current structures of both kinds. Such effects of an external field on the
expansion of plasma subjected to particle injection are analyzed in the next
section.

##  5 Effect of an external field on the generation of strong small-scale
magnetic fields in the course of plasma injection into a cold plasma layer

###  5.1 Particle injection into a magnetized background plasma

Let us now consider the case of combined presence of both background plasma
and an external magnetic field, which is realized, as in the previous section,
for a laser plasma created by ablation of the target with a femtosecond pulse
having a prepulse. This section is focused on the effect of an external
homogeneous magnetic field $B_{0}$ on a similar process of filamentation of
the current density $j(r,t)$ and generation of small-scale quasi-magnetostatic
structures. The number density of cold background plasma with a temperature
$T_{0}<1$ keV is considered to decrease with distance from the target surface
(along the $y$ axis) according to the exponential law $n_{0}\exp(-y/L)$ with a
characteristic scale $L\sim 10$ $\mu$m and a near-surface value of the order
of $n_{0}=10^{21}$ cm-3. Again, we have in mind the cylindrical focusing of a
femtosecond laser pulse onto a target. When heated, a narrow strip of a width
$2x_{0}$ (from several to tens of microns) for a long time $t_{\mathrm{inj}}$
(picoseconds or more) ejects a Maxwellian plasma which contains fairly cold
ions (with the temperature $T_{0}$) and very hot electrons with a temperature
$T$ up to hundreds of keV. On the considered scales of the order of tens of
microns, for these electrons Coulomb collisions are rare and the plasma is
actually collisionless. To be specific, we assume that its number density is
distributed along the $x$ axis according to the law
$0.3n_{0}\exp[-(x/x_{0})^{2}]$.

We choose the external field in the range 5–500 T and consider it being
parallel to the target $xz$ plane and oriented either along the hot strip ($z$
axis) or orthogonal to it ($x$ axis), according to the end of subsection 3.4.
Above the specified maximum value of field, satisfying the condition
$B_{0\mathrm{max}}\sim(n_{0}T)^{1/2}$, i.e., lying near the magnetization
boundary of hot electrons, the expansion of plasma is strongly suppressed and
the generation of currents and magnetic fields practically does not occur. For
the field values less than the specified minimum value, the hot electron
gyroradius significantly exceeds the scale of plasma inhomogeneity and the
influence of the magnetic field turns out to be insignificant.

The PIC-modeling is carried out with several billion macroparticles in a
numerical box with the side lengths $L_{x}$, $L_{y}$, $L_{z}$ in the range of
60–200 $\mu$m for a duration of $t_{\mathrm{sim}}=5$–$10$ ps. For uniformity
in all calculations, the injection time $t_{\mathrm{inj}}=2$ ps is taken equal
by the order of magnitude to the time of flight of a hot electron through the
numerical box, the walls of which across the $x$ and $y$ axes are open (along
the $z$ axis the boundary conditions are periodic). In this case, the total
number of injected electrons is of the order of the initial number of cold
electrons in the background plasma layer. Below, to illustrate rich physics of
the processes under consideration, both simplified 2D3V (with the spatial
dependence on the $z$ coordinate being neglected) and fully 3D3V calculations
are used; their results are qualitatively similar for each case examined.

Let’s briefly list the mechanisms and main stages of the formation of the
current configurations and their magnetic fields, indicating the reasons for
their dependence on the external magnetic field directed along the target
surface. Its orientation significantly affects the resulting structures and in
the next two subsections two qualitatively different, orthogonal orientations
are considered, similar to section 3.

The currents of the largest scale observed in this problem are the well-known
”fountain” currents (see, for example, [4, 55, 103]) of hot electrons escaping
from a strip of width $\sim 2x_{0}$ from the target surface mainly upward
along the $y$ axis and at moderate angles to it. At a weak external magnetic
field and a not very thick layer of the background plasma, they form a current
sheet that expands along the $x$ axis with distance from the target and
creates a magnetic field, which in the regions $x<0$ and $x>0$ is directed
against and along the $z$ axis, respectively.

###  5.2 The case of an external field parallel to the injection strip

If the electrons are high-energy and the injection strip is narrow, then very
quickly, as shown in Fig. 23 where $T=300$ keV and $2x_{0}=8$ $\mu$m, this
kind of current sheet is formed in the presence of strong field oriented along
the injection strip ($B_{0z}=100$ T). This field to a certain extent even
narrows the current sheet, of course, bending it by the Lorentz force, and
creates in the regions $x<0$ and $x>0$ oppositely directed large-scale fields
which exceed the external one markedly, even several times in magnitude in
some localized regions (especially near the target, where current eddies can
form). In the region with accumulating and anisotropically cooling electrons,
due to Weibel instability, $z$-pinches and associated transverse magnetic
field, also strong and oriented predominantly in the $xy$ plane, quickly form
and exist for quite a long time (Fig. 24).

Figure 23:  Distribution of the field component $B_{z}-B_{0z}$ (in Teslas) in
the 3D3V simulation at the time moment $t=2$ ps with an external field
$B_{0z}=100$ T. For clarity, the color scale is limited by the value $\pm 1$
kT; in some regions the field value reaches $2$ kT.  Figure 24:  Left panel:
Distribution of the absolute value of transverse field,
$\left(B_{x}^{2}+B_{y}^{2}\right)^{1/2}$ (in Teslas), and its field lines as
per 3D simulation at $t=2.5$ ps in the plane $z=30$ $\mu$m. Right panel:
Distribution of the magnetic field component $B_{z}-B_{0z}$ in the 2D
simulation at $t=4$ ps. In both simulations, the external field is
$B_{0z}=100$ T.

At the same external field for less energetic electrons and a wider injection
strip, according to calculations, the depth of penetration of the fountain
current upward from the target turns out to be smaller, this current has a
solenoidal shape and is concentrated closer to the boundary of the region out
of which the external field is displaced. However, this region is much wider
along the $x$ axis and inside it not only the above $z$-pinches emerge, but
due to eddy currents, including currents of cold electrons of the background
plasma, quite far from the target even stronger small-scale magnetic fields
are now formed, directed along the $z$ axis and exceeding the external field
by an order of magnitude.

