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Sherlock, Mark, Brodrick, Jonathan Peter and Ridgers, Christopher Paul (2017) A comparison of non-local electron transport models for laser-plasmas relevant to inertial confinement fusion. Physics of Plasmas. pp. 1-10. ISSN 1089-7674

https://doi.org/10.1063/1.4986095

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A comparison of non-local electron transport models for laser-plasmas relevant to

inertial con�nement fusion

M.Sherlock∗

Lawrence Livermore National Laboratory, P.O. Box 808, Livermore, CA 94551, USA

J.P.Brodrick and C.P.RidgersYork Plasma Institute, University of York, York, YO10 5DD, UK

(Dated: 1st Nov 2016)

We compare the reduced non-local electron transport model developed by Schurtz et al. (Phys.Plasmas 7, 4238 (2000)) to Vlasov-Fokker-Planck simulations. Two new test cases are considered:the propagation of a heat wave through a high density region into a lower density gas, and a 1-dimensional hohlraum ablation problem. We �nd the reduced model reproduces the peak heat �uxwell in the ablation region but signi�cantly over-predicts the coronal preheat. The suitability ofthe reduced model for computing non-local transport e�ects other than thermal conductivity isconsidered by comparing the computed distribution function to the Vlasov-Fokker-Planck distribu-tion function. It is shown that even when the reduced model reproduces the correct heat �ux, thedistribution function is signi�cantly di�erent to the Vlasov-Fokker-Planck prediction. Two simplemodi�cations are considered which improve agreement between models in the coronal region.

I. INTRODUCTION

In laser-produced plasmas relevant to Inertial Con�ne-ment Fusion (ICF) the electron temperature gradientsare often so strong that the scale-length of the tempera-ture variation becomes comparable to the mean-free-pathof the electrons which transport thermal energy. As a re-sult, the heat �ux can not be determined by the local con-ditions in the plasma and the �nite electron mean-free-path must be taken into account (i.e. the thermal energytransport across the gradient is no longer purely di�u-sive). In these situations, the heat �ux can be calculatedwith Vlasov-Fokker-Planck (VFP) models [1], which re-solve the motion, scattering and acceleration of electrons,but with much greater computational cost in comparisonto local thermal transport models.A widely used alternative to the full VFP models is the

reduced multi-group model proposed by Schurtz, Nico-laï and Busquet (the �SNB� model) [2]. Marrocchino et

al. have also compared the SNB model to a VFP modelwhich retains higher order terms in the expanded dis-tribution function [3]. The purpose of this paper is tocompare the SNB model to two further test cases rel-evant to ICF: (1) a �burn-through� problem in whicha heat-front breaks out of a high density region into alower density gas and (2) a physically realistic simula-tion of a laser-ablated plasma over hydrodynamic time-scales. These scenarios are typically more challenging forVlasov-Fokker-Planck simulations codes (than the usualhotspot relaxation problems studied previously) due tothe presence of large density gradients.We are also interested in checking the applicability of

SNB-like models to study transport e�ects other thanheat �ow, for example non-local Nernst advection of

∗ sherlock3@llnl.gov

magnetic �eld and non-local corrections to electron Lan-dau damping rates (which are important for determin-ing Stimulated Raman Scattering gain spectra). Thesee�ects rely on a correct representation of the electrondistribution function, so we also present some compar-isons of the SNB and VFP distribution functions for thesimple case of a linear temperature gradient. We �ndthat even when the SNB model predicts the correct heat�ow, it may not predict the correct underlying electrondistribution function (somewhat paradoxically).

II. COMPUTATIONAL MODELS

The VFP model used in this paper (�K2�) solvesthe Vlasov-Fokker-Planck equation for the elec-tron distribution function f (t, x, y, px, py, pz) =∑

n,m fmn (t, x, y, p)Y m

n (θ, ϕ) expanded in spherical

harmonics in momentum-space (p, θ, ϕ) to arbitraryorder. The details of the K2 model are described inthe Appendix, here we give a brief overview. Theelectron VFP equation is coupled to the equations ofradiation hydrodynamics which account for ion motion,PdV work, electron-ion equilibration and radiativecooling. These e�ects are coupled to the electron VFPequation via heating and cooling operators, and theelectron distribution function is advected with the ionbackground to maintain quasineutrality. The electronVFP part of the model accounts for thermal transport,including the determination of a self-consistent electric�eld and return current. Laser absorption is modelledby solving a simple ray equation, with the heatingof electrons determined by an inverse-bremsstrahlungheating operator (which accounts for the distortionof the electron distribution function due to electronoscillation in the laser �eld).

