# Updated Magnetized Transport Coefficients: Impact on Laser-Plasmas with
Self-Generated or Applied Magnetic Fields

C. A. Walsh  Lawrence Livermore National Laboratory  J. D. Sadler  Los Alamos
National Laboratory  J. R. Davies  Laboratory for Laser Energetics, University
of Rochester<EMAIL_ADDRESS>
###### Abstract

Errors in the Epperlein & Haines [PoF (1986)] transport coefficients were
recently found at low electron magnetizations, with new magnetic transport
coefficients proposed simultaneously by two teams [Sadler, Walsh & Li, PRL
(2021) and Davies, Wen, Ji & Held, PoP (2021)]; these two separate sets of
updated coefficients are shown in this paper to be in agreement. The
importance of these new coefficients in laser-plasmas with either self-
generated or applied magnetic fields is demonstrated. When an external
magnetic field is applied, the cross-gradient-Nernst term twists the field
structure; this twisting is reduced by the new coefficients in the low
magnetization regime. For plasmas where only self-generated magnetic fields
are present, the new coefficients are found to result in the magnetic field
moving with the Righi-Leduc heat-flow, enhancing the impact of MHD.
Simulations of Biermann Battery magnetic fields around ICF hot-spot
perturbations are presented, with cross-gradient-Nernst transport increasing
spike penetration.

##  I Introduction

Magnetic fields exist in all plasmas: in laser-driven ablation fronts Gao et
al. ; Campbell ; Manuel et al. (b), ICF capsules Igumenshchev et al. ; Hill
and Kingham ; Walsh et al. (c), shock fronts Li et al. , hohlraums Farmer et
al. ; Sherlock and Bissell  and Z-pinches Gomez et al. . Magnetic fields can
also be purposefully applied to ICF implosions to improve fusion performance
Chang et al. ; Perkins et al. ; Walsh et al. (b); Slutz et al. . The transport
of these magnetic fields within the plasma is critically important to the
design and interpretation of experiments. Non-local VFP simulations capture
the transport processes innately by resolving the distribution of electron
energies Sherlock and Bissell ; Hill and Kingham ; Joglekar et al. . However,
these calculations are prohibitively expensive and unstable, so simplified
models with prescribed transport coefficients are used for spatial and
temporal scales of interest. These transport coefficients are calculated by
fitting to VFP data. The seminal work in this area is that of Braginskii, who
laid the groundwork for the anisotropic transport of thermal energy and
magnetic fields within the magneto-hydrodynamic (MHD) framework Braginskii .
Following on from Braginskii, Epperlein & Haines improved the transport
coefficients, allowing for closer agreement to VFP calculations Epperlein and
Haines . For 35 years the transport coefficients of Epperlein & Haines were
the agreed-upon standard to be implemented into extended-MHD codes Walsh et
al. (c); Farmer et al. ; Perkins et al. ; Watkins and Kingham ; Igumenshchev
et al. .

Figure 1: The cross-gradient-Nernst coefficient as a function of Hall
Parameter calculated using different sources. All coefficients are attempting
to replicate those given by VFP calculations.

Recently, simultaneous work by two research teams demonstrated errors in two
of the transport coefficients at low electron magnetization Sadler et al. (c);
Davies et al. (c). While Epperlein & Haines stated low errors in fitting the
transport coefficients to VFP data, they did not consider that their transport
coefficients were ill-formed. From their perspective the magnetic field
evolution was captured by:

$\displaystyle\frac{\partial\underline{B}}{\partial{}t}=\nabla\times(\underline{v}\times\underline{B})-\nabla\times\frac{\underline{j}\times\underline{B}}{n_{e}e}-\nabla\times\frac{\underline{\underline{\alpha}}\cdot\underline{j}}{n_{e}^{2}e^{2}}$
$\displaystyle+\nabla\times\frac{\underline{\underline{\beta}}\cdot\nabla{}T_{e}}{e}+\nabla\times\frac{\nabla
P_{e}}{n_{e}e}$ (1)

where $\underline{\underline{\alpha}}$ and $\underline{\underline{\beta}}$ are
tensor transport coefficients calculated by fitting to 0D VFP simulations.
While accurate, equation 1 does not make clear the impact of each term on the
movement or generation of magnetic fields. Recent work re-arranged this
equation to make the different physical processes more obviousWalsh et al.
(a); Davies et al. (b); Sadler et al. (a):