If the external field directed along $z$ axis is many times weaker, say,
$B_{0z}=20$ T (Fig. 25), then both the fountain currents of hot electrons and
corresponding large-scale $B_{z}$ field and even more numerous ensemble of
$z$-pinches and their small-scale transverse fields are no longer very
different from those existing in the absence of external field. However,
according to the simulations performed, the violation of symmetry and
distortion of the overall structure of the fountain currents and the ensemble
of $z$-pinches, for the selected parameters, are quite noticeable down to
small values of external field $\sim B_{0z}=5$ T. In all cases, the plasma
region with hot electrons and the displaced external magnetic field expands
from the injection strip at approximately the ion-acoustic speed.

Figure 25:  The absolute value of transverse field
$\left(B_{x}^{2}+B_{y}^{2}\right)^{1/2}$ (in Teslas) as per the 2D simulation
with an external magnetic field $B_{0z}=20$ T at the time moment $t=4$ ps.

Moreover, for all considered values of this field, $B_{0z}=5$–$500$ T, the
slope of the current filaments, formed due to the filamentation Weibel
instability, is apparent from the very start of the injection of hot electrons
into the cold background plasma (see Fig. 26). The cold electrons, under the
influence of the electric field of ions, are forced to move towards the target
until the supply of electrons in the background plasma layer runs out or the
injection of plasma with hot electrons ends.

Figure 26:  The magnetic field’s component $B_{z}-B_{0z}$ (in Teslas) in the
2D simulation with an external field $B_{0z}=20$ T at the time moment $t=0.6$
ps.

Note that in the simplified 2D3V modeling the pattern of these currents has
the form of a ”fan” of sheets emerging from the injection strip, while in the
full 3D3V modeling, according to Fig. 27, above this strip a dense system of
filaments is observed, elongated along the $y$ axis, partially ordered into
sheets along the $z$ axis and having different transverse scales – larger for
hot electrons and smaller for cold ones.

Figure 27:  Distribution of the current density $j_{y}/(en_{0}c)$ in the plane
$y=6$ $\mu$m in the 3D simulation with an external field $B_{0z}=100$ T at the
time moment $t=0.6$ ps.

###  5.3 The case of an external field orthogonal to injection strip

Turning to the case of an external field directed along the $x$ axis, let us
note that in all calculations performed the external field does limit
significantly the injection of plasma with hot electrons, change fountain
currents, and suppress the formation of small-scale current filaments and
magnetic fields more strongly, other things being equal, than in the case
discussed in subsection 5.2. This can be seen from comparing Figs. 25 and 28
at a moderate field $B_{0x}=20$ T and a high temperature $T=100$ keV.
Moreover, in Fig. 28 all small-scale current filaments of hot electrons have
one sign of projection onto the $z$ axis (in contrast to filaments with
alternating signs in Fig. 25), since they are formed due to Weibel instability
dragged by the background of a wider current which is directed along this axis
and has arisen out of the fountain currents under the influence of external
field in the process of its displacement. Large-scale current filaments near
the boundary of the plasma region with the displaced magnetic field are not
shown in Fig. 28.

Figure 28:  Current density $j_{z}/(en_{0}c)$ in the 2D simulation with an
external field $B_{0x}=20$ T at the time moment $t=4$ ps.

In a stronger field $B_{0x}=100$ T with the same other parameters of plasma,
the formation of small-scale current filaments is suppressed. However, even
after a long time after the end of injection, according to Fig. 29, hot
electrons still form two pairs of large-scale stratified $z$-pinches, ensuring
displacement of external field from the region pressed to the target and
stretched along it.

Figure 29:  Absolute value of transverse field minus the external one,
$\left([B_{x}-B_{0x}]^{2}+B_{y}^{2}\right)^{1/2}$ (in Teslas), as per 2D
simulation at the moment $t=9$ ps. The external magnetic field is $B_{0x}=100$
T.

During the formation of such a region in the case of injection of plasma with
more energetic electrons ($T=300$ keV) out of a less wide band ($2\,x_{0}=8$
$\mu$m) at the same external field $B_{0x}=100$ T, as shown in Fig. 30, small-
scale current filaments exist not for long and only at the very start of the
injection process, mainly at the edges of injection strip and not far from the
target, where the roles of both hot and cold electrons are substantial.

Figure 30:  Current density $j_{z}/(en_{0}c)$ in the 3D simulation with an
external field $B_{0x}=100$ T at the time moment $t=2$ ps.

It is necessary to emphasize that, as in the previous sections, a few examples
of the electron current filamentation in the course of the injection of plasma
containing hot electrons into an inhomogeneous layer of cold plasma in
external magnetic field cannot address all important kinetic phenomena
observed in numerical modeling. Their dependence on various parameters of the
problem, even in the considered configurations, remain to be studied.

##  6 Discussion and conclusions

Thus, the electron Weibel-type instability turns out to be decisive for the
creation of small-scale strong magnetic fields in all cases of expansion of a
collisionless laser plasma analyzed above. This statement is valid for the
boundary-value problem related to the injection of a plasma containing hot
electrons into a cold plasma layer, including the magnetized one, as well as
the initial-value problem related to the decay of a strong discontinuity
between a plasma containing hot electrons and a cold plasma or vacuum
subjected to a magnetic field. Yet, it remains to study the effect of particle
collisions, albeit rare but existing in an actual laser plasma, especially
near the target where the plasma is dense, on the aforesaid mechanism of the
magnetic field generation.

###  6.1 Basic scenarios of plasma discontinuity decay and current
filamentation

During the decay of a sharp discontinuity between a dense plasma containing
hot electrons and a rarefied cold plasma (sec. 2), the Weibel instability
takes place in an expanding plasma owing to a considerable, by a factor of two
or more, cooling of hot electrons along the direction of their expansion,
while in the transverse plane they cool much weaker. Consequently, in a large
downstream region of the emerging quasi-electrostatic shock [81] and in a
fairly wide layer in the shock’s upstream, a strong small-scale magnetic
fields quickly form. They are oriented mostly along the front, have a spatial
modulation in the orthogonal direction and bear an energy of up to an order of
magnetude less than the energy of hot electrons. In laser plasma, the mega-
Gauss magnitudes of the generated fields, their micrometer spatial scales, and
picosecond growth times are feasible.

