# Exploring extreme magnetization phenomena in directly-driven imploding
cylindrical targets

C. A. Walsh1 R. Florido2 M. Bailly-Grandvaux3 F. Suzuki-Vidal4 J. P.
Chittenden4 A. J. Crilly4 M. A. Gigosos5 R. C. Mancini6 G. Pérez-Callejo7 C.
Vlachos7 C. McGuffey8 F. N. Beg8 J. J. Santos7 1Lawrence Livermore National
Laboratory  2iUNAT - Departamento de Física, Universidad de Las Palmas de Gran
Canaria, 35017 Las Palmas de Gran Canaria, Spain  3Center for Energy Research,
University of California, San Diego  4Blackett Laboratory, Imperial College
London, London SW7 2AZ, UK  5Departamento de Física Teórica, Atómica y Óptica,
Universidad de Valladolid, 47071 Valladolid, Spain  6Department of Physics,
University of Nevada, Reno, NV 89557, USA  7Université de Bordeaux-CNRS-CEA,
Centre Lasers Intenses et Applications (CELIA),
UMR 5107, F-33405 Talence, France  8General Atomics, San Diego

###### Abstract

This paper uses extended-magnetohydrodynamics (MHD) simulations to explore an
extreme magnetized plasma regime realisable by cylindrical implosions on the
OMEGA laser facility. This regime is characterized by highly compressed
magnetic fields (greater than 10 kT across the fuel), which contain a
significant proportion of the implosion energy and induce large electrical
currents in the plasma. Parameters governing the different magnetization
processes such as Ohmic dissipation and suppression of instabilities by
magnetic tension are presented, allowing for optimization of experiments to
study specific phenomena. For instance, a dopant added to the target gas-fill
can enhance magnetic flux compression while enabling spectroscopic diagnosis
of the imploding core. In particular, the use of Ar K-shell spectroscopy is
investigated by performing detailed non-LTE atomic kinetics and radiative
transfer calculations on the MHD data. Direct measurement of the core electron
density and temperature would be possible, allowing for both the impact of
magnetization on the final temperature and thermal pressure to be obtained. By
assuming the magnetic field is frozen into the plasma motion, which is shown
to be a good approximation for highly magnetized implosions, spectroscopic
diagnosis could be used to estimate which magnetization processes are ruling
the implosion dynamics; for example, a relation is given for inferring whether
thermally-driven or current-driven transport is dominating.

##  I Introduction

Magnetic fields are of interest in astrophysics, existing in the Universe on
all scales: from kilometers or less for compact neutron stars up to millions
of light years for clusters of galaxies. Natural magnetic fields can exceed
the laboratory magnetic fields achieved by many orders of magnitude, with
white dwarfs sustaining magnetic fields of up to $10^{5}$ T and the most
extreme neutron stars up to $10^{9}$ T [1]. In such compact stars the magnetic
fields are strong enough to modify the structure and composition of the star,
as well as its radiation properties [2, 3].

In the laboratory, laser-driven plasma compression has been used to amplify
initial (seed) magnetic fields [4]; to first order the magnetic field is
frozen into the plasma as it compresses. Proton deflectometry has provided a
direct measurement of these compressed magnetic fields up to $4$ kT in
dedicated cylindrical experiments [5, 6]. Novel techniques to generate strong
magnetostatic fields $\sim 1-10$s T by either capacitor-bank pulsed discharges
[7, 5, 8, 9] or intense laser-driven discharges [10, 11, 12, 13, 14, 15] have
stimulated a growing interest in laser-driven, high-energy-density (HED)
systems with embedded magnetic fields.

In Inertial Confinement Fusion (ICF), highly compressed magnetic fields have
been demonstrated to contribute to increase implosion performance [16]. The
compressed B-field strengths can be enough to magnetize the electron thermal
conduction, reducing the energy losses from the hot-spot and resulting in
hotter fuel [17, 18]. The B-fields can also effectively confine D-T ions and
thermonuclear $\alpha$-particles [19], enhancing collisionality and fusion
yield. In addition, large magnetic fields can reduce hydrodynamic
instabilities through magnetic tension [20, 21, 22, 17, 18], although there
are also mechanisms that increase perturbation growth with magnetization [18,
23].

Cylindrical geometry provides a simple testbed for exploring magnetized
phenomena in HED plasmas, as magnetic fields can increase asymmetry in
spherical implosions [18, 23]. In this scheme, the cylinder axis is parallel
to the applied B-field direction. Using this geometry, the MagLIF concept was
first proposed at Sandia National Laboratories (SNL) in 2010 [24]. It consists
of laser pre-heat of cold fuel inside a Z-pinch cylindrical liner (initially
with an axial B-field) which is magnetically compressed by a $\sim 20$ MA, 150
ns electrical current. Due to fuel magnetization and pre-heating, MagLIF aims
at near-adiabatic stable cylindrical compression with lower implosion velocity
($\sim 100$ km/s, instead of $>300$ km/s) and lower convergence ratios than
conventional ICF. MagLIF has shown promising results through different
experimental campaigns by demonstrating thermonuclear neutron generation at
fusion-relevant conditions, a high enough fuel magnetization for
$\alpha$-particles trapping [25, 26, 27] and reporting first performance
scaling studies [28].

A laser-driven approach to MagLIF is being explored at the OMEGA Laser
facility [29, 30, 31, 32, 33] to provide a test for MagLIF scaling. The
driving laser energy is $1000\times$ less than that delivered on the Z-pinch
facility at SNL and therefore the laser targets are $10\times$ smaller than on
Z. Yet, OMEGA provides a significantly higher repetition rate and better
diagnostic access, facilitating the parametric study of the underlying basic
physics. Cylindrical implosions are easier to perform and offer a clear axis
for diagnostics, as well as for a laser preheat [30] or a fast ignition beam.
A recent series of laser-driven experiments [33] used the improved MIFEDS [5,
9] to deliver a seed B-field of nearly 30 T, which was high enough to
demonstrate the effects of magnetic pressure decreasing plasma compression.
There are prospects to deliver even higher B-field values ($\sim 50$ T) within
the next few years.

The design and interpretation of all above experiments and applications
strongly rely on magneto-hydrodynamic (MHD) codes. Simulations must include
extended-MHD effects, which represent the transport of energy and magnetic
flux in a plasma [34]. Magnetized plasmas are thought to exhibit complex
behavior in the electron population and, above resistive-MHD, extended-MHD
additionally accounts for temperature-gradient-driven transport –such as the
Nernst term moving magnetic fields down electron temperature gradients– and
electric-current-driven transport –such as the Hall term moving magnetic
fields with the flow of charge–. The Nernst term, in particular, is of great
interest to the scientific community, with importance in a broad range of
experiments [16, 35, 36, 28, 37] with expectations of plasma demagnetization
[38, 39, 40, 23, 41]. Even with its wide applicability, Nernst advection of
magnetic field is yet to be directly measured.

This paper explores the development of extreme magnetization phenomena in
directly-driven imploding cylindrical targets using extended-MHD simulations.
The starting point for the design is a previous laser-driven implosion on
OMEGA (without any fuel pre-heat), which found evidence of significant
magnetic pressure during target stagnation [33]. Distinct magnetization
phenomena are outlined for this high B-field regime, which can be categorised
as related to magnetic field amplification, electron magnetization or induced
electrical currents; metrics governing which phenomena are significant are
summarized in Table 1. Initial target conditions are varied to transition
between different regimes; for example, the concentration of a dopant added to
the imploding fuel can be increased to amplify magnetic field compression at
the expense of electron magnetization, while enabling a spectroscopic
characterization of plasma conditions. In particular, the feasibility of Ar
K-shell spectroscopy [42, 43, 44, 45, 46, 47, 48] for diagnosing the core
conditions in magnetized cylindrical implosions is demonstrated by post-
processing the extended-MHD simulations and performing non-LTE atomic kinetics
and radiative transfer calculations. By using the frozen-in-flow
approximation, a spectroscopic diagnosis of core conditions would allow for
estimates of the magnetization metrics.

This paper is organized as follows. Section II briefly summarizes the main
features of the Gorgon code used to perform the extended-MHD simulations.
Section III discusses the platform for magnetized direct-drive cylindrical
implosions and the corresponding nominal target design. Section IV discusses
key magnetization phenomena that may occur in a magnetized compressed hot-
spot. Non-dimensional parameters to quantify the relative importance of
different physical processes for any given implosion are introduced. Section V
uses the metrics defined in Sec. IV to modify the point-of-design to explore
specific physics regimes and scenarios; e.g. maximizing the magnetic field
strength requires a different design to maximizing the electron magnetization.
Synthetic images of time-resolved Ar K-shell spectra are presented in Section
VI in an effort to mimic potential measurements recorded by a streaked
spectrometer. Lastly, the appendix compares synthetic proton radiographs
produced with Gorgon to experimental data recorded in the cylindrical
implosions described in Refs. [5, 6], thus validating the basic magnetic
transport processes in the code. Sensitivity of the radiographs to magnetic
transport by bulk plasma motion and Nernst demagnetization is observed.