$\displaystyle\begin{split}\frac{\partial\underline{B}}{\partial
t}=&-\nabla\times\frac{\alpha_{\parallel}}{\mu_{0}e^{2}n_{e}^{2}}\nabla\times\underline{B}+\nabla\times(\underline{v}_{B}\times\underline{B})\\\
&+\nabla\times\Bigg{(}\frac{\nabla
P_{e}}{en_{e}}-\frac{\beta_{\parallel}\nabla T_{e}}{e}\Bigg{)}\end{split}$ (2)

where the first term is diffusive, the second term is advection of magnetic
field at a velocity $\underline{v}_{B}$ and the final term is a source of
magnetic flux. The advection velocity is then a combination of the bulk plasma
motion, thermally-driven terms and current-driven terms:

$\underline{v}_{B}=\underline{v}-\gamma_{\bot}\nabla
T_{e}-\gamma_{\wedge}(\underline{\hat{b}}\times\nabla
T_{e})-\frac{\underline{j}}{en_{e}}(1+\delta_{\bot}^{c})+\frac{\delta_{\wedge}^{c}}{en_{e}}(\underline{j}\times\underline{\hat{b}})$
(3)

where the transport coefficients have been re-written from the form given by
Braginskii. The dimensionless forms (denoted by a super-script c) depend only
on Hall Parameter and effective Z Walsh et al. (a):

$\gamma_{\bot}^{c}=\frac{\beta_{\wedge}^{c}}{\omega_{e}\tau_{e}}$ (4)

$\gamma_{\wedge}^{c}=\frac{\beta_{\parallel}^{c}-\beta_{\bot}^{c}}{\omega_{e}\tau_{e}}$
(5)

$\delta_{\bot}^{c}=\frac{\alpha_{\wedge}^{c}}{\omega_{e}\tau_{e}}$ (6)

$\delta_{\wedge}^{c}=\frac{\alpha_{\bot}^{c}-\alpha_{\parallel}^{c}}{\omega_{e}\tau_{e}}$
(7)

The dimensional forms of the thermally-driven coefficients are given by:

$\gamma=\gamma^{c}\frac{\tau_{e}}{m_{e}}$ (8)

Once the transport coefficients have been re-written into this physically-
motivated form, the fits of Epperlein & Haines to the VFP simulations have
large errors in $\gamma_{\wedge}^{c}$ and $\delta_{\wedge}^{c}$ at small
$\omega_{e}\tau_{e}$. While the errors in fitting to $\beta_{\bot}^{c}$ are
small, the $\gamma_{\wedge}$ numerator
($\beta_{\parallel}^{c}-\beta_{\bot}^{c}$) and denominator
($\omega_{e}\tau_{e}$) both tend to zero for small $\omega_{e}\tau_{e}$. This
resulted in both $\gamma_{\wedge}^{c}$ and $\delta_{\wedge}^{c}$ being maximum
for small $\omega_{e}\tau_{e}$, while the VFP simulations calculated them as
exactly zero. Figure 1 shows how $\gamma_{\wedge}^{c}$ calculated using the
values from Epperlein & Haines deviate from the desired VFP data.

New fits to the VFP data incorporate the knowledge of how the transport
coefficients should be formulated to capture their limiting behavior at low
Hall Parameter Sadler et al. (c); Davies et al. (c). It should be reiterated
that this newer work did not modify the VFP calculations, only changing how
the transport coefficients were fit to the VFP data. Figure 1 includes
$\gamma_{\wedge}^{c}$ calculated by Sadler, Walsh & Li as well as by Davies,
Wen, Ji & Held; note that these two references give near-identical results.

Figure 2: Ratio of thermal to magnetic transport coefficients as a function of
electron Hall Parameter for the old and new coefficients. These are plotted
for $Z=1$.

