# Magnetic Field Transport in Propagating Thermonuclear Burn

B. Appelbe<EMAIL_ADDRESS>The Centre for Inertial Fusion Studies, The
Blackett Laboratory, Imperial College, London, SW7 2AZ, United Kingdom  A. L.
Velikovich  Plasma Physics Division, Naval Research Laboratory, Washington, DC
20375, USA  M. Sherlock  Lawrence Livermore National Laboratory, P.O. Box 808,
Livermore, California 94551-0808, USA  C. Walsh  Lawrence Livermore National
Laboratory, P.O. Box 808, Livermore, California 94551-0808, USA  A. Crilly
The Centre for Inertial Fusion Studies, The Blackett Laboratory, Imperial
College, London, SW7 2AZ, United Kingdom  S. O Neill  The Centre for Inertial
Fusion Studies, The Blackett Laboratory, Imperial College, London, SW7 2AZ,
United Kingdom  J. Chittenden  The Centre for Inertial Fusion Studies, The
Blackett Laboratory, Imperial College, London, SW7 2AZ, United Kingdom

###### Abstract

High energy gain in inertial fusion schemes requires the propagation of a
thermonuclear burn wave from hot to cold fuel. We consider the problem of burn
propagation when a magnetic field is orthogonal to the burn wave. Using an
extended-MHD model with a magnetized $\alpha$ energy transport equation we
find that the magnetic field can reduce the rate of burn propagation by
suppressing electron thermal conduction and $\alpha$ particle flux. Magnetic
field transport during burn propagation is subject to competing effects: field
can be advected from cold to hot regions by ablation of cold fuel, while the
Nernst and $\alpha$ particle flux effects transport field from hot to cold
fuel. These effects, combined with the temperature increase due to burn, can
cause the electron Hall parameter to grow rapidly at the burn front. This
results in the formation of a self-insulating layer between hot and cold fuel
that reduces electron thermal conductivity and $\alpha$ transport, increases
the temperature gradient and reduces the rate of burn propagation.

††preprint: AIP/123-QED

##  I Introduction

The ultimate goal of inertial fusion energy is to produce high energy gain, in
which the energy liberated in thermonuclear reactions is orders of magnitude
larger than the energy of the driver used to compress the fusion fuel. High
energy gain can be achieved by using the driver energy to ignite a small
fraction of the fuel, the hotspot, which then drives a propagating
thermonuclear burn wave into a surrounding layer of dense, cold
fuel.Lindl_1995 ; Slutz_PRL2012  The burn wave consists of significant energy
transport due to $\alpha$ particles, electron thermal conduction and radiative
processes.Tong_NF2019

Magnetic fields are a central feature of a broad class of inertial fusion
schemes referred to as Magneto-Inertial Fusion (MIF)Wurden_MIF2016 . In MIF, a
magnetic field is applied to the fuel during the compression phase. This
magnetic field provides magnetothermoinsulationLindemuth_1983  during the
compression phase by reducing electron thermal conduction losses from hot
plasma. This reduces the implosion velocity required to reach the ignition
temperature. The magnetic field also confines $\alpha$ particles within the
fuel during thermonuclear burn but with a Larmor radius that is significantly
larger than that of the thermal electrons.

A number of MIF schemes under current investigation, such as MagLIF at Sandia
National LaboratoriesSlutz_POP2010 ; Gomez_PRL2020 , aim to achieve volumetric
thermonuclear burn. Volumetric burn schemes are designed to heat all of the
fuel to fusion temperatures during the compression phase. If the fuel ignites
then a net energy gain can be achieved without the need for thermonuclear burn
propagation (even though the theoretical maximum energy gain is smaller for
volumetric burn compared with propagating burn). For volumetric burn schemes
it is desirable to have a magnetic field which maximizes the
magnetothermoinsulation, as long as the magnetic pressure remains small
relative to thermal pressure.

In the case of MIF schemes involving propagating burn, such as high gain
MagLIFSlutz_PRL2012  and magnetized indirect-drive Inertial Confinement
Fusion(ICF)Perkins_POP2013 ; Perkins_POP2017 , the magnetothermoinsulation
effect is required during the compression phase to aid the formation of an
igniting hotspot. However, the role of the magnetic field during the
propagating burn phase is less clear. The suppression of thermal conduction
and $\alpha$ particle transport reduces the rate of burn propagation into the
cold fuel.Slutz_PRL2012 ; Jones_NF1986 ; Velikovich_ICOPS2012  This can limit
the energy gain achieved before the target disassembles. While some magnetic
confinement of $\alpha$ particles during burn propagation could be desirable,
to allow for lower areal densities, it is clear that a magnetic field which is
too large could be detrimental to the achievable yield. This question of how a
magnetic field affects thermonuclear burn propagation motivates the present
work. In addition to propagating burn MIF schemes, this question may also be
relevant to conventional ICF since simulations have shown that the Biermann
battery mechanismBiermann_1950  can generate magnetic fields at the surface of
the hotspot in perturbed ICF implosions.Walsh_PRL2017

In this work we study magnetic field dynamics in a propagating burn wave and
the effect of the magnetic field on the evolution of burn propagation. The
principal novelty of the work is our detailed treatment of the magnetic field
transport. We utilize an MHD model (described in detail in section II) in
which the induction equation includes advection, resistive diffusion, electron
temperature gradients (the Nernst effect) and the flux of $\alpha$ particles.
This last term is typically not included in simulations of MIF schemes. We
find that the various terms in the induction equation result in competing
dynamics of the magnetic field at a propagating burn front: field is advected
from cold to hot regions as the burn wave causes ablation of cold fuel whilst
temperature gradients and $\alpha$ particle fluxes transport field in the
opposite direction. These dynamics have a significant effect on the shape and
rate of propagation of the burn wave.