##  II Gorgon extended-MHD code

The simulations presented in this paper utilize the Gorgon extended-MHD code
[49, 50, 41]. Gorgon includes magnetic transport by temperature gradients
(such as the Nernst term), as well as by electrical currents (such as the Hall
effect), and the analogous terms advecting electron energy [34]. These
processes will be discussed in more detail in Section IV. The transport
coefficients preserve physical behaviour at low magnetization [51], which can
give significant differences [52] to the coefficients calculated by Epperlein
& Haines [53]. The thermal conduction algorithm is specifically designed for
anisotropic heat-flow [54, 55]. Biermann battery production of magnetic fields
is also taken into account [41, 36]. Importantly for this work, the plasma
dynamics include the Lorentz force, resistive diffusion and Ohmic dissipation.

Gorgon results have been compared with magnetized cylindrical implosion
experiments on OMEGA [5, 6] using synthetic proton radiography, matching the
peak proton deflection (and therefore the compressed magnetic field) –details
are shown in the appendix. This provides a crucial benchmark for the code and
lends additional confidence in using the Gorgon code for the experimental
design addressed here, where the main differences are in the increased applied
magnetic field and more sophisticated diagnostics (presence of a dopant for
spectroscopic purposes). Gorgon simulations of pre-magnetized capsules with 3D
perturbations have also been used to study how differing magnetization effects
impact on short and long wavelength perturbations, as well as the impact of
magnetization on stagnation temperatures and yields [18, 23].

The laser drive is modelled using a ray-trace package and inverse
Bremsstrahlung absorption. The simulations in this paper are predominantly 2D,
with two demonstrative 3D simulations in Sec. IV. The 2D simulations are in
cylindrical geometry $r-z$, such that the axial plasma end losses are
captured.

Gorgon uses a $P\frac{1}{3}$ automatic flux-limiting scheme for the radiation
transport [56], which captures both the free-streaming and diffusive limits.
As Ar-doped D2 is used to fill the targets, 100 energy groups are considered
in order to capture the complex dependence of argon opacity and emissivity on
radiation energy, for different concentrations of dopant. It is assumed that
the dopant only affects the radiative properties of the fuel, with any changes
to the equation of state or electron transport coefficients not incorporated.

Thermal and Nernst flux limiters of 0.1 are used throughout this paper. The
simulations are run with a radial resolution of $\frac{1}{2}\,\mathrm{\micro
m}$, which is required in order to resolve the small converged hot-spot size,
with final hot-spot diameters on the order of $10-20\,\mathrm{\micro m}$ [33].
Gradients in the axial direction are much shallower, allowing for a converged
resolution of $2\,\mathrm{\micro m}$.

##  III Magnetized cylindrical implosions on OMEGA

Figure 1:  VISRAD image showing the intensity radiation pattern delivered by
40 OMEGA beams on a cylindrical target with an outer radius of 290 µm (see
text for details). A nearly uniform irradiation close to $700$ TW/cm2 is
expected on the central $\sim 650\,\mathrm{\micro m}$-length portion along the
cylinder axis. Figure 2:  Density (top) and electron temperature (bottom) for
2D MHD Gorgon simulations of deuterium-filled cylindrical implosions with no
magnetic field applied (left) and a 50 T seed magnetic field applied (right).
The deuterium fuel has an argon dopant with a concentration of 0.325% by atom.
The data is shown at 1.45 ns, which is approximately 50 ps before the time of
peak neutron production.

While spherical laser-driven implosions are used to obtain high yields, the
application of a magnetic field results in anisotropic implosions [18, 23],
with the magnetic field normal to the ablator at the pole and in the plane of
the ablator at the waist; i.e. a magnetized spherical implosion is
intrinsically two-dimensional in its evolution. It was also found that a
magnetic field normal to the ablator surface can increase perturbation growth
from laser non-uniformities [23]. Therefore, by using a cylindrical target
with applied field along the axis, these complications are avoided; the
magnetic field is allowed to be everywhere in the plane of the ablator,
resulting in a one-dimensional implosion geometry before realistic asymmetries
are taken into account. Also, while there is more mass compression in a
spherical implosion, the magnetic field is not compressed at the poles, so
that effective magnetic field compression is similar between the two
geometries.

The nominal target design from recent laser-driven MagLIF implosions (mini-
MagLIF campaign) performed at OMEGA is chosen [33], i.e. targets are
cylindrical shells made of parylene-N with a density of $1.11$ g/cm3, with an
outer radius of $290\,\mathrm{\micro m}$ and a wall-thickness of
$17\,\mathrm{\micro m}$. The targets are filled with deuterium at a pressure
of $11$ atm, i.e. a mass density of $1.81\,\mathrm{kg/m^{3}}$. Targets are
imploded by 40 OMEGA beams, delivering a total of 14.5 kJ over 1.5 ns square-
shaped pulses. The laser pointing has been empirically optimized to achieve a
uniform implosion velocity over an axial extent greater than 600 µm [31, 57].
Figure 1 shows a VISRAD [58] image with the expected laser power irradiation
pattern on target; this illumination is used as an input to the Gorgon
simulations.

While the present simulation study originates from the mini-MagLIF platform it
is worth noting that, in contrast, the goal here is not to find the regime of
maximum fusion performance. The approach is to demonstrate that a regime of
extreme magnetization is attainable and investigate how the variation of
different target parameters can be used to test complex MHD phenomena. For
instance, in this work fuel pre-heating is intentionally not considered as it
increases the thermal pressure of the hot-spot at the cost of reducing B-field
compressibility. Pre-heating is otherwise essential in the MagLIF design to
enhance the neutron yield [28]. Therefore, the proposed research is distinct
–albeit complementary– to the current mini-MagLIF studies.

As illustration, Fig. 2 shows simulation results for this setup using a 0.325%
argon dopant by deuterium atom. On the left is the density and electron
temperature without an applied magnetic field, while the right shows the same
properties when a 50 T seed B-field is applied. The snapshot is at 1.45 ns,
which is 50 ps before the time of peak neutron production. The hot-spot
temperature is greatly enhanced by magnetization, as the electron thermal
losses are suppressed perpendicular to the applied field. The core hot-spot
density is also greatly decreased by magnetization; this is due to two
factors. Firstly, changes in thermal conduction do not change the overall hot-
spot thermal pressure [59]. Therefore, an increased temperature by electron
magnetization results in a lower density. Secondly, the final total pressure
is the sum of thermal and magnetic components; as the magnetic pressure
increases, the stagnated thermal pressure decreases, reducing hot-spot
compression.

Simulations show that the irradiation pattern of Fig. 1 drives an implosion
with a central $\sim 400\,\mathrm{\micro m}$-long portion for which core
conditions show very little variations in the axial direction. This can be
seen in Fig. 2 at the corresponding time of 1.45 ns, but the same behaviour is
found over the entire implosion evolution.

As the magnetic flux is compressed along with the plasma, the B-field is
strongest in the stagnated fuel. The field strength is greater than 10 kT
throughout the D2 plasma when an initial 50 T field is applied to the target.
While the outer core is denser than its central portion, i.e. the hot-spot,
the magnetic field is able to diffuse over the short radial core length-
scales, resulting in a relatively uniform magnetic field throughout the fuel.

Figure 3 shows the temporal evolution of the implosions by plotting the
instantaneous neutron-averaged ion temperature and mass density, e.g. the
burn-averaged ion temperature is the ion temperature that a fusing D-D
reaction sees (a spatial and temporal average). The burn-average is chosen as
it weights the properties to those occurring in the hot-spot, where the
extreme magnetization phenomena typically take place. Cases with and without a
50 T seed applied field are shown, as well as for two argon dopant
concentrations. Clearly the compressed magnetic field enhances the temperature
but significantly decreases the hot-spot density. The increased dopant also
results in more radiated energy, lowering the temperature but enhancing the
compression of hot-spot mass. When compared to unmagnetized implosions, the
compression of the imposed B-field has a significant impact on the conditions
of the imploding core, which leads to different regimes of extreme
magnetization. This raises the question about how to measure and characterize
the corresponding magnetization scenario in an implosion experiment. A
detailed assessment about the use of Ar K-shell spectroscopy for this purpose
is addressed in Sec. VI.

Figure 3: Time history of core conditions from 2D Gorgon simulations of
cylindrical implosions as a function of different argon dopant concentrations
and applied field strength (see Fig. 2). Shown values are calculated as
instantaneous neutron-averaged quantities, which results in a weighting
towards the hot-spot core.

##  IV Magnetization phenomena and associated metrics

This section outlines the processes of interest in a magnetized implosion.
These phenomena can be split into three categories: the effectiveness of
magnetic flux compression, magnetization of the electron population, and
induced electrical currents. Metrics for the relative importance of each
phenomenon are presented, with design changes explored in Sec. V.

Firstly, the process of hot-spot magnetic flux compression is examined. As the
work here is primarily interested in the field strength in the compressed hot-
spot, discussion of magnetic transport in the laser absorption region is
avoided. For an in-depth discussion on this issue, see Refs. [23, 60].