The updated transport coefficients make clear the similarities between thermal
and magnetic transport. Looking specifically at the transport due to
temperature gradients:

$\displaystyle\underline{v}_{N}$ $\displaystyle=$
$\displaystyle-\gamma_{\bot}\nabla_{\bot}T_{e}$
$\displaystyle-\gamma_{\wedge}\underline{\hat{b}}\times\nabla T_{e}$ (9)
$\displaystyle\underline{q}_{\kappa}$ $\displaystyle=$
$\displaystyle-\kappa_{\parallel}\nabla_{\parallel}T_{e}$
$\displaystyle-\kappa_{\bot}\nabla_{\bot}T_{e}$
$\displaystyle-\kappa_{\wedge}\underline{\hat{b}}\times\nabla T_{e}$ (10)

where equation 9 is magnetic transport and equation 10 is thermal conduction.
These equations are split into components parallel ($\parallel$) to the
magnetic field, perpendicular to the field ($\bot$) and perpendicular to both
the magnetic field and temperature gradient ($\wedge$). In equation 9 the
$\bot$ term is Nernst advection and the $\wedge$ term is cross-gradient-
Nernst. For thermal conduction the terms are the unrestricted thermal
conduction along field lines, suppressed thermal conduction perpendicular to
the field and the Righi-Leduc heat-flow.

The analogy between Nernst ($\gamma_{\bot}^{c}$) and perpendicular heat-flow
($\kappa_{\bot}^{c}$) was originally noted by Haines Haines . Haines found
that if the kinetic collision operator uses an artificial $v^{-2}$ dependence,
where $v$ is the electron speed, there is an exact equivalence between heat
flow and magnetic field motion, such that
$\kappa_{\perp}^{c}/\gamma_{\perp}^{c}=\kappa_{\wedge}^{c}/\gamma_{\wedge}^{c}=5/2$.
The physical interpretation is that magnetic field is only frozen into the
faster electrons in the distribution, since it can easily diffuse through the
slower, more collisional electrons. The field therefore follows the faster
electrons, which tend to move down temperature gradients with the heat flow.
For the artificial collision operator, this intuition holds exactly for all
$Z$ and $\omega_{e}\tau_{e}$.

However, this physical picture is unaffected by changing the $v$ dependence of
the collision operator, so the character of magnetic field movement with the
electron heat flow should still approximately hold. For example, if instead
the collisions use the realistic Fokker-Planck $v^{-3}$ dependence, magnetic
field should still move with the heat flow, although they are no longer
directly proportional. With the Fokker-Planck collision operator, the ratios
are no longer fixed at $5/2$, and they have dependence on $Z$ and
$\omega_{e}\tau_{e}$. However, for fixed $Z$, the ratios remain approximately
constant. Figure 2 shows the ratio $\kappa_{\bot}^{c}/\gamma_{\bot}^{c}$
against Hall Parameter for $Z=1$; this result is similar using the old or new
coefficients, giving little variation.

Figure 2 also plots the ratio $\kappa_{\wedge}^{c}/\gamma_{\wedge}^{c}$
against Hall Parameter for $Z=1$. Using the old Epperlein & Haines
coefficients the Righi-Leduc ($\kappa_{\wedge}^{c}$) coefficient is much
smaller than cross-gradient-Nernst ($\gamma_{\wedge}^{c}$) at low
magnetization. With the new transport coefficients, however, it can be seen
that the cross-gradient-Nernst term is analogous to advection of magnetic
field with the Righi-Leduc heat-flow; for $Z=1$, the ratio
$\kappa_{\wedge}^{c}/\gamma_{\wedge}^{c}$ varies by less than 3% across
magnetization.

Figure 3:  Azimuthal magnetic field component generated by cross-gradient-
Nernst advection for an under-dense magnetized plasma Walsh et al. (a) using
the Epperlein & Haines transport coefficients Epperlein and Haines  (left) and
the updated coefficients (right) Sadler et al. (c); Davies et al. (c). These
profiles are 1.0ns after the laser turns on.

The ratio of thermal to magnetic transport coefficients does vary with plasma
ionization. For low Hall Parameter, $\kappa_{\bot}^{c}/\gamma_{\bot}^{c}$
ranges between 3.63 for $Z=1$ to 1.39 for $Z=100$. This ratio is fundamental
for systems where thermal conduction drives plasma ablation, where equation 9
will move magnetic fields into the colder regions, while ablation driven by
equation 10 will counter-act that transport Betti et al. . The balance of
Nernst demagnetization to plasma ablation has been found to be critically
important in magnetized plasma conditions Manuel et al. (a); Hill and Kingham
; Walsh et al. (c); Farmer et al. ; Campbell ; Sherlock and Bissell ; Walsh et
al. (d); Hill . The decrease of $\kappa_{\bot}^{c}/\gamma_{\bot}^{c}$ with Z
suggests that low-Z plasmas will be less impacted by Nernst de-magnetization
than high-Z plasmas.