The magnetic field profile at the onset of burn in MIF is determined by the
implosion phase and there has been a number of studies showing the complex
field transport occurring in that phase.Velikovich_POP2015 ;
Velikovich_POP2019 ; GarciaRubio_PRE2018 ; GarciaRubio_POP2017  For
simplicity, we do not study the implosion phase in the current work. Instead
we take as our starting point a region of hot fuel that is undergoing
significant $\alpha$ heating adjacent to a cold fuel layer. In order to
elucidate the field dynamics during burn, we focus on three simple cases for
the initial magnetic field profile: a spatially uniform field, a spatially
uniform electron Hall parameter and a magnetic field that is non-zero only in
a small region at the burn front. Results for each of these cases are
discussed in section III.

This is not an exhaustive search of magnetic field strengths and profiles but
the cases considered show that there is a complex interplay between the
magnetic field dynamics and the evolution of the propagating burn wave. The
suppression of thermal conduction and $\alpha$ transport by the magnetic field
depends on both the magnitude of the magnetic field and the collisionality of
the plasma, where the relevant metric is the Hall parameter. As thermonuclear
burn heats the plasma, collisionality decreases rapidly. Combining this with
the magnetic field transport effects leads to enhanced suppression of energy
transport from hot to cold fuel. In our case studies we identify scenarios in
which a self-insulating magnetized layer can form between hot and cold fuel
due to feedback between decreasing collisionality in the burn wave and the
magnetic field transport. This self-insulating layer significantly reduces
thermal conductivity and $\alpha$ transport across it and leads to the
development of large temperature gradients. In section IV we formulate an
equation for the growth rate of the Hall parameter in order to better
understand the conditions leading to formation of the self-insulating layer.
This equation illustrates, inter alia, the dependence of the growth rate on
the Hall parameter value, $\alpha$ heating, and the temperature and $\alpha$
energy density profiles.

Finally, we present some conclusions in section V. While recent
theoreticalVelikovich_POP2019  and experimentalGomez_PRL2020  work has
demonstrated the importance of magnetic field transport during the implosion
phase, our results show that field transport effects during the burn phase are
also of interest.

##  II Model Outline

Our model is a one-dimensional, planar, classical DT plasma consisting of
semi-infinite regions of cold and hot fuel, separated by a smooth transition
region with an orthogonal magnetic field. This idealized geometry is not
intended to model a specific MIF scheme, but instead allows us to study how
thermonuclear burn propagates from hot to cold fuel. The system is governed by
an equation of continuity, the isobaric condition, an induction equation and
energy equations for the fuel and the $\alpha$ particles as follows:

$\displaystyle\frac{\partial
n}{\partial\hat{t}}+\frac{\partial}{\partial\hat{x}}\left(n\hat{u}\right)$
$\displaystyle=$ $\displaystyle 0,$ (1)
$\displaystyle\frac{\partial}{\partial\hat{x}}\left(2nT+\frac{B^{2}}{2\mu_{0}}\right)$
$\displaystyle=$ $\displaystyle 0,$ (2) $\displaystyle\frac{\partial
B}{\partial\hat{t}}+\underbrace{\frac{\partial}{\partial\hat{x}}\left(\hat{u}B\right)}_{\text{advect.}}$
$\displaystyle=$
$\displaystyle\frac{\partial}{\partial\hat{x}}\left[\underbrace{\hat{\alpha}\frac{\partial
B}{\partial\hat{x}}}_{\text{res.
diff.}}+\left(\underbrace{\hat{\beta}\frac{\partial\ln
T}{\partial\hat{x}}}_{\text{Nernst}}+\underbrace{\hat{\gamma}\frac{\partial\ln\mathcal{E}_{\alpha}}{\partial\hat{x}}}_{\text{$\alpha$-e
coll.}}\right)B\right],$ (3) $\displaystyle
3n\frac{DT}{D\hat{t}}+2nT\frac{\partial\hat{u}}{\partial\hat{x}}$
$\displaystyle=$
$\displaystyle\frac{\partial}{\partial\hat{x}}\left[\underbrace{\hat{\kappa}\frac{\partial
T}{\partial\hat{x}}}_{\text{cond.}}+\underbrace{\hat{\beta}\frac{B}{\mu_{0}}\frac{\partial
B}{\partial\hat{x}}}_{\text{Ettings.}}\right]+\underbrace{\hat{q}_{\alpha}}_{\text{$\alpha$
heat.}}-\underbrace{\hat{P}}_{\text{brem.}},$ (4)
$\displaystyle\frac{D\mathcal{E}_{\alpha}}{D\hat{t}}+\frac{5}{3}\mathcal{E}_{\alpha}\frac{\partial\hat{u}}{\partial\hat{x}}$
$\displaystyle=$
$\displaystyle\underbrace{\frac{\partial}{\partial\hat{x}}\left(\hat{\delta}_{\mathcal{E}}\frac{\partial\mathcal{E}_{\alpha}}{\partial\hat{x}}\right)}_{\text{$\alpha$
energy diff.}}-\hat{q}_{\alpha}+\underbrace{\hat{Q}}_{\text{reac.}},$ (5)

where $\hat{t}$, $\hat{x}$ and $\hat{u}$ represent dimensionless time,
position and fluid velocity of the fuel and

$\frac{D}{D\hat{t}}=\frac{\partial}{\partial\hat{t}}+\hat{u}\cdot\frac{\partial}{\partial\hat{x}},$

is the convective derivative. The normalizing constants are $L_{T}$,
representing the width of the transition region between hot and cold fuel and
$\mathcal{T}_{h}$, the stopping time for a $3.45\,MeV$ $\alpha$ particle in
the initial hot fuel. Other variables include fuel density, $n$, magnetic
field, $B$, fuel temperature, $T$, and $\mathcal{E}_{\alpha}$ is the energy
density of $\alpha$ particles. SI units are used for quantities that have not
been made dimensionless.

