# Design and Modeling of Indirectly-Driven Magnetized Implosions on the NIF

D. J. Strozzi<EMAIL_ADDRESS>H. Sio  G. B. Zimmerman  J. D. Moody  C. R.
Weber  B. Z. Djordjević  C. A. Walsh  B. A. Hammel  B. B. Pollock  A. Povilus
Lawrence Livermore National Laboratory, Livermore, CA 94550  J. P. Chittenden
S. O’Neill  Centre for Inertial Fusion Studies, The Blackett Laboratory,
Imperial College, London SW7 2AZ, United Kingdom

###### Abstract

The use of magnetic fields to improve the performance of hohlraum-driven
implosions on the National Ignition Facility (NIF) is discussed. The focus is
on magnetically insulated inertial confinement fusion (ICF), where the primary
field effect is to reduce electron-thermal and alpha-particle loss from the
compressed hotspot (magnetic pressure is of secondary importance). We
summarize the requirements to achieve this state. The design of recent NIF
magnetized hohlraum experiments is presented. These are close to earlier shots
in the three-shock, high-adiabat (BigFoot) campaign, subject to the
constraints that magnetized NIF targets must be fielded at room-temperature,
and use $\lesssim 1$ MJ of laser energy to avoid risk of optics damage from
stimulated Brillouin scattering. We present results from the original
magnetized hohlraum platform, as well as a later variant which gives higher
hotspot temperature. In both platforms, imposed fields (at the capsule center)
of up to 28 T increase the fusion yield and hotspot temperature. Integrated
radiation-magneto-hydrodynamic (rad-MHD) modeling with the Lasnex code of
these shots is shown, where laser power multipliers and a saturation clamp on
cross-beam energy transfer (CBET) are developed to match the time of peak
capsule emission and the $P_{2}$ Legendre moment of the hotspot x-ray image.
The resulting fusion yield and ion temperature agree decently with the
measured relative effects of the field, although the absolute simulated yields
are higher than the data by $2.0-2.7\times$. The tuned parameters and yield
discrepancy are comparable for experiments with and without an imposed field,
indicating the model adequately captures the field effects. Self-generated and
imposed fields are added sequentially to simulations of one BigFoot NIF shot
to understand how they alter target dynamics.

††preprint: AIP/123-QED

##  I Introduction

Magnetized inertial confinement fusion (ICF) is an old idea [1, 2, 3] that has
received significant interest in recent years. The basic concept is to reduce
the loss of energy (carried by electrons and fusion-generated charged
particles) from the implosion hotspot, and thereby relax the requirements for
ignition and energy gain. This thermal insulation and resulting improved
fusion performance have been demonstrated in the three major approaches to
ICF: laser direct drive, magnetic direct drive, and indirect (x-ray) drive.
Laser-driven cylindrical and spherical implosions at the OMEGA laser facility
have compressed externally-imposed B fields to large values [4, 5, 6] and
demonstrated increased fusion yield and ion temperature $T_{i}$. The
magnetically-driven MagLIF concept uses the pulsed-power Z Machine at Sandia
National Laboratories to implode a cylindrical metal liner that contains
fusion fuel with an imposed axial magnetic field and preheated by a laser [7,
8]. The imposed field is needed to confine DT fusion alpha particles, and has
been shown experimentally to improve performance. An imposed magnetic field
can also improve the fast ignition concept [9] by guiding relativistic
electrons to the hotspot and mitigate their angular divergence. This was
proposed in a modeling study [10] and has recently been explored in
experiments with kilotesla laser-driven fields [11].

This paper focuses on magnetized indirect-drive ICF, which has been researched
at Lawrence Livermore National Lab (LLNL) for over a decade [12, 13]. Recent
experiments at the National Ignition Facility (NIF) [14] have studied
hohlraum-driven implosions of gas-filled capsules, with imposed initial axial
magnetic field up to 28 tesla [15, 16]. These have demonstrated increased
fusion yield and $T_{i}$, in line with radiation-magneto-hydrodynamic (rad-
MHD) modeling. We continue to build on this work, with the goal of using
magnetic fields to enhance high-yield designs in the ignition regime.

The role of magnetic fields in fusion devices can be understood by considering
two dimensionless measures: the electron Hall parameter
$H_{e}\equiv\omega_{ce}\tau_{ei}$ ($\omega_{ce}\equiv eB/m_{e}$ is the
electron cyclotron frequency and $\tau_{ei}$ the electron-ion collision time),
and $\beta\equiv 2\mu_{0}p/B^{2}$ ($p$ is the thermal pressure). For all
systems currently under study, the minimum requirement to be magnetized is
$H_{e}>1$. This means the electron thermal conductivity $\kappa$ perpendicular
to $\mathbf{B}$ is significantly reduced. Within this class, systems can be
divided based on $\beta$. For $\beta\gg 1$, magnetic pressure has a small
direct effect, and we call this regime “magnetically-insulated” ICF.
Magnetized ICF schemes for NIF and other laser facilities fall in this domain.
$\beta\sim 1$ is the “magneto-inertial fusion” regime, where magnetic and
thermal pressures are comparable [17]. This includes the magnetic compression
stage of pulsed-power driven MagLIF, though the imploded fuel has $\beta\gg 1$
(so the laser-driven “mini-MagLIF” scheme at OMEGA [18] is magnetically-
insulated ICF). Other interesting MIF approaches include the sheared-flow
stabilized Z-pinch [19] and the dense plasma focus [20]. $\beta\ll 1$, where
magnetic pressure dominates the confinement, is the regime of classic
magnetic-fusion systems like tokamaks and stellarators.

We now estimate the magnetic field requirements for magnetizing a typical NIF
hohlraum-driven hotspot. High-performing capsules with a cryogenic DT ice
layer have a measured convergence ratio $CR=$ (initial capsule radius) /
(stagnated hotspot radius) $\sim 30$. We consider an initial axial $B$ field
of a uniform 40 T in the capsule, and make the optimistic assumption that
magnetic flux is conserved in the implosion according to the frozen-in law of
ideal MHD. The field on the equator is then enhanced by $CR^{2}\sim 900$,
giving $B\sim$ 36 kT. A marginally igniting hotspot has density $\sim$ 100
mg/cm3 and temperature $\sim$ 6 keV, which gives Hall parameter $H\sim$ 9.5.
To estimate the effect on $\kappa$, for a purely axial field the effective
$\kappa/\kappa_{||}\sim 1/3+(2/3)\kappa_{\perp}/\kappa_{||}$ with
$\kappa_{||}$ the unmagnetized, axial value. $\kappa$ is reduced in the two
perpendicular directions. For $Z=1$ and $H=9.5$, the Epperlein and Haines [21]
fits for magnetized $\kappa$ give almost full suppression:
$\kappa_{\perp}/\kappa_{||}=0.012$. Even $H=1$ gives significant reduction:
$\kappa_{\perp}/\kappa_{||}=0.32$. These parameters thus give substantial
margin for processes that reduce the benefits of magnetization. These include
flux loss due to effects like resistive diffusion and the Nernst effect, or
non-axial imploded field geometries. From this simple estimate and detailed
rad-MHD modeling, the LLNL program aims to impose axial fields up to 100 T on
layered cryogenic capsules driven by NIF hohlraums.

This paper is organized as follows. Section 2 describes the constraints on,
design of and results from magnetized NIF hohlraum-driven gas-filled capsule
experiments. Rad-MHD modeling of these shots with the Lasnex code and Lasnex
Hohlraum Template (LHT) common model are presented in Sec. 3. Sec. 4 contains
a modeling study of MHD and imposed-field effects on one NIF shot from the
BigFoot campaign, upon which the magnetized platform is based. Conclusions and
future work toward magnetized ignition on NIF are given in Sec. 5. Two
appendices give details on the Lasnex LHT model and Lasnex’s MHD package.

##  II Magnetized Hohlraum-Driven Implosions of Gas-Filled Capsules on NIF

This section presents the rationale behind and results of NIF experiments with
hohlraum-driven gas-filled capsules, with and without imposed magnetic field.
This work has demonstrated increased hotspot ion temperature and yield with a
field, and has been partially reported elsewhere [15, 16]. The new aspects
here are a detailed discussion of the “WarmMag” platform’s design rationale
and early results from recent experiments with increased hotspot temperature
(with or without field).

###  II.1 “WarmMag” Magnetized Room-Temperature NIF platform constraints

Our “WarmMag” platform is designed to demonstrate magnetic insulation of an
indirect-drive ICF hotspot, subject to various practical constraints. The
platform has three designs: the initial low-T design with prolate implosion
shape, the final low-T design with round shape when magnetized, and the high-T
design with increased capsule hotspot temperature. We magnetize a NIF hohlraum
by wrapping it with an external coil and running current through it via a
pulsed-power system. The most salient present constraint is that one can only
magnetize room-temperature (293 K) NIF experiments. A room temperature pulsed
power system consisting of a 4 $\mu$F capacitor and a spark gap trigger system
has been in use for a variety of NIF target experiments since 2016. Pre-
magnetizing cryogenic DT-layered implosion targets requires integration of a
pulsed power system with a cryogenic target positioner. NIF has designed this
system but the completion date for construction is currently under evaluation.
Room-temperature hohlraum implosions are fielded somewhat routinely on NIF
[22]. They are limited to gas-filled capsules or “symcaps,” with no ice layers
and no keyhole shock-timing experiments (these require a re-entrant gold cone
filled with liquid D2 inserted into the capsule).