We use the iterative/implicit algorithm proposed by

2

Cao et al [11] to solve the SNB multi-group energy trans-port equations, with only two iteration cycles. We see nosigni�cant di�erence in results when using more iterationcycles in a few test runs, and this is probably due to thefact that the timestep used is signi�cantly shorter thantypical hydrodynamic simulation timesteps (as we resolvethe electron-ion collision frequency). We have developedtwo SNB models independently and benchmarked themagainst each other, for a number of test cases.Our implementation of the SNB model contains a

subtle modi�cation to the mean-free-path which is dif-ferent to that in the original model. For a particlewith energy ǫg (where �g� is the energy group index),we use the geometrically averaged mean-free-path λg =

2√2 (ǫg/Te)

2λ0,where (in cgs units):

λ0 =T 2e

4πne

√Ze4 lnΛ

1√φ

(1)

and φ = (Z∗ + 4.2) / (Z∗ + 0.24). Our de�nition canbe compared to Equation 23 in the original SNB model[2]. We �nd that use of this mean-free-path improvesagreement between the SNB and VFP models in high-Zplasmas. See [14] for a full discussion of this choice ofmean-free-path (in the notation used in [14], our modelis equivalent to the choice of �r=2�). The reason for thismodi�cation stems from the fact that the original SNBmodel employs a Krook electron-electron collision oper-ator. Since a Krook operator is only an approximationto the Fokker-Planck operator, there is some degree offreedom in choosing the exact form of the collision fre-quency. The factor φ, originally introduced in [15], isa common feature in VFP simulation models, allowingthem to reproduce Spitzer's thermal conductivity in thelow-Z limit without the need for an anisotropic collisionoperator. This allows the SNB model to more accuratelyaccount for the e�ect of electron-electron collisions in thenon-local contribution to the heat �ow.All of the following test problems use f0 and f1 only

(i.e. higher order harmonics are not included), exceptthose in section IV, which also include f2. We veri�edin each case that the inclusion of higher order terms didnot produce any signi�cant di�erences in the results. Al-though the code is 2-dimensional, the test cases consid-ered in this paper are 1-dimensional (and only require aLegendre expansion in momentum-space). Note that thesame Coulomb logarithm is used in both the SNB modeland the K2 VFP model in each simulation (see below).

III. SIMPLE HEAT-BATH PROBLEM

We are interested in comparing the electron velocitydistribution function (EDF) in the SNB model to thatgiven by the K2 VFP model, as this gives us insightinto how useful the SNB model could be in determin-ing transport e�ects other than non-local heat �ow inthe unmagnetized regime. For example, heat �ow in

300 400 500 6000

200

400

600

800

1000

T e (e

V)

x (microns)

VFP Initial (t=0) SNB

Figure 1. Heat Bath Problem: The electron temperatures att=80ps. The initial temperature pro�le is also shown. (Notethat the simulation box extends from 0µm to 700µm, butwe reduced the range of the x-axis in the plot to allow thegradient's features to be more visible).

magnetized plasmas involves thermoelectric coe�cientswhich are not the same velocity moments of the EDF asthe thermal conductivity coe�cients, and the dampingof electrostatic waves is determined by the shape of theEDF near the wave phase velocity. These thermoelectriccoe�cients, along with other moments of the EDF, alsocontribute to the electric �eld (which in turn a�ects themagnetic �eld) through an e�ective Ohm's Law.Our �rst test case is a simple non-linear heat-bath

problem, in which the initial temperature (plotted in Fig-ure 1) is 1000eV for x < 400µm, with a linear ramp over100µm down to 100eV (over 100 computational spatialcells). The typical ratio of the electron mean-free-pathto temperature-gradient scale-length is λei/LT ≈ 0.013.The total computational box size is 700µm. The electrondensity and charge state are �xed at 2.47 × 1026m−3,Z∗ = 50, and the coulomb logarithm was held �xedthroughout, logΛ = 7.1. The relatively high value of Z∗

was chosen to compliment the comparisons in [3] (whichused the values Z∗ = 1, 2, 4). We plot the electron tem-peratures after 80ps in Figure 1 and the VFP heat �uxis compared to the SNB heat �ux in Figure 2, showingreasonably good agreement for the characteristic reduc-tion of the peak relative to Spitzer. In this test-case, wedo not include harmonics above f1 (f2 is set to zero).Simulations including f2 do not show any signi�cant dif-ferences in the heat �ux. The temperature pro�les inFigure 1 indicate that the SNB model tends to overpre-dict the preheating. This type of simulation comparisonis not new (hotspot relaxation was simulated and com-pared to SNB in [3]), but we address the question of howwell the SNB model predicts the energy distribution ofheat-�ux-carrying electrons in simple scenarios such asthis.The distribution of heat �ux for this test case, dqx ∝

v5f1 (v), is plotted in Figure 3 at x = 500µm, where

3

0 100 200 300 400 500 600 7000

1x1016

2x1016

3x1016

4x1016

q x (W

m-2

)

x (microns)