The induction, (3), and fuel energy, (4), equations contain classical
magnetized transport coefficients for resistivity, thermoelectricity and
thermal conductivityBraginskii_1965

$\displaystyle\hat{\alpha}$ $\displaystyle=$
$\displaystyle\frac{\mathcal{T}_{H}}{L_{T}^{2}}\frac{m_{e}}{\mu_{0}e^{2}n\tau_{ei}}\alpha_{\bot}^{c},\qquad\hat{\beta}=\frac{\mathcal{T}_{H}}{L_{T}^{2}}\frac{\tau_{ei}T}{m_{e}}\frac{\beta_{\wedge}}{\chi_{e}},$
$\displaystyle\hat{\kappa}$ $\displaystyle=$
$\displaystyle\frac{\mathcal{T}_{H}}{L_{T}^{2}}\frac{nT\tau_{ei}}{m_{e}}\left(\kappa_{\bot
e}^{c}+\sqrt{\frac{2m_{e}}{m_{i}}}\kappa_{\bot i}^{c}\right),$

The coefficients $\alpha_{\bot}^{c}$, $\beta_{\wedge}$, $\kappa_{\bot e}^{c}$
and $\kappa_{\bot i}^{c}$ are functions of electron Hall parameter,
$\chi_{e}=eB\tau_{ei}/m_{e}$, where
$\tau_{ei}=6\varepsilon_{0}^{2}\sqrt{2\pi^{3}m_{e}}T^{\frac{3}{2}}/\left(e^{4}\ln\Lambda_{ei}n\right)$.
We use the tabulated values of Epperlein and HainesEpperlein_1986  for these
coefficients. The chosen geometry of our problem means that the Hall
termFarmer_2019  and Biermann battery termBiermann_1950  do not need to be
considered in the induction equation.

The final term on the rhs of (3) represents the collisionally-induced current
arising from the interaction of the $\alpha$ particle flux with the thermal
electrons. Similar fast ions effects have previously been studied for
collisionless plasmas in current drive in tokamaksFisch_RevModPhys1987 ,
cosmic rays in astrophysical plasmasBai_ApJ2015  and relativistic particles in
laser-plasma interactions.Sherlock_PRL2010  Here we follow the procedure of
Appelbe et alAppelbe_POP2019  to include this effect in a collisional,
magnetized plasma.

An analogy may be drawn between the Nernst and the $\alpha$-e collisional
terms in (3). The Nernst effect is a result of the effect of a magnetic field
on the thermal force.Braginskii_1965  Electrons being driven by a temperature
gradient and experiencing friction due to the background ions are deflected by
a magnetic field that is orthogonal to the temperature gradient, generating an
electron current orthogonal to both. In the case of the $\alpha$-e collisional
term the electron current is instead driven by a flux of $\alpha$ particles.
Both the electron temperature gradient and the $\alpha$ flux are themselves
dependent on the magnetic field. The interplay between temperature gradient,
magnetic field and driven electron current has been well
studiedVelikovich_POP2019  and is often included in simulations. However, the
equivalent interplay between the magnetic field, the $\alpha$ flux and the
driven electron current has not received such attention.

In our geometry the $\alpha$-e collisional effect can be expressed as

$\displaystyle\hat{\gamma}$ $\displaystyle=$
$\displaystyle\mathcal{E}_{\alpha}\left(\frac{\partial\mathcal{E}_{\alpha}}{\partial\hat{x}}\right)^{-1}\frac{Z_{\alpha}}{n}\left(\gamma_{\bot}F_{\bot}+\gamma_{\wedge}F_{\wedge}\right),$
(6) $\displaystyle\gamma_{\bot}$ $\displaystyle=$ $\displaystyle-
j_{\bot}^{\nu}\left(1+\frac{\alpha_{\wedge}^{c}}{\chi_{e}}\right)+j_{\wedge}^{\nu}\frac{\alpha_{\bot}^{c}}{\chi_{e}},$
(7) $\displaystyle\gamma_{\wedge}$ $\displaystyle=$ $\displaystyle
j_{\wedge}^{\nu}\left(1+\frac{\alpha_{\wedge}^{c}}{\chi_{e}}\right)+j_{\bot}^{\nu}\frac{\alpha_{\bot}^{c}}{\chi_{e}},$
(8)

where $F_{\bot}$, $F_{\wedge}$ are components of $\alpha$ particle flux in the
directions of burn propagation and orthogonal to burn propagation and $B$
field, respectively, and $j_{\bot}^{\nu}$, $j_{\wedge}^{\nu}$ are functions of
$\chi_{e}$.Appelbe_POP2019

The $\alpha$ particle fluxes can be estimated from the gradients of
$\mathcal{E}_{\alpha}$ and $\alpha$ particle diffusion coefficients as
followsliberman_velikovich_1984

$\displaystyle F_{\bot,\wedge}$ $\displaystyle=$
$\displaystyle-\frac{\hat{\delta}_{p\bot,\wedge}}{\langle
E_{\alpha}\rangle}\frac{\partial\mathcal{E}_{\alpha}}{\partial\hat{x}},$ (9)
$\displaystyle\hat{\delta}_{p\bot}$ $\displaystyle=$
$\displaystyle\frac{\mathcal{T}_{h}}{L_{T}^{2}}\frac{E_{\alpha
0}\tau_{\alpha}}{3m_{\alpha}\left(1+\chi_{\alpha}^{2}\right)},\quad\hat{\delta}_{p\wedge}=\chi_{\alpha}\hat{\delta}_{p\bot}$

where $\langle E_{\alpha}\rangle\approx E_{\alpha 0}/2$ is the mean $\alpha$
particle energy.