There are limits on the room-temperature pressures and compositions of capsule
and hohlraum gas fills. Our diamond or high density carbon (HDC) capsule can
hold very high pressures of H and He isotopes without permeation, so this does
not affect our gas choice. The hohlraum gas fill is limited by not bursting
the laser entrance hole (LEH) windows, for which the probability gradually
increases with pressure. To minimize the likelihood of bursting, we typically
use $\sim 1\ \mu$m thick windows vs. $\sim 1/2\ \mu$m for cryo experiments. We
also follow the common NIF practice of using neopentane (C5H12) for room-
temperature hohlraum fills, with the same fully-ionized electron density as
the He in the analogous cryo design [22]. This corresponds to much lower
initial fill pressure.

Another constraint is having the field permeate the hohlraum and capsule
before the laser fires, without significantly perturbing either. The field
must soak through the high-Z hohlraum with acceptably small $J\times B$ force,
Joule heating, and temporal delay due to resistive diffusion [23]. All these
effects increase with hohlraum conductivity. The most limiting for our current
gas-filled experiments is the $J\times B$ force. We require the resulting
radial wall motion to be less than $50\ \mu$m, which is comparable to NIF
laser beam pointing error. Pure gold hohlraums are too conductive to meet
this, though pure uranium hohlraums might be acceptable. Future experiments
with ignition-quality cryogenic DT ice layers have more stringent constraints
which require lower conductivity. For instance, the hohlraum and coil must
allow adequate thermal control and spatial uniformity, and radiation from a
Joule-heated hohlraum must not disturb the ice layer. Uranium is probably not
adequate for layered targets with 60-70 T imposed field, so instead we use a
novel, resistive hohlraum alloy AuTa4 [24, 25]. Our HDC capsule is non-
conductive so field soak-through poses no challenges, though it may for a
conductive capsule made of e.g. beryllium.

The final constraint due to magnetization is the risk of laser optics damage
due to SBS [26]. Magnetic fields introduce risk like any new target, and laser
energy and power must be increased in safe increments. The new risk with
fields is that the pulsed-power system could prefire before the laser. The
current melts the coils and induces $J\times B$ motion, all of which can leave
the target in an unknown state (with unknown SBS risk) when the laser fires.
Pulser prefire has not been observed on any NIF shot to date, but its risk is
unacceptably high. Every magnetized NIF shot must therefore pass a stringent
SBS risk rule, which can only be relaxed when the prefire risk is made small
enough. The SBS rule is a limit on the “equivalent fluence” of laser power vs.
time, which is a complicated functional somewhat between total energy and peak
power [27]. This usually limits hohlraum pulses to $\sim$ 5 kJ/beam (or 1 MJ
full NIF). A low-power 1 TW/beam “caboose” at the end of the pulse for a
standard hohlraum is considered to not add risk. The planned cryo pulser will
include prefire mitigation and remove the SBS limit.

Figure 1: Total laser power (top) and cone fraction (bottom: inner power /
total power) vs. time for four gas-filled capsule NIF experiments. BigFoot in
black, WarmMag (WM) initial low-T in blue, WM low-T in orange, WM high-T in
green. Cone fraction is only plotted until laser power falls below 50 TW.

###  II.2 “WarmMag” initial low-T platform: BigFoot analog, prolate hotspot

The room-temperature magnetized hohlraum project aims to understand
magnetization with $\lesssim$ 5 shots per year which each use $\lesssim$ 1 MJ
of laser energy. We therefore start from an existing NIF platform that has
both high performance and robustness, meaning shots are likely to be
reproducible and not depart much from expectations. The BigFoot platform
stands out in this regard [28, 29]. It was deliberately designed for
robustness, with a high implosion adiabat and high density HDC capsule. Both
of these lead to short laser pulses, which require a relatively low density
hohlraum gas fill of 0.3 mg/cm3 of He to adequately limit inward hohlraum wall
motion. This has been demonstrated to produce much less backscatter than
longer-pulse, higher-fill platforms like the earlier LowFoot and HighFoot.
BigFoot achieved round implosions without using wavelength shifts
$\Delta\lambda$ between the NIF lasers. This is done to control cross-beam
energy transfer (CBET)[30, 31], a nonlinear laser-plasma interaction that
transfers power to the longer-wavelength laser in the plasma flow frame.
$\Delta\lambda=0$ for all BigFoot and WarmMag shots. BigFoot has also given
the highest fusion yield (29.9 kJ) of any NIF platform with $\leq$ 1.3 MJ of
laser energy, on NIF shot N190721-2 [32]. The record yield NIF shots use the
Hybrid E platform [33] and have given much more $\sim 60\times$ yield than
BigFoot, but this platform uses at least 1.8 MJ of laser which our SBS rule
precludes. Besides that, new ideas are more affordably tested with less laser
energy and power, since optics damage increases rapidly with both. N190721-2
is an appealing BigFoot layered target for a baseline magnetized design. This
was used in the alpha-heating campaign [32] and gave an inferred yield
amplification due to alpha heating of $\sim 2.5\times$. Adding a initial B
field $B_{z0}\geq 20$ T (quoted at the capsule center) should measurably
enhance the yield due to both reduced electron thermal loss and enhanced alpha
heating.

N190721-2 unfortunately used 1.3 MJ of laser energy and is beyond our SBS risk
limit. We base our initial design on an earlier BigFoot series with slightly
less laser energy, specifically gas-filled capsule shot N161204-3. Table 1 and
Figs. 1-3 present details on the shots discussed here. The first two WarmMag
shots, N201228-1 ($B_{z0}=0$) and N210301-1 ($B_{z0}=26$ T), stayed as close
to N161204-3 as our constraints allowed (except for the reduced capsule fill
density). The differences of this initial low-T platform vs. N161204-3 are:

  * •

Reduced late-time laser power to meet SBS risk limit, see Fig. 1.

  * •

Changed capsule gas fill from 6.74 mg/cm3 of D3-3He7 to 5.10 mg/cm3 of D3-4He7
to give more convergence and B-field compression.

  * •

Changed hohlraum gas fill from 0.3 mg/cm3 of He to 0.258 mg/cm3 of C5H12 to
prevent LEH window bursting.

  * •

Changed hohlraum wall from 30-50 $\mu$m thick Au to 15 $\mu$m thick AuTa4 to
reduce electrical conductivity and field soak-thru issues.

The goals of the WarmMag NIF shots were to demonstrate: 1. that hohlraum x-ray
conversion was similar with or without a B field, and with an AuTa4 instead of
pure Au hohlraum; 2. a repeatable comparison of nuclear performance in shots
with and without B, with a roughly round hot spot. Our starting point was to
hew as closely as possible to BigFoot shot N161204-3 given our constraints,
and tuning implosion shape came later. The first shot N201228-1 had no imposed
field, and no capsule fill due to a fielding issue. It was essentially a
hohlraum coupling test. The AuTa4 hohlraum and modified laser pulse gave x-ray
drive as measured by DANTE comparable to N161204-3 until the time of the
caboose [16]. The backscatter was very low. The next shot N210301-1 had
imposed field $B_{z0}=26$ T and capsule fill. It gave similar x-ray drive and
backscatter to the prior shot, and produced a very prolate implosion:
$P_{2}/P_{0}=63\%$.

Figure 2: Data for NIF shots in Table 1, same color scheme as Fog. 1. Cone
fraction is (inner-beam energy) / (total laser energy). Figure 3: Simulation
tuned parameters (left) and comparison to experimental results (right) for NIF
shots from Fig. 2.

###  II.3 “WarmMag” final low-T platform: round hotspot

Our next goal was to make the magnetized implosion shape round and finalize
the low-T platform. We did this on the third shot N210607-2 by reducing the
laser cone fraction, or ratio of inner-beam to total laser power. We only
reduced the inner beam power: the outer beams were already at their SBS limit,
so we could not raise them to maintain total power. We also changed the
capsule fill to 3.99 mg/cm3 of pure D, to increase yield via increased D
density and reduced hotspot radiative loss. This design was modeled in Lasnex
to give a round implosion, which was basically achieved: $P_{2}/P_{0}=5.7\%$.
Our next goal was an unmagnetized shot of this design. Several attempts had
fielding issues, with the best one being N210912-1 (which still dropped 2 of
the 48 NIF laser quads). We subsequently shot an intermediate $B_{z0}=12$ T on
N220912-1, which had almost the same yield and $T_{i}$ as $B_{z0}=26$ T. This
validates the expected performance scaling for gas-filled targets where
reduced electron thermal conduction is the main B-field effect [34].