In order to reach a large magnetization regime in the hot-spot, efficient
magnetic flux compression is required. If the magnetic field is completely
frozen into the plasma motion, and the plasma motion is one dimensional, then
the magnetic field in the fuel would exactly follow
$B_{z}=B_{0}\frac{\rho}{\rho_{0}}$. Therefore, to quantify how close the
magnetic transport is to being frozen into the plasma during the implosion, a
_frozen-in-flow factor_ is defined:

$\Gamma=\frac{|\underline{B}|}{B_{0}}\frac{\rho_{0}}{\rho},$ (1)

where a value of $\Gamma<1$ indicates that some magnetic flux has been lost
from the hot-spot. The primary factors that change the magnetic flux
compression from frozen-in-flow are resistive diffusion, Nernst advection of
magnetic fields down temperature gradients and axial plasma motion. It will be
seen in Sec. V that $\Gamma\approx 1$ for highly magnetized implosions. In
some cases the magnetic flux compression actually exceeds the frozen-in-flow
approximation.

While $\Gamma$ indicates how much magnetic flux is lost from the hot fuel,
maximizing this value does not necessarily help to enhance the magnetic field
strength in the hot-spot. For this purpose, the field compression factor is
utilized:

$\Lambda=\frac{|\underline{B}|}{B_{0}}.$ (2)

For an implosion with effective magnetic flux compression ($\Gamma=1$) the
field compression factor is simply:

$\Lambda=\frac{\rho}{\rho_{0}},\quad\text{if }\Gamma=1,$ (3)

i.e. the more the plasma is compressed, the more the magnetic field is
compressed.

Once the magnetic field is compressed, the electron population can become
magnetized, with each electron taking a curved path between collisions. The
Hall Parameter $\omega_{e}\tau_{e}$ quantifies the number of orbits a typical
electron takes between collisions. While linked to the magnetic field
strength, the magnetization of electrons is also dependent on the electron
temperature and density [61]:

$\omega_{e}\tau_{e}\propto\frac{|\underline{B}|T_{e}^{3/2}}{n_{e}}.$ (4)

Again, using the frozen-in-flow assumption, the Hall parameter can be
expressed in terms of the initial fill density of the target as well as the
applied field strength:

$\omega_{e}\tau_{e}\propto\frac{B_{0}}{\rho_{0}}T_{e}^{3/2},\quad\text{if
}\Gamma=1.$ (5)

Magnetizing the electron population typically requires much lower magnetic
field strengths than magnetic pressure effects [18]. In magnetized fusion
experiments, magnetizing the electron population is a primary goal, with
thermal conduction suppression resulting in lower energy losses from the
fusing plasma [62]. The effect of thermal conduction suppression can be
clearly seen in the hot-spot temperature in Fig. 2 increasing from
approximately 1000 eV to 6000 eV when magnetized (although it will be seen
later that a fraction of this increase is due to Ohmic dissipation). In the
hot-spot the simulated Hall parameter is in excess of 500, reducing the
electron thermal conduction perpendicular to the field lines by a factor of
more than $2\times 10^{6}$.

The Hall parameter also impacts the transport of magnetic fields by the Nernst
term. In an extended-MHD plasma, Nernst can be approximated as the advection
of magnetic field with the electron heat-flow [63, 34, 52]. Therefore, the
extreme magnetizations simulated in Fig. 2 result in the suppression of Nernst
by a factor greater than $2\times 10^{6}$, allowing for reduced magnetic flux
losses from the hot-spot.

Now the focus turns to induced electrical currents in magnetized implosions.
The current takes the form $\underline{j}=\nabla\times\underline{B}/\mu_{0}$,
where the electric field is assumed to vary slowly. Associated with this
current is an electron drift velocity $v_{e}=-\underline{j}/en_{e}$. This
drift is expected to result in: the transport of thermal and magnetic energy
at velocity $v_{e}$; a dynamically important force; an additional heating term
as the finite resistivity of the plasma converts magnetic energy into electron
thermal energy. Observing these phenomena experimentally would be an
interesting outcome of this platform.

The Lorentz force is recovered by noting that the electron drift is a movement
of electrons in a magnetic field
$\underline{f}=(\underline{v}_{e}\times\underline{B})en_{e}$; a component of
this force is the magnetic pressure. In an unmagnetized target the implosion
kinetic energy is converted into stagnated thermal energy, with the plasma
retaining some residual kinetic energy depending on implosion symmetry. When a
large magnetic field is introduced, some of that implosion kinetic energy
converts to magnetic energy instead; the imploding shell decelerates on the
magnetic pressure rather than the hot-spot pressure. To characterize this, the
ratio of the thermal pressure to the magnetic pressure is typically used:

$\beta=\dfrac{P_{\mathrm{th}}}{P_{\mathrm{mag}}}=\frac{2(P_{e}+P_{i})\mu_{0}}{|\underline{B}|^{2}}\approx\frac{2(n_{e}T_{e}+n_{i}T_{i})\mu_{0}}{|\underline{B}|^{2}},$
(6)

where an ideal gas equation of state is assumed for the gas-fill.

For the simulation with a 50 T applied magnetic field in Fig. 2, the magnetic
pressure is approximately equal to the thermal pressure ($\beta\approx 1$). A
combination of the substantial magnetic pressure and the heat-flow suppression
accounts for the lower hot-spot density in the simulation when a 50T magnetic
field is applied.

Equation 6 can be recast in terms of the initial applied field and gas fill
density by assuming $\Gamma=1$ (i.e frozen-in-flow). Also incorporating the
assumption that the ion and electron temperatures are similar gives:

$\beta=\frac{2(1+Z)\rho_{0}^{2}T_{e}\mu_{0}}{m_{i}B_{0}^{2}\rho},\quad\text{if
}\Gamma=1,$ (7)

where $Z$ is the average ionization of the core plasma ($Z=1$ for deuterium)
and $m_{i}$ is the corresponding ion mass. According to Eq. 7, a higher
temperature hot-spot enhances the thermal pressure, but more compression of
the plasma (and therefore magnetic field) enhances the magnetic pressure
contribution.

Implosions dominated by magnetic pressure have previously been indirectly
inferred in an experiment similar to the configuration discussed here [33].
The neutron yield was seen to increase by approximately 50% going from 0 T to
10 T, primarily due to hot-spot thermal losses being reduced. However, going
from 10 T to 27 T decreased the yield by around 50%. Simulations attributed
the decrease to the magnetic pressure increasing in the hot-spot at the
expense of the thermal pressure [33]. While the decrease in thermal pressure
is undesirable for reaching high neutron yields, the large magnetic pressure
also results in a highly magnetized regime with large induced electrical
currents; these currents are of interest to the research proposed here.

Simulations anticipate that an initial 50 T field is amplified to greater than
$2\times 10^{4}$ T. Compressing a magnetic field in excess of $10^{4}$ T in a
hot-spot of around 10 $\mu m$ radius results in a large electric current. For
a cylindrical implosion the magnetic field remains predominantly axial, giving
a purely azimuthal current that can be approximated as:

$j_{\theta}\approx-\frac{B_{z}}{R_{hs}\mu_{0}},$ (8)

where $R_{hs}$ is the hot-spot radius. The final hot-spot radius can also be
related to the compressed density with the convergence relation
$R_{hs}/R_{0}=\sqrt{\rho_{0}/\rho}$, i.e. mass conservation assumption.
Combining this with the frozen-in-flow assumption:

$j_{\theta}\approx-\frac{B_{0}}{\mu_{0}R_{0}}\Big{(}\frac{\rho}{\rho_{0}}\Big{)}^{3/2},\quad\text{if
}\Gamma=1.$ (9)

Clearly, a larger initial applied magnetic field and greater plasma
compression result in a larger magnitude induced current.

The electrical resistance of the plasma leads to diffusion of the magnetic
field. The associated drop in magnetic energy results in additional energy
into the electron population (through Ohmic dissipation). For the magnetized
simulation in Fig. 2 this contributes to an enhanced electron temperature that
exceeds the ion temperature. When Ohmic dissipation is switched off in the
code, the peak electron temperature at bang time drops from 6.0 keV to 4.2
keV.

The resistive diffusion timescale is:

$t_{\mathrm{diff}}=\frac{R^{2}\mu_{0}}{\eta},$ (10)

where $\eta=\alpha_{\perp}^{c}m_{e}/(\tau_{ei}e^{2}n_{e})$ is the magnetic
diffusivity and $\alpha_{\perp}^{c}$ is the dimensionless resistivity that
only depends on plasma magnetization and ionization [51]. The timescale
$t_{\mathrm{diff}}$ defines both the rate of magnetic field diffusion as well
as the Ohmic heating. For plasmas with $\beta\approx 1$ and resistive
diffusion time-scales on the order of the hot-spot stagnation time
($t_{\mathrm{diff}}\approx t_{\mathrm{stag}}$), the energy exchange from
magnetic to electron thermal energy will be significant. Therefore, an Ohmic
heating factor $\Pi$ is defined:

$\Pi=\frac{t_{\mathrm{stag}}}{t_{\mathrm{diff}}}\frac{1}{\beta}.$ (11)

Using the convergence relation $R/R_{0}=\sqrt{\rho_{0}/\rho}$ and Eqs. 7, 10
and 11, the Ohmic heating factor can be re-written:

$\Pi=\frac{B_{0}}{\rho_{0}^{3}}\frac{t_{\mathrm{stag}}\rho}{\tau_{ei}T_{e}}\alpha_{\perp}^{c}\frac{m_{i}^{2}m_{e}Z}{2(Z+1)e},\quad\text{if
}\Gamma=1.$ (12)

Note that the diffusion is greatest at the edge of the core, where the fuel is
more dense and lower in temperature.