When the Weibel instability is saturated, a stage of gradual decay of the
self-consistent quasi-static fields and currents starts. Their slow nonlinear
deformation and increasing spatial scales are consistent with the expansion of
both the shock wave and the region of pronounced anisotropy of the electron
velocity distribution. Using typical model examples, we establish a spatial
correlation between the degree of this anisotropy and the magnitude of
magnetic field, describe the dynamics of the spatial spectra of the field and
current density, and point out the dependence of the developing electron
anisotropy and the features of the emerging magnetic field structure on the
parameters of the initial discontinuity between the cold rarefied plasma and
dense plasma with hot electrons.

At the same time, the numerical modeling and qualitative analysis, described
both in section 2 and in the subsequent sections, represent only the beginning
of a detailed study of the outlined class of processes occurring in the course
of decay of strong discontinuity. In particular, the question remains open
about (i) a possibility of effective generation of strong magnetic fields due
to counter flows of cold electrons from the background plasma and (ii) plasma
parameters on opposite sides of the discontinuity required for this generation
to occur. Another important question concerns the optimal parameters of the
discontinuity, including the ratio of the initial temperatures and number
densities of cold and hot electrons, that are necessary for the utmost
efficient generation of magnetic field. It is interesting also to what extent
the created field can affect the dynamics of the nonequilibrium plasma’s
expansion, structure of the compacted layer at the shock front, temper of the
anisotropic cooling of the hot electrons as well as heating of the cold ions.
Also intriguing is the analysis of the field structure and the related spectra
of electron and ion currents in the upstream region of the shock wave.
Analysis of a nonlinear phase of Weibel instability faces the important
problem of particle energy exchange and the problem of an accelerated
diffusion, particularly if the decay of a discontinuity in a plasma is
supplemented with the injection, i.e., the presence of a constant flow of
plasma with hot electrons coming from the region of dense plasma (or from a
heated target in the problem of laser ablation).

According to section 3, already a weak external field, with the pressure
significantly less than that of a cloud containing hot electrons, expanding in
the $xy$ plane and homogeneous along the $z$ axis significantly affects plasma
spreading into vacuum and spatio-temporal pattern of the generated quasi-
magnetostatic formations (turbulence). The most important point here is the
anisotropic cooling of the ejected electrons. It significantly depends on the
magnitude of external transverse magnetic field through a wide (several orders
of magnitude) range. This dependence persists down to a fairly small field
magnitudes not affecting the density dynamics of expanding plasma. On the
basis of detailed numerical PIC-simulations, we reveal formation, long-term
existence and pronounced motion of localized current structures ($z$-pinches
and sheets) in the region where the external field becomes weakened. The
associated small-scale fields can be stronger than the external field by many
times. The density of plasma inside $z$-pinches can be notably increased.
These outcomes are the result of the nonlinear development of Weibel-type
instability getting saturated when the energy of the quasistatic magnetic
field approaches several percent of the energy of hot electrons.

At the same time, there are fountains currents generated by the most energetic
escaping electrons and the quasi-2D or volumetric currents generated by hot
electrons inside the expanding plasma cloud. They both are associated with the
large-scale magnetic fields displacing the external field inside the cloud.
Symmetry of the expansion can be violated by an external field directed across
the hot electron flow. In this case, the current structures of all scales will
be inhomogeneously deformed and cumulation of the plasma flow could take
place.

In fact, the above analysis of a femtosecond laser ablation is based on the
simplest model of an inhomogeneously heated plasma of a uniform density in a
boundary layer of a target. The other factors could make the electron velocity
distribution non-Maxwellian and anisotropic as well as the spatial
distribution of the effective temperature and density of heated electrons non-
quasi-1D already from the very start of expansion. These factors include a
complex polarization, cross section and temporal profile of the laser pulse,
detuning of the optical frequency from the plasma frequency, non-planar
geometry and finite thickness of the discontinuity layer, a significant influx
of hot electrons continuously coming from deeper layers of the heated target,
where particle collisions are important. The analysis of how these and other
factors of the initial creation of a nonequilibrium plasma influence the
generated currents and fields just begins and is beyond the scope of this
review.

The above also applies to the features of the emergence, development and decay
of the magnetic and current structures of various scales, described in
sections 4 and 5 on the basis of a boundary-value problem. Namely, the problem
when a rarefied plasma with hot electrons is injected into a layer of cold,
denser plasma which has a monotonically decreasing number density and may be
subjected to an external magnetic field. The considered model corresponds to
the collisionless expansion of a high-energy-density laser plasma created as a
result of femtosecond ablation of a flat target by a cylindrically focused
laser beam, but can be characteristic of a number of other problems in the
physics of space and laboratory plasma.

In a wide range of the problem parameters, there coexist two qualitatively
distinct Weibel mechanisms of formation of mutually orthogonal structures of
currents, creating strong small-scale magnetic fields in neighboring plasma
layers. When crossing the boundary between these layers, the fraction of cold
electrons in the total plasma number density jumps considerably.
Correspondingly, the type of the electron velocity anisotropy, which is
responsible for the Weibel-type instability, changes. In the denser layer the
anisotropy axis is parallel to the direction of electron injection, while in
the less dense layer it is orthogonal to that direction.

In the former layer the electron distribution contains a gradually relaxing
hot flow and relatively cold thermal background. In the latter layer it
reminds a bi-Maxwellian distribution of hot electrons. The currents in these
layers are parallel to the direction of anisotropy axis, having the form of
wedge-shaped inhomogeneous sheets in 2D simulations (filamentation type of
Weibel instability) and filaments resembling deformed $z$-pinches (thermal
type), respectively. When injection of hot electrons ceases, the filaments
decay much slower than current sheets in virtue of the nonlinear trapping of
electrons. A lifetime of the filaments considerably exceeds the injection
duration.