The new fits make intuitive sense. The $\wedge$ terms come from electrons
having curved orbits; if the magnetization is low, the electron trajectories
should be straight on average. The Epperlein & Haines fits captured this
behavior for the heat-flow ($\kappa_{\wedge}$) but not for the magnetic
transport ($\gamma_{\wedge}$). At large magnetization the electrons go through
many orbits before colliding, meaning that there is no preferential direction
for transport; this is captured in both the old and new coefficients, with
$\kappa_{\wedge}$ and $\gamma_{\wedge}$ going to zero at large magnetization.
For the wedge terms to be important, the plasma must be in a moderately
magnetized regime ($\omega_{e}\tau_{e}\approx 1$).

The error in the Epperlein & Haines transport coefficients can alternatively
be explained by their focus on the electric field. They did not consider the
coefficients in terms of the induction equation, where the gradients of the
coefficients are important. The fitting function in the Hall parameter that
Epperlein & Haines chose has a derivative that does not tend to zero as the
magnetic field tends to zero, which is a physical requirement. The errors in
the fits become more apparent when the coefficients are reformulated in a
manner that more clearly demonstrates their physical effects in the induction
equation.

This paper demonstrates the importance of the new transport coefficients in
laboratory plasmas, focusing on the thermally-driven cross-gradient-Nernst
term. In section II the new coefficient is shown to reduce magnetic field
twisting in under-dense systems relevant to MagLIF preheat Gomez et al. . This
setup was proposed in order to make the first measurements of the cross-
gradient-Nernst transport Walsh et al. (a). Section III then shows how the
cross-gradient-Nernst term can result in field twisting in pre-magnetized ICF
capsules Walsh ; again, the new coefficients result in reduced twisting at low
plasma magnetizations. However, the new coefficients do not universally reduce
the impact of cross-gradient-Nernst transport. Section IV compares ICF hot-
spot simulations without MHD, with MHD but no cross-gradient-Nernst, full
extended-MHD with the Epperlein & Haines coefficients and with the new
coefficients. Cross-gradient-Nernst transport is shown to be important in hot-
spot cooling, increasing the penetration depth of a cold spike. An example of
how the new $\gamma_{\wedge}$ coefficient affects perturbation growth in
direct-drive ablation fronts was already given in one of the original updated
coefficients publications Sadler et al. (c)

Simulations in this paper use the Gorgon extended-MHD code Ciardi et al. ;
Chittenden et al. ; Walsh et al. (c). The code can use either the old
Epperlein & Haines coefficients Epperlein and Haines  or the updated
coefficients (see the supplementary material of reference Sadler et al. (c)).
The coefficients calculated by Sadler, Walsh & Li are used in this paper,
although the coefficients from Davies, Wen, Ji & Held have been implemented
and give no noticeable differences Davies et al. (c) (as expected from figure
1). Epperlein & Haines provided tabulated data for the coefficients at
specific values of Z; these values are interpolated in the simulations. The
new coefficients provide a polynomial fit to ionization, which is preferable
for reducing computations and eliminating discontinuities when gradients are
taken in Z.

##  II Underdense Plasmas

This section investigates the impact of cross-gradient-Nernst on magnetized
under-dense gases heated by a laser. This setup is relevant to MagLIF Slutz et
al.  and mini-MagLIF preheat Barnak et al. , but was also investigated as a
means of measuring specific extended-MHD transport terms for the first time
Walsh et al. (a). Particularly relevant was the suggestion that quantification
of a twisted magnetic field component could be used to measure the cross-
gradient-Nernst advection velocity Walsh et al. (a). Simulations in that
experiment design paper used the erroneous Epperlein & Haines coefficients
Epperlein and Haines , which are shown here to give excessive magnetic field
twisting at low magnetization.

The configuration used as demonstration here is as follows. A low density
($5\times 10^{19}$ atoms/cm3) deuterium gas is irradiated by a $5J$ beam with
a $0.5ns$ square temporal pulse. The beam has a Gaussian spatial profile with
a standard deviation $\sigma=100\mu$m at best focus. The beam is tapered along
its propagation direction such that at $r=\sigma$ the angle of the rays to the
axis satisfies $\sin\theta=1/12$. A 1T magnetic field is applied along the
laser propagation axis. The beam tapering is purposefully used to induce
magnetic field twisting by cross-gradient-Nernst transport Walsh et al. (a).