The $\alpha$ energy transport equation, (5), represents a single group
diffusion approximation.liberman_velikovich_1984  It contains terms for
magnetized diffusion, slowing on electrons and a source due to DT reactions
BoschHale_1992

$\displaystyle\hat{\delta}_{\mathcal{E}}$ $\displaystyle=$
$\displaystyle\frac{\mathcal{T}_{h}}{L_{T}^{2}}\frac{E_{\alpha
0}\tau_{\alpha}}{m_{\alpha}\left(9+\chi_{\alpha}^{2}\right)},\quad\hat{\nu}=\frac{\mathcal{T}_{h}}{\tau_{\alpha}},\quad\hat{Q}=\mathcal{T}_{h}\frac{n^{2}}{4}\langle\sigma
v\rangle_{DT}E_{\alpha 0},$ $\displaystyle\chi_{\alpha}$ $\displaystyle=$
$\displaystyle\frac{Z_{\alpha}eB}{m_{\alpha}}\tau_{\alpha},\quad\tau_{\alpha}=3\sqrt{\frac{\pi^{3}}{2m_{e}}}\frac{m_{\alpha}\varepsilon_{0}^{2}}{e^{4}\ln\Lambda_{\alpha
e}}\frac{T^{\frac{3}{2}}}{n},$

where $\chi_{\alpha}$ is the $\alpha$ Hall parameter, $\tau_{\alpha}$ is
slowing time on electrons and $E_{\alpha 0}=3.45\,MeV$. Electron and $\alpha$
Hall parameters are related by
$\chi_{\alpha}/\chi_{e}=Z_{\alpha}\ln\Lambda_{ei}/\left(4\ln\Lambda_{\alpha
e}\right)$. Finally, $\hat{q}_{\alpha}=2\hat{\nu}\mathcal{E}_{\alpha}$ and
bremsstrahlung lossesNRL_form  are $\hat{P}=1.69\times
10^{-38}n^{2}\sqrt{\frac{T}{e}}\mathcal{T}_{h}\,J\,m^{-3}$.

Returning to the induction equation, (3), we note that it has been written
such that $\hat{\alpha}$, $\hat{\beta}$ and $\hat{\gamma}$ can all be treated
as diffusion coefficients. In the burning plasma regime the magnetic Lewis
number,Velikovich_POP2015  defined as
$Lm=\left.\hat{\kappa}/\left(3n\hat{\alpha}\right)\right|_{\chi_{e}=0},$ is
large, meaning that thermal diffusivity is much larger than magnetic
diffusivity and so resistive diffusion plays only a minor role in $B$ field
transport. Using (9) for $\alpha$ particle fluxes we can write (6) more
compactly as

$\hat{\gamma}=\frac{\mathcal{T}_{h}}{L_{T}^{2}}\frac{Z_{\alpha}\mathcal{E}_{\alpha}E_{\alpha
0}\tau_{\alpha}}{3m_{\alpha}n\langle
E_{\alpha}\rangle}\underbrace{\left[\frac{-\gamma_{\bot}-\chi_{\alpha}\gamma_{\wedge}}{1+\chi_{\alpha}^{2}}\right]}_{\text{$\Gamma_{\chi}$}},$

where the dependency on magnetization is now fully contained in the square
brackets which we now denote as $\Gamma_{\chi}$. We note that even though (7)
and (8) contain terms $\propto 1/\chi_{e}$, the term $\Gamma_{\chi}$ does not
diverge at $\chi_{e}=0$ since
$\alpha_{\wedge}^{c},j_{\wedge}^{\nu},F_{\wedge}\rightarrow 0$ as
$\chi_{e}\rightarrow 0$.

The $\Gamma_{\chi}$ term is compared with the dependence of $\hat{\beta}$
(which determines the magnitude of the Nernst effect) on $\chi_{e}$ in fig. 1.
Comparing the unmagnetized components of $\hat{\beta}$ and $\hat{\gamma}$ then
gives

$\frac{\hat{\gamma}}{\hat{\beta}}\sim\frac{E_{\alpha
0}}{6T}\frac{n_{\alpha}}{n}\frac{\ln\Lambda_{ei}}{\ln\Lambda_{\alpha
e}}\Gamma_{\chi}\left(\frac{\chi_{e}}{\beta_{\wedge}}\right),$ (10)

where $n_{\alpha}$ is the number density of non-thermalised $\alpha$
particles. We can assume that the ratio of Coulomb Logarithms is of order
unity and $E_{\alpha 0}/\left(6T\right)\sim 10^{2}$ for an igniting plasma.
Therefore, a relative density of $\alpha$ particles of $n_{\alpha}/n\sim
10^{-2}$ could be sufficient for the $\hat{\gamma}$ term to dominate.

It is also worth noting that $\hat{\beta}\geq 0$ for all values of $\chi_{e}$.
This is because in a plasma the friction force exerted by background ions is
lower for faster electrons. However, $\Gamma_{\chi}<0$ for very large and very
small values of $\chi_{e}$. This is due to $j_{\bot}^{\nu}<0$ at small values
of $\chi_{e}$, while for large values the $-\chi_{\alpha}\gamma_{\wedge}$ term
dominates.

The system of equations (1)-(5) is solved using an implicit Lagrangian model
in order to accurately resolve the steep gradients that exist at the burn
front.Velikovich_POP2015 ; Velikovich_POP2019  We use the isobaric condition,
(2), in place of a momentum equation for simplicity as our principal interest
is the transport processes occurring in (3)-(5). However, this limits our
studies to burn propagation regimes driven by deflagration rather than
detonation.

Figure 1: The variation of the diffusion coefficients $\hat{\beta}$ and
$\hat{\gamma}$ with $\chi_{e}$, where we assume $\chi_{\alpha}=\chi_{e}/2$.
Note that $\beta_{\wedge}/\chi_{e}\geq 0\,\forall\,\chi_{e}$ while
$\Gamma_{\chi}<0$ for $\chi_{e}\ll 1$ and $\chi_{e}\gg 1$. For $\chi_{e}=0$ we
have $\Gamma_{\chi}\approx-1.7$. The scale of the horizontal axis of the inset
plot represents $\log_{10}\chi_{e}$.