In addition to the resistive diffusion of the magnetic field, the electrical
current drives magnetic field advection [64, 34]. Primarily, this is the
transport of the magnetic field at the electron drift velocity
$\underline{v}_{e}=\underline{j}/en_{e}$, but with collisional corrections.
The advection of magnetic field with the electron drift velocity is often
called the Hall term. There is a similar transport of the electron energy at
the electron drift velocity, which here will be called the thermal Hall term.
The similarities in the advection velocities can be made clear by writing
[34]:

$\displaystyle\underline{v}_{jB}$ $\displaystyle=$
$\displaystyle-(1+\delta_{\bot}^{c})\frac{\underline{j}_{\bot}}{en_{e}}+\delta_{\wedge}^{c}(\frac{\underline{j}}{en_{e}}\times\underline{\hat{b}})$
(13) $\displaystyle\underline{v}_{jU_{e}}$ $\displaystyle=$
$\displaystyle-(1+\beta_{\parallel}^{c})\frac{\underline{j}_{\parallel}}{en_{e}}$
$\displaystyle-(1+\beta_{\bot}^{c})\frac{\underline{j}_{\bot}}{en_{e}}+\beta_{\wedge}^{c}(\frac{\underline{j}}{en_{e}}\times\underline{\hat{b}}),$
(14)

where $\underline{v}_{jB}$ is the advection of magnetic field due to
electrical currents and $\underline{v}_{jU_{e}}$ is the advection of electron
energy due to electrical currents, $\underline{j}_{\parallel}$ represents the
component of the current along the magnetic field, and $\underline{j}_{\bot}$
is the component perpendicular. $\underline{\hat{b}}$ is the magnetic field
unit vector. The $\beta_{\wedge}^{c}$ term is known as the Ettingshausen heat-
flow. Note that there is no advection of magnetic field parallel to the
magnetic field itself, as this has no effect on the magnetic field. The
dimensionless transport coefficients $\delta^{c}$ and $\beta^{c}$ [34] provide
collisional corrections to the collisionless current-driven phenomena. The
$\delta_{\wedge}$ coefficient calculated from Epperlein & Haines [53] gives
erroneous behaviour at low magnetization; updated polynomial fits are given in
Ref. [51]. Transport driven by electrical currents has been discussed
theoretically in pulsed power plasma experiments [65, 66] and in low density
plasma jets [67] but to the best of the authors’ knowledge has not been
measured experimentally. Current-driven transport has not been shown to be
significant in dense laser-driven implosions such as those discussed here.

The Hall (and thermal Hall) velocities are typically larger than the
collisional corrections. An approximate expression for this velocity can be
found by using Eq. 9, which assumed a purely axial magnetic field:

$v_{j}\approx\frac{B_{0}}{\mu_{0}R_{0}}\frac{\rho^{1/2}}{\rho_{0}^{3/2}}\frac{m_{i}}{Ze}.$
(15)

The drift velocity is larger for a greater applied magnetic field strength. A
lower initial fill density $\rho_{0}$ also enhances the velocity, as the
magnetic field is more easily compressed. Note that these relations must be
adapted if studying spherical implosions, as the magnetic field lines are not
everywhere perpendicular to the implosion velocity.

A dimensionless number $\Xi$ was defined in Ref. [34] as the ratio of the
Nernst velocity to the Hall velocity:

$\Xi=\frac{\gamma_{\perp}^{c}\tau_{ei}T_{e}en_{e}\mu_{0}}{|\underline{B}|m_{e}}.$
(16)

This ratio is a metric for whether the transport is dominated by temperature
gradients or electrical currents. Using burn-averaged quantities the implosion
shown in Fig. 2 has $\Xi<0.002$ for an applied field of 50 T. While this shows
the current-driven transport dominating over the thermal transport, this is
mainly due to the extreme suppression of Nernst at such high magnetizations.

For 2D $r-z$ simulations, the induced electrical current is purely in the
azimuthal direction $\theta$, which means there is no compression of magnetic
field by advection with the current and exactly no change in the electron
energy profile. The collisional corrections to the Hall and thermal Hall can
be in the simulation plane (such as the Ettingshausen $\beta_{\wedge}^{c}$
term in Eq. 14), but these are smaller than the collisionless component. With
variations in the axial direction, the Hall term can induce a twist in the
magnetic field profile. This can be seen by decomposing the change in magnetic
field due to an advection velocity directed in $\theta$:

$\Bigg{[}\frac{\partial B_{\theta}}{\partial
t}\Bigg{]}_{\mathrm{advection}}=B_{r}\frac{\partial v_{j\theta}}{\partial
r}-B_{z}\frac{\partial v_{j\theta}}{\partial z}.$ (17)

Noting that the magnetic field is dominated by the axial component, a time-
scale for twisting the magnetic field can be obtained:

$t_{\mathrm{twist}}=\frac{L_{hs}}{2v_{j}},$ (18)

where $L_{hs}$ is the axial length over which the hot-spot is uniformly
driven. Using Eq. 8, Eq. 18 becomes:

$t_{\mathrm{twist}}=L_{hs}R_{hs}\frac{1}{2}\frac{\rho_{0}}{B_{0}}\frac{\mu_{0}eZ}{m_{i}},$
(19)

i.e. both a small hot-spot radius $R_{hs}$ and length $L_{hs}$ are required
for significant magnetic field twisting to occur.

For 2D simulations such as those shown in Fig. 2, the Hall term has been
included. Due to the large uniformly-driven length ($L_{hs}\approx
600\,\mathrm{\micro m}$) only marginal twisting of the magnetic field occurs.
Over the uniformly-driven central length the field twisting
$|B_{\theta}|/|\underline{B}|\leq 0.01$. Towards the edges, where 2D effects
are most important $|B_{\theta}|/|\underline{B}|\leq 0.1$.

Once the magnetic field is twisted by the Hall term, a component of the
induced current emerges in the 2D $r-z$ simulation plane. The electron energy
is then affected by the thermal Hall term. A component of the Lorentz force
becomes in the $\theta$ direction, which may result in rotation of the plasma.
Simulations where the uniform hot-spot length $L_{hs}$ is reduced have been
attempted, but once significant $B_{\theta}$ field emerges, the simulations
become unstable. This may be related to the natural instability of
Braginskii’s MHD description [68].

The stabilization of perturbation growth by large induced electrical currents
is of great interest to magnetized ICF implosions [17, 18, 22]. The
cylindrical implosions discussed in this paper could be used as a platform for
investigating this behaviour. To demonstrate this, 3D simulations are
initialized from 1D once the first shock converges onto the axis of the
implosion. At the time of initialization the implosion velocity is
artificially modulated by $\pm 1\%$ sinusoidally in both $\theta$ and $z$. A
mode 5 perturbation is used in $\theta$, which matches the mode number of the
baseline laser asymmetry [33]. In $z$ a perturbation of 20 $\mu$m wavelength
is applied.

Figure 4: Areal density at time of peak neutron production for 3D Gorgon
simulations of cylindrical implosions with an without a 50 T seed magnetic
field with artificial velocity perturbations applied. In the axial direction
($z$, top panel) the 50 T seed magnetic field lowers perturbation growth due
to magnetic tension stabilization [20, 22, 18]. However, in the azimuthal
direction ($\theta$, bottom panel) the perturbation growth is enhanced by
magnetization, due to decreased thermal ablative stabilization [18]. The
proposed experimental platform could be used to investigate the trade-off
between these two effects for varying perturbation wavelengths.

The top panel in Fig. 4 shows the radially-integrated density (areal density)
as a function of axial position at the time of peak neutron production, with
and without a 50T magnetic field applied. The average areal density is lower
in the magnetized case, as the significant magnetic pressure decreases the
compression. It can be seen that the size of the axial areal density
perturbation is decreased with the magnetic field applied, as the magnetic
tension stabilizes the perturbation [20].

In the azimuthal direction, however, the perturbation size is enhanced by
magnetization (see bottom panel in Fig. 4). In this orientation the magnetic
tension does nothing to suppress the perturbation growth. Instead, suppression
of electron thermal conduction decreases the stability, as thermal ablative
stabilization is reduced; there is less ablation of cold plasma into the hot-
spot to stabilize the growth [18]. This process is still occurring in the
axial direction, but the magnetic tension stabilization dominates.

The effect of an applied magnetic field on the perturbation growth is a
balance between the magnetic tension stabilizing axial perturbations and the
decreased thermal ablative stabilization in all directions [18]. The balance
between the two processes is highly dependent on magnetic field strength and
the perturbation wavelength. Shorter wavelengths and higher field strengths
are thought to be dominated by magnetic tension, while intermediate
wavelengths and smaller magnetic fields are mostly affected by suppression of
thermal conduction [18]. If an artificial perturbation is applied to these
cylindrical implosions (for example through contouring of the shell) then
these predictions could be tested experimentally, which would be of great
benefit to the magnetized ICF community.