2-D $\underline{r}$,$\underline{z}$ simulations are used, with the laser drive
and applied magnetic field along $\underline{z}$. The laser is treated as
individual rays that are traced through the plasma and deposit their energy by
inverse Bremsstrahlung. A $460\mu$m axial extent is used, with radial and
axial resolution of $1\mu$m.

As the gas density is low, very little energy couples to the system. At
$0.1$ns the total absorption along the $460\mu$m axial extent is 2.5%,
reducing to 1% by the end of the laser pulse when the plasma temperature is
higher ($>100$eV). As the Hall Parameter is low for the chosen setup
($\omega_{e}\tau_{e}<0.2$ everywhere) thermal conduction effectively
transports energy radially. Simultaneously, the Nernst term advects magnetic
field away from the beam center, reducing the field strength to 0.3T by 0.5ns.
The magnetic flux piles up at the thermal conduction front, peaking at 2T.

The impact of cross-gradient-Nernst can be seen by decomposing the advection
term in equation 2:

$\bigg{[}\frac{\partial\underline{B}}{\partial
t}\bigg{]}_{N\wedge}+(\underline{v}_{N\wedge}\cdot\nabla)\underline{B}=-\underline{B}(\nabla\cdot\underline{v}_{N\wedge})+(\underline{B}\cdot\nabla)\underline{v}_{N\wedge}$
(11)

As the temperature gradient is predominantly radial and the magnetic field is
predominantly axial, the cross-gradient-Nernst velocity is purely in
$\underline{\theta}$. Therefore, equation 11 reduces to $\big{[}\frac{\partial
B_{\phi}}{\partial t}\big{]}_{N\wedge}=B_{z}\frac{\partial
v_{N\wedge\theta}}{\partial z}$. This term represents magnetic field twisting.
Following a single axial magnetic field line, if $\underline{v}_{N\wedge}=0$
at the bottom and $\underline{v}_{N\wedge}\neq 0$ at the top, a $B_{\theta}$
field will be generated by twisting in between. By tapering the laser beam,
this setup purposefully makes this happen; near the laser edge a magnetic
field line will pass between an unheated and a heated regime, resulting in
twisting of the magnetic field Walsh et al. (a). This process can be viewed as
a dynamo driven by electron heat-flow.

Figure 3 compares the twisted magnetic field component at $t=1.0$ns using the
old Epperlein & Haines transport coefficients and the updated coefficients
Sadler et al. (c); Davies et al. (c). The Biermann Battery generation has been
neglected in these simulations, which means that $B_{\theta}$ can only be
generated by cross-gradient-Nernst twisting. As the experiment here is in the
low magnetization regime, the difference in the $\gamma_{\wedge}$ coefficient
is substantial; at the laser spot edge, where much of the twisting takes
place, $\omega_{e}\tau_{e}\approx 0.05$, resulting in a factor of 4 reduction
in the cross-gradient-Nernst coefficient. By 1.0ns the peak
$B_{\theta}\approx$0.08T for the old coefficients, and $\approx$0.01T using
the new coefficients.

The new coefficients change the regime where significant $B_{\theta}$ is
generated. Designs using the old coefficients aimed for the underdense plasma
being as low magnetization as possible. Now, however, it is clear that the
plasma should be balanced in the moderately magnetized regime, with the cross-
gradient-Nernst velocity peaking at around $\omega_{e}\tau_{e}\approx 0.8$ for
deuterium Sadler et al. (c). This is still realizable, with the density, laser
power and applied field all acting as free variables. An experiment with the
plasma parameters defined here but scanning applied field strength is expected
to observe the true $\gamma_{\wedge}$ dependence.

However, with the effect of cross-gradient-Nernst reduced, the Biermann
Battery contribution to $B_{\theta}$ becomes more important. At 0.1ns for the
simulations shown here the $B_{\theta}$ field from Biermann Battery is of the
same order as from cross-gradient-Nernst. Biermann requires a density gradient
in the plasma, which only develops later in time, at time-scales on the order
of the laser radius divided by the sound speed. Therefore, probing early in
time before the plasma hydrodynamically expands is important to distinguish
Biermann from cross-gradient-Nernst.

##  III Pre-magnetized ICF Capsules

Figure 4:  $B_{\phi}$ at 6.3ns for a warm indirect-drive implosion with an
applied axial field of 5T. Left is a simulation with cross-gradient-Nernst
included (using the new coefficients), while right shows a case with both
cross-gradient-Nernst and Biermann included.