##  III Burn propagation dynamics for various $B$ profiles

In this section we apply the model to a variety of different initial $B$ field
profiles. In each case the initial temperature profile is chosen to be
sigmoidal

$T_{0}\left(\hat{x}\right)=T_{c}+\left(T_{h}-T_{c}\right)\left(1+\exp\left(-10\hat{x}\right)\right)^{-1},$

where $T_{c}$ ($T_{h}$) is the temperature of the cold (hot) fuel. The factor
of $10$ ensures that $98$% of the temperature change takes place within a
region of unity length. The normalizing constant, $L_{T}$, is the burn front
scale length. Its value is chosen to be some multiple of the stopping distance
of a $3.45\,MeV$ $\alpha$ particle at the conditions of the midpoint of the
initial temperature profile.

###  III.1 Uniform $B$ field

We begin with the case of a $B$ field whose strength is initially uniform
across the burn front. Since $B$ is constant, $\chi_{e}\propto\tau_{ei}$, and
the initial value of $\chi_{e}$ will vary significantly across the burn front
with the largest value in the hot fuel. Figure 2 shows results for three
different initial values of $B$ in which $L_{T}$ is $0.3$ times the $\alpha$
stopping distance, corresponding to $\sim 15\,\mu m$. The initial values of
$\chi_{e}$ in the hot fuel were $0.1,6,40$, corresponding to $B_{0}\approx
80,500,3500\,T$. The corresponding values in the cold fuel were a factor of
$\sim 800$ smaller. We note that at $\chi_{e}=0.1,\,6,\,40$, the electron
thermal conductivity is a factor of $\sim 1,\,10^{-1}\,10^{-3}$ lower than its
unmagnetized value. The maximum mean free path of thermal electrons in this
system is $\lambda_{ei}\approx 0.009$, ensuring validity for our MHD model.

As burn propagates the evolution of the burn front is highly sensitive to the
$B$ field. The $\chi_{e}=0.1$ case (which closely resembles an unmagnetized
burn front) develops a temperature profile that is far less steep than the
more magnetized cases and propagates furthest into the cold fuel. However,
this burn front also contains a localized region of steep temperature gradient
near the cold fuel, which we refer to as the foot of the burn front. This is
caused by rapid $\alpha$ heating of the cold, dense fuel. In the more
magnetized cases, the burn propagates less far due to suppression by the
magnetic field of thermal conduction and $\alpha$ transport from hot to cold
fuel.

Transport of the $B$ field during burn is also dependent on $\chi_{e}$. The
middle diagram of fig. 2 illustrates competing transport effects. The Nernst
and $\alpha$ flux effects transport $B$ field from hot to cold plasma. This is
most clearly evidenced by the spikes at the foot of the burn front where
temperature gradients rapidly increase. This is similar to the Nernst waves
that have previously been studied in MagLIF and magnetized laser
ablation.Velikovich_POP2019 ; Nishiguchi_PRL1984  Magnetic resistivity limits
the sharpness of these spikes. Meanwhile, ablation of cold fuel by the burn
front leads to $B$ field being advected from cold to hot fuel. Evidence of the
ablative effect is shown in the $\chi_{e}=6$ case, where a dip in $B$ field
centered at $\hat{x}=1.7$ coincides with a shallower temperature gradient.

The dynamics at an evolving burn front can cause both compression and
rarefaction of a magnetic field. Due to the isobaric condition and the
increasing temperature of the plasma, advection does not cause any compression
of $B$ field, only rarefaction. However, the Nernst effect also contributes to
rarefaction of the $B$ field in regions behind the compression wave. This
effect has previously been observed in the compression phase of
MagLIFSlutz_POP2010  and will be more clearly shown in fig. 4.

The bottom diagram of fig. 2 shows profiles for $\sigma_{\chi}$, the
exponential growth rate for $\chi_{e}$ at $\hat{t}=5\mathcal{T}_{h}$. The
value of $\chi_{e}$ can increase due to both heating of the plasma (which
increases $\tau_{ei}$) and $B$ field transport. In the cases of $\chi_{e}=0.1$
and $6$, peak values of $\sigma_{\chi}$ occur near the foot of the burn front
and are significantly larger than the $\sigma_{\chi}$ values in the hot fuel.
In the $\chi_{e}=40$ case, the temperature at the foot of the burn front is
increasing at a slower rate and so the value of $\sigma_{\chi}$ is
approximately the same as that in the hot fuel.

The effects of these transport phenomena on burn propagation are illustrated
in fig. 3. There is a significant drop in the rate of burn propagation as the
initial value of $\chi_{e}$ increases. The hot fuel is insulated from the cold
fuel and, since $\sigma_{\chi}$ is large at the burn front, this insulating
effect increases as burn evolves. Similar behaviour is found for a wide range
of initial parameters ($n_{h}\sim\left[10^{28},5\times 10^{31}\right]\,m^{-3}$
and $T_{h}>\sim 5\,keV$).

We conclude this subsection with the results of an integrated simulation, see
fig. 4, carried out using the Chimera codeTong_NF2019  which includes
radiation transport, extended-MHD and $\alpha$ transport package. However, the
$\alpha-e$ collisional term has not yet been included in this model. These
simulations used a similar set-up and initial conditions to those shown in
fig. 2 but with an initial value of $\chi_{e}\approx 1$ in the hot fuel.