As the simulation results reveal, the presence of external field in an
inhomogeneous layer of cold plasma, into which the plasma with hot electrons
is injected, in a wide range of system parameters significantly influences
dynamics and spatial structure of the appearing small-scale currents and
magnetic fields. These fields can be much stronger than the external field
and, simultaneously, substantially depend on it. Such current structures and
fields are generated not only by the hot electrons, but also by the cold
electrons of background plasma, and, therefore, significantly depend on its
number density and the scale of inhomogeneity. Thus, the characteristics of
not only the femtosecond laser pulse, which determine the size of the heated
spot on the target and the duration of the injection of hot electrons, are
important, but also those of the prepulse, which implicitly sets the scale of
the inhomogeneity and the initial number density of the background plasma.
These characteristics affect the time of formation, number of filaments and
current density in the emerging current systems as well as their subsequent
restructuring, decay rate and total lifetime after injection of plasma
containing hot electrons ceases.

In laser ablation experiments, it is possible to control the plasma
discontinuity parameters, i.e., the properties of its hot and cold parts, by
tuning the powerful femtosecond pulse that creates the hot part, and its
prepulse that determines the cold one. Modern technologies also make it
possible to create fairly uniform magnetic fields with different orientations
and induction ranging from units to hundreds of Tesla (in pulsed mode).
Finally, there are methods for diagnostics of the plasma number density and
magnetic fields with high spatial and temporal resolution (see, e.g., [110,
148, 9, 15, 120, 121, 90, 28]). All this gives reason to count on the
possibility of experimentally studying the predicted phenomena in laser
plasma. One can hope that the aforementioned patterns of transient phenomena
of the formation of current structures and related magnetic fields of
different scales and orientations in the course of the expansion of plasma
with hot electrons take place for more complex geometries and laser ablation
models and will prove useful for predicting and studying similar phenomena in
astrophysical and magnetospheric plasmas.

###  6.2 Some open problems in the theory of Weibel-type instabilities and
magnetic turbulence

Note that even in the approximation of a homogeneous plasma, without
accounting for its expansion, a number of features of Weibel-type aperiodic
instabilities of initialy anisotropic particle velocity distributions and
associated spatio-temporal dynamics of emerging quasi-magnetostatic turbulence
remain very poorly studied. Very fragmentary is the knowledge of even the
linear dispersion properties of these instabilities, i.e., the dependencies of
the growth rates of unstable modes of ordinary and extraordinary types on the
direction and magnitude of their wave vectors for a somewhat representative
class of particle velocity distribution functions (see, e.g., [48, 17, 132,
53, 114, 89]). Finding these dependencies turns out to be especially difficult
in the presence of external field, when one has to use numerical methods [13,
42, 137, 12, 133, 66, 131, 78, 62] or be limited to approximate analytical
solutions for a very narrow range of system’s parameters and wave vectors’
directions or lengths [59, 74, 64, 37, 92, 39, 25, 124].

Even less studied are the nonlinear properties of these instabilities, for
example, the criteria for their saturation, which establish maximum of the
root-mean-square value of the magnetic field in the emerging Weibel turbulence
depending on the problem parameters. The known criteria are very approximate,
quite partial and sensitive to the specific type of particle velocity
distribution function (see, e.g., [18, 138, 1, 126, 53, 57]). The patterns of
evolution of the Weibel turbulence after its growth has saturated are only
beginning to be studied, in extremely simple cases and mainly by numerical
methods. Examples of recently identified general patterns include partial
self-similarity of the evolution of the field turbulence spectrum and the
weakly nonlinear character of Weibel turbulence, largely governed by
quasilinear dynamics (see [18, 100, 57, 58] and references therein). In
accordance with the latter, the direct 3- and 4-wave interaction of individual
spatial modes is weak compared to their implicit, integral interaction due to
the joint deformation of the spatially averaged particle velocity distribution
function, which determines the instantaneous growth/decay rates of each mode.
The effectiveness of the quasilinear approximation for a number of specific
problems of the evolution of Weibel turbulence is shown using examples of bi-
Maxwellian and Kappa distributions in [57, 58]. These and some other works,
say [31], also present numerical simulations demonstrating a direct nonlinear
interaction of Weibel modes, which in certain cases is resonant and goes
beyond the quasilinear approximation at limited stages of the turbulence
evolution, including the possibility of super-exponential or power-law growth
of resonant harmonics and their power-law (non-exponential) decay.

As one of the few analytical achievements in the theory of nonlinear evolution
of Weibel turbulence, formally valid for an arbitrary particle distribution
function, let us point out the recently obtained [82] universal expression for
the plasma magnetization level. By definition, the latter is the ratio of the
current value of the energy density of turbulent magnetic field $w_{B}$ to the
initial thermal energy density of particles with an effective temperature
$T_{z0}$ along a certain axis $z$ (determined, for example, by external
field). It can be expressed as a function of the average wave number $k$,
dominating in the spectrum of magnetic turbulence, instantaneous anisotropy
parameter $A(t)$ and its initial value $A_{0}$:

$\frac{2\,w_{B}(t)}{n_{0}T_{z0}}=\frac{2k^{2}}{1+k^{2}}\,\frac{\left(A_{0}+1\right)^{-1}\left(A_{0}-A\right)}{A+2k^{2}+3}\leq\frac{2}{2\sqrt{6}+5}\,\frac{A_{0}}{A_{0}+1}.$
(4)

This approximate expression is obtained under the assumption of a weak
inhomogeneity of the plasma number density ($n_{0}$) and weak turbulence along
the specified spatial coordinate $z$, i.e., under the assumption of a two-
dimensional spectrum of turbulence, including only wave vectors orthogonal to
$z$ axis. It also follows from this equation that the magnetization value is
limited by approximately $0.2$, which was verified by various 2D3V simulations
presented in [82, 58].