Axial magnetic fields can be applied to ICF capsules to reduce thermal cooling
of the fuel during stagnation Chang et al. ; Perkins et al. ; Walsh et al.
(b). Throughout the implosion the magnetic field compresses along with the
plasma; this happens dominantly at the capsule waist, where the implosion
velocity is perpendicular to the magnetic field lines. At the poles the
magnetic field remains uncompressed, as the implosion velocity is along field
lines.

It has been suggested that the cross-gradient-Nernst term should twist the
applied magnetic field during the implosion Davies et al. (b). Preliminary
simulations showed significant twisting by bang-time, with the yield increased
by as much as 10% due to the extra path length required for heat to travel
along field lines out of the hot-spot Walsh . Again, these calculations used
the old Epperlein & Haines coefficients. Here, as with the twisting seen in
the underdense configuration in section II, the twisting is found to be lower
when the new transport coefficients are used.

The simulated configuration is a $D_{2}$ gas-filled HDC indirect-drive capsule
that is used on the National Ignition Facility, although these results broadly
apply to all magnetized spherical implosions. A 5T magnetic field is applied
axially. $1\mu$m radial resolution is used throughout the implosion and 180
polar cells are used in the simulation range $\theta=0,\pi$. Before the first
shock converges on axis the simulations are transferred to a cylindrical mesh.

Moderate HDC shell thickness variations are initialized Casey et al. , such
that the capsule is mildly perturbed by neutron bang-time. 1600 perturbations
with random amplitude and mode are used, with each chosen in the range
$\epsilon=0,0.5$nm and $k=0,180$ respectively.

Figure 5:  Density and $B_{\phi}$ at neutron bang-time ($t=7.3ns$) using the
old Epperlein & Haines coefficients (left) and the updated versions (right).

While for the underdense configuration a tapered beam was used to induce
twisting, a spherical implosion naturally has twisting peaking at
$\theta=\pi/4,3\pi/4$. This happens because the cross-gradient-Nernst velocity
$-\underline{\hat{b}}\times\nabla T_{e}$ is maximum at the capsule waist and
zero at the poles. Therefore, the axial magnetic field at the waist is
displaced in $\phi$ but at the poles it is stationary; in between the waist
and the poles the magnetic field lines are twisted out of the plane.

Twisting of magnetic field lines introduces a closed field line component,
$B_{\phi}$. This is of great interest to the magneto-inertial fusion
community, as this component is effectively compressed during the implosion
and results in additional thermal energy containment. Schemes have been
considered to apply a purely $B_{\phi}$ component to a capsule Hohenberger et
al. , although none have been demonstrated for spherical implosions.

The cross-gradient-Nernst velocity is significant both in the in-flight shock-
compressed gas and in the stagnating hot-spot. For the configuration chosen
here the in-flight phase is not changed significantly by the new coefficients,
as the low density and high temperature gas is magnetized even with
$B_{0}=5$T. For lower applied fields the coefficients may make a significant
difference, but these regimes are not of interest for magneto-inertial fusion.

Figure 4 shows the $B_{\phi}$ field profile induced by the cross-gradient-
Nernst effect at 6.3ns, which is before the first shock converges onto the
axis. While $B_{\phi}$ only reaches 3T at this time, the field twisting is
significant: $B_{\phi}/|\underline{B}|=1/3$ in locations near the shock-front.

Also shown in figure 4 is a simulation with the Biermann Battery self-
generated magnetic fields included. This term requires an asymmetric implosion
to be important, of which there are two sources in this implosion. First of
all, the applied field results in anisotropy of the heat-flow; here that
results in a hotter compressed gas at the waist compared with the poles. Also,
the HDC thickness variations generate magnetic fields. The interest here is in
the fuel $B_{\phi}$ component, which is dominated by cross-gradient-Nernst
twisting for $B_{0}=5$T. Larger applied fields result in suppression of cross-
gradient-Nernst and enhancement of the thermal conductivity anisotropy,
enhancing the impact of Biermann.

Figure 5 shows the density and out-of-plane magnetic field component at bang-
time. Biermann Battery generation of magnetic field has been turned off in the
code for these simulations, which means that all of $B_{\phi}$ is from cross-
gradient-Nernst twisting. On the left is a case using the old cross-gradient-
Nernst coefficients from Epperlein & Haines, while on the right uses the
updated coefficients. The new coefficients lower the peak $B_{\phi}$ from 500T
to 250T. Nonetheless, the twisting is significant, with
$B_{\phi}/|\underline{B}|$ up to 0.3 throughout the hot-spot.