NIF Shot | N161204-3 | N161205-3 | N210301-1 | N210912-1 | N220912-1 | N210607-2 | N230612-1 | N230212-2
---|---|---|---|---|---|---|---|---
Shot parameters |  |  |  |  |  |  |  |
Platform | BF | BF | WM first low-T | WM low-T | WM low-T | WM low-T | WM high-T | WM high-T
$B_{z0}$ [T] | 0 | 0 | 26.1 | 0 | 11.8 | 26.1 | 0 | 28.0
Laser energy [kJ] | 1092 | 1065 | 926 | 840 | 874 | 883 | 940 | 989
Cone fraction (inner/total) | 27.3 | 26.7 | 27.2 | 21.2 | 22.1 | 21.9 | 25.7 | 27.0
LEH diameter [mm] | 3.45 | 3.45 | 3.46 | 3.48 | 3.45 | 3.48 | 3.15 | 3.13
Capsule fill gas | D3-3He7 | D1-T1 | D3-4He7 | D | D | D | D | D
Capsule fill density [mg/cm3] | 6.74 | 4.01 | 5.10 | 3.99 | 3.99 | 3.99 | 3.00 | 3.00
Experimental data |  |  |  |  |  |  |  |
x-ray bangtime [ns] | 7.1 | 7.04 | 7.48 | 8.00 | 7.58 | 7.65 | 7.53 | 7.40
x-ray $P_{0}$ [$\mu$m] | 60 | 74.2 | 51 | 58.5 | 55.8 | 53.3 | 47 | 37.3
x-ray $P_{2}/P_{0}$ [%] | 18.7 | 8.1 | 63 | -16.6 | -17 | 5.7 | -24 | -5.8
DD neutron yield $Y$ | 8.69E11 | 2.60E12 | 5.00E11 | 6.72E12 | 1.71E13 | 1.95E13 | 1.03E13 | 2.26E13
DD $T_{i}$ [keV] | 3.07 | 3.38 | 4.2 | 2.72 | 3.6 | 3.8 | 3.44 | 4.4
Lasnex modeling |  |  |  |  |  |  |  |
Peak power multiplier | 0.85 | 0.86 | 0.9 | 0.85 | 0.92 | 0.916 | 0.9 | 0.873
Peak CBET clamp $\delta n_{e}/n_{e}$ | 1E-4 | 5E-5 | 2E-3 | 1E-4 | 2E-3 | 3.46E-3 | 5E-4 | 1.5E-3
Inner $E_{post}/E_{inc}$ \- 1 [%] | 3.7 | 4.0 | 15.7 | 6.9 | 18.4 | 27.9 | 8.2 | 13.7
Modeling vs. experiment |  |  |  |  |  |  |  |
bangtime: sim - expt [ps] | 20 | 20 | 10 | 30 | 20 | -20 | 30 | -10
$P_{2}/P_{0}$: sim - expt [%] | 2.3 | 1.5 | 0 | 2.9 | -0.2 | 0.3 | 1.3 | 1.8
$Y$: sim / expt | 0.826 | 0.935 | 1.95 | 2.72 | 2.22 | 2.50 | 2.13 | 2.29
DD $T_{i}$: sim / expt | 0.863 | 0.935 | 0.888 | 1.022 | 1.042 | 1.066 | 0.945 | 0.995
Effect of B field |  |  |  |  |  |  |  |
$Y[B]/Y[B=0]$: expt | n/a | n/a | n/a | 1: $[B=0]$ | 2.54 | 2.90 | 1: $[B=0]$ | 2.19
$Y[B]/Y[B=0]$: sim | n/a | n/a | n/a | 1 | 2.08 | 2.67 | 1 | 2.37
$T_{i}[B]/T_{i}[B=0]$: expt | n/a | n/a | n/a | 1 | 1.32 | 1.40 | 1 | 1.28
$T_{i}[B]/T_{i}[B=0]$: sim | n/a | n/a | n/a | 1 | 1.35 | 1.46 | 1 | 1.35
Table 1: Experimental and Lasnex modeling data for NIF gas-filled capsule
shots discussed in this paper. For each platform, shots are ordered by
$B_{z0}$ not shot date. BigFoot (BF) shots all had Au hohlraum walls and
hohlraum fills of 0.3 mg/cm3 of He, while all WarmMag (WM) shots had AuTa4
walls and fills of 0.258 mg/cm3 of C5H12. “Cone fraction” is the inner-beam
energy divided by total laser energy. “Inner $E_{post}/E_{inc}$” is the post-
CBET inner-beam energy divided by the incident and measures CBET from outer to
inner beams. Typical measurement error bars are $\pm 50$ ps for bangtime, $\pm
3\ \mu$m for $P_{0}$ and $P_{2}$, $\pm 5\%$ for DD yield, and $\pm 0.15$ keV
for $T_{i}$.

###  II.4 “WarmMag” high-T platform: increased hotspot temperature

In 2023 our goal was to increase the capsule hotspot temperature while still
respecting our constraints. This WarmMag “high-T” platform entailed four
changes vs. the low-T one:

  * •

Extended caboose to increase laser energy and x-ray drive, see Fig. 1.

  * •

Reduced LEH diameter from 3.45 to 3.13 mm, to reduce hohlraum x-ray loss.

  * •

Reduced capsule fill density from 3.99 to 3.00 mg/cm3 to increase capsule
convergence for a given x-ray drive.

  * •

Increased laser cone fraction to maintain a round implosion shape.

These changes increased the hotspot temperature both with and without a B
field, as demonstrated in our most recent shots N230212-2 and N230612-1.

Figures 2-3 provide a summary of the WarmMag NIF shots to date, with $B_{z0}$
the abscissa in all plots. The delivered laser energy and cone fraction
slightly increase with $B_{z0}$, which accounts for some of the decrease in
bangtime with $B_{z0}$ (besides any field effects). It also somewhat
complicates drawing clear conclusions from the increase in $P_{2}$, yield and
$T_{i}$ with $B_{z0}$. A compelling demonstration of the benefits of
magnetization is the higher yield and $T_{i}$ in the low-T, $B_{z0}=12$ T shot
vs. the unmagnetized high-T shot: the field over-compensates for the higher
laser energy, x-ray drive and capsule convergence.

##  III Lasnex Rad-MHD Modeling of WarmMag NIF Experiments

We now describe our Lasnex rad-MHD modeling of the WarmMag NIF shots. Hohlraum
modeling at LLNL has traditionally not included MHD effects, though work is
underway to address this deficiency. Biermann-battery fields are known to be
present in the hohlraum and capsule, and experiments are ongoing to understand
their role. Earlier hohlraum MHD modeling with the Hydra rad-MHD code [35]
includes studies of self-generated fields [36] and imposed fields [37].
Compared to the earlier imposed-field work, the present work uses a more
complete MHD model with current best practices, and considers targets with low
hohlraum gas fill, HDC ablator and short laser pulses (the prior work
considered a “LowFoot” design with plastic ablator, high hohlraum gas fill,
and long laser pulses). Appendix B presents details on MHD modeling with
Lasnex and the LHT. Imposed field is done through an initial condition on the
vector potential $A_{\phi}[r,z]$, which produces $\mathbf{B}_{rz}$ but no
$B_{\phi}$. We use the analytic solution for a solenoid of finite axial length
and no radial thickness [38]. This idealization compares well to detailed
calculations of the real NIF coils.

Figure 4: Solid: incident laser power for BigFoot keyhole shot N161115-2
(black) and symcap shot N161204-3 (blue). The circles indicate times for which
ANTS finds power multipliers. Dashed: ANTS power multipliers.

Figure 5: Shock speed for BigFoot keyhole shot N161115-2, as measured along
the capsule waist (top) and pole (bottom). Blue is measured data. Orange is
LHT simulation with no power multipliers (i.e. a constant value of 1). Green
is LHT simulation with tuned power multipliers given in Fig. 4.

Our modeling goals are to design the WarmMag NIF shots, and to understand how
well we can model MHD effects. We therefore follow the standard practice
outlined in Appendix A of tuning laser power multipliers and the CBET
saturation clamp to match data, namely shock velocity, bangtime and hotspot
$P_{2}$ shape. This provides what we consider to be a reasonably realistic
x-ray drive for the capsule. Subsequent capsule performance, such as yield and
ion temperature, are untuned and thus an indication of simulation quality.