###  6.3 Relevant problems in the cosmic plasma physics

The results stated above open up new possibilities for using the concept of
the electron Weibel instability and nonlinear dynamics of currents and
magnetic fields produced by it to solve problems of laboratory modeling and
physical interpretation of a number of expected or observed phenomena in
cosmic nonequilibrium plasma. The study of similar processes of current
microstructuring in the course of the decay of a strong discontinuity in a
partially magnetized cosmic plasma with hot electrons has not been carried out
yet. The studied transient phenomena may occur during explosive processes in
cosmic plasma, for example, at the initial stage of a rapid local heating of
density ducts in the atmospheres of stars or the magnetospheres of exoplanets.
Particularly interesting in this regard are the problems of stellar (or solar)
flares and winds, including the problem of injection of high-energy electrons
and reconnection of magnetic field lines inside the coronas of active stars
and inside magnetic clouds of stellar wind [142, 22, 126, 94, 140, 61].

Thus, in a stellar (solar) wind [22, 19, 139, 24] one can expect the
occurrence of a strong discontinuity of nonequilibrium plasma and the
subsequent restructuring of magnetic fields as a result of contact
interactions of individual plasma filaments containing hot electrons with
extended magnetic clouds filled with much more rarefied and cold plasma. In
the atmospheres of stars and the magnetospheres of planets (exoplanets), in
particular, in the areas of high density [8, 79, 112, 135, 23, 50, 141, 124],
as a result of various explosive processes, a significant local change in
plasma number density and rapid heating of its electrons, for example by X-ray
radiation, is possible. Such a heating, as well as an increase in plasma
density, can lead to the displacement of the surrounding magnetic field,
followed by small-scale structuring of the newly generated field throughout
the entire region occupied by the plasma with anisotropically cooling, but
still hot, electrons.

For the Sun and late-type stars [142, 93, 27, 123, 26, 68] it is not difficult
to imagine a situation, where in an inhomogeneous coronal loop (a bundle of
twisted magnetic tubes), filled with a sufficiently cold plasma, a region with
hot electrons is formed, elongated along magnetic field and localized across
it. This can be caused by various explosive processes, e.g., the injection of
high-energy electrons by the convective fields of the photosphere or powerful
Alfven waves rising from it, heating of the plasma by a longitudinal electric
current when the loop moves in the chromosphere, or the reconnection of some
of magnetic field lines of intersecting loops in the corona. Then, as our
preliminary analytical estimates as well as numerical simulations reveal, a
rapidly developing Weibel-type instability creates small-scale magnetic
turbulence, which could not only change the balance of magnetic and kinetic
pressures in the coronal loop, but also increase the effective (anomalous)
conductivity of the plasma by many orders of magnitude. As a result, a
restructuring of the original large-scale currents and an inhomogeneous
magnetic field in the coronal loop may occur, multiple regions of magnetic
reconnection may arise, and significant deformation of the loop structure or
even its local ruptures may begin.

In other words, small-scale magnetic turbulence caused by the Weibel
instability can initiate almost simultaneous (within tens of seconds)
occurrence of a large number (up to billions) of so-called nanoflares
associated with local processes of reconnection of magnetic field lines and
heating of the coronal plasma in a separate loop. These solar nanoflares, as
the attributes of statistical models of flares, have long been discussed and
observed [86, 130, 95, 134, 52], but their origin remained mysterious and was
not associated with the Weibel-type instability or an anisotropic distribution
of hot electrons elongated along the large-scale magnetic field of a loop (or
a bundle of twisted flux tubes); see, e.g., [140, 141, 142, 143, 43, 65, 149].
It is the result of this small-scale instability that will be the rapid
dissipation of both the current of injected energetic (keV) electrons and the
existing large-scale current of background (rather cold) electrons of the
loop, explaining the ”nanostructure” of this type of solar flares.

The aforementioned and other problems of space physics and astrophysics
related to (i) the transient processes of plasma expansion in the presence of
the hot electrons as well as (ii) the modification of the kinetic and dynamic
properties of plasma due to microstructuring of the self-consistent currents
and magnetic fields seem relevant and promising for detailed studies.

## Aknowledgements

The work was supported partially by RSF grant n. 21-12-00416. The computer
time for simulations were provided by the Joint Supercomputer Center of the
Russian Academy of Sciences and by the Keldysh Institute of Applied
Mathematics.

## Declarations

Conflict of interest The authors declare no conflict of interests.

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#  Collection Space Navigator: An Interactive Visualization Interface for
Multidimensional Datasets

Tillmann Ohm*‡1,2 Mar Canet Solà*‡1,3
Andres Karjus1,4 Maximilian Schich1,3
1ERA Chair for Cultural Data Analytics, Tallinn University
2School of Digital Technologies, Tallinn University
3Baltic Film, Media and Arts School, Tallinn University
4School of Humanities, Tallinn University
‡Corresponding authors<EMAIL_ADDRESS><EMAIL_ADDRESS>
*equal contribution as first authors

(May 10, 2023 )

###### Abstract

We introduce the Collection Space Navigator (CSN), a browser-based
visualization tool to explore, research, and curate large collections of
visual digital artifacts that are associated with multidimensional data, such
as vector embeddings or tables of metadata. Media objects such as images are
often encoded as numerical vectors, for e.g. based on metadata or using
machine learning to embed image information. Yet, while such procedures are
widespread for a range of applications, it remains a challenge to explore,
analyze, and understand the resulting multidimensional spaces in a more
comprehensive manner. Dimensionality reduction techniques such as t-SNE or
UMAP often serve to project high-dimensional data into low dimensional
visualizations, yet require interpretation themselves as the remaining
dimensions are typically abstract. Here, the Collection Space Navigator
provides a customizable interface that combines two-dimensional projections
with a set of configurable multidimensional filters. As a result, the user is
able to view and investigate collections, by zooming and scaling, by
transforming between projections, by filtering dimensions via range sliders,
and advanced text filters. Insights that are gained during the interaction can
be fed back into the original data via ad hoc exports of filtered metadata and
projections. This paper comes with a functional showcase demo using a large
digitized collection of classical Western art. The Collection Space Navigator
is open source. Users can reconfigure the interface to fit their own data and
research needs, including projections and filter controls. The CSN is ready to
serve a broad community.

##  1 Introduction

Large collections of digital artifacts with multidimensional associated
metadata can be studied using browsable interactive visualizations that
reflect or at least resonate with the intrinsic shape of their data [20].
Mapping the multidimensional topology of a collection into a multidimensional
space can help to better understand the overall structure of a dataset and can
uncover patterns hinting at underlying trends and dynamics. For example, a
researcher or curator may visually explore the space, as constituted by some
measure of artifact similarity, looking at different groups of similar objects
to identify regions of interest for further quantitative and qualitative
investigation.