The twisting introduces several additional effects. Firstly, thermal
conduction and Nernst are moderately reduced, as these components become
partially out of the simulation plane; Cross-gradient-Nernst and Righi-Leduc
then have components within the simulation plane. If the magnetic tension
becomes significant then the twisted field would also be expected to induce
plasma motion in $\phi$; however, as cross-gradient-Nernst is suppressed for
the large applied fields required for tension to be important Perkins et al. ;
Walsh et al. (b), this may not be a realizable regime. For $B_{0}=5$T the
plasma $\beta$ is too large for significant motion.

While the simulations here focused on indirect-drive implosions, cross-
gradient-Nernst also twists magnetic fields in pre-magnetized direct-drive
ablation fronts Walsh et al. (d), where the temperature gradients drive
extreme heat-flows. However, the $B_{\phi}$ component is effectively advected
by the plasma and Nernst velocities, lowering its impact.

##  IV Self-Generated Fields in ICF Hot-spots

Figure 6:  Self-generated magnetic fields at bang-time around a cold spike
pushing into a hot-spot. 3 cases are shown. Left: a simulation without cross-
gradient-Nernst advection included. Middle: a simulation using the Epperlein &
Haines transport coefficients Epperlein and Haines . Right: a simulation using
the updated transport coefficients Sadler et al. (c); Davies et al. (c).

This section looks into the impact of cross-gradient-Nernst on self-generated
magnetic field profiles in regular ICF hot-spots, where magnetic fields have
not been externally applied.

Magnetic fields are generated by the Biermann Battery mechanism around
perturbations, with estimated field strengths up to 10kT Walsh et al. (c).
These fields are predominantly generated in the stagnation phase, when the
temperature and density gradients are largest Walsh and Clark . Recent
advances have been made in the theoretical understanding of magnetic flux
generation in these systems, with more flux being generated around high mode
and large amplitude perturbations Walsh and Clark .

While it was expected that self-generated magnetic fields would reduce heat
loss from ICF hot-spots, research found that magnetization of the electron
population introduced Righi-Leduc heat-flow, which enhanced cooling Walsh et
al. (c). As cross-gradient-Nernst is the magnetic transport analogue of Righi-
Leduc heat-flow, it is important in these systems.

The simulations here use the indirect-drive HDC design N170601 is used,
although the physics is also applicable to direct-drive hot-spots. An isolated
200nm HDC shell thickness variation is imposed at the capsule pole, causing a
cold spike to push into the hot-spot. Shell thickness variations have been
found to be a significant degradation mechanism for HDC implosions Casey et
al. ; the variation applied here is not based on any target fabrication
specifications, and is instead used as a demonstration of magnetized heat-flow
in ICF implosions.

The simulations here are 2-D and are in spherical geometry until the first
shock converges onto the axis ($t\approx 7.6$ns). The resolution up to this
time is $1\mu$m radially, with 180 cells in the polar direction from
$\theta=0,\pi$. The simulations are then remapped into cylindrical geometry
for the stagnation phase, when the resolution is $\frac{1}{2}\mu$m.

Figure 6 shows the magnetic field distribution at neutron bang-time
($t=8.5$ns) for simulations with different cross-gradient-Nernst physics
included. The magnetic field profiles are in green (into the page) and purple
(out of the page), plotted over the density so that the proximity to the spike
can be seen. The dense fuel is at the top of the figure and the hot-spot at
the bottom.

Figure 7:  Electron temperature along the axis of a hot-spot spike at bang-
time. Temperature profiles are shown for cases: without any self-generated
fields included; with extended-MHD but no cross-gradient-Nernst advection;
using the Epperlein & Haines transport coefficients Epperlein and Haines ;
using the updated coefficients Sadler et al. (c); Davies et al. (c).

First in figure 6 is a case with no cross-gradient-Nernst advection. The
Righi-Leduc heat-flow direction is shown with white arrows. Righi-Leduc acts
to cool the spike tip, which results in regular Nernst compressing the field
onto the simulation axis. This process causes numerical issues, advecting all
of the magnetic field into a single line of cells. Once the cross-gradient-
Nernst is included, the magnetic field moves in the same direction as Righi-
Leduc, preventing the field profile from compressing onto the axis. It can be
seen in figure 6 that the new cross-gradient-Nernst coefficient does not
advect as rapidly as the Epperlein & Haines version.