Shock velocity data is obtained from keyhole experiments, which can only be
done cryogenically. We therefore use the BigFoot keyhole shot N161115-2 to
develop laser power multipliers. This shot directly applies to the two BigFoot
gas-filled shots we model, but not to the WarmMag platform (with different
e.g. wall material and hohlraum fill). We use the Automated NIF Tuning Suite
(ANTS) [39] to develop our tuned parameters. Figure 4 plots the delivered
laser power and multipliers, and Fig. 5 shows the resulting shock velocities.
The power multipliers clearly improve the agreement between simulated shocks
and data, but it is not perfect. We deem this to be adequate for the present
study of gas-filled capsules, which should be less sensitive to shock dynamics
than DT ice-layered implosions. The power multipliers for N161115-2 are
applied to all other shots, at equivalent times found by ANTS. Figure 4
illustrates this for N161204-3. A single peak power multiplier is chosen for
each shot other than N161115-2 to match the implosion bangtime. For N161204-3
this starts around 4 ns. The CBET clamp $\delta n_{e}/n_{e}$ is 0.005 until
3.8 ns (near the end of the rise to peak power), which is a typical value used
for similar shots. After 3.8 ns a new clamp is chosen to match the hotspot
$P_{2}$. For all shots, the simulated bangtime is within 30 ps of data, and
$P_{2}/P_{0}$ is within 2.9%.

Run: MHD model | 1: No MHD | 2: self-gen, $f_{N}=1$ | 3: self-gen, $f_{N}=0.1$ | 4: 3 plus $B_{z0}=30$ T | 5: 3 capsule-only | 6: 4 capsule-only
---|---|---|---|---|---|---
Peak $T_{rad}$ [eV] | 287.91 | 288.81 | 288.97 | 288.20 | n/a | n/a
Peak M-band fraction [%] | 18.60 | 18.62 | 18.68 | 18.44 | n/a | n/a
x-ray bangtime [ns] | 7.25 | 7.20 | 7.19 | 7.21 | 7.16 | 7.16
x-ray $P_{0}$ [$\mu$m] | 58.15 | 58.68 | 58.39 | 56.35 | 58.54 | 57.88
x-ray $P_{2}/P_{0}$ [%] | 2.06 | 12.90 | 19.97 | 16.17 | -0.20 | 9.73
DD neutron yield | 9.94E11 | 7.34E11 | 7.20E11 | 1.12E12 | 7.42E11 | 1.05E12
DD $T_{i}$ [keV] | 2.96 | 2.63 | 2.64 | 3.53 | 2.67 | 3.40
Table 2: Lasnex modeling of BigFoot shot N161204-3 with different MHD models.
“Self-gen” means run only had self-generated $B_{\phi}$ but no imposed fields.
$f_{N}$ is the Nernst multiplier outside the capsule ($f_{N}=1$ inside the
capsule). $B_{z0}$ is the initial imposed field at capsule center. “Capsule-
only” runs 5 and 6 do not include the hohlraum and are driven by symmetrized
versions of run 3’s x-ray drive. Figure 6: BigFoot shot N161204-3 Lasnex
modeling from Table 2: hohlraum $|\mathbf{B}|$ (top) and $T_{e}$ (bottom) at
several times. Left half $r<0$ is for run 3 with self-generated field and
$f_{N}=0.1$, and right half $r>0$ is for run 4 with initial $B_{z0}=30$ T
field. $|\mathbf{B}|$ is scaled to 30 T. The imposed-field run is hotter in
the LEH at 2 ns and 4 ns, but at 6 ns it is hotter in the self-generated run.

We return to Fig. 3 to discuss the modeling results. The peak power
multipliers vary from 0.85 to 0.92, without a clear trend with $B_{z0}$: the
multiplier increases toward unity for the low-T shots but decreases for the
high-T ones. All WarmMag shots have multipliers equal to, or closer to unity
than, BigFoot. The peak CBET clamp increases substantially with $B_{z0}$, from
5E-5 – 5E-4 for unmagnetized shots to 1.5E-3 – 3.5E-3 for magnetized ones
(recall a larger clamp means less artifical modification to the physical CBET
model). This results in significantly more CBET to the inner beams with
increasing $B_{z0}$: see Table 2. Physical ion-wave nonlinearities can limit
CBET for amplitudes $\delta n_{e}/n_{e}\gtrsim 0.01$. All of this indicates
the WarmMag platform and B fields do not make hohlraum modeling less accurate
(and thus requiring more tuning to match data). With tuning parameters chosen,
the capsule yield and $T_{i}$ are untuned and reflect simulation quality. The
$T_{i}$ simulated / experiment ratio is closer to unity (at most 1.066) for
all but one WarmMag shot than the average of the two BigFoot shots (0.904).
The exception is N210301-1 (the blue point: 0.888) which was extremely prolate
and could have $T_{i}$ compromised by substantial plasma flow. The relative
yield increase due to magnetization (see Table 1) is $2.19-2.90\times$ in the
data, and is similar but slightly lower on average in the modeling
$(2.08-2.67\times)$. For $T_{i}$ the relative increase due to magnetization is
$1.28-1.4\times$ in the data and slightly higher $(1.35-1.46\times)$ in
modeling. The yield and $T_{i}$ increases are roughly in line with modeling,
especially given measurement error bars.

The major shortcoming of the modeling is that the absolute simulated yield is
$1.95-2.72\times$ the data in the WarmMag platform, while it agrees well for
BigFoot $(0.83-0.94\times$). The discrepancy is slightly less for the
magnetized shots, which indicates the field is at least not introducing new
mysteries. This suggests the field may reduce capsule mix, but this is
speculative. There are many differences between the BigFoot and WarmMag
platform which could account for the difference in yield modeling. We are
exploring these now and will report results in the future. In particular,
modeling of capsule mix due to hydro instabilities, the fill tube, etc.
remains to be done.

##  IV Rad-MHD Study of NIF BigFoot Experiment N161204-3

This section studies the role of MHD and imposed B fields in BigFoot shot
N161204-3. This is closer to a typical cryogenic NIF implosion than our
WarmMag shots: it uses a pure Au hohlraum, He hohlraum fill, and no caboose on
its laser pulse. We use our Lasnex modeling methodology and only vary the MHD
physics included. We consider four integrated hohlraum modeling runs: 1: a
base run with no MHD; 2: run 1 with self-generated azimuthal field and the
full Nernst effect (Nernst multiplier $f_{N}=1$, see Eq. 20); 3: run 2 but
with our standard $f_{N}=0.1$ (except $f_{N}=1$ inside the capsule); and 4:
run 3 but with a finite-solenoid imposed $\mathbf{B}_{rz}$ field and
$B_{z0}=30$ T. All runs use the same laser power multipliers and CBET clamp as
given in Table 1 for this shot, which corresponds to run 3.

Figure 6 shows $|\mathbf{B}|$ and $T_{e}$ in the hohlraum at several times.
Self-generated field is produced mostly by the Biermann battery effect
(misaligned $\nabla n_{e}$ and $\nabla T_{e}$). This leads to highly localized
field, such as at the LEH window boundary (2 ns) and where the laser is
absorbed near the hohlraum wall. The imposed $B_{z0}=30$ T field roughly
follows the frozen-in law: the field increases where plasma compresses normal
to $\mathbf{B}$ (like the hohlraum fill) and decreases where it expands (like
the ablated hohlraum wall and capsule). The imposed $\mathbf{B}_{rz}$ field
roughly “adds on” to the self-generated field. $T_{e}$ is hotter for the
imposed-field run in the LEH at 2 ns, as expected due to reduced electron
thermal conduction. This difference is reduced at 4 ns, and the “gold bubble”
where the outer beams hit the wall is hotter in the self-generated run. At 6
ns the hohlraum fill is generally hotter in the self-generated run. This goes
against the expectation that the main effect of the imposed field is to reduce
thermal conduction. It also seems to contradict $T_{e}$ measurements in
hohlraums at the OMEGA Laser Facility [40]. Those hohlraums were much smaller
in scale than our NIF experiments: they were driven by $\sim 1/40$ the laser
energy, had equivalently shorter spatial dimensions and laser pulses, and had
no capsule. In absolute terms, those experiments resemble the LEH early in
time of the NIF shots, which are hotter with the imposed field. Analysis of
the later times when the imposed-field run is cooler is ongoing. One
possibility is the higher fill pressure, from both higher $T_{e}$ and magnetic
pressure, reduces compression and $pdV$ heating.

The hohlraum x-ray drive as seen by the capsule is presented in Fig. 7. There
is very little difference in total x-ray flux or the M-band fraction (photon
energy $>1.8$ keV) between the four runs. MHD and imposed field thus have
little effect on overall hohlraum energetics. Fig. 8 shows the $P_{2}$
Legendre moment of the x-ray flux. We follow the standard sign convention with
$P_{2}>0$ for a pole-hot x-ray drive, which produces an oblate ($P_{2}<0$)
hotspot shape. The four runs are similar but not identical: the no-MHD run has
the most pole-hot drive, the self-generated with $f_{N}=0.1$ the least, and
the other two runs are in between. The peak-power CBET is very small in all
four runs since we used the very low clamp of 1E-4 from our tune for this
shot. A better CBET model without the need for a clamp may reveal differences
between the runs.