Multidimensional feature vectors further describe artifact properties in a
heterogeneity of ways. This can include both categorical and numerical
information. Numerical properties can be derived directly from metadata, such
as in the case of artifact creation dates, or constructed through various
feature extraction techniques. Neural network methods, for example, can encode
measures of complex text semantics [9], of visual image properties [16, 34,
13, 15], of joint image-text pair embeddings [36, 29, 31], or of spectral
audio features [30]. Hand-crafted feature engineering approaches [41] and more
straightforward algorithmic approaches such as compression ensembles [14]
further promise to offer more interpretable vector representations.

Dimensionality reduction techniques can be used to reduce high-dimensional
data to a more manageable number of dimensions by remapping or projecting the
multidimensional topology into a lower-dimensional coordinate space [26, 38,
23, 2, 37]. For visual interpretation of multidimensional embedding spaces,
such projection methods are used to present the data in two or three
dimensions, which essentially can function as a more or less distorted
reference topography of the original multidimensional topology. The challenge
for dimensionality reduction techniques is to preserve complex relationships,
while removing information. For example, objects close to each other in high-
dimensional topological space should ideally also be close to each other in
the low-dimensional topographic projection space.

Multiple projection views can help to better understand multidimensional data
[2]. Two-dimensional static projections are visually comprehensible, but can
only provide a limited view into multidimensional data. Alternatively, to gain
intuition over a high-dimensional vector space, it can be helpful to interpret
and compare many different projections. Interactive components in graphical
user interfaces (GUI) can further help to gain intuition over the complex
interactions between many dimensions. Graphical user interface elements such
as range sliders are particularly useful to navigate through multiple features
and dimensions, and to query and filter the data [40, 1].

##  2 Related Work

Most closely related to the Collection Space Navigator are multiple
interactive visualization experiments and prototypes which have emerged in
recent years, and which mediate high-dimensional embeddings through low-
dimensional projections. They either try to provide an intuitive understanding
of multidimensionality and embedding methods by visualizing datasets commonly
used for Machine Learning tasks [35, 17, 8], or offer explorative interfaces
to similarity spaces of cultural collections [10, 11, 25, 12]. These
interactive visualization projects aim to provide overview and deeper insight
into their collections, tailored to specific datasets. The VIKUS Viewer [27,
28] offers a more general framework for exploring cultural collections. It
allows not only to view a collection as a similarity map of its image
embeddings, but also to dynamically filter metadata such as time and
categories. The Selfiexploratory of the Selfiecity project [21] similarly uses
a number of range sliders to filter a multidimensional dataset of images.

The CSN user interface aligns with classic conventions of cultural cartography
and modern examples of interactive data visualization and scholarly figure
design. Following long-established conventions, the CSN combines a cartesian
projection with an auxiliary index and call-out details [24].

Following a modern paradigm of interactive figure design [5, 39], the CSN
further allows for a deeper functional user experience (UX) and eventually
understanding of multidimensional data. The navigation paradigm of the CSN
range sliders, which function as Dimension Filters, further resonates, in
particular, with the recent state-of-the-art of understanding mathematical
multidimensionality via interactive animation [32]. The CSN combines these
foundations with the paradigm of a scatter plot of images as made popular in
Cultural Analytics since 2012 [19]. The CSN is also functionally similar to
network visualization applications, such as Cytoscape [33] or Gephi [3], which
focus on depicting another (yet related) form of multidimensionality in node-
link diagrams of complex networks [6].

Even more broadly, the authors of the TensorFlow Embedding Projector [35]
suggest to include multipanel projections, i.e. more than one simultaneous
projection panel. This would also make sense as a possible extension of the
CSN and would be in line with the prevalence of “multi-chart” figure panels in
multidisciplinary science journals [18, 4].

##  3 The Collection Space Navigator

Figure 1:  The Collection Space Navigator (CSN). The central _Projection Area_
displays a x-y scatter plot of images based on the selected projection (e.g.
UMAP, t-SNE), with filtered images greyed out, and mouse-over highlight. The
_Object Panel_ (left) shows a larger _Object Preview_ of the highlighted
image, together with _Object Info_ based on selected metadata; _Object
Appearance_ visualizes clusters (optional), sets the projection thumbnail size
(zoomed-out and zoomed-in). The _Control Panel_ (right) allows for selection
of _Data and Projections_ ; custom interactive _Dimension Filters_ and
_Advanced Filters_ facilitate dataset exploration, analysis, and understanding
(see text); the filtered object metadata and current projection view can be
downloaded via _Export Filtered Data_.

###  3.1 Motivation

We developed the Collection Space Navigator as a flexible browser-based
research tool applicable across various use cases and research domains:

  1. 1.

Researching large collections of digital objects (e.g. images, videos, audio,
text, 3D models) with the ability to identify patterns and similar groups
based on metadata and vector embeddings;

  2. 2.

Understanding multidimensionality and projection methods by comparing
different embedding spaces and dimensionality reduction techniques through
intuitive navigation;

  3. 3.

Presenting entire media collections online and communicating research findings
with diverse audiences.

While prototypes and use cases exist for each of these aspects as discussed
above, we are not aware of a tool that meets all of these requirements. Our
contribution therefore lies in the combination and extension of existing
interfaces to work towards a more universal and open modular research and
curation system.

Figure 2:  Examples of various 2D projections and visualization features in
the CSN tool. a) UMAP projection with large thumbnails, providing a
comprehensive view of the image content; b) UMAP projection with medium size
thumbnails and cluster colors of categorical data selected in _Object
Appearance_ ; c) UMAP projection with medium size thumbnails and filtered-out
objects in grey; d) UMAP projection with small thumbnails and cluster colors,
providing a more compact representation; e) t-SNE projection, showing an
alternative dimensionality reduction technique; f) Simple x/y plot, here
showing PC1 over time for temporal analysis. The flexibility of the CSN tool
provides the ability to effectively explore and compare data across different
visualization methods via displaying multiple different 2D projections,
flexible based on data import configuration, selectable in _Data &
Projections_, combined with thumbnail sizes and cluster highlights as
adjustable via the _Object View Settings_.