It is posited that the updated coefficients will always be important in
systems with self-generated magnetic fields and thermal conduction; there is
always a null-point of magnetic field, resulting in no electron magnetization.
Even if the peak magnetization is large enough to be in a regime where the
Epperlein & Haines coefficients are valid, the magnetization will decrease to
zero in the surrounding regions, passing through the regime where the old
coefficients are invalid.

Figure 7 shows the impact of the imposed spike on the hot-spot temperature for
simulations with various MHD packages included. The electron temperature is
plotted along the spike axis. Without any MHD included the spike does not
propagate as far; this suggests that current design calculations of ICF
implosions (both directly and indirectly driven) are underestimating the
impact of perturbations.

Simulations with the new coefficients included result in the greatest
discrepancy with the estimates that do not include Biermann generation; at the
$T_{e}=3.5$keV contour the spike has penetrated $7\mu$m further due to the
magnetization of heat-flow. The simulations without cross-gradient-Nernst are
less affected by the magnetic fields, as the fields are transported by the
standard Nernst term into a one-cell width (see figure 6).

The new coefficients give a greater impact of MHD on spike penetration
compared with the old coefficients. This can be understood from the fact that
the new coefficients result in the magnetic field moving with the Righi-Leduc
heat-flow. In contrast, the old coefficients allowed the magnetic flux to move
ahead of the Righi-Leduc heat-flow into regions where it would have a lower
impact on the plasma magnetization. This can be seen in figure 2, which shows
that the ratio of Righi-Leduc coefficient to cross-gradient-Nernst coefficient
decreasing to zero for low magnetization.

Closer analysis of the impact of MHD on capsule perturbation growth for a
variety of mode numbers and amplitudes will be the subject of a future
publication.

##  V Conclusions

In summary, updated magnetic transport coefficients that accurately replicate
kinetic simulations at low electron magnetizations Sadler et al. (c); Davies
et al. (c) have been shown to be important across a range of laboratory plasma
conditions. These coefficients are the new standard for implementation into
extended-MHD codes. While the two references give different fits for the
coefficients, they are found to be practically equivalent.

With an external magnetic field applied, the cross-gradient-Nernst term tends
to twist the magnetic field Davies et al. (a); Walsh ; using the new
coefficients reduces twisting in the low magnetization
($\omega_{e}\tau_{e}<1$) regime. This result impacts attempts to measure
cross-gradient-Nernst for the first time Walsh et al. (a) as well as the
design of pre-magnetized capsule implosions Chang et al. ; Perkins et al. ;
Walsh ; Walsh et al. (b). Twisting is still possible in the moderate
magnetization regime ($\omega_{e}\tau_{e}\approx 1$).

For systems with significant self-generated magnetic fields Farmer et al. ;
Walsh et al. (c); Sadler et al. (b) the new coefficients result in the
magnetic fields moving with the Righi-Leduc heat-flow. This is found to
enhance the impact of Righi-Leduc, as shown in direct-drive ablation fronts
(where Righi-Leduc reduces perturbation growth) Sadler et al. (c) and in ICF
hot-spots (where Righi-Leduc enhances perturbation growth).

In addition to being more physically accurate, the new coefficients have been
found to increase numerical stability, as they introduce fewer discontinuities
into the simulations.

## Acknowledgements

This work was performed under the auspices of the U.S. Department of Energy by
Lawrence Livermore National Laboratory under Contract DE-AC52-07NA27344 and by
the LLNL-LDRD program under Project Number 20-SI-002. The simulation results
were obtained using the Imperial College High Performance Computer Cx1..

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.

Research presented in this article was also supported by the Laboratory
Directed Research and Development program of Los Alamos National Laboratory,
under the Center for Nonlinear Studies project number 20190496CR. This
research was supported by the Los Alamos National Laboratory (LANL) through
its Center for Space and Earth Science (CSES). CSES is funded by LANL’s
Laboratory Directed Research and Development (LDRD) program under project
number 20180475DR.

Finally, the information, data, or work presented herein was funded in part by
the Advanced Research Projects Agency-Energy (ARPAE), U.S. Department of
Energy, under Award No. DE-AR0001272, by the Department of Energy Office of
Science, under Award No. DE-FG02-04ER54746, by the Department of Energy
National Nuclear Security Administration under Award No. DENA0003856, the
University of Rochester, and the New York State Energy Research and
Development Authority.

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