We now discuss the capsule physics of the four hohlraum runs, as seen in Table
2 and Fig. 9. The most salient effect of self-generated fields, as seen in
comparing runs 2 and 3 vs. 1, is the less pole-hot x-ray flux $P_{2}$. This is
consistent with the hotspot $P_{2}$ for runs with self-generated fields being
prolate vs. the round no-MHD run. The very low capsule Hall parameter < 0.01
in Fig. 9 suggests that self-generated capsule fields have very little effect.
This is as expected: high-mode asymmetries are required to generate
significant magnetic flux in capsules [41], and are not included here. The
bangtimes are all close (within 60 ps), consistent with the close x-ray
fluxes. The lower yield and $T_{i}$ vs. no-MHD is likely due to the prolate
shape. The only difference between the two self-generated runs with different
Nernst multiplier is in hotspot $P_{2}$, again consistent with the x-ray flux
$P_{2}$.

The main effect of the imposed $B_{z0}=30$ T field in run 4 is to increase
yield by 56% and $T_{i}$ by 34% vs. run 3. The x-ray flux $P_{2}$ is slightly
more pole hot, which is consistent with the slightly lower hotspot $P_{2}$.
This is in contrast to a naïve expectation that an axial field leads to a more
prolate hotspot, since electron thermal conduction is unmagnetized along $z$.
To separate the effects of the imposed field on hohlraum x-ray drive and
capsule physics, we performed runs 5 and 6. These are capsule-only runs both
driven by the x-ray flux from hohlraum run 3. This is done via a frequency-
dependent source (FDS) imposed outside the capsule vs. time and photon energy,
but symmetric in space. Any asymmetry is the result of capsule physics. Run 5
with no imposed field has $P_{2}/P_{0}=-0.2\%$, which is very small and
indicates residual numerical asymmetry in either the simulation or post-
processing. The imposed-field run is prolate ($P_{2}/P_{0}=9.7\%$), as
expected. In the imposed-field hohlraum run, the more pole-hot x-ray flux
over-compensates for this effect.

Figure 7: BigFoot shot N161204-3 Lasnex modeling: x-ray drive along a contour
enclosing the capsule. $T_{rad}$ is the spectrally-integrated radiation
temperature, and M-band fraction is the spectral fraction with $h\nu\geq 1.8$
keV. Blue: run 1 with no MHD. Orange: run 2 with self-generated B fields and
no Nernst multiplier ($f_{N}=1$). Green: run 3: run 2 but standard LHT
$f_{N}=0.1$ outside capsule. Red: run 4: run 3 plus initial solenoidal
$B_{r}/B_{z}$ field with $B_{z0}=30$ T at capsule center. There is very little
difference between the four runs. Figure 8: BigFoot shot N161204-3 Lasnex
modeling: $P_{2}$ Legendre moment of the hohlraum x-ray flux at a sphere of
radius 100 $\mu$m larger than the time-dependent capsule ablation front. Same
simulations and coloring as Fig. 7.

Figure 9: BigFoot shot N161204-3 Lasnex modeling: simulated conditions at 7.25
ns (slightly after bangtime) for run 3 (left: no imposed field) and 4 (right:
imposed field). Magenta contour is the capsule gas - ablator interface. Top:
$|\mathbf{B}|$ including all three field components; left panel:
$|B_{\phi}|\times 100$ $(B_{r}=B_{z}=0)$, right panel arrows indicate
direction of $(B_{r},B_{z})$. Middle: electron temperature $T_{e}$. Bottom:
electron Hall parameter $\omega_{ce}\tau_{ei}$, left panel $\times 1000$.

##  V Conclusions and Future Work

This paper reviewed the magnetized NIF hohlraum experiments to date with room-
temperature, gas-filled capsules, which we call the WarmMag platform. The
emphasis was on the design considerations and modeling with the Lasnex rad-MHD
code and the LHT common model. The WarmMag platform is based on BigFoot gas-
filled capsules driven by $1.05-1.1$ MJ of laser energy. Changes were made to
satisfy the constraints of allowing the field to soak thru the hohlraum wall
without excessive $J\times B$ motion (novel AuTa4 hohlraum instead of standard
Au or DU), room-temperature fielding (hohlraum fill gas of C5H12 instead of
He), and a stringent laser energy limit $\lesssim 1$ MJ to avoid SBS optics
damage risk. The two most recent shots in the “high-T” campaign on the WarmMag
platform, N230212-2 and N230612-1, were presented for the first time. These
increased the capsule compression and hotspot temperature with or without an
imposed field, and showed an imposed field gives a hotter hotspot. A
compelling demonstration of the benefits of magnetization is that the lowest
imposed field of 11.8 T in the “low-T” shot N220912-1 gave a higher neutron
yield and ion temperature $T_{i}$ than the best-performing unmagnetized shot
N230612-1 in the high-T campaign.

Lasnex modeling captures most of the NIF observations, with the notable
exception of the absolute neutron yield being $2.0-2.7\times$ higher than the
data. Hohlraum modeling required similar laser power multipliers to match the
implosion bangtime for WarmMag shots with and without imposed field. The
needed clamp $\delta n_{e}/n_{e}$ on cross-beam energy transfer (CBET) to
match hotspot $P_{2}$ increased greatly with $B_{z0}$, from 5E-5 – 5E-4 for
unmagnetized shots to 1.5E-3 – 3.5E-3 for magnetized ones. This is approaching
the level of 0.01 when physical ion-wave nonlinearities may play a role. With
hohlraum tuning parameters (power multiplier and CBET clamp) chosen, the
capsule outputs of yield and $T_{i}$ are untuned and indicate simulation
quality. The modeling is close to the measured $T_{i}$ and relative yield and
$T_{i}$ increase due to magnetization, but the modeled yields are too high. We
will study yield degradation due to mix, hydro instabilities and the fill tube
in the future, all of which our current modeling neglects.

A modeling study of effects from self-generated (mostly Biermann battery) and
imposed fields ($B_{z0}=30$ T) on BigFoot shot N161204-3 showed both cause
little change in the total x-ray drive and its M-band fraction. There is some
effect on the x-ray drive’s $P_{2}$ moment, with the imposed field making it
slightly more pole-hot. This leads to a slightly less prolate hotspot vs. the
self-generated run, though capsule-only modeling with the same, symmetric
x-ray drive shows the imposed field makes the hotspot more prolate. The
$P_{2}$ variations due to imposed field in both the x-ray drive and hotspot
shape are minor, and should be tunable through standard techniques like laser
cone fraction and wavelength shift (to control CBET). The optimal shape of a
magnetized implosion is an open question, but it is likely not too far from
round. Non-spherical implosion comes at a substantial cost in reduced
compression: the limit of a cylindrical implosion gives a density increase
$\sim CR^{2}$ not $\sim CR^{3}$ for a spherical one (CR is convergence ratio).

To close we discuss ongoing work at LLNL towards magnetized indirect-drive
ignition on NIF. Current experiments are limited by being room-temperature and
having $\lesssim 1$ MJ of laser energy, so magnetized ignition work is mostly
theory and modeling. This is broadly geared toward finding the best uses of
imposed field for ignition designs. The main thrust is to increase the maximum
achievable yield and gain on NIF. We start by imposing fields with $B_{z0}\leq
100$ T on the best performing NIF implosions, in the Hybrid E campaign. Early
modeling results show up to tripling the yield with $B_{z0}\sim 60-70$ T. We
are also investigating magnetized target designs for use on NIF “Enhanced
Yield Capability” (EYC) which is a proposal for upgrading NIF to $2.6-3$ MJ of
laser energy. As mentioned above, a cryogenic pulsed-power system which
mitigates the SBS optics risk due to prefire has been designed, but its
construction timeline is uncertain. Ongoing research on advanced hohlraum
coils and pulser architecture are showing promise in achieving higher magnetic
fields and easier target fabrication.

Future work will consider variants on these designs optimized to take
advantage of the field, and other designs entirely. For instance, we speculate
that a field may give the most benefit for designs that do not have a hot
enough hotspot to self-heat and ignite without a field. We have also studied
non-axial fields, such as magnetic mirrors or even closed field lines, which
can give better performance. A major challenge is how to impose such field of
reasonable magnitude without seriously disrupting the implosion (say by
running a wire through the capsule). Besides increasing the maximum yield,
another goal is to achieve the same yield but with relaxed requirements on
laser energy or target quality. This is appealing for “users” of ignition, who
desire a $1-10$ MJ yield to conduct other high-energy-density (HED)
experiments.