Description | Symbol | Definition |  $\Gamma=1$ Approximation
---|---|---|---
Frozen-in-flow factor | $\Gamma$ | $\frac{|\underline{B}|}{B_{0}}\frac{\rho_{0}}{\rho}$ | 1
Field Compression Factor | $\Lambda$ | $\frac{|\underline{B}|}{B_{0}}$ | $\frac{\rho}{\rho_{0}}$
Electron Hall Parameter | $\omega_{e}\tau_{e}$ | $\propto\frac{|\underline{B}|T_{e}^{3/2}}{n_{e}}$ | $\propto\frac{B_{0}}{\rho_{0}}T_{e}^{3/2}$
Ratio of Thermal to Magnetic Pressure | $\beta$ | $\frac{2(P_{e}+P_{i})\mu_{0}}{|\underline{B}|^{2}}$ | $\frac{2(1+Z)\rho_{0}^{2}T_{e}\mu_{0}}{m_{i}B_{0}^{2}\rho}$
Ohmic Dissipation Factor | $\Pi$ | $\frac{t_{\mathrm{stag}}}{t_{\mathrm{diff}}}\frac{1}{\beta}$ | $\frac{B_{0}}{\rho_{0}^{3}}\frac{t_{\mathrm{stag}}\rho}{\tau_{ei}T_{e}}\alpha_{\perp}^{c}\frac{m_{i}^{2}m_{e}Z}{2(Z+1)e}$
Maximum Unstable Magnetic Tension Mode | $L_{hs}/\lambda_{B}$ | $\frac{\mu_{0}(\rho_{h}-\rho_{l})gL_{hs}}{|B^{2}|}$ | $\frac{L_{hs}\rho_{0}^{2}\mu_{0}(\rho_{\mathrm{shell}}-\rho)V_{\mathrm{imp}}}{\rho^{2}B_{0}^{2}t_{\mathrm{stag}}}$
Twisting Factor | $t_{\mathrm{stag}}/t_{\mathrm{twist}}$ | $\frac{2|\underline{j}|t_{\mathrm{stag}}}{L_{hs}en_{e}}$ | $\frac{2}{L_{hs}}\frac{B_{0}}{R_{0}}\frac{\rho^{1/2}}{\rho_{0}^{3/2}}\frac{m_{i}}{\mu_{0}Ze}$
Ratio of Thermal to Current-Driven Transport | $\Xi$ | $\frac{\gamma_{\perp}^{c}\tau_{ei}T_{e}en_{e}\mu_{0}}{|\underline{B}|m_{e}}$ | $\frac{\gamma_{\perp}^{c}\tau_{ei}T_{e}e\mu_{0}\rho_{0}Z}{m_{i}m_{e}B_{0}}$
Table 1: Non-dimensional numbers determining the relative importance of
different physical processes in magnetized cylindrical implosions. These
factors can be used to optimize an experiment to explore specific phenomena
(see Sec. V) or can be inferred by using the frozen-in-flow approximation
along with spectroscopy (see Sec. VI)

The significance of the magnetic tension stabilizing perturbations can be
quantified by using linear magnetized Rayleigh-Taylor theory [20]. The minimum
unstable length-scale due to the magnetic field alone (i.e. neglecting thermal
conduction) can be written as:

$\lambda_{B}=\frac{|B^{2}|}{\mu_{0}(\rho_{h}-\rho_{l})g},$ (20)

where $g$ is the acceleration of an unstable interface between a heavy and
light fluids of densities $\rho_{h}$ and $\rho_{l}$ respectively. For a
stagnating hot-spot the light fluid is the hot-spot density $\rho$, while the
heavy fluid is the peak density $\rho_{\mathrm{shell}}$. Again, by assuming
frozen-in-flow, the minimum unstable wavelength due to magnetic tension can be
written in terms of initial experiment conditions and stagnation quantities:

$\lambda_{B}=\frac{\rho^{2}B_{0}^{2}t_{\mathrm{stag}}}{\rho_{0}^{2}\mu_{0}(\rho_{\mathrm{shell}}-\rho)V_{\mathrm{imp}}},\quad\text{if
}\Gamma=1.$ (21)

All the key parameters for quantifying the relevance of different magnetized
processes are summarized in Table 1; namely, _(a)_ $\Gamma$ determines the
effectiveness of magnetic flux compressing with the plasma, _(b)_ $\Lambda$ is
a measure of how compressed the magnetic field becomes, _(c)_ the
magnetization of the electron orbits is captured by $\omega_{e}\tau_{e}$,
_(d)_ the importance of magnetic pressure to the plasma dynamics is
characterized by $\beta$, _(e)_ $\Pi$ determines the significance of Ohmic
dissipation, _(f)_ $L_{hs}/\lambda_{B}$ is the axial mode number above which
magnetic tension stabilizes the hot-spot to linear Rayleigh-Taylor growth,
_(g)_ the twisting factor $t_{\mathrm{stag}}/t_{\mathrm{twist}}$ determines
whether the Hall term is expected to cause significant twists in the magnetic
field profile, and finally, _(h)_ $\Xi$ then determines if thermal or current-
driven transport is more important.

##  V Magnetization Regimes in Cylindrical Implosions

In this section, modifications to the nominal experiment are explored through
the non-dimensional factors in Table 1. It is shown that the metrics can be
used to design implosions for different regimes where certain processes
dominate magnetic flux compression, thermal transport and bulk hydrodynamics.

Throughout this section, metrics are taken as their definition and do not
utilize the frozen-in-flow ($\Gamma=1$) approximation. The 2D simulations are
reduced to 0D metrics by taking the burn-averaged values.

Figure 5: The frozen-in-flow factor $\Gamma=|\underline{B}|\rho_{0}/B_{0}\rho$
for 2D simulations over a range of applied magnetic field strengths and
initial gas fill pressures. $\Gamma=1$ means that the magnetic flux has
perfectly compressed with the plasma density. Low applied field strengths
result in extra Nernst de-magnetization of the hotspot. $\Gamma>1$ can be
obtained due to 2D motion reducing the plasma compression while the magnetic
flux continues to compress.

First, the topic of magnetic flux compression is explored. Fig. 5 plots
$\Gamma$ for a series of 2D simulations with various fill densities and
applied field strengths. Note that varying the fuel density is used to modify
the stagnation conditions, with all densities quoted as a multiple of the
nominal deuterium fill density of 1.81 kg/m3. While higher densities may not
be practical from a target fabrication standpoint, lowering the fill density
increases the implosion convergence ratio to an extent that is difficult to
resolve with current simulations. Increased convergence may also be
undesirable for the onset of hydrodynamic instabilities. Higher fill densities
could be achieved by using a foam fuel instead. However, the current study
keeps the fill composition as $D_{2}$ in order to differentiate the impact of
changing ionization state from changing fill density. The calculation of
$\Gamma$ uses the burn-averaged density and magnetic field strength.

As shown in Fig. 5, low applied field strengths result in the flux compression
being far below the frozen-in-flow approximation. This is primarily due to the
Nernst term, which transports magnetic field in the same direction as the
electron heat-flow. Like the heat-flow, the Nernst term is suppressed at
higher magnetizations. The lower fill density cases have larger contributions
from Nernst, which is due to a combination of the hot-spot being hotter and
having a smaller radius, both amplifying the temperature gradients.

For higher applied fields, the Nernst term is almost completely suppressed. At
50 T, $\Gamma>1$ for all fill densities simulated, meaning that the magnetic
field has compressed more than the plasma itself. At first this seems counter-
intuitive, as Nernst moves magnetic field out of the hot-spot and resistive
diffusion acts to lower the field strength. However, here it is actually found
that resistive diffusion can increase the field strength in the hot-spot and
that axial motion allows for additional magnetic flux compression, as
explained hereafter. It is important to remember that the compressed field and
density values used to calculate $\Gamma$ plotted in Fig. 5 are burn-averaged
quantities, which means they sample the hot core at the center more than the
hot-spot edge. However, the hot-spot is lower density than the core periphery,
meaning that the compressed field in the former is lower than in the
surrounding plasma. Therefore, resistive diffusion acts to move magnetic flux
from the compressed low temperature fuel to the low density hot-spot,
increasing the plasma magnetization in the process.

The axial plasma motion also enhances the magnetic flux compression. As this
is a cylindrical implosion, there are significant end losses out of the hot-
spot [33]. These unconfined hot-spots can be compressed to a smaller radius.
While the axial motion removes mass from the hot-spot, it does not remove
magnetic flux, which is predominantly axial itself. Therefore, more axial
motion of plasma allows higher compression of the magnetic field. In theory,
the greatest magnetic flux compression can be reached by driving extreme axial
flows. If the aim of the experiment is to reach large electron magnetizations,
however, large axial flows result in lower hot-spot temperatures and low burn
durations. The low fill 50T simulation has the largest $\Gamma$ of greater
than 1.4, as the axial flows driven are larger than the higher fill densities.

Note that $\Gamma<1$ is always expected in experiments where the fuel is laser
preheated, such as MagLIF [28]. The fuel becomes hot even before the magnetic
flux is compressed, resulting in low electron magnetizations and large
temperature gradients, allowing Nernst demagnetization to dominate.

Figure 6: Burn-averaged field compression factor
$\Lambda=|\underline{B}|/B_{0}$ for 2D simulations over a range of argon
dopant concentrations and with applied magnetic fields of 5 T, 30 T and 50 T.
Lower applied field strengths are easier to compress, as electron heat-flow
magnetization results in a high temperature, low density hot-spot. In
addition, large magnetic pressures result in lower compression. Introducing a
dopant to the fuel increases magnetic field compression. For a large initial
field strength, the magnetic pressure sets the magnetic field compression
(rather than the plasma thermal pressure), reducing the compression factor
dependence on dopant concentration.