Spitzer K2 SNB

Figure 2. Heat Bath Problem: The electron heat �uxes at10ps.

the K2 and SNB heat �uxes are in very good agreement.However, the underlying distributions that give rise tothis heat �ux are signi�cantly di�erent. This indicatessome modi�cations to the SNB model may be requiredin order for it to reproduce other aspects of transport(mentioned above). In particular, we note the heat �uxaccording to K2 is carried by higher energy electrons onaverage, and K2 predicts a signi�cant return heat �ux(associated with the return current). Although returnelectrons do not contribute signi�cantly to delocalization,their population is enhanced when the heat �ux exceedsthe local value due to the enhanced current carried bythe energetic forward electrons.In the original paper [2], no way of recovering f1 is

given explicitly. In general, if f1 is known, the heat �uxcan be calculated with the following relation:

q = x̂4π

3

m

2

∫

∞

0

f1 (v) v5dv (2)

where m is the electron mass and v the velocity magni-tude. This integral is numerically represented as a sum:

q = x̂4π

3

m

2

∑

g

f1 (vg) v5g∆vg (3)

(where g numbers the velocity group vg and ∆vg is thevelocity bin size of group g). The SNB model expressesthe heat �ux, qSNB , in terms of the Spitzer value (qSH)and a non-local correction:

qSNB = qSH −∑

g

λ′

g

3∇Hg (4)

where Hg is the principle function to be solved for in

the SNB model, de�ned in Equation 27 of [2], and λ′

g is

the electron mean-free-path of energy group g, de�ned inEquation 23 of [2] (and adjusted in our formulation, asmentioned in the previous chapter).The Spitzer heat �ux qSH appearing in the above equa-

tion can be expressed in terms of the Spitzer expressionfor f1 (denoted fmb

1 ) as:

qSH = x̂4π

3

m

2

∫

∞

0

fmb1 (v) v5dv (5)

where

fmb1 (v) =

v

3νei (v)

{

mv2

2Te− 4

}

fmb0 (v)

∇Te

Te(6)

In the above, Te is the electron temperature, fmb0 (v)is

the Maxwell-Boltzmann distribution and νei is theelectron-ion collision frequency[15].Again, the integral in Equation 5 is represented nu-

merically as a sum:

qSH = x̂4π

3

m

2

∑

g

fmb1 (vg) v

5g∆vg (7)

This expression can be substituted into Equation 4,allowing us to rewrite it as:

qSNB =∑

g

(

x̂4π

3

m

2fmb1 (vg) v

5g∆vg −

λ′

g

3∇Hg

)

(8)

In order to �nd an expression for f1 according to theSNB model, we note that Equations 8 and 3 refer to thesame heat �ux and can therefore be equated:

x̂4π3

m2

∑

g f1 (vg) v5g∆vg

=∑

g

(

x̂4π3

m2 f

mb1 (vg) v

5g∆vg −

λ′

g

3 ∇Hg

)

(9)

and we therefore infer a possible de�nition for f1 (vg)could be the expression:

x̂4π3

m2 f1 (vg) v

5g∆vg

= x̂4π3

m2 f

mb1 (vg) v

5g∆vg −

λ′

g

3 ∇Hg

(10)

In other words, we have assumed that the quantitywhich is integrated to get the heat �ux is ∝ f1v

5. Thisde�nition for f1 is consistent with the Vlasov-Fokker-Planck derivation of the SNB equations, which givex̂ν

′

ei∆f1 = −v∇ (∆f0) /3 (see Equation 56 of [2] and thetext preceding it).

4

0.00 0.05 0.10 0.15 0.20 0.25 0.30-1.0x1017

-5.0x1016

0.0

5.0x1016

1.0x1017

1.5x1017

2.0x1017v5 f 1

(arb

itrar

y un

its)

p/mc

K2 Spitzer SNB

x=500um

Figure 3. Heat Bath Problem: The �heat �ux distributionfunction�, dqx ∝ v5f1 (v) , at x = 500µm and t = 10ps, whichindicates how the heat �ux is carried by each velocity group.Although K2 and the SNB model predict the same total heat�ux, the underlying distribution function is di�erent.