The Nernst term causes a significant spike in the $B$ field at the burn front,
with a corresponding spike in $\chi_{e}$, by compressing the $B$ field. A
comparison of the $B$ field profiles with and without Nernst, shown in the
middle panel, demonstrates how the Nernst also causes rarefaction of the $B$
field behind this compression wave. The bottom panel of fig. 4 shows the
velocity components of $B$ field transport with $v_{fluid}$ ($v_{Nernst}$)
equivalent to $\hat{u}$ ($\hat{\beta}\partial\ln T/\partial\hat{x}$) in (3).
From this we can see that the spike in $B$ field corresponds to a region in
which the absolute value of $v_{Nernst}$ has a local minimum. Although the
isobaric condition no longer applies, the evolution of the profiles of $\rho$,
$T$ and $B$ is qualitatively similar to the results obtained from the model
outlined in section II and suggest that observations from this model are not
unduly compromised by the isobaric condition.

Figure 2: Profiles at $\hat{t}=5\mathcal{T}_{h}$ for initially uniform $B$
field with $\chi_{e}=0.1,6,40$ in the hot fuel. Top: Temperatures. Initial
temperature, $T_{0}$, and density, $\rho_{0}$, are also shown. Middle: $B$
fields where $B_{0}\approx 80,500,3500\,T$ for $\chi_{e}=0.1,6,40$,
respectively. Bottom: Growth rates of $\chi_{e}$. Parameter values of
$T_{c}=500\,eV$, $T_{h}=10\,keV$, $n_{h}=10^{31}\,m^{-3}$ and $L_{T}=0.3$ were
used. Note that the isobaric condition means that the self-heating hot fuel is
allowed to freely expand, resulting in $B$ decreasing uniformly in the hot
fuel as $T$ increases. Figure 3: Ratio of burn rate in cold fuel to the
unmagnetized burn rate at $\hat{t}=5\mathcal{T}_{h}$. Cold fuel is defined as
all fuel with an initial temperature less than $\left(T_{c}+T_{h}\right)/2$.
The $B$ field is initially uniform and $\chi_{e}$ denotes the initial value in
the hot fuel. Values of $T_{c}$, $T_{h}$ and $n_{h}$ are as in fig. 2. The
dashed black line indicates $L_{T}=0.3$, the cases shown in fig. 2. Figure 4:
Results from an integrated simulation with an initially uniform $B$ field.
Top: Initial and final profiles of temperature and density. Middle: Final
profiles of $B$ field with and without the Nernst term included. Also, initial
and final $\chi_{e}$ profiles. Bottom: $B$ field advection velocity
($v_{fluid}$), Nernst velocity ($v_{Nernst}$) and their sum.

###  III.2 Uniform electron Hall parameter

We next consider the case of a burn front in which $\chi_{e}$ is initially
uniform. Figure 5 shows an example of burn propagation where initially
$\chi_{e}=0.1$. To achieve uniformity of $\chi_{e}$, the initial $B$ field
needs to be significantly larger in the cold fuel and so we choose a parameter
value of $n_{h}=10^{30}\,m^{-3}$ to ensure that the magnetic pressure is small
compared with thermal pressure.

The top diagram of fig. 5 illustrates how $\chi_{e}$ grows rapidly in the burn
front region due to both increasing temperature and increasing magnetic field.
The bottom diagram shows different terms contributing to magnetic field growth
in the burn front region.

The final $T$ profile has a distinctive shape with very steep temperature
gradients developing at two locations. First, at the foot of the burn pulse
where $\alpha$ heating is largest (see fig. 6) and secondly in the region
where $\chi_{e}$ reaches a maximum. This second steep temperature gradient
evolves to compensate for the drop in electron thermal conductivity caused by
increasing $\chi_{e}$. This temperature profile is reminiscent of double
ablation front structuresDrean_POP2010 ; Sanz_POP2009  which occur in direct-
drive ICF. In that scenario radiation energy and electron heat flux are
absorbed at different locations, leading to the formation of the double
ablation front. In our case the first front is formed by $\alpha$ heating
while the second front is due to the suppression of energy transport.

At the location of the second steep temperature gradient, advection is causing
$B$ to grow while $\alpha-e$ collisional and Nernst terms are preventing $B$
from penetrating further into the hot fuel, as can be seen from the bottom
diagram of fig. 5. We describe the region where $\chi_{e}$ grows rapidly as
self-insulating since the growth is driven by the dynamics of burn
propagation. The top diagram of fig. 5 also shows the final profiles of
$\chi_{e}$ that are obtained if these two terms are neglected in the induction
equation. Clearly, these terms play a crucial role in establishing the self-
insulating region.

Finally, fig. 6 shows profiles of quantities related to the $\alpha$
particles. It is found that $\chi_{\alpha}$ evolves in a similar way to
$\chi_{e}$ with a spike in $\chi_{\alpha}$ occurring at the burn front. This
causes a sharp drop in $F_{\bot}$ and a spike in $F_{\wedge}$ in the self-
insulating region, while $\alpha$ heating propagates less far into the cold
fuel compared with the unmagnetized case.

Figure 5: Top: Evolution of $\chi_{e}$ over a period of $8\mathcal{T}_{h}$
given an initial value $\chi_{e}=0.1$. Also shown are initial and final $T$
profiles, and the final $\chi_{e}$ profiles when the (i) $\alpha-e$ collision
term (labelled $\gamma=0$) and (ii) $\alpha-e$ and Nernst terms (labelled
$\gamma,\beta=0$) are neglected. Bottom: Components of $B$ field growth rate
at the final time in the self-insulating region. Also shown are initial and
final $B$ field profiles. Parameters: $T_{c}=500\,eV$, $T_{h}=10\,keV$,
$n_{h}=10^{30}\,m^{-3}$, $L_{T}=0.3$. Figure 6: Profiles of $\chi_{\alpha}$,
$\alpha$ heating and $\alpha$ flux terms after $8\mathcal{T}_{h}$. Also shown
are the profiles for the case with no $B$ field. The Larmor radius for
$\alpha$ particle with energy $\langle E_{\alpha}\rangle$ at maximum
$\chi_{\alpha}$ is $\approx 4$ and its mean free path is $\approx 0.11$.