Mid- or high-Z dopant mixed into the capsule fuel helps to increase magnetic
field compression. In particular, Fig. 6 plots the compression factor
$\Lambda$ as a function of argon concentration –the percentage quoted is per
deuterium atom– for 2D simulations with 5 T, 30 T and 50 T applied fields.
While applying a 50 T field invariably results in higher compressed field
strengths than a 30 T field, the compression factor is always lower. This is
because the higher applied field strengths result in lower hot-spot
compression. The reasons for this are twofold. Firstly, suppression of thermal
conduction reduces the hot-spot compressibility, increasing the temperature at
the cost of the density (giving the same stagnated pressure) [59]. Secondly,
the final total pressure is the same; higher magnetic pressure results in a
lower thermal pressure [33].

For the 5 T case in Fig. 6, increasing the dopant concentration greatly
increases the magnetic flux compression. This is because the hot-spot
radiatively cools, reducing the thermal pressure and allowing for more
compression. The same is true in principle for the 50 T case. However, the
hot-spot deceleration is not just dominated by the thermal pressure when 50T
is applied, since the magnetic pressure is significant –as will be seen later
in Fig. 8(a), the ratio of thermal to magnetic pressure is $\sim 1$.
Therefore, the dopant has less of an impact on the magnetic field compression
and leads to the magnetic field compression factor being relatively
independent of the dopant concentration, as seen in Fig. 6.

Figure 7 shows the burn-averaged electron and ion temperatures of 2D
simulations scanning both applied field strength and fill density. For these
cases, a dopant concentration of 0.325% is used. As the magnetic field
strength is increased, the thermal conduction from the hot-spot into the cold
fuel is suppressed, raising the core temperature. For the 50 T low fill
density case the electron Hall parameter reaches a burn-averaged value of over
300, reducing the thermal conductivity perpendicular to the magnetic field to
$\kappa_{\bot}/\kappa_{\parallel}=5\times 10^{-5}$.

The higher fill densities reach lower temperatures, reducing the electron
magnetization. Therefore, the enhancement in temperature is lower for a higher
initial fill density.

Figure 7: Burn averaged electron and ion temperatures for 2D simulations for a
range of gas fill pressures and applied field strengths. Larger magnetic
fields reduce the electron thermal conduction losses from the hot-spot,
raising the temperature. Higher fill pressures result in lower plasma
magnetizations, reducing the temperature enhancement. In the regime of large
magnetic pressure, the electron temperature can exceed the ion temperature due
to Ohmic dissipation.

An interesting feature in Fig. 7 is the electron temperature being larger than
the ion temperature in the highly magnetized low fill density cases. This is
due to the Ohmic dissipation converting magnetic energy into electron thermal
energy. In unmagnetized implosions the electron population loses energy faster
than the ions due to their additional mobility. Typically it is through
electron-ion collisions that the ion population loses its energy, but in the
magnetized case a significant proportion of the implosion kinetic energy is
used to compress the magnetic field ($\beta\approx 1$). Once the field starts
to diffuse and relax, this potential energy is transferred to the electrons,
which are resistively heated. For $B_{0}=50\,\mathrm{T}$, the Ohmic heating
factor is $\Pi=0.13$ at the hot-spot edge, indicating that a significant
proportion of the magnetic energy is being transferred into electron thermal
energy.

Measuring a higher electron temperature than ion temperature would be a
definitive observation of this process taking place. However, there may be
complications due to the ion temperature typically being inferred by the
spectra of fusion-produced neutrons [69]. Doppler shifting of the spectra by
fluid motion can increase the apparent ion temperature, which may be
comparable in size to the $\approx\,500~{}\mathrm{eV}$ difference in electron
and ion temperatures reported here.

Figure 8(a) displays the dependence of a range of metrics from Table 1 on the
applied field strength. Universally, these all show the importance of the
induced current increasing with applied field strength. For an applied field
of 50 T the hot-spot magnetic pressure is approximately the same size as the
thermal pressure, giving a $\beta\approx 1$. Along with this, the axial mode
numbers that are suppressed by magnetic tension ($L_{hs}/\lambda_{B}$)
decreases to 12, meaning that all high mode features are expected to be
suppressed in the axial direction. The importance of thermal transport of
magnetic field and electron energy is almost completely suppressed by electron
magnetization, which can be seen from $\Xi<0.001$. While the magnetic field
twisting factor is still small for a seed B-field of 50 T, this could be
enhanced by reducing the driving length of the target.

Figure 8(b) then shows how the induced-current metrics vary with the argon
dopant percentage for an applied magnetic field of 30 T. Again, the added
dopant results in additional radiative losses, allowing the hot-spot to become
more compressed. Similarly, the magnetic field compresses further and gives
larger induced currents. As a result, the magnetic pressure is greater, the
Ohmic heating becomes more important, the magnetic tension stabilizes more
wavelengths, the importance of thermal transport is reduced and the Hall term
twists the magnetic field more effectively.

(a)

(b)

Figure 8: Magnetization metrics from Table 1 for: (a) varying the applied
field strength while keeping the argon concentration at 0.325% by atom; (b)
varying the argon fuel dopant concentration while keeping the applied field
strength at 30 T.

##  VI Argon K-shell spectroscopy

Here we address the application of X-ray spectroscopy for the characterization
of core conditions. The Gorgon MHD simulations show that the compression of
the B-field changes the hydrodynamic behavior, thus producing a variety of
magnetization phenomena whose underlying physics demand a deeper understanding
as discussed in Sec. IV and V. Since changes in the hydrodynamics translate
into significant variations in the radial profiles of core temperature and
density throughout the implosion collapse compared to the unmagnetized case, a
way to bring information about the associated physics is by diagnosing the
conditions of the imploding core. In this regard, given the success in past
experimental campaigns of spherical implosions at OMEGA [42, 43, 44, 45, 46]
and also in laser-heated cylindrical plasma experiments performed at Z related
to the MagLIF preheat stage [47, 48], here we propose to rely on time-resolved
Ar K-shell spectroscopy. This diagnostic exploits two basic properties of the
Ar K-shell spectrum emitted from hot, dense plasmas: (1) the Stark-broadened
line shapes strongly depend on density and are relatively insensitive to
variations in electron temperature, (2) the relative intensity distribution of
K-shell lines and their associated satellites are sensitive to variations in
electron temperature and density through the dependence to these parameters on
the atomic level population kinetics. The Ar K-shell spectroscopic technique
works properly for plasma electron densities and temperatures typically in the
ranges of $n_{e}\sim 10^{23}-10^{24}$ cm-3 and $T_{e}\sim 600-2000$ eV, since
for these conditions the Stark effect is the dominant line broadening
mechanism, and He-like and H-like Ar are significantly populated so that their
corresponding line emission becomes bright and useful for analysis. In this
regard, Fig. 9 shows the time histories of mass-weighted average electron
temperature and electron density of the imploded core for different argon
concentrations obtained from 2D Gorgon simulations. Overall, average core
conditions for the non-magnetized implosions fit well within the ranges above,
while for the magnetized case the temperature can go above 2 keV around
stagnation. Still, since the new physics and main goal here focuses on
exploring and providing evidence on extreme magnetization phenomena, we
consider appropriate to use the well-established Ar K-shell spectroscopic
technique and perform an assessment of its potential application, in order to
find out how far the technique can be pushed for the characterization of core
conditions in the magnetized scenario. It is also worth noting that despite
the high B-field values achieved in the compressed core ($>10$ kT around
stagnation), Stark-Zeeman spectroscopy cannot be used to obtain an estimation
of compressed B-field in an experimental scenario. Given the range of electron
densities achieved, we performed systematic calculations of Stark-Zeeman line
profiles based on molecular dynamics simulations [70] and checked that the
broadening of Ar K-shell lines produced by the Stark effect blurs the
characteristic Zeeman pattern even in the case of the expected achievable
B-field. Zeeman features will be further washed out by radiation transport
effects through the imploded core and instrumental broadening. Details of
these calculations are beyond of the scope of this paper and will be addressed
in a forthcoming publication.

Figure 9: Time history of core conditions from 2D Gorgon simulations of
cylindrical implosions as a function of different argon dopant concentrations.
Shown values are obtained as mass-weighted averages within the radial core
boundary and over the central 400 µm-long portion of the imploded cylinder.
Top and bottom panels show electron temperature and electron density from non-
magnetized and magnetized cases, respectively. Figure 10: Comparison of
synthetic images of Ar K-shell time-resolved spectra for unmagnetized and
magnetized ($B_{0}=50$ T) implosions. For the non-magnetized case, changes in
the spectra due to different Ar concentrations are shown. For the magnetized
case, the sensitivity with Ar concentration is lost and only the case for Ar
at 0.175% is displayed –see the text for details–.