IV. BURN-THROUGH PROBLEM

In this test case we consider a thin �target� with peakdensity = 2 × 1028m−3 (∼ 2 × ncrit for 3ω light) whichis heated on the RHS by a laser (0.35µm wavelength),with a linear 1ns rise time up to a peak intensity of1015Wcm−2 . On the LHS of the target, the electron den-sity drops to a constant low value = 1.45×1026m−3, rep-resenting a gas �ll. The density is held �xed throughoutthe simulation (hydrodynamic motion is turned o�). Theinitial electron temperature was 50eV , and the coulomblogarithm logΛ = 7.1 and charge state Z∗ = 2 wereheld �xed throughout the simulation. Harmonics up toand including f2 were used in this test case. The ini-tial density pro�le is shown in Figure 4, along with thetemperature (according to K2) at 70ps, showing the hotcoronal plasma on the RHS. Again we use 100 computa-tional spatial cells.

The temperatures according to K2 and the SNB modelare plotted at 170ps (after the heat front has penetratedthe shell and entered the gas) in Figure 5.

The SNB model overpredicts the preheat of the shell(similarly to the �linear ramp� test-case), as is evidentfrom the increased temperature at x ≈ 220µm. The ini-tial preheat of the gas, close to the gas-shell interface atx ≈ 170µm, is also overpredicted by the SNB model. Thedip in temperature at x ≈ 200µm, apparent in both mod-els, is due to hot electrons traversing this region withoutdepositing much energy. Interestingly, the K2 model pre-dicts enhanced long-range heating of the gas (in the re-gion x . 100µm), probably indicating a breakdown ofdi�usive transport. This is consistent with the fact thatwe �nd f2 makes signi�cant contributions to the heat�ux in this region, which will allow for a strong depar-ture from equilibrium. As a result of this, the heat �ux

0 100 200 300 4000.0

5.0x1027

1.0x1028

1.5x1028

2.0x1028

x (microns)

n e (m

-3)

"shell"

0

500

1000

1500

2000

T e (e

V)

t=70ps

"gas" density

Figure 4. Burn-Through Problem: The initial density pro-�le and electron temperature (according to the K2 code) att=70ps.

0 100 200 300 40010

100

1000

x (microns)

T e (e

V) K2 SNB

Figure 5. Burn-Through Problem: The temperatures att=170ps (after the heat front has penetrated the shell andentered the gas region). The SNB model overpredicts thepreheat of the shell, and the initial preheat close to the gas-shell interface, but it underpredicts the long-range heating ofthe gas. The initial temperature was 50eV .

in this region (x ≈ 100µm) is about twice as large asthat predicted by the SNB model (at t = 170ps). Con-tributions from f3 have been checked-for and found to benegligible.

Although this test case bears many similarities withthe ICF scenario of capsule gas preheat, we cautionagainst drawing too many conclusions regarding theirsimilarity because the two scenarios di�er in importantways: (a) the ICF shell has a much higher ρR, sothe spectrum of hot electrons (generated by LPI in thecorona) that reach the gas may be di�erent, and (b)the gas is initially much colder than in our simulation,so a Fokker-Planck/Spitzer transport treatment is notvalid (which is the reason we did not choose to tackle

5

the full problem with our models). However, this prob-lem represents an important benchmark for reduced non-local models, and highlights the interesting discrepancyin both short- and long-range heating.

V. HOHLRAUM ABLATION PROBLEM

In this test case, we consider the ablation of a Au-likewall next to a He gas, including hydrodynamics. We usean arti�cal material to represent the Au wall, which has areduced solid mass density of 3.9g/cc (to enable us to runwith a shorter timestep) and the ionization state is arti-�cially capped at Z∗ ≤ 40 (which allows us to take intoaccount the fact that the Thomas-Fermi ionization levelis too high compared to non-LTE models). The solid Aumaterial initially occupies the region x < 100µm, and Heoccupies the region 100 ≤ x ≤ 700µm. The laser pulseshape is shown in Figure 6. The hydrodynamic part ofthe K2 model is run alone (i.e. with VFP and SNB trans-port turned o� and Spitzer transport turned on) for the�rst 1500ps to produce realistic density and temperatureconditions (shown in Figure 7), after which we allow thehydrodynamics to evolve with the heat �ux calculatedfrom the VFP solver. The SNB heat �ux and temper-atures are self-consistently evolved separately from theVFP temperature, although both �temperature� mod-els use the same hydrodynamics heating/cooling terms(PdV work, radiation cooling etc). The previous testcases were somewhat arti�cial/simpli�ed in their design- here we attempt to give an indication of how the mod-els compare in a more realistic scenario. This scenariois similar to the 2D test case compared against in theoriginal SNB work (taken from Epperlein's work[12]),except we include the e�ects of hydrodynamic motion,electron-ion equilibration, ionization and radiation cool-ing etc, and simulate the Au-gas interface and longerpulse lengths, which should be more directly relevant tohohlraum transport. Since simulations in this section aredesigned to be more physically realistic than the previ-ous cases, the Coulomb logarithm is calculated locally ateach time-step using the NRL prescription [13].The heat �ux pro�les are compared at t=1640ps (i.e.