An important topic we are starting to address is the role of magnetic field in
capsule degradation (“mix”) due to hydro instabilities or engineering features
like the fill tube or capsule support tent. Some experimental [42] and
simulation [43] work exists, but this topic is little studied in ignition-
relevant implosions. We are conducting more gas-filled capsule implosions on
the NIF WarmMag platform to study how imposed field alters mix. These use the
dual crystal backlighter imager (CBI) platform [44] to image high-Z tungsten
x-ray emission from the implosion hotspot. Experiments at the OMEGA laser are
underway to study magnetized mix, as well as magnetized CBET. Recent 3D MHD
modeling of the Rayleigh-Taylor instability [45] in idealized settings shows
reduced growth for k-vectors $\mathbf{k}||\mathbf{B}$ but not for
$\mathbf{k}\perp\mathbf{B}$. In this case, growth can be enhanced by reduced
thermal conductivity, which reduces ablative stabilization [46]. We plan to
extend this work to integrated implosion modeling. A very desirable goal is to
develop reduced mix models like those available in Lasnex (discussed in
Appendix A and Refs. 47, 48, 49) to include magnetization.

We look forward to this and related work culminating in magnetized ignition
experiments on the NIF.

###### Acknowledgements.

It is a pleasure to thank C. A. Thomas and K. L. Baker for fruitful
discussions about the BigFoot platform. We are grateful to the NIF and OMEGA
operations teams for fielding our nonstandard experiments, and to the target
fabrication team for providing AuTa4 hohlraums. This work was performed under
the auspices of the U.S. Department of Energy by Lawrence Livermore National
Laboratory under Contract No. DE-AC52-07NA27344. Partially supported by LLNL
LDRD projects 20-SI-002 and 23-ERD-025.

## Data Availability

The data that support the findings of this study are available from the
corresponding author upon reasonable request.

##  Appendix A Lasnex Hohlraum Template (LHT) Rad-MHD Simulation Model

The rad-MHD modeling presented here uses the Lasnex simulation code [50, 51],
which is a 2D axisymmetric ($rz$), arbitrary-Lagrangian-Eulerian (ALE) code.
We use the Lasnex Hohlraum Template (LHT), a “best-effort,” version-controlled
common model used by the LLNL ICF program. An earlier version of this model is
described in Ref. 52. We describe the current recommended practice for
“design-quality” modeling of NIF hohlraums. A typical LHT 2D hohlraum run
takes 12 to 48 wall-clock hours on 36 Intel CPUs. The LHT also supports 1D and
2D capsule-only runs, driven either directly by a laser, an x-ray frequency-
dependent source with no angular spatial variation, or an x-ray “tallied flux”
source with such variation. The external laser entrance hole (LEH) hardware is
included, namely the retainer ring and washer that secure the main window and
thin secondary “storm window” which prevents NIF chamber gas from condensing.
The LEH hardware can affect LEH plasma conditions and thus cross-beam energy
transfer (CBET), which transfers energy among the NIF lasers [53]. The storm
window itself is not currently included, due to mesh management challenges.

Laser model: The LHT treats the lasers with 3D ray tracing with an effectively
infinite speed of light: at each time step, rays propagate through the target
until the remaining power is $<$ 1E-4 of incident. This includes refraction
and inverse bremsstrahlung absorption as reduced by the Langdon effect [54].
The incident ray positions, k-vectors and relative powers are chosen to
accurately represent the near-field square lens aperture, far-field intensity
profile due to phase plates, and pointing of each of the 192 NIF beams (or 96
in the more common one-sided runs shown here). The NIF beams are grouped into
“quads” of 4 beams. The LHT follows this in that rays for the 4 beams in a
quad are treated as a single laser with one power history, and function as a
unit for CBET. The inline CBET model is used [55], which currently treats the
lasers as unpolarized quads and finds their 3D intensity on an auxiliary 3D
mesh (the plasma conditions are still axisymmetric). The quad treatment
involves $\sim 16\times$ fewer couplings than the beam treatment and is thus
much faster. The inline CBET model currently does not handle polarization,
which an accurate beam model needs. CBET between two lasers transfers power to
the one with the longer wavelength in the plasma-flow frame, which depends on
both flow-induced Doppler shifts and differences in the incident (lab-frame)
wavelengths. All four NIF cones (located at polar angles of 23, 30, 44 and 50
degrees) can have separate wavelengths. All shots discussed in this paper have
one wavelength for all cones. Inline stimulated Brillouin (SBS) and Raman
(SRS) backscatter models are available [55], but not used in this paper:
backscatter for the relevant shots is generally low.

Atomic and plasma models: Material zones are treated with local thermodynamic
equilibrium (LTE) models for properties like x-ray opacity and equation of
state, until they exceed a critical electron temperature $T_{cr}$ of 0.3 keV.
We raise this to 1 keV “inside the capsule”, meaning all capsule regions other
than the outermost one: the latter ablates into the hohlraum, is heated by the
laser, and becomes part of the hohlraum fill. The higher $T_{cr}$ reduces
computer cost and agrees with current LLNL practice for high-resolution
capsule instability modeling. Above $T_{cr}$, all properties come from LLNL’s
detailed configuration accounting (DCA) non-LTE atomic models [52, 56]. DCA is
not a single model but a framework which accommodates models of widely varying
complexity and cost. The results shown here use the “November 2020” DCA models
with detail considered acceptable for evolving atomic populations “inline” in
current ICF rad-hydro simulations. We also use a tabulated, steady-state non-
LTE approach based on the Linear Response Method in the high-Z hohlraum wall
[57]. These tables are generated with substantially more detailed models than
the inline ones, and greatly reduce the non-LTE computational cost.

The multi-ion species hydrodynamics package is used [53], run here in an
intermediate mode where each ion species has its own density and flow velocity
but a single ion temperature (to reduce computer time and numerical issues).
The package supports per ion species temperatures and heat fluxes. The plasma
anisotropic stress tensor is also used. Electrons are not treated as an
independent species, instead the plasma is assumed to be quasi-neutral:
$n_{e}=\sum_{i}Z_{i}n_{i}$. Separate electron and ion temperatures are used,
and the net current in the MHD package is equivalent to a separate electron
flow velocity. Ion-ion transport and coupling coefficients come from Ref. 58,
while electron-ion (e-i) coefficients come from either the analytic dense-
plasma model of Lee and More [59], specifically the GMS-6 model from Table I
of Ref. 60, or a more advanced LTE table if available.

Mix models: Reduced models of hydrodynamic mix due to Richtmyer-Meshkov (RM)
and Rayleigh-Taylor (RT) instabilities are available. These are designed to
capture mix physics without spatially resolving it, and involve some
parameters that must be tuned. The methodology is detailed in Ref. 49, which
demonstrates good agreement between Lasnex mix modeling and experimental data
for layered implosions with plastic capsules and separated nuclear reactants.
RT growth is calculated via a buoyancy-drag model [48], which creates zones of
mixed material at user-specified interfaces. These mixed zones entail 3
subzones: one for each pure material, and a third that is atomically mixed. We
use an RM model that limits the growth to keep the perturbation “finger tips”
between the shocked material interface and the transmitted shock [47]. Mix
models are not used in the present work, but will be to understand the over-
predicted yields on WarmMag gas-filled capsules, and for modeling magnetized
ignition designs.

Drive deficit: laser power multipliers: LHT modeling differs in important ways
from NIF ICF data. We discuss three discrepancies and physics modifications to
reduce them. The x-ray drive deficit refers to the fact that the total
hohlraum x-ray drive is higher in modeling than data. This is seen in x-ray
measurements with the DANTE detector, shock velocity measurements in “keyhole”
shots with the VISAR detector, and in the time of peak capsule emission or
“bangtime” [52]. In this paper, we modify the laser power by a time-dependent
factor (the same on all beams) to bring modeled shock velocity and bangtimes
in line with measurements. We also use an electron heat flux limiter $f$,
which limits the magnitude of the electron heat flux $q$ to some fraction of
the free-streaming value: $q=\min(fn_{e}T_{e}v_{Te},\kappa\nabla T_{e})$. This
is an old technique in HED modeling [61], intended to mimic physical flux
inhibition by things like nonlocal transport, microturbulence, or magnetic
fields. We follow the “high flux model” [62] and use $f=0.15$ outside the
capsule (as defined above), which generally has little effect on ICF
hohlraums: similar results obtain with a larger value or the nonlocal Schurtz
model [63]. Inside the capsule we use $f=0.1$, which has little effect and is
done to agree with high-resolution capsule modeling practice. Jones et al.
[52] revisited a low $f\approx 0.03$ to explain the drive deficit by allowing
more unabsorbed inner-beam laser light to escape the opposite LEH as “glinted
light” [64]. Dedicated experiments to examine this show much less glint than
$f=0.03$ modeling and generally support a high $f\approx 0.15$ [65, 66].

