining in the
region (after the displacement) average magnetic field $B_{\mathrm{r}}$ and
the inverse gyroradius of energetic electrons in this field are sufficiently
(logarithmically) less than the maximum growth rate and the corresponding
wavenumber of the instability in the anisotropic expanding plasma,
respectively. Otherwise, the instability is influenced by the remaining large-
scale inhomogeneous magnetic field which depends on the orientation of the
external field and is comparable to the Weibel saturation field. As a result,
the structure of the self-generated magnetic field during the decay of the
discontinuity will differ considerably from that in the absence of an external
field (fig. 6).

The resulting profile of the ejected plasma cloud is almost independent of
even stronger external fields for any of their orientations, e.g., for the
values $B_{0z}$, $B_{0x}$ under $13$ T at the density of $n_{0}=1.7\cdot
10^{22}$ cm-3. However, upon reaching the indicated (see figs. 5 and 7 for
$n_{0}=10^{20}$ cm-3) and larger values (for example, $B_{0z}$, $B_{0x}=250$
T, as in figs. 3 and 4 for $n_{0}=1.7\cdot 10^{22}$ cm-3), the cases of
orientation of the external field along and across the semi-cylinder of heated
electrons begin to differ significantly. In the first case, the aforementioned
cumulative effect takes place and, as a result, the rate of the displacement
of this field is significantly higher (for example, almost by $1.5$ times at
$B_{0z}=250$ T) and the plasma cloud is narrower (also almost by $1.5$ times
at $B_{0z}=250$ T) than in the case of the same-magnitude field $B_{0x}$
orthogonal to the heated semi-cylinder. In the latter case, the field $B_{0x}$
flattens the plasma cloud via a formation of a completely different system of
the large-scale electric currents. In addition, only when the external field
is oriented along the $z$-axis there is a noticeable violation of the symmetry
of expansion caused by a systematic displacement of a large fraction of
electrons to the left under the action of the Lorentz force (directed mainly
opposite to the $x$-axis), see figs. 3 and 5. At the same time, only when the
external field is oriented along the $x$-axis, a formation of the large-scale
currents predominantly directed along the $z$-axis takes place at the top of
the ejected plume. As a result, they stretch and smoothly bend the magnetic
field lines in the $xy$-plane and significantly enhance the external magnetic
field in vacuum above the plasma plume; see figs. 4b and 7b.

Our calculations reveal small-scale structures of electron currents flowing
mainly parallel to the $z$-axis in the form of $z$-pinches with a wide range
of transverse scales $\sim 2-50$ $\mu$m (deformed by the plasma density
gradient). They gradually drift along with the plasma in the course of its
expansion and give rise to an inhomogeneous set of dipole spots in the
structure of the transverse magnetic field components $B_{x}$, $B_{y}$ (minus
the external field $B_{0x}$, if any; see figs. 3b and 4b). The observed
phenomenon is due to the Weibel instability owing to the anisotropically
cooling hot electrons, whose effective temperature rapidly decreases along the
$x$\- and $y$-axes during the discontinuity decay and changes more slowly
along the $z$-axis because of an unlimited length of the initially heated
plasma semi-cylinder (cf. Thaury et al., 2010). The typical patterns of the
electron anisotropy in the cases with and without the external magnetic field
$B_{0z}$ are shown in fig. 8. The anisotropy degree is quite strong, $A\sim
1$, and the external field $B_{0z}=13$ T does not inhibit the formation of a
set of $z$-pinches, though introduces an asymmetry and slows down a bit the
plasma expansion. The appearance of the small-scale current filaments, similar
to $z$-pinches, can be prevented by the strong external field $B_{0x}$ during
the displacement of which the large-scale currents along the $z$-axis are
generated in the plasma cloud and, as a result, the growth of the temperature
anisotropy of the cooling electron distribution is suppressed.

On the contrary, the presence of the external field $B_{0z}$ does not prevent
the anisotropic cooling of electrons, so that, from the very beginning of the
discontinuity decay, a formation of the multiple-pinch-like current structures
occurs (see fig. 4b for the $x$-component of the quasi-magnetostatic turbulent
field created by them as well as fig. 5b for the effective electron
temperature $T_{z}$ along the $z$-axis associated with such a turbulence). The
thickness of these $z$-pinches turns out to be of the order of the electron
gyroradius, and their transverse magnetic fields can be of the order of or
exceed the external field $B_{0z}$. In particular, in the case of fig. 5,
where $n_{0}=10^{20}$ cm-3 and $B_{0z}=13$ T, for a typical filament we
observe the magnetic field magnitude $\sim 5-10$ T, radius $\sim 3-5$ $\mu$m
and total current $\sim 100$ A which are an order of magnitude weaker, by two-
three times larger and of the same order of magnitude than, respectively,
those in the case of fig. 3 (or fig. 2), where both $n_{0}=1.7\cdot 10^{22}$
cm-3 and $B_{0z}=250$ T are larger. The simulations for the intermediate value
of the initial plasma density, $n_{0}=10^{21}$ cm-3, show qualitatively the
same results.

Additional simulations (not provided in this short article) show that an
increase of the ion-to-electron mass ratio decreases the ion-acoustic speed,
slows down the process of the plasma expansion and proportionally reduces the
scale of the current structures being formed, but does not qualitatively
change their small-to-large scale hierarchy and evolution. Also, as expected,
an insufficient heating of electrons, i.e., the presence of a significant
fraction of cold electrons, notably changes the density profile and geometry
of the plasma ejection, the type of deformation and the rate of the
displacement of the external field, the number and spatial distribution of the
formed small-scale $z$-pinches.