140ps after kinetic e�ects were turned on) in Figure 8.The SNB model predicts the peak heat �ux very wellnear the ablation surface (x ≈ 110µm), but overpredictsthe heat �ux into the corona by a factor of ≈ 2. We �ndthis type of behavior persists over 100's of picoseconds.The overestimation of heat �ux in coronal regions waspredicted and commented on by Schurtz et al. in theiroriginal paper[2] and also observed in [14].The coronal (He gas) temperature in this simulation is

relatively low (∼ 2keV ), so we also performed a simula-tion over a longer timescale and at higher laser intensity(∼ 4ns foot, peaking at ∼ 1015Wcm−2), giving typicalpeak gas temperatures ∼ 3.5keV . The initial conditionsfor the K2 simulation (i.e. just after the thermal trans-port is switched from the hydrodynamic model to the

0 500 1000 15000

1x1014

2x1014

3x1014

4x1014

inte

nsity

(W

cm-2

)

time (ps)

Figure 6. Hohlraum Ablation Problem: The laser pulse shapefor the hohlraum ablation test case.

0 100 200 300 400 500 600 7001026

1027

1028

1029

ne Te

x (microns)

n e (m

-3)

0

500

1000

1500

2000

2500

T e (e

V)

material boundary

Figure 7. Hohlraum Ablation Problem: The density and tem-perature pro�les at the time (t=1500ps) at which the VFPthermal transport model is turned on in the K2 code (for thehohlraum ablation test case). This represents the initial con-ditions (as provided by pure radiation-hydrodynamics) for thestudy of thermal transport with the SNB and VFP models.The vertical line shows the location of the Au-He boundary(He is on the right).

Fokker-Planck model) in this case are shown in Figure 9.In this case, we �nd reasonable agreement between theK2 and SNB heat �uxes at the ablation surface (the SNBpeak ablation heat �ux is around 10-20% higher than K2throughout the simulation), though not as good as atlower laser intensity (see Figure 10). Again, the peakheat �ux into the corona predicted by K2 is a factor of≈ 2 lower than the SNB prediction.In all hohlraum-relevant cases, we �nd the peak �ux

is typically ≈ 0.035 of the free-streaming value, as il-lustrated in Figure 11 (also at t=5300ps). However, wecaution against interpreting the value of q/qFS as corre-sponding to the value of the �ux-limiter that could be

6

0 100 200 300 400 500 600 700-2x1018

-1x1018

0

1x1018

2x1018

3x1018q x

(Wm

-2)

x (microns)

Spitzer K2 SNB

Figure 8. Hohlraum Ablation Problem: The heat �uxes att=1640ps. The SNB model predicts the peak �ux very wellnear the ablation surface, but overpredicts the �ux into thecorona.

0 500 1000 15001026

1027

1028

1029 ne Te

x (microns)

n e (m

-3)

0

500

1000

1500

2000

2500

3000

3500

4000

T e (e

V)

material boundary

Figure 9. Hohlraum Ablation Problem (higher intensity): Thedensity and temperature pro�les at the time (t=5160ps) atwhich the VFP thermal transport model is turned on inthe K2 code (for the higher-intensity hohlraum ablation testcase). This represents the initial conditions (as provided bypure radiation-hydrodynamics) for the study of thermal trans-port with the SNB and VFP models. The vertical line showsthe location of the Au-He boundary (He is on the right). Thelaser intensity has a ∼ 4ns foot, peaking at ∼ 1015Wcm−2 att = 5ns.

used with a �ux-limited Spitzer model (necessary to re-produce the actual VFP heat �ux). We also mention inpassing that we have postprocessed data from fully inte-grated 2D radiation-hydrodynamics simulations of NIF-scale hohlraums and �nd there that the peak �ux maybe a higher fraction of the free-streaming value than thefactor 0.035 in these simulations.

Note that in this section we computed the SNB heat�ux from the VFP temperature pro�le, since the aim

0 500 1000 1500-3.0x1018

0.0

3.0x1018

6.0x1018

q x (W

m-2

)

x (microns)

Spitzer K2 SNB

Figure 10. Hohlraum Ablation Problem (higher intensity):The heat �uxes at t=5300ps. The SNB model predicts thepeak �ux reasonably well near the ablation surface, but over-predicts the �ux into the corona by a factor of around 2.

0 500 1000 15000.00

0.02

0.04

0.06

0.08

0.10

x (microns)

qx / free-streaming

Figure 11. Hohlraum Ablation Problem (higher intensity):The ratio of the VFP heat �ux to the free-streaming value(q/qFS), at t=5300ps.

was to focus on the di�erence between the predicted heat�uxes for a given temperature and density pro�le. In thenext section we explore how the heat �uxes actually a�ectthe temperature pro�les.

