Self-Similar Solutions with Electro-Thermal Processes for Plasmas of Arbitrary Beta*

John Giuliani & Alexander (Sasha) Velikovich Plasma Physics Division, Naval Research Laboratory

Magnetic Fields in Laser Plasmas Laboratory for Laser Energetics, University of Rochester Rochester, New York April 23 – 24, 2018

*WorksupportedbyDOE/NNSA

Electro-thermal effects in 1-D

Nernst

Bzx

y ΔTee-

RTy = −β∧uT b̂ ×∇Te

Ettinsghausen

Bzx

y Bz

Δ

RT

qx

qx = −Te

β∧uT

eneb̂ ×J

Thomson

2

left side slower e- experience stronger opposing collisional drag than right side faster e-

conduction E accelerates e- on right and decelerates e- on left

ET ⋅J = + β∧

uT

ene

∂Te∂x

c4π

∂Bz∂x

heating (or cooling) effects in a current carrying plasma with a �Te

Braginskii, Rev. Plasma Phys. 1, ed. Leontovich, 205, 1965.

e-

β∧uT = ne

χe( ′′β1χe2 + ′′βo )

χe4 +δ1χe

2 +δ o

, χe =ω eτ e

Jy = − 4πc

∂Bz∂x

T

B / B∞steady state self-similar solution with Braginskii coefficients and constant total pressure;

3

Early studies of the Nernst effect in heated plasmas bounded by liners

H plasma

B∞

n∞ T∞ non-conducting, insulating

wall

Garanin, JAppMechTechPhy, 28, 816, 1987.

Garanin & Mamyshev, JAMTP, 31, 28, 1990.

As above, but included wall vaporization.

x

y

For effective magnetic diffusivity with Nernst

Deff(B) ~ cTe ∞

10