Implosion hotspot shape: CBET clamp: A second discrepancy with NIF data is
with the compressed hotspot shape, as measured by x-ray or neutron images near
peak emission. Specifically the second Legendre moment or $P_{2}$ is a key
quantity. A major factor in $P_{2}$ is CBET between the inner and outer laser
cones. This relies on accurate plasma conditions especially in the LEH. This
region has many overlapping laser beams and weakly collisional plasma.
Accurate modeling likely needs better accounting for the lasers and kinetic or
other effects beyond standard rad-hydro. LHT modeling generally produces too
much CBET to be consistent with implosion shape data - assuming the non-CBET
parts of the model that affect shape are correct. LLNL has for over 10 years
used a time-dependent saturation clamp $\delta n_{e}/n_{e}$ on the amplitude
of the CBET-driven ion acoustic waves. Physical ion-wave nonlinearities can
set in for $\delta n_{e}/n_{e}\sim 0.01$. Values between 0.001 and 0.01 are
typically needed to match shape data in hohlraums with low gas fill density
$\leq$ 0.6 mg/cm3. We stress that the $\delta n_{e}$ clamp at least partly
stands in for incorrect plasma conditions instead of ion-wave nonlinearity.

Outer-beam bubble conditions: Nernst multiplier: The final discrepancy we
discuss is with NIF measurements of flows and $T_{e}$ in the “gold bubble”
plasma that expands from the hohlraum wall heated by the outer beams [67]. LHT
modeling with a low flux limit $f=0.03$ is consistent with the data, while
$f=0.15$ is not. This is at odds with the inner-beam glint results. It
suggests a model with $f=0.15$ where the inners hit the wall but $f=0.03$
where the outers hit. Recently, it was found that the MHD model with self-
generated fields does this, if a Nernst multiplier $f_{N}=0.1$ is used to
greatly reduce Nernst advection [68]. The Nernst effect advects $B$ field from
hot to cold electrons. In the hohlraum wall, this advects self-generated
“Biermann battery” fields farther out in radius, to colder and denser plasma
where the field is too small to magnetize the electron heat flow. The LHT
adopts $f_{N}=0.1$ outside the capsule but $f_{N}=1$ (no modification) inside
the capsule.

##  Appendix B Lasnex MHD Model

We describe the MHD package in detail since it is a key part of this work.
Lasnex has had a “toroidal”, or azimuthal, field package for four decades
[69]. This includes a relatively complete implementation of Braginskii’s
classical fluid model [70]. The “poloidal”, or $rz$, MHD model is a more
recent addition and does not yet implement all of Braginskii [71, 72]. Lasnex
is an $rz$ code in that all quantities only vary in $r$ and $z$, but azimuthal
$\phi$ vector components are of course allowed. We write a vector
$\mathbf{v}=\mathbf{v}_{rz}+v_{\phi}\hat{\bm{\upphi}}$ with
$\mathbf{v}_{rz}=v_{r}\hat{\mathbf{r}}+v_{z}\hat{\mathbf{z}}$ the 2D vector in
the $rz$ plane. The relevant fields are the magnetic field $\mathbf{B}$,
vector potential $\mathbf{A}$, electric field $\mathbf{E}$, net current
$\mathbf{J}$ and center-of-mass or bulk flow velocity $\mathbf{u}$.

We start with Maxwell’s equations. $\mathbf{B}$ is written

$\displaystyle\mathbf{B}$
$\displaystyle=\mathbf{B}_{rz}+B_{\phi}\hat{\bm{\upphi}},$ (1)
$\displaystyle\mathbf{B}_{rz}$
$\displaystyle=\nabla\times\mathbf{A}=\left(-\partial_{z}A_{\phi},r^{-1}\partial_{r}(rA_{\phi})\right).$
(2)

$\mathbf{A}=A_{\phi}\hat{\bm{\upphi}}$ is only the vector potential for
$\mathbf{B}_{rz}$, the full $\mathbf{A}$ would involve $A_{r}$ and $A_{z}$ for
$B_{\phi}$. $\mathbf{B}$ evolves in time by Faraday’s law:

$\partial_{t}\mathbf{B}=-\nabla\times\mathbf{E}.$ (3)

The $\phi$ component gives

$\partial_{t}B_{\phi}=-(\nabla\times\mathbf{E}_{rz})\cdot\hat{\bm{\upphi}}=\partial_{r}E_{z}-\partial_{z}E_{r}.$
(4)

The $r$ component gives $\partial_{z}(\partial_{t}A_{\phi}+E_{\phi})=0$, which
we integrate over $z$ and set the boundary term to zero to find

$\partial_{t}A_{\phi}=-E_{\phi}.$ (5)

The $z$ component gives the same result. Faraday’s law implies
$\partial_{t}(\nabla\cdot\mathbf{B})=0$. We assume $\nabla\cdot\mathbf{B}=0$
initially so this remains true for all time, which gives

$\partial_{r}B_{r}+\partial_{z}B_{z}=-B_{r}/r.$ (6)

The current is given by Ampère’s law with no displacement current:
$\mu_{0}\mathbf{J}=\nabla\times\mathbf{B}$ which in components gives

$\displaystyle\mu_{0}\mathbf{J}_{rz}$
$\displaystyle=\left(-\partial_{z}B_{\phi},r^{-1}\partial_{r}(rB_{\phi})\right),$
(7) $\displaystyle\mu_{0}J_{\phi}$
$\displaystyle=\left(r^{-2}-r^{-1}\partial_{r}-\partial_{rr}-\partial_{zz}\right)A_{\phi}.$
(8)

$\mathbf{E}$ is given by Ohm’s law below. It is convenient to label fields in
the poloidal package as “P” fields: $\mathbf{B}_{rz}$, $A_{\phi}$, $E_{\phi}$
and $J_{\phi}$. Similarly the toroidal or “T” fields are $B_{\phi}$,
$\mathbf{E}_{rz}$ and $\mathbf{J}_{rz}$. We call a field “P+T” if it is
nonzero only if both P and T fields are nonzero. $\mathbf{u}$ is discussed
below.

The continuity and bulk momentum equations are

$\displaystyle d_{t}\rho+\rho
u_{r}/r+\rho(\partial_{r}u_{r}+\partial_{z}u_{z})$ $\displaystyle=0,$ (9)
$\displaystyle\rho
d_{t}\mathbf{u}_{rz}-\frac{\rho}{r}u_{\phi}^{2}\hat{\mathbf{r}}+\nabla
p+\nabla\cdot\overline{\overline{\tau}}-\left(\mathbf{J}\times\mathbf{B}\right)_{rz}$
$\displaystyle=\mathbf{F}_{rz},$ (10) $\displaystyle\rho
d_{t}u_{\phi}+\frac{\rho}{r}u_{r}u_{\phi}-\left(\mathbf{J}_{rz}\times\mathbf{B}_{rz}\right)_{\phi}$
$\displaystyle=F_{\phi}.$ (11)

The gradient symbol $\nabla$ contains no $\phi$ derivative. These fields are
summed over electron and ion species: mass density $\rho$, isotropic pressure
$p$, anisotropic stress tensor $\overline{\overline{\tau}}$, and other forces
$\mathbf{F}$ not of current interest, from e.g. laser, radiation, and charged-
particle deposition ($F_{\phi}$ is currently zero). Terms with $u_{\phi}$ and
no derivatives are centripetal forces and are only nonzero if $u_{\phi}$ is.
$\mathbf{u}_{rz}$ is always present and $(\mathbf{J}\times\mathbf{B})_{rz}$ is
nonzero if either P or T fields are present:

$\left(\mathbf{J}\times\mathbf{B}\right)_{rz}=\mathbf{J}_{rz}\times(B_{\phi}\hat{\bm{\upphi}})+J_{\phi}\hat{\bm{\upphi}}\times\mathbf{B}_{rz}.$
(12)

The only force that produces $u_{\phi}$ from an initial state with
$u_{\phi}=0$ is

$\left(\mathbf{J}_{rz}\times\mathbf{B}_{rz}\right)_{\phi}=\mu_{0}^{-1}(B_{z}\partial_{z}B_{\phi}+r^{-1}\partial_{r}(rB_{\phi})B_{r}).$
(13)

This force is a “P+T” field and is nonzero only if both P and T magnetic
fields are nonzero.
$d_{t}\equiv\partial_{t}+u_{r}\partial_{r}+u_{z}\partial_{z}$ is the
Lagrangian derivative in the $rz$ plane, but does not act on basis vectors:
$d_{t}\mathbf{u}_{rz}=(d_{t}u_{r})\hat{\mathbf{r}}+(d_{t}u_{z})\hat{\mathbf{z}}$.
If one starts with $\mathbf{u}\cdot\nabla\mathbf{u}$ in Cartesian coordinates
and transforms to cylindricals, one must account for spatial derivatives of
the basis vectors $\hat{\mathbf{r}}$ and $\hat{\mathbf{z}}$. This gives rise
to the centripetal terms. There is no electric force since Lasnex assumes
quasi-neutrality and thus the charge density $\sigma$ is zero.