VI. INLINE COMPARISON

We showed in the previous section that in hothohlraums, the SNB peak heat �ux near the ablation sur-face is overpredicted by 10-20%. However, simply com-puting heat �uxes from given temperature pro�les maynot be a fair comparison of how well the SNB model per-forms when run inline (compared to an inline simulationwith VFP). The argument can be made that if the SNB

7

0 500 1000 1500 2000 2500

0.1

1

10

100

x (microns)

n e/n

crit

0

10

20

30

40

Z*

Z*

ne

Figure 12. The electron density (in units of critical densityfor 3ω light) and ionization state pro�les at t=11.8ns for theinline hohlraum VFP and SNB simulations.

model overpredicts the heat �ux in a region, the temper-ature will drop there at a faster rate, which will in turnlead to a reduction in the local temperature gradient andhence a reduction in the heat �ux. Therefore small er-rors in the model may be �self-correcting�. To explorethis possibility we have made a comparison between fullyinline SNB and VFP (by �inline� we mean that the heat�ux is used to update the temperature during the simula-tion). For this case we freeze the hydrodynamic motionafter a signi�cant amount of ablation has occured, re-sulting in the density pro�le shown in Figure 12. Afterhydrodynamic motion is turned o�, the laser continues toheat the plasma which then evolves to an approximatelysteady-state via a combination of thermal transport andradiative cooling over ∼ 100ps. The corona becomes hotenough in this steady-state (∼ 5keV ) to drive signi�cantnon-local transport.

The temperature pro�les corresponding to this steady-state are shown in Figure 13 for each of the two mod-els. As can be seen, the temperature pro�les are in goodagreement in the ablation/absorption region and only dif-fer by ∼ 300eV in the coronal region. This suggests thattypical errors on the order of 10-20% in the peak ablativeheat-�ux predicted by SNB can e�ectively �self-correct�.We expect that the improvements discussed in the nextsection will further reduce the coronal error.

We have not performed all SNB calculations inlinein this paper due to the very large computational de-mands of resolving the electron collision time in the highdensity regions of the plasma over the long timescalesneeded for an inline comparison - the results in this sec-tion simply serve to bring attention to the possibility of�self-correction� in a relatively simple scenario. Improve-ments to the standard computational techniques used inVFP codes will be necessary in order to allow them toovercome the timestep limitation. This highlights thefact that one of the major design advantages of the SNB

Figure 13. The electron temperatures at t=11.8ns for theinline hohlraum simulations, according to the SNB model andthe K2 VFP model.

model is its natural ability to relax to Spitzer transportin regions of high collisionality (which is due to its ex-pansion of f about the Spitzer distribution function). Asimilar expansion (or equivalent relaxation process) ofthe VFP equation should be possible. A related problemis the di�culty of maintaining quasineutrality in the highdensity regions in VFP simulation models without resolv-ing the plasma period: implicit electric �eld solvers donot guarantee quasineutrality over long (hydrodynamic)timescales, but this problem can be largely overcome bythe fully-implicit simulation technique employed by theIMPACT model [17].

VII. IMPROVEMENTS TO THE SNB MODEL

We have found it is possible to increase the agreementbetween the SNB model and the VFP model in the coro-nal regions by simply modifying the mean-free-path def-initions. The SNB model employs a somewhat arti�cialcombination of electron-electron (λee) and electron-ion(λei) mean-free-paths into a single mean-free-path (�λe�).This is only possible when the ionization Z∗ is assumedto be constant. The use of separated mean-free-paths(considered in detail in [14] and alluded to in [16]) is astraightforward modi�cation of the original model, and itcorrectly accounts for ionization gradients. The resultingheat �ux is shown in Figure 14 for the higher intensitycase described in Section V. The heat �ux is signi�cantlyimproved in the coronal region, with the error in the peakcoronal heat �ux reducing from about 90% to about 27%.

A very similar improvement in the heat-�ux is obtainedby simply neglecting the �electric �eld correction� to themean-free-path. The original SNB paper [2] distinguishesbetween the standard collisional mean-free-path λg and

an electric-�eld-corrected mean-free-path λ′

g related via:

8

Figure 14. Hohlraum Ablation Problem (higher intensity):The improved heat �uxes at t=5120ps. The curve labelled�SNB no E� is the SNB model without the electric �eld correc-tion to the mean-free-path. Neglect of this correction signi�-cantly improves the agreement of the SNB model with VFPin the coronal region. The dashed curve labelled �SNB sep.�corresponds to the SNB model rewritten to make use of sep-arate electron-electron and electron-ion mean-free-paths.