###  III.3 Local Magnetization

Integrated simulations of unmagentized ICF implosions have shown that self-
generated $B$ fields can develop at the interface of hot and cold
fuel.Walsh_PRL2017  In this section, we consider how such a local $B$ field
could evolve during burn by assuming that the initial $\chi_{e}$ has a
Gaussian profile that is narrower than $L_{T}$. An example of such a scenario
is shown in fig. 7 where the FWHM of the initial $\chi_{e}$ profile is
$1/3L_{T}$ with a peak value of $\chi_{e}=1$.

The $B$ field is rapidly advected from cold to hot fuel by the propagating
burn. However, the Nernst and $\alpha-e$ collision terms cause this advected
$B$ field to maintain a steep front. This coincides with the front of the
rapidly growing $\chi_{e}$ profile and a region of very steep temperature
gradient. As in the case of initially uniform $\chi_{e}$, the growth in
$\chi_{e}$ leads to the formation of a self-insulating layer between hot and
cold fuel. This behaviour is rather counter-intuitive. Our usual expectation
would be that the increasing temperature would increase the rate at which
transport processes act to smooth out the steep gradients. Instead we see that
the increasing temperature leads to increasing $\chi_{e}$ and a reduction in
energy transport, such that the gradients become even steeper.

Figure 7 also shows the reduction in burn rate of cold fuel as a function of
time. We can see from this that even though the majority of fuel is
unmagnetized the growth of $\chi_{e}$ during burn can significantly reduce the
rate of burn propagation.

We note that the value of $\kappa_{\bot e}^{c}$, the electron thermal
conductivity coefficient, is about an order of magnitude smaller at
$\chi_{e}=7$ compared with $\chi_{e}=1$. Theoretical models of burn in
unmagnetized ICFGuskov_1976 ; Hurricane_PPCF2018 ; Christopherson_POP2020
typically do not account for such rapid changes in conductivity but this may
be necessary if the presence of self-generated $B$ fields is verified. Such
fields may also alter the ablative stabilization of Rayleigh-Taylor
instabilities during burn.Lobatchev_PRL2000

Figure 7: Top: Profiles of $\chi_{e}$, $B$ and $T$ at $\hat{t}=0$ and
$\hat{t}=8\mathcal{T}_{h}$ for an initially local $B$ field profile. Also
shown are $B$ and $T$ profiles for when Nernst and $\alpha-e$ collision terms
are neglected in the induction equation ($\gamma,\beta=0$) and the temperature
profile for an unmagnetized plasma. Bottom: Ratio of cold fuel burn rate to
unmagnetized rate as a function of time. Parameter values of $T_{c}=500\,eV$,
$T_{h}=10\,keV$, $n_{h}=10^{31}\,m^{-3}$ and $L_{T}=0.4=24\,\mu m$ were used.

##  IV Hall parameter growth rates

The formation of a self-insulating layer with two steep temperature gradients
during burn propagation is observed across a broad range of simulation
parameters and initial $B$ field profiles. It is a result of the Hall
parameter growing faster at the burn front than in the hot fuel. We can
investigate its formation by considering the factors on which $\sigma_{\chi}$
depends. Starting with the definition $\chi_{e}=eB\tau_{ei}/m_{e}$ and
assuming an isobaric, high-beta plasma with thermal pressure $P_{0}=2nT$ we
obtain

$\sigma_{\chi}=\frac{D\ln\chi_{e}}{D\hat{t}}=\frac{1}{B}\frac{DB}{D\hat{t}}+\frac{g}{T}\frac{DT}{D\hat{t}},$
(11)

where $g=\left(5-3\left(\ln\Lambda_{ei}\right)^{-1}\right)/2\approx 5/2$. If
we assume that $\chi_{e}$ is uniform, neglect bremsstrahlung losses
($\hat{q}_{\alpha}\gg\hat{P}$) and neglect resistive diffusion and
Ettingshausen heat flow then using (3) and (4) results in

$\displaystyle\sigma_{\chi}=$
$\displaystyle\frac{2\left(g-1\right)}{5P_{0}}\hat{q}_{\alpha}+\hat{\beta}\left(\frac{\partial^{2}\ln
T}{\partial\hat{x}^{2}}+\left(\frac{\partial\ln
T}{\partial\hat{x}}\right)^{2}\right)+\frac{2\left(g-1\right)}{5P_{0}}T\hat{\kappa}\left(\frac{\partial^{2}\ln
T}{\partial\hat{x}^{2}}+\left(g+1\right)\left(\frac{\partial\ln
T}{\partial\hat{x}}\right)^{2}\right)$ (12)
$\displaystyle+\hat{\gamma}\left(\frac{\partial^{2}\ln\mathcal{E}_{\alpha}}{\partial\hat{x}^{2}}+\left(\frac{\partial\ln\mathcal{E}_{\alpha}}{\partial\hat{x}}\right)^{2}+\frac{\partial\ln
T}{\partial\hat{x}}\frac{\partial\ln\mathcal{E}_{\alpha}}{\partial\hat{x}}\right),$

The first term on the right-hand-side (the $\alpha$ heating term) will always
be positive but the three remaining terms could be positive or negative,
depending on the profiles of $T$ and $\mathcal{E}_{\alpha}$. This equation
illustrates why the steep, opposing gradients in $\chi_{e}$ and $T$ are
observed in figs. 5 and 7: locations with $\partial^{2}\ln
T/\partial\hat{x}^{2}<0$ lead to $\sigma_{\chi}<0$ and vice versa.