In an effort to have a realistic representation of the time-resolved Ar
K-shell spectra to be measured in a real experiment, we have post-processed
the 2D MHD Gorgon simulations and performed detailed atomic-kinetics and
radiation transport calculations throughout the evolution of the imploding
cylinder. Line of sight is assumed to be perpendicular to the cylinder axis,
so that to compute the emergent intensity the radiation transport equation
must be solved along chords inside the plasma source parallel to the line of
sight. The solution to this problem can be well approximated by a system of
effective slabs [71] resulting from the discretization of the cylindrical
plasma in a finite number of cylindrical shells, which here was chosen to
exactly match the radial grid used in the Gorgon MHD simulations. Temperature
and density conditions of each cylindrical shell are taken as the spatial
average for a given time over the central 400 µm-long portion of the imploded
core along the axial direction. This is a reasonable simplification, since
according to MHD simulations –see Fig. 2–, conditions of the imploded cylinder
show little dependence on the axial direction in the central region. The
required emissivities, opacities and atomic level population distributions
were calculated with the collisional-radiative model ABAKO [72] for each pair
of core electron temperature and density over the corresponding radial grid as
obtained from the Gorgon MHD output, i.e. no interpolation over pre-calculated
emissivity and opacity databases was applied. Bound-bound, bound-free and
free-free contributions from the Ar-doped deuterium plasma within the spectral
range of interest were included. In particular, relevant bound-free cross
sections [73] were computed with the LANL suite of codes [74]. Energy levels
and line transitions rates for the required Ar ion stages were computed with
the atomic structure code FAC [75]. For given plasma conditions, ABAKO
explicitly includes all non-autoionizing and autoionizing states consistent
with the corresponding ionization potential depression, which is estimated
according to the Stewart-Pyatt model [76], modified to account for the argon-
deuterium mixture. An escape factor model was used to account for radiation
trapping effects on the atomic-level population kinetics [77]. Importantly for
this application, detailed Stark-broadened line shapes for Ar He$\alpha$,
He$\beta$, He$\gamma$, Ly$\alpha$, Ly$\beta$, and Ly$\gamma$, and associated
satellites were calculated with the MERL line-shape code [78], including ion-
dynamics effects [79]. Recently, MERL Stark-broadened line shapes of Ar
K-shell have been further validated for spectroscopic diagnosis applications
against full molecular dynamics simulations [80].

Figure 10 shows the time evolution of the computed Ar K-shell emission, i.e.
synthetic streaked images, for unmagnetized and magnetized cases assuming a
seed B-field of $50$ T. Besides Doppler and Stark broadening mechanisms,
synthetic spectra also accounts for instrumental broadening. Consistently with
available streaked spectrometers on OMEGA for the photon energy range of
interest, a spectral resolution of $E/\Delta E\approx 500$ was assumed. Main
line transitions in He-like and H-like ions are labeled, i.e. He$\alpha$
(1s2p$\rightarrow$1s2), He$\beta$ (1s3p$\rightarrow$1s2) and He$\gamma$
(1s4p$\rightarrow$1s2) in He-like Ar, and Ly$\alpha$ (2p$\rightarrow$1s),
Ly$\beta$ (3p$\rightarrow$1s) and Ly$\gamma$ (4p$\rightarrow$1s) in H-like Ar.
There are significant differences between the expected time-resolved spectra
of a non-magnetized implosion and a magnetized one. Moreover, the metrics
analysis of Sec. V demonstrated that a extreme magnetization regime is already
attainable for a seed B-field of 30 T, thus also expecting noticeable
differences –similar to those shown in Fig. 10– in the spectral output when
compared to the unmagnetized case. Therefore, based on our simulations, in a
potential experimental realization the observation of the time-resolved Ar
K-shell emission should be able to distinguish between the unmagnetized and
magnetized scenarios (at least for $B_{0}>30$ T). A detailed analysis of
spectroscopic data would bring further information about the magnetization
state, as pointed out below.

For interpretation, it is useful to correlate the synthetic streaked images in
Fig. 10 with the time histories of electron temperature and density shown in
Fig. 9. In the unmagnetized case, line emission turns on at $t\gtrsim 1.35$ ns
and lasts for $\lesssim 200$ ps. Figure 10 shows that, overall, as expected,
signal intensity increases with time until stagnation ($t=1.5$ ns), when the
average temperature and density reach their maximum values, and decreases
after. Lines are narrower early in time, then gradually broaden until
stagnation time and narrowing down afterwards, mainly due to Stark-broadening
which increases with increasing electron density. When comparing the results
for different Ar concentrations, the ratio of He-like to H-like for the same
type of transition increases with the Ar amount. This is a result of radiative
cooling, i.e. core temperature decreases and He-like ion population is favored
relative to that of H-like ion. Also, core compressibility is higher for the
case of Ar at 0.65%, so we observe broader lines when compared to the results
for Ar at 0.175%. For Ar at 0.65% the overall spectral intensity drops
compared to Ar at 0.175%, which, despite the density increasing, can be
explained because the plasma cooling remains as the dominant effect; i.e. for
Ar at 0.65%, the core average temperature stays below $700$ eV throughout the
implosion and, relative to the case of Ar at 0.175%, the ion population
distribution shifts to lower ionization stages, so that the number of K-shell
ion emitters decreases accordingly.

In the magnetized case, line emission lasts longer ($\sim 250$ ps) and the
overall signal intensity is lower than compared to a non-magnetized implosion.
Moreover, since the effect of magnetic pressure reduces the core
compressibility, electron density and therefore the impact of Stark-broadening
decrease, which ultimately leads to narrower lines compared to the
unmagnetized scenario. Interestingly, contrary to what was found for
unmagnetized implosions, streaked images of K-shell emission show little
sensitivity to the Ar concentration –only the case for Ar at 0.175% is
displayed in Fig. 10 due to this reason–. This is due to the large gradients
in core conditions along the radial direction throughout the collapse of the
implosion for the magnetized case. For illustration, Fig. 11 shows the radial
profiles at stagnation time of core electron temperature and density averaged
in the axial direction over the central 400 µm-portion of the imploding
cylinder. The temperature radial profiles show that, regardless of the Ar
amount, a lower bound of $\sim 700$ eV is found at the core periphery.
Temperature rises radially inwards to values far above $\sim 2$ keV, where Ar
ions will be fully ionized. Thus, it is clear that K-shell emission for the
magnetized case will be only representative of the core outermost portion.
When temperature spatial profiles are correlated with the corresponding
density profiles, one realizes that, no matter the Ar concentration, the
radiation transport comes across basically the same ranges and combinations of
core conditions relevant for K-shell line emission. Accordingly, since a
similar behavior is found over the implosion collapse, the resulting time-
resolved spectra show little dependence on the Ar amount. No significant
gradients are observed in the radial profiles for the unmagnetized case. In
Sec. V, the variation of Ar concentration was suggested as a fine-tuning knob
for achieving different magnetization regimes for a given seed B-field value.
However, since the sensitivity with Ar concentration is lost, our simulations
here show that Ar K-shell spectroscopy would not be useful to distinguish
between such regimes. Still, if ideally multiple spectrometers were available
in the experimental facility to simultaneously measure different spectral
ranges, we could keep using the dopant concentration as a control knob and
speculate with doping the targets with small amounts of different dopants, for
instance mixing Ar and Kr. Ar K-shell spectroscopy seems to work properly for
the characterization of core conditions in the unmagnetized case and,
potentially, Kr K-shell and L-shell spectroscopy could provide a better
characterization of magnetized implosions, even enabling to retrieve the
sensitivity of the emission spectra to the dopant concentration. This subject
is under investigation and will be published elsewhere.

Figure 11: Radial profiles of core conditions at stagnation time ($t=1.5$ ns)
for a magnetized cylindrical implosion (with $B_{0}=50$ T). Results for
different Ar concentrations are shown.

Figure 12 shows synthetic time-integrated spectra for the unmagnetized and
magnetized cases to evaluate potential data from this diagnostic on OMEGA.
Spectra are shown for a photon energy from 3500 to 4300 eV, since temperature
and density diagnosis based on K-shell spectroscopy usually relies on the
$\beta$ and $\gamma$ lines region of the spectrum –$\alpha$ lines are much
more sensitive to spatial gradients and absorption effects than others due to
their characteristic large optical depths–. As already discussed, there is
little variation with Ar concentration for the magnetized case, so that only
the case for Ar at 0.175% is displayed. The comparison of time-integrated
spectra obtained at different Ar concentrations for a non-magnetized implosion
allows us to answer the question about the optimum dopant concentration for
spectroscopic purposes. For Ar at 0.175%, strong He$\beta$ and Ly$\beta$
emissions are observed, which is critical for a reliable spectroscopic
analysis. For Ar at 0.325% the Ly$\beta$ emission is almost lost and the
situation is even worse for Ar at 0.650%. The same conclusion is found when
doing the comparison for time-resolved spectra. Besides the difference in the
absolute signal strength, time-integrated spectra from unmagnetized and
magnetized cases mostly differ in the broadening of the lines. The higher
compressibility of the unmagnetized implosion ultimately leads to a
significant impact of Stark-broadening, which is not so severe in the
magnetized case. Regarding the overall signal strength, results for non-
magnetized and magnetized implosions differ in a factor of $\sim 2$, and both
compare within one order of magnitude with calculations done in the past for
the campaign of spherical implosions on OMEGA, where the Ar K-shell emission
was successfully recorded. Therefore, for all the reasons above, an argon
concentration of 0.175% seems to be a good choice for application of K-shell
spectroscopy for the study of the referred direct-drive cylindrical
implosions.