1

λ′

g

=1

λg+

| eE |ǫg

(11)

which serves to further limit the particle stopping dis-tance in regions where the electric �eld may be the domi-nant cause of electron stopping (see the discussion in [2]).In the above expression, E is the electric �eld and ǫg isthe energy of the electron (in energy group �g�). Thegreen curve in Figure 14 shows the heat �ux when theelectric �eld is neglected in the de�nition of the mean-free-path (i.e. by simply setting λ

′

g = λg) - the level ofimprovement is similar to the use of separated mean-free-paths.

SUMMARY

We have compared the SNB non-local transport modelto fully kinetic Vlasov-Fokker-Planck simulations in a va-riety of realistic situations relevant to inertial con�ne-

ment fusion. In hohlraum-like situations, we �nd goodagreement between the SNB model and VFP simulationsat the ablation front (where the heat �ux peaks). Theheat �ow out into the coronal region, however, is typi-cally overpredicted by the SNB model by a factor of ≈ 2.This overprediction can be substantially reduced by ig-noring electric �eld corrections to the mean-free-path orproperly separating the electron-electron and electron-ion mean-free-paths. However, the distribution functionpredicted by the SNB model may be signi�cantly di�er-ent to the VFP distribution function even when the totalheat �uxes are in good agreement, suggesting improve-ments need to be made in order to make the SNB modelmore generally useful as a predictive tool for a wider va-riety of kinetic e�ects. In the more demanding case ofhot electron penetration through a thin shell into a gas,the SNB model signi�cantly overpredicts the short-rangeheat �ux while underestimating the long-range heat �ux.

ACKNOWLEDGEMENTS

This work was performed under the auspices of theU.S. Department of Energy by LLNL under ContractDE-AC52-07NA27344.CPR and JB's contribution to this work was funded by

EPSRC grant EP/K504178/1. CPR's contribution wasalso funded by EPSRC grant EP/M011372/1. CPR'scontribution has been carried out within the frameworkof the EUROfusion Consortium and has received fund-ing from the Euratom research and training programme2014-2018 under grant agreement No 633053 (project ref-erence CfP-AWP17-IFECCFE-01). The views and opin-ions expressed herein do not necessarily re�ect those ofthe European Commission.We would like to thank M.Rosen for stimulating dis-

cussions regarding �ux limiters.

APPENDIX

The VFP model used in this paper (�K2�) solvesthe Vlasov-Fokker-Planck equation for the elec-tron distribution function f (t, x, y, px, py, pz) =∑

n,m fmn (t, x, y, p)Y m

n (θ, φ) expanded in spherical

harmonics in momentum-space (p, θ, φ) to arbitraryorder. The expansion coe�cients fm

n (t, x, y, p) evolve as[4]:

9

∂fm

n

∂t +(

n−m2n−1

)

v∂fm

n−1

∂x +(

n+m+12n+3

)

v∂fm

n+1

∂x

+(

12n−1

)

v2

{

∂fm−1

n−1

∂y − (n−m) (n−m− 1)∂fm+1

n−1

∂y

}

+(

12n+3

)

v2

{

−∂fm−1

n+1

∂y + (n+m+ 1) (n+m+ 2)∂fm+1

n+1

∂y

}

−Fx

{(

n−m2n−1

)

Gmn−1 +

(

n+m+12n+3

)

Hmn+1

}

−Fy

2

{(

12n−1

)

[

Gm−1n−1 − (n−m) (n−m− 1)Gm+1

n−1

]

+(

12n+3

)

[

−Hm−1n+1 + (n+m+ 1) (n+m+ 2)Hm+1

n+1

]

}

− iFz

2

{(

12n−1

)

[

−Gm−1n−1 − (n−m) (n−m− 1)Gm+1

−1

]

+(

12n+3

)

[

Hm−1n+1 + (n+m+ 1) (n+m+ 2)Hm+1

n+1

]

}

+i eBx

me

mfmn +

ieBy/me

2

{

(n−m) (n+m+ 1) fm+1n + fm−1

n

}

+ eBz/me

2

{

(n−m) (n+m+ 1) fmn − fm−1

n

}

= −n(n+1)2 νei (v) f

mn + Cee + CIB + CMHD

(12)

where v is the electron velocity magnitude relative tothe ion velocity, νei (v) is the electron-ion scattering fre-quency [10], Fx, Fy, Fz are the components of the elec-tric force, Bx, By, Bz are the components of the magnetic�eld and the functions Gm

n and Hmn are de�ned as:

Gmn = pn

∂

∂p

(

p−n ∂fmn

∂p

)

(13)

and

Hmn = p−(n+1) ∂

∂p

(

pn+1 ∂fmn

∂p

)