Assuming that the gradient length scales of $T$ and $\mathcal{E}_{\alpha}$ are
similar, we can obtain the following estimates of the strengths of the
different terms

$\displaystyle\frac{2\left(g-1\right)}{5P_{0}}\hat{q}_{\alpha}$
$\displaystyle\approx$ $\displaystyle\mathcal{T}_{H}6.29\times
10^{-5}\ln\Lambda_{\alpha
e}\frac{\mathcal{E}_{\alpha}}{T_{keV}^{\frac{5}{2}}},$
$\displaystyle\hat{\beta}$ $\displaystyle\approx$
$\displaystyle\mathcal{T}_{H}\frac{6.14\times
10^{12}}{\ln\Lambda_{ei}}\frac{T_{keV}^{\frac{7}{2}}}{L_{T\mu}^{2}P_{0G}}\frac{\beta_{\wedge}}{\chi_{e}},$
$\displaystyle\frac{2\left(g-1\right)}{5P_{0}}T\hat{\kappa}$
$\displaystyle\approx$ $\displaystyle\mathcal{T}_{H}\frac{1.84\times
10^{12}}{\ln\Lambda_{ei}}\frac{T_{keV}^{\frac{7}{2}}}{L_{T\mu}^{2}P_{0G}}\left(\kappa_{\bot
e}^{c}+\sqrt{\frac{2m_{e}}{m_{i}}}\kappa_{\bot i}^{c}\right),$
$\displaystyle\hat{\gamma}$ $\displaystyle\approx$
$\displaystyle\mathcal{T}_{H}\frac{4.09\times 10^{-2}}{\ln\Lambda_{\alpha
e}}\mathcal{E}_{\alpha}\frac{T_{keV}^{\frac{7}{2}}}{L_{T\mu}^{2}P_{0G}^{2}}\Gamma_{\chi},$

where $T_{keV}$ is fuel temperature in $keV$, $P_{0G}$ is fuel pressure in
$Gbar$ and $L_{T\mu}$ is temperature length scale in $\mu m$.

From these relations we can see that the Nernst and thermal conductivity terms
are very similar in magnitude. Their dependence on $\chi_{e}$ is shown in fig.
8. This figure illustrates that we expect the largest magnetization growth
rates to occur at small values of $\chi_{e}$. It is likely that the cryogenic
fuel layers of many MIF schemes will have initially low $\chi_{e}$ values due
to the high density and so $\chi_{e}$ can increase rapidly in such regions as
burn propagates.

Finally, we can take the ratio of the $\alpha$ heating term to the $\alpha$-e
collisional term to obtain a criterion (which is indepedent of
$\mathcal{E}_{\alpha}$) for when the latter term is larger

$T_{keV}^{6}\left|\Gamma_{\chi}\right|>\frac{\left(\L_{T\mu}P_{0G}\ln\Lambda_{\alpha
e}\right)^{2}}{650}.$ (13)

From this formula we can, for given values of $\chi_{e}$, $L_{T\mu}$ and
$P_{0G}$, calculate the temperature above which the $\alpha$-e collisional
term is larger. An example of this is shown in fig. 9.

Figure 8: The variation of the dimensionless thermoelectricity and thermal
conductivity coefficients with $\chi_{e}$. For $\chi_{e}>\sim 6$, ion thermal
conductivity is larger than electron thermal conductivity. Figure 9: A plot of
the minimum temperature for which the $\alpha$-e collisional term dominates
the $\alpha$ heating term in the growth rate of $\chi_{e}$ for $L_{T\mu}=10$
and $P_{0G}=100$. This result may be scaled according to (13) for other values
of $L_{T\mu}$ and $P_{0G}$. The dashed vertical lines indicate the values of
$\chi_{e}$ for which $\Gamma_{\chi}=0$.

##  V Conclusions

In this work we have studied the propagation of a subsonic thermonuclear burn
wave across a magnetic field in a high beta plasma using an MHD model. We can
summarize our findings as follows:

  * (i)

Ablation of cold fuel due to thermonuclear burn advects $B$ field from cold to
hot fuel at the burn front, while the Nernst and $\alpha-e$ collision terms
transport $B$ field from hot to cold fuel. These combined transport effects
confine $B$ field in the burn front region and cause both compression and
rarefaction of $B$.

  * (ii)

The electron and $\alpha$ Hall parameters, $\chi_{e}$ and $\chi_{\alpha}$ ,
can grow rapidly at the burn front due to $\alpha$ heating, magnetic field
transport and thermal conduction processes. The growth rate at the burn front
can be significantly larger than in the hot fuel, causing a self-insulating
layer to form at the burn front.

  * (iv)

The temperature profile of a magnetized burn front is steeper than the
unmagnetized case due to the suppression of thermal conduction. Furthermore,
the formation of a self-insulating layer can lead to very steep temperature
gradients in localized regions of the burn front.

  * (iv)

A magnetized burn front reduces the rate at which cold fuel is burnt due to
the self-insulating effect.

These results demonstrate that careful modelling of the $B$ field transport
during burn is required to accurately simulate the propagating burn wave and
predict the energy gain achieved. However, some of the assumptions in our
idealized model prevent us from making quantitative predictions for how $B$
field transport could affect specific MIF schemes. The isobaric condition
restricts us to regimes of subsonic burn (deflagration). Our results prompt
the question of whether the formation of a self-insulating layer at the burn
front could cause a transition to detonation of the hot fuel. Furthermore, the
planar geometry used here is suitable for idealized studies of burn front
dynamics since the hot fuel at infinity is unaffected by the cold fuel. It is
likely that there will be a more complex interplay between hot fuel and burn
front in finite sized targets. Questions such as these will be addressed in
future work using integrated simulations.

The results also provide further evidenceSlutz_PRL2012 ; Velikovich_ICOPS2012
that an optimum magnetic field strength and profile shape could exist for a
given MIF scheme. Such a field would need to provide the correct amount of
magnothermoinsulation during compression and ignition, followed by sufficient
confinement of $\alpha$ particles during burn propagation without unnecessary
slowing of the burn wave. The strength and shape of this optimum magnetic
field and how it can be generated remain open questions.

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