Figure 12: Comparison of synthetic Ar K-shell time-integrated spectra for non-
magnetized and magnetized ($B_{0}=50$ T) implosions. For the unmagnetized
case, changes in the spectrum due to different Ar concentrations are shown.
For the magnetized case, the sensitivity with Ar concentration is lost and
only the case for Ar at 0.175% is displayed –see the text for details.

Lastly, it is worth noting that, similarly to previous work, a spectroscopic
modelling and analysis of the measured time-resolved Ar K-shell emission would
allow the determination of the temperature and density of the imploded core
throughout the collapse of the implosion. From these spectroscopic
measurements and if the magnetic field is assumed to be frozen, estimations
for the magnetization metrics in Table 1 can be obtained. However, in the
magnetized scenario, the spectroscopic diagnosis will be more complicated due
to the severe gradients in core conditions. A feasibility assessment of the
spectroscopic technique based on synthetic data is on progress and will be
published in a forthcoming paper.

##  VII  Conclusions

This paper outlines a cylindrical implosion platform for studying highly
magnetized plasmas on the OMEGA laser facility. In the highly magnetized
regime the imploding shell decelerates on the magnetic field, converting
kinetic energy into magnetic. The highly compressed magnetic field induces
large electrical currents in the hot-spot; finite plasma resistivity results
in Ohmic dissipation, diffusing the magnetic field and heating the electron
population. This is expected to result in a higher temperature in the electron
population than in the ions, which may be diagnosable.

The large induced electrical currents can also cause Hall physics to be
important, which has not been reported in laser-driven implosions to date. The
calculations here estimate that a typical electron in the hot-spot will
undergo a 1/3 rotational drift around the cylindrical axis within the fusion
time frame, advecting the magnetic field and electron energy distribution.

The electrical currents also result in perturbation stabilization in the axial
direction, which could be studied by using a shell with applied corrugations.
Experiments scanning applied field strength and perturbation wavelength would
be of great interest to the magnetized ICF community.

All of these magnetization processes have been summarized as non-dimensional
metrics in Table 1.

Care has been taken to optimize future potential experiments to reach peak
magnetic field strengths. By using a low gas fill, the magnetic field is more
easily compressed. However, as the convergence ratio of the capsule increases,
the implosion will be more susceptible to perturbation growth, limiting the
minimum fill density. Dopants added to the imploding fuel can also be used to
radiatively cool the hot-spot, allowing the magnetic field to be more
compressible. More axial motion, too, allows for greater flux compression.

X-ray spectroscopy can be used to diagnose the core electron temperature and
density. The core temperature is anticipated to increase through the
application of a magnetic field, while the density will decrease. The
measurement of lower overall core pressure when a large magnetic field is
applied would indicate that the magnetic pressure has become significant. By
assuming frozen-in-flow, which has been shown with simulations to be a good
approximation in the highly magnetized regime, many of the metrics governing
which magnetization phenomena are important can be indirectly inferred.

##  VIII  Appendix

Magnetic flux compression has been measured in previous cylindrical implosions
on the OMEGA Laser Facility using proton radiography [5, 6]. Here, 2D Gorgon
simulations are post-processed to create synthetic proton radiographs to
compare with these experiments. Good agreement is found when the Nernst term
is included in the simulations. These results build confidence in the
simulation results in this publication and also (to a lesser extent) in other
non-cylindrical Gorgon simulations where magnetic flux compression is an
important process.

There are significant differences between the target dimensions used in the
bulk of this paper and the target used for this magnetic flux compression
validation. Here the target radius is $430\,\mathrm{\mu m}$, filled with $3$
atm of $D_{2}$ gas [6]. A $5$ T magnetic field was applied, which is low
enough to have relatively little contribution from magnetic pressure in the
hot-spot. The experiment discussed in this appendix was also carried out
before the laser non-uniformity was optimized [31, 57], resulting in a lack of
axial hot-spot confinement. While this lowers the electron magnetization, the
Gorgon simulations actually predict an enhancement to the magnetic flux
compression, because the hot-spot is more compressible.

Figure 13: Proton radiographs of a cylindrical magnetic flux compression
experiment. The experimental data is taken from Ref. [6], showing here only
the data from the protons that have slowed through the target (energy ranges
14.4-14.6 MeV and 14.6-14.8 MeV). The experimental radiographs show a bump at
around 1.8 cm, which is consistent with the simulation including Nernst
advection.

The proton source in the experiment was from an exploding pusher, which
provides a relatively monoenergetic proton population ($\Delta E/E\sim 0.03$)
[6]. This population is born at 14.7 MeV, but is accelerated to 15.2 MeV due
to charging of the backlighter [6]. The hot-spot being imaged is small
($\approx 10\,\mathrm{\micro m}$), meaning that very few protons hitting the
radiograph are significantly deflected by the compressed magnetic field. On
the face of it, this makes imaging the peak field very difficult. However, the
protons passing through the compressed plasma are slowed; by looking only at
the protons that have had their energies lowered by the compressed plasma, the
signal to noise ratio is improved. In the experiment analysis, the proton
populations in the energy bands
$<14.4\,\mathrm{MeV}$,$14.4-14.6\,\mathrm{MeV}$ and $14.6-14.8\,\mathrm{MeV}$
all show bumps at around $1.8\,\mathrm{cm}$ deflection [6]. When the applied
field was reversed in direction, the bump moved to $-1.8\,\mathrm{cm}$, which
indicates that the feature is from magnetic field deflections, not from
electric field effects. While the signals for each proton range are relatively
small compared with the background, the consistency of the feature across
several proton energy ranges and across multiple experiments is encouraging.

Synthetic proton radiographs of 2D Gorgon simulations have been plotted in
Fig. 13. These are produced using a proton tracking code without electric
field effects or collisional scattering. A monoenergetic source is assumed.
The protons are reported to be emitted over a $150\,\mathrm{ps}$ burst [6], so
a Gaussian emission profile with standard deviation of $75\,\mathrm{ps}$ is
used for the synthetic radiographs.

When the Nernst term is included in the simulations a bump is observed in the
radiograph at 1.9 cm, similar to that seen experimentally. This bump is not
found to be due to the peak field strength reached at the centre of the
target, but by the magnetic flux that develops at the hot-spot edge, $\approx
10\,\mathrm{\micro m}$ from the axis. While the field strength at this radius
is lower, deflecting the protons less, the total flux of protons into this
region is greater, increasing the signal. The magnetic field strength at the
hot-spot edge is approximately $4000$ T, which is consistent with the values
inferred by the original experiment analysis [5].

Without the Nernst term included, the synthetic proton radiograph does not
compare favourably with the experiment, with a wide spread of proton
deflections and no clear bump –see Fig. 13. This is the first direct evidence
of Nernst in magnetized implosions; Nernst has been indirectly inferred in
MagLIF through observation that, above a certain laser preheat energy, the
neutron yield plateaus [28]. No sensitivity of the radiograph to a Nernst flux
limiter in the range 0.04-0.1 was found. While resistive diffusion smooths the
radiograph, the cases with and without compare equally favourably to the
experiment.

##  IX Acknowledgements

This work performed under the auspices of the U.S. Department of Energy by
Lawrence Livermore National Laboratory under Contract DE-AC52-07NA27344. This
document was prepared as an account of work sponsored by an agency of the
United States government. Neither the United States government nor Lawrence
Livermore National Security, LLC, nor any of their employees makes any
warranty, expressed or implied, or assumes any legal liability or
responsibility for the accuracy, completeness, or usefulness of any
information, apparatus, product, or process disclosed, or represents that its
use would not infringe privately owned rights. Reference herein to any
specific commercial product, process, or service by trade name, trademark,
manufacturer, or otherwise does not necessarily constitute or imply its
endorsement, recommendation, or favoring by the United States government or
Lawrence Livermore National Security, LLC. The views and opinions of authors
expressed herein do not necessarily state or reflect those of the United
States government or Lawrence Livermore National Security, LLC, and shall not
be used for advertising or product endorsement purposes.

This work has also been carried out within the framework of the EUROfusion
Consortium and has received funding from the Euratom research and training
programme 2014-2018 and 2019-2020 under grant agreement No. 633053. The views
and opinions expressed herein do not necessarily reflect those of the European
Commission. The involved teams have operated within the framework of the
Enabling Research Projects: CfP-AWP17-ENR-IFE-CEA-02 Towards a universal
Stark-Zeeman code for spectroscopic diagnostics and for integration in
transport codes and CfP-FSD-AWP21-ENR-01-CEA-02 Advancing shock ignition for
direct-drive inertial fusion.

This material is based upon work supported by the US National Nuclear Security
Administration under Award No. DE-NA0003940. The work has also been supported
by the Research Grant No. GOB-ESP2019-13 from ULPGC; and by Research Grant No.
PID2019-108764RB-I00 from the Spanish Ministry of Science and Innovation.

F.S.-V. acknowledges funding from The Royal Society (UK) through a University
Research Fellowship. C.V. acknowledges the support from the LIGHT S&T Graduate
Program (PIA3 Investment for the Future Program, ANR-17-EURE-0027). G. P.-C.
acknowledges funding from the French Agence Nationale de la Recherche
(ANR-10-IDEX-03-02, ANR-15-CE30-0011).

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