(14)

where p = γmv is the magnitude of electron momen-tum. The electron-electron collision term (Cee) is com-posed of isotropic (Ciso

ee ) and anisotropic (Caniee ) compo-

nents:

1

ΓeeCiso

ee =1

v2∂

∂v

{

C0 (v) f00 +D0 (v)

∂f00

∂v

}

(15)

where D0 (v) = v3

{

I2(

f00

)

+ J−1

(

f00

)}

and C0 (v) =

I0(

f00

)

, and (using Tzoufras' notation [5]):

1Γee

Caniee = 8πf0

0 fmn +

I2(f00 )+J−1(f0

0 )3v

∂2fm

n

∂v2 +−I2(f0

0 )+2J−1(f00 )+3I0(f0

0 )3v2

∂fm

n

∂v − n(n+1)2

−I2(f00 )+2J−1(f0

0 )+3I0(f00 )

3v2 fmn

+ 12v {C1In+2 (f

mn ) + C1J−n−1 (f

mn ) + C2In (f

mn ) + C2J1−n (f

mn )} ∂2f0

0

∂v2

+ 1v2 {C3In+2 (f

mn ) + C4J−n−1 (f

mn ) + C5In (f

mn ) + C6J1−n (f

mn )} ∂f0

0

∂v(16)

where

C1 =(n+ 1) (n+ 2)

(2n+ 1) (2n+ 3)

C2 = − n (n− 1)

(2n+ 1) (2n− 1)

C3 = −n (n+ 1) /2 + (n+ 1)

(2n+ 1) (2n+ 3)

C4 = −n (n+ 1) /2 + (n+ 2)

(2n+ 1) (2n+ 3)

C5 =n (n+ 1) /2 + (n− 1)

(2n+ 1) (2n− 1)

C6 =n (n+ 1) /2− n

(2n+ 1) (2n− 1)

and

Ij (fmn ) =

4π

vj

∫ v

0

fmn (u)uj+2du

Jj (fmn ) =

4π

vj

∫

∞

v

fmn (u)uj+2du

Γee is de�ned in [5]. The term CIB represents theinverse bremsstrahlung heating term, which is solved inthe form given by Weng [6], and can be added in to C0

ee.The laser intensity I (x) is modelled in 1D by solving the1D ray equation:

dI (x)

dx= −κ (x) I (x) (17)

10

where κ (x) is the inverse bremsstrahlung absorptioncoe�cient, which is self-consistently calculated from theenergy deposited by the term CIB . Similarly, in 2D theray equations are solved by actually propagating rays,but without the self-consistent calculation of κ.The term CMHD represents energy exchange between

the MHD model and the VFP model, and includes hydro-dynamic PdV work, electron-ion equilibration, ioniza-tion energy loss/gain and radiation emission/absorption.These are computed by the MHD model and assumedto heat the electrons while maintaining a thermal (i.e.Maxwellian) distribution: deviations from this assump-tion for ionization have been discussed by Robinson [7],and for PdV work by Matte [8]. We have found no de-viation from a Maxwellian distribution when includingthe full electron-ion energy exchange terms (except in ex-treme conditions not relevant to this study). Our implicitassumption is that all hydrodynamic terms operate ontime-scales signi�cantly longer than the electron-electronthermalization time. In non-Cartesian geometry (cylin-drical or spherical), the equations need to be modi�ed toinclude the correct geometric factors as in [9].We follow the KALOS formulism [4], time-splitting the

VFP equation and integrating the advection and acceler-ation terms to second-order accuracy in time, space andmomentum, including polynomial expansions of the fm

n

close to p = 0. The electron-ion scattering, isotropicelectron-electron collision operator, and magnetic �eldterms are solved fully implicitly. The anisotropic colli-sion operator is solved semi-implicitly. The irrotationalelectric �eld is found from an implicit solution of Gauss'Law and the solenoidal component from an implicit solu-tion of the fm

1 equations (i.e. a generalized Ohm's Law).We �nd the use of Gauss' Law improves quasineutralityin the presence of large density gradients. Faraday's Lawis solved for the magnetic �eld.In this paper, we do not use the anisotropic electron-

electron collision operator, but instead implement thesimple electron-ion collision frequency multiplier givenby Epperlein [15], since this is used in many VFP modelsas well as the SNB model. We use 180 energy groups inboth the SNB and VFP models, and re�ective boundaryconditions in all cases.K2 di�ers from SPARK [10] and IMPACT [17] by be-

ing fully explicit in space (which allows e�cient paral-lelization), being capable of using an arbitrary numberof polynomials in the momentum-space expansion, andby its inclusion of laser propagation and MHD. It alsocontains a number of other features, such as anisotropiccollisions, EM wave propagation, 3D magnetic �elds andcoupling to Vlasov ions, but they are not relevant to thework presented here.

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