##  5 Main qualitative results and some open physical problems

It is clear from the above that the process of the decay of a plasma–vacuum
discontinuity with a semi-cylindrical region of heated electrons elongated
along the plasma surface largely depends on the magnitude and direction of the
external magnetic field parallel to the discontinuity surface, even if the
pressure of this field is much less than the plasma pressure. The hot
electrons make a decisive contribution to the latter and strongly affect the
evolution and spatial structure of emerging quasi-magnetostatic perturbations
of various scales, especially due to the anisotropic cooling during the
expansion.

Along with the obvious influence of the sufficiently strong external magnetic
field on the density profile of the expanding plasma cloud, it turns out that
even a relatively weak external magnetic field can both suppress and promote,
depending on its orientation, the formation of various current filaments,
sheets and large-scale structures. The performed simulations and estimates
allow us to determine the conditions under which the multiple formation,
prolonged existence and significant shift of the localized current filaments
(similar to $z$-pinches) and more complex current configurations take place in
the most part of the region swept from the external magnetic field by the
expanding plasma.

The resulting current filaments of the $z$-pinch type can have a significantly
increased plasma density and create small-scale magnetic fields which turn out
to be of the order of or even stronger than the external field and contain
about several percent of the initial energy of hot electrons. This occurs in
those spatio-temporal regions where the plasma density and the anisotropy of
the electron velocity distribution are sufficiently high and promote the
development of the Weibel-type instability until its nonlinear saturation as
well as a long-term existence of the entire current structures.

The formation of magnetic fields with a larger scale, of the order of the
characteristic transverse size of the elongated heated plasma region, is
caused by both the initially strongest ”fountain” currents of the fastest
escaping electrons as well as by the rapidly generated volumetric or quasi-
two-dimensional currents of the hot electrons moving inside the
inhomogeneously expanding plasma cloud and along its boundary with the
unperturbed external magnetic field. Interacting with this larger-scale
current structure, the unidirectional external magnetic field oriented
transverse to the main flow of the hot electrons can lead to not just a
violation of the symmetry of the expansion, but also to a cumulation of the
plasma flow and an inhomogeneous deformation of the emerging multi-scale
current structures.

We considered just the simplest possible model of the expansion of an
inhomogeneously heated magnetized plasma with a uniform density, bearing in
mind an ideal cylindrical lens and an ideal target subjected to an ablation by
a femtosecond laser beam. Obviously, rather different (not quasi-one-
dimensional) distributions of the density and effective temperatures of the
rapidly heated electrons are possible if they are exposed to ultrashort laser
pulses of various duration, cross section, polarization, and optical
frequency. Moreover, the velocity distribution function of electrons could be
non-Maxwellian and anisotropic at the very beginning of the expansion. In the
actual experiments, a non-planar initial geometry of the interface between a
heated plasma and vacuum or a background plasma, particle collisions, and an
inflow of hot electrons from deeper heated regions of the target — all of
which we did not take into account — could also be important. These and other
factors related to the planned experiments on the plasma discontinuity decay
in an external magnetic field will be addressed elsewhere.

The obtained results can find applications in the topical studies of the
astrophysical objects as well as the high-energy density laboratory plasma
systems. The described transient phenomena of the self-consistent growth and
nonlinear evolution of the small- and large-scale magnetic field structures
are particularly relevant to the dynamics of the solar flares, stellar wind
formation and nonstationary structure of planetary magnetosheath regions; see,
e.g., (DeForest et al., 2018; Dudík et al., 2017; Viall & Borovsky, 2020;
Lazar et al., 2022; Vörös et al., 2017; Kelley, 2009; Balogh et al., 2018).
Discussions of the related kinetic phenomena in the physics of cosmic plasma
are beyond the scope of the present Letter.

##  6 Conclusions

The results presented above show that the character of the expansion of a
collisionless electron-ion plasma into vacuum and the currents and magnetic
fields of various scales generated in the transition layer depend
significantly on the geometry of the region of initially heated electrons and
the magnitude and direction of the external magnetic field oriented along the
plasma surface. We carried out the detailed numerical (PIC) analysis for a
region with initially isotropically heated electrons in the form of a long
semi-cylinder, the axis of which is located on the plasma surface. This
analysis discloses a crucial role of the Weibel-type instabilities that are
associated with the emerging anisotropy of the electron velocity distribution
and strongly depend on the orientation of the external magnetic field.

We reveal, under certain conditions, a formation and a rapid expansion of
highly inhomogeneous electron currents in the form of filaments (similar to
$z$-pinches) parallel to the external magnetic field as well as a formation
and a slow evolution of current sheets oriented at different angles to the
boundary between the plasma and the deformed magnetic field. We find that
these currents can create fields significantly exceeding in magnitude the
external magnetic field and indicate qualitatively the conditions on the
magnitude and orientation of the latter as well as on the plasma parameters
and electron heating required for this to occur. The predicted phenomena of
the decay of the plasma discontinuity are feasible not only in the laser
plasma but also in the coronal arches, stellar wind, and explosive processes
in the planetary magnetospheres.

##  7 Acknowledgments

The laboratory-astrophysics part of the work, especially related to the 3D
simulations, was supported by the Russian Science Foundation, project no.
21-12-00416. Simulations were carried out in the Joint Supercomputer Center of
the Russian Academy of Sciences.

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