###  3.2 Design principles

To make complex interactions in multifaceted datasets comprehensible while
meeting the diverse needs of researchers, we formulated three design
principles for the tool:

  1. 1.

Providing an open modular system that adapts to different research needs,
domains and datasets while preventing information overload;

  2. 2.

Providing a complete overview of the collection while encouraging immersive
exploration of the objects;

  3. 3.

Providing a multitude of interaction mechanisms and modalities to foster
intuition, such as zooming, panning, hovering, and sliding through feature
dimensions.

###  3.3 Components

####  3.3.1 Projection Area

The central part of the interface is the _Projection Area_ (Figure 1 center).
It maps the given input collection in its entirety as miniature images in an
interactive 2D scatter plot, with coordinates defined by the chosen projection
method. Basic navigation operations such as zooming or “drag and move” allow
free exploration of the projection space. While the user sees only 2
dimensions in the central projection area, the CSN technically includes a
third axis for depth. Moving along the depth axis (by zooming) effectively
reveals overlapping objects. The appearance of the thumbnails can be adjusted
in the _Object Panel_.

####  3.3.2 Object Panel

The _Object Panel_ (Figure 1 left) has three collapsible sub-menus: _Object
Preview_ , _Object Info_ and _Object Appearance_. The _Object Preview_ section
displays a larger version of the miniature thumbnail currently hovered on in
the _Projection Area_.

By default it simply shows a larger version of the same thumbnail, but it can
be set to display a higher resolution version of the hovered image, stored
either locally or remotely.

The _Object Info_ section provides detailed information on the currently
selected object. This aspect of the CSN is highly flexible, as the metadata
fields that provide this information can be easily defined in the separate
configuration file. Minimally it can display just the file name, but it can
equally well include extensive metadata — for example in the case of art
collections, the author, production year, location, genre, style, and other
details.

The _Object Appearance_ section contains options to control the visual
appearance of the objects in the _Projection Area_. A predefined group (from
categorical metadata) can be selected from a drop-down list to show clusters.
Objects of the same category are depicted with the same color border around
their thumbnails. The size and scale of the miniature images are adjustable
with convenient sliders. Size determines how large the thumbnails should be
when fully zoomed out, while Scale affects the size when zoomed in.

####  3.3.3 Control Panel

The _Control Panel_ (Figure 1 right) has four collapsible sub-menus: _Data &
Projections_, _Dimension Filters_ , _Advanced Filters_ , and _Export Filtered
Data_. The _Data & Projections_ section contains a _Dataset_ drop-down list of
selectable datasets. For very large collections, we recommend providing a
smaller subset by default and offering the entire set on demand (such subsets
can be conveniently produced using the CSN configuration Python notebook). The
section also contains a _Projection_ drop-down list of selectable projections
and mappings. Switching between different projections, e.g. different
embedding or reduction methods, smoothly rearranges the positions of the
objects in the _Projection Area_. These intuitive animations can provide new
insights into the intermediate state between two projections and expose their
differences.

The _Dimension Filters_ are optional interactive elements that function to
filter the objects in the _Projection Panel_ (Figure 3). They control the
range of the assigned variables, which could be dimensions of the embedding,
metadata such as date or year of creation of an artwork, or inferred
properties of the image such as colorfulness or contrast. Histograms above the
range sliders provide additional statistical information and feedback on how
changes affect the distribution of the mapping. They are constantly updated to
reflect the distribution of the entire dataset as well as the distribution of
filtered and unfiltered objects. Clicking on a histogram activates the _Bin
Mode_ : moving the cursor over the bars of the histogram temporarily displays
only objects within the narrow range of the bar. A second click terminates
this function and sets the filters back to the previous state. Additionally,
hovering over the thumbnails in the _Projection Area_ highlights the
corresponding vertical bar in each histogram.

The _Advanced Filters_ section is a text field to construct and apply search
and filter queries. By default, it handles basic query operators such as AND,
OR, equals (==), does not equal (!=), as well as custom operators. Nested and
complex search queries are also supported using round brackets. In the setup,
using the configuration Python notebook, each metadata field can be defined as
a Free Text Entry (enabling queries) or as a Categorical Selection (generating
drop-down lists, i.e. GUI elements that allow simple search and selection).

The _Export Filtered Data_ section in the _Control Panel_ allows downloading
the metadata of the currently filtered objects as a CSV file, and the current
projection view as PNG file.

Figure 3:  Interactive Dimension Filters. Left: Unfiltered Dimension Filters,
consisting of range sliders with interactive histograms above them, showing
the distribution of all objects along the slider’s dimension with the bin of
the currently selected object highlighted in red; Center: Reducing the range
of one slider affects the distribution of all dimensions, reflected by the
histograms; Right: _Bin Mode_ functionality is activated by clicking on a
histogram, allowing the user to activate one bin at a time, with the
Projection Area displaying the corresponding objects within the active bin.
These interactive Dimension Filter features allow in-depth exploration and
visualization of multi-dimensional data distributions allowing users to gain a
deeper understanding of the relationships between data points.

##  4 Applications

An important aspect of the CSN is the flexibility to combine different data
types and different methods for feature extraction and dimensionality
reduction. To initially test the CSN towards many potential applications in
other disciplines, we utilized the application in several use cases within our
multidisciplinary research lab. All these cases vary in terms of research
questions, approaches, and datasets. Here we briefly describe three examples
which (i) analyze embedding spaces, (ii) study multidimensional metadata, and
(iii) explore and curate large sets of generated images respectively. Mixed
applications are of course possible. A growing diversity of application demos
will be added to the project website111https://collection-space-
navigator.github.io/.