We now discuss Ohm’s law for $\mathbf{E}$, which is used to evolve
$\mathbf{B}$. Gauss’s law $\nabla\cdot\mathbf{E}=\sigma/\epsilon_{0}$ is not
used. $\mathbf{E}$ is given by combining ion and electron momentum equations
to find $\partial_{t}\mathbf{J}$ and setting it to zero. This is equivalent to
setting $\partial_{t}\mathbf{u}_{e}=0$ in just the electron momentum equation.
We also neglect advective terms $\mathbf{u}\cdot\nabla\mathbf{u}$ as higher
order. Using $m_{e}\ll m_{i}$, we find

$\displaystyle\mathbf{E}$
$\displaystyle=-\mathbf{u}\times\mathbf{B}+\mathbf{E}_{\text{nouB}},$ (14)
$\displaystyle\mathbf{E}_{\text{nouB}}$
$\displaystyle=(en_{e})^{-1}\left(\mathbf{J}\times\mathbf{B}-\nabla
p_{e}+\mathbf{F}_{ep}+\mathbf{R}\right).$ (15)

We have separated the ideal-MHD term $\mathbf{u}\times\mathbf{B}$ and dropped
the anisotropic electron stress tensor. In $\mathbf{E}_{\text{nouB}}$,
$\mathbf{J}\times\mathbf{B}$ and $-\nabla p_{e}$ are the Hall and Biermann
battery terms. $\mathbf{F}_{ep}$ is the laser ponderomotive force on
electrons, which is the only force from the full $\mathbf{F}_{e}$ currently in
Ohm’s law. These 3 terms are fully implemented and not discussed further.
$\mathbf{R}$ is the e-i friction force and described below. We start with Eq.
4 for $\partial_{t}B_{\phi}$ and rewrite the ideal
$\mathbf{u}\times\mathbf{B}$ term to find

$d_{t}B_{\phi}+\left(\partial_{r}u_{r}+\partial_{z}u_{z}\right)B_{\phi}-r\mathbf{B}_{rz}\cdot\nabla\frac{u_{\phi}}{r}=-(\nabla\times\mathbf{E}_{\text{nouB},rz})\cdot\hat{\bm{\upphi}}.$
(16)

The first two terms on the left side reflect conservation of toroidal flux, or
$B_{\phi}$ times the $rz$ area of a spatial zone. The final left-side term is
a source of $B_{\phi}$ due to twisting of the poloidal field by spatially
varying azimuthal flow, or differential rotation. By similar logic, for the
poloidal fields

$d_{t}\left(rA_{\phi}\right)=-\frac{r}{en_{e}}\left(\underbrace{\left(\mathbf{J}_{rz}\times\mathbf{B}_{rz}\right)\cdot\hat{\bm{\upphi}}}+R_{\phi}\right).$
(17)

The poloidal Hall term with the underbrace is not currently implemented.

We now discuss the friction force $\mathbf{R}$, which is partly implemented.
We give the rationale for dropping various terms, though there is no rigorous
justification. All terms are included for a purely T field. The full
Braginskii form is

$\displaystyle\mathbf{R}$
$\displaystyle=(en_{e})^{-1}\mathbf{R}_{\alpha}-\mathbf{R}_{\beta},$ (18)
$\displaystyle\mathbf{R}_{\alpha}$
$\displaystyle=(\alpha_{||}-\alpha_{\perp})(\hat{\mathbf{b}}\cdot\mathbf{J})\hat{\mathbf{b}}+\alpha_{\perp}\mathbf{J}+\alpha_{\wedge}\mathbf{J}\times\hat{\mathbf{b}},$
(19) $\displaystyle\mathbf{R}_{\beta}$
$\displaystyle=(\beta_{||}-\beta_{\perp})(\hat{\mathbf{b}}\cdot\nabla
T_{e})\hat{\mathbf{b}}+\beta_{\perp}\nabla
T_{e}+f_{N}\beta_{\wedge}\hat{\mathbf{b}}\times\nabla T_{e}.$ (20)

$\hat{\mathbf{b}}=\mathbf{B}/|\mathbf{B}|$, $\mathbf{R}_{\alpha}$ is the
resistive force with different units from $\mathbf{R}$, and
$\mathbf{R}_{\beta}$ is the thermal force (note its sign). $f_{N}$ is the
Nernst multiplier described above, and multiplies $\beta_{\wedge}$ in all
equations. Components of the collisional tensors $\alpha$, $\beta$ and
$\kappa$ (see Eq. (26) below) are found from Sadler, Walsh and Li [73], which
is a corrected version of Epperlein and Haines [21]. The components are
functions of the effective charge state
$Z_{\text{eff}}\equiv\sum_{i}Z_{i}^{2}n_{i}/\sum_{i}Z_{i}n_{i}$ and the
electron Hall parameter $\omega_{ce}\tau_{eI}$ with $\omega_{ce}\equiv
eB/m_{e}$ and $\tau_{eI}$ the unmagnetized e-i collision time averaged over
ion species (found as described above for e-i transport coefficients). Lasnex
currently does not include P+T terms in $\mathbf{R}_{\alpha}$ which leaves

$\displaystyle\mathbf{R}_{\alpha,rz}$
$\displaystyle=\alpha_{\perp}\mathbf{J}_{rz}+\alpha_{\wedge}\left(\mathbf{J}\times\hat{\mathbf{b}}\right)_{rz},$
(21) $\displaystyle R_{\alpha,\phi}$ $\displaystyle=\alpha_{\perp}J_{\phi}.$
(22)

In $\mathbf{R}_{\beta}$, Lasnex drops $\left(\hat{\mathbf{b}}\cdot\nabla
T_{e}\right)\hat{\mathbf{b}}$: it is zero for a purely T field. This leaves

$\displaystyle\mathbf{R}_{\beta,rz}$ $\displaystyle=\beta_{\perp}\nabla
T_{e}+f_{N}\beta_{\wedge}\left(\hat{\mathbf{b}}\times\nabla
T_{e}\right)_{rz},$ (23) $\displaystyle R_{\beta,\phi}$
$\displaystyle=f_{N}\beta_{\wedge}\left(\hat{\mathbf{b}}\times\nabla
T_{e}\right)_{\phi}.$ (24)

The electron energy density per volume $E_{e}$ obeys

$\displaystyle
d_{t}E_{e}+(E_{e}+p_{e})\nabla\cdot\mathbf{u}+\nabla\cdot\mathbf{q}_{e}+\overline{\overline{\tau}}_{e}:\nabla\mathbf{u}-\nabla\cdot\left(\mathbf{J}E_{e}/en_{e}\right)$
(25) $\displaystyle-
p_{e}\nabla\cdot(\mathbf{J}/en_{e})=(en_{e})^{-1}\mathbf{J}\cdot\mathbf{R}-\omega_{ei}(T_{e}-T_{i}).$

$\mathbf{R}$ includes the same terms as in Ohm’s law, and $\omega_{ei}$ is the
e-i temperature equilibration rate. The electron heat flux $\mathbf{q}_{e}$
has units (speed * energy / volume) and is

$\displaystyle-\mathbf{q}_{e}$
$\displaystyle=\frac{T_{e}}{en_{e}}\left(\underbrace{(\beta_{||}-\beta_{\perp})\mathbf{J}\cdot\hat{\mathbf{b}}\hat{\mathbf{b}}_{rz}}+\beta_{\perp}\mathbf{J}_{rz}-f_{N}\beta_{\wedge}(\mathbf{J}\times\hat{\mathbf{b}})_{rz}\right)$
(26)
$\displaystyle+(\kappa^{e}_{||}-\kappa^{e}_{\perp})(\hat{\mathbf{b}}_{rz}\cdot\nabla
T_{e})\hat{\mathbf{b}}_{rz}+\kappa^{e}_{\perp}\nabla
T_{e}+\kappa^{e}_{\wedge}b_{\phi}\hat{\bm{\upphi}}\times\nabla T_{e}.$

Lasnex does not currently include the term with the underbrace, which is P+T.
$\kappa^{e}$ is the electron thermal conductivity tensor, with units
1/(length*time). The ion energy equation is

$d_{t}E_{i}+(E_{i}+p_{i})\nabla\cdot\mathbf{u}+\nabla\cdot\mathbf{q}_{i}+\overline{\overline{\tau}}_{i}:\nabla\mathbf{u}=\omega_{ei}(T_{e}-T_{i}).$
(27)

The ion heat flux is analogous to $\mathbf{q}_{e}$ with $\beta=0$:

$-\mathbf{q}_{i}=(\kappa^{i}_{||}-\kappa^{i}_{\perp})(\hat{\mathbf{b}}_{rz}\cdot\nabla
T_{i})\hat{\mathbf{b}}_{rz}+\kappa^{i}_{\perp}\nabla
T_{i}+\underbrace{\kappa^{i}_{\wedge}b_{\phi}\hat{\bm{\upphi}}\times\nabla
T_{i}}.$ (28)

Lasnex does not currently include the term with the underbrace. Usually
$|\mathbf{q}_{i}|\ll|\mathbf{q}_{e}|$ but $\mathbf{q}_{i}$ can matter when
magnetization strongly reduces $\mathbf{q}_{e}$.

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