###  4.1 Analyzing aesthetic complexity in art collections

Karjus et al. (2022) propose quantifying visual aesthetic complexity using
“compression ensembles”, an information theory driven method based on
transforming images using visual filters and repeatedly compressing them. This
approach produces embeddings analogous to those from deep learning based image
embeddings, but with explainable and arguably more interpretable results in
the aesthetic domain. This paper uses the same dataset as our examples in
figures 1 to 3 and our associated initial demo application, i.e. a corpus of
digitized images of visual art such as paintings and drawings, a subset of
74,028 works of the art500k project corpus [22], in turn primarily sourced
from Wikiart, covering mainly Western art of the last 400 years.

Here and in similar projects focusing on visual art, the CSN can serve both as
a research and a presentation tool in several capacities. The visual and
interactive overviews of the embedded artworks allow for a closer examination
of art historical trends over time and in relation to their aesthetic
complexity. Using multiple representations and filters can further help to
build an intuitive understanding of the given embedding method. In the case of
compression ensembles, the variables or dimensions are interpretable [14], and
can be decorrelated and grouped, for example using Principal Component
Analysis. A concrete usage of the CSN in this circumstance is for example to
display the dataset in the projection area using two principal components or
specific compression dimensions as x and y coordinates, while filtering
another compression dimension with the range sliders to visualize the
respective interaction.

Figure 4:  Two distinct use cases of the Collection Space Navigator. Left:
Historical newsreels with rich metadata; Right: Investigating text-to-image
generation through Stable Diffusion with linguistic basic concepts as prompts.
By using domain specific metadata filters and projections, the CSN tool allows
for deeper exploration providing insight into visual and thematic patterns.

###  4.2 Studying represented worldviews in historical newsreels

Oiva et al. (forthcoming) propose a multidisciplinary framework for the study
of historical newsreels, which can be extended to news broadcasts and
audiovisual collections more broadly. The initially analyzed dataset consists
of 1747 newsreels produced weekly over 50 years in the Soviet Union.

Within the project, the digitized video clips, about 8-10 minutes each, are
split into individual frames, i.e. static images, which form the unit of
analysis either in their full granularity or coarse-grained, for example as a
single frame per shot. The material consists of predominantly black and white
footage of variable quality, transitioning to color towards the end of the
period. Rich metadata accompanies the visual information. Taking all this into
account, the authors aim to analyze and understand the material, including in
particular the variation of intrinsic worldviews.

In this project, the CSN is used both as a presentation tool to communicate to
the audience, and a research tool for the authors (Figure 4), allowing for
exploration of the film footage, to identify recurring patterns and shots, and
to generate hypotheses based on a combination of the visual representation of
movie frames based on image embedding vectors, and accompanying metadata. In
this case, indeed, a stronger focus lies on the latter explicit metadata-
dimensions, which can be mapped to range-sliders in a similar way as the
numerical vectors in example 4.1.

###  4.3 Exploration and curation of text-to-image models

The advent of easily accessible text to image models such as DALL-E,
Midjourney, and Stable Diffusion [31]. has brought artistic and photo-
realistic image generation to the mainstream. This is very likely to transform
creative industries in the near future. Yet the effectively near-infinite
content capacity will require understanding of the models to be made use of,
and efficient curation to work with already generated images.

The CSN is well suited for such tasks, allowing for insight into large image
sets using various visual similarity and metadata representations and
filtering options. For the demo, we generated of images with Stable Diffusion
1.5 using the basic word list of the Automated Similarity Judgement Program
(ASJP) [7] as the input text prompts, and embedded them using the ResNet50
pretrained image model.

This approach, querying simple terms such as colors, body parts and animals,
allows for probing the “defaults” of the text2image models, while generating
multiple versions of each prompts yields insight into variation in its
capacity. The same approach could be used to compare different models, as well
as investigate effects of complexity of the prompts. Naturally, more complex
prompts such as those common in artistic text2image creation could be explored
as well.

##  5 Conclusion

The Collection Space Navigator (CSN) as introduced in this paper is a powerful
and flexible data visualization tool working with large sets of images
associated with multidimensional data, including embedding vectors and tabular
metadata. The CSN is publicly available on Github, including a functioning
demo as well as a How-To configuration and usage guide in form of an
interactive Python notebook. As described above, the interface can be
conveniently and readily tailored to the needs of a particular research
project and dataset. An important aspect of the CSN in this regard is its
flexibility to combine different vector and metadata types, a variety of
projection methods, and a diversity of representations. This includes the
bespoke configuration of different representation and filter controls,
resulting in an elegant user interface that allows the user to focus on the
material under investigation, without unnecessary distractions in the form of
”default” controls.

We have exemplified the CSN here using visual data. Yet, while input data is
by default represented by small images or thumbnails in the Projection Area,
the input does not need to be vectors derived from images. For example,
embeddings or metadata of audio or text could be visualized equally well,
represented by suitable thumbnails or labels. We imagine, such ”unorthodox”
usage could facilitate the exploration and further research on such non-visual
spaces of meaning too. By making the code fully open source, we encourage
further development in this and other directions, for example adding playback
functionality to images associated with audio or video files.

In sum, the Collection Space Navigator (CSN) is a new research tool that
facilitates the work of artists, curators, scholars, and scientists, who study
large collections of digital visual artifacts associated with metadata and/or
vector embeddings. We release the CSN as open source under MIT License, to
enable broad reuse and further development without restriction. We hope it
will invigorate a diverse ecology of multidisciplinary research and curatorial
practice, and help to deepen our collective understanding of multidimensional
meaning spaces.

## Code availability and demo

The CSN is released as open-source (MIT licence), with code and documentation
available on GitHub at https://github.com/Collection-Space-Navigator/CSN
A live demo, using the the same example data as in figures 1 to 4 in this
paper, is available at https://collection-space-navigator.github.io/CSN

## Author contributions, acknowledgments and funding

T.O. and M.C. designed, co-authored, and developed the Collection Space
Navigator. T.O., M.C., A.K. and M.S. contributed to the research design and
co-wrote the manuscript. T.O., M.C. and A.K. collected data. T.O. and M.C.
contributed equally to this work as first authors. The authors thank Sebastian
Ahnert, Mila Oiva, and the entire CUDAN team for useful conversations and
input. All authors are supported by the CUDAN ERA Chair project, funded
through the European Union’s Horizon 2020 research and innovation program
(Grant No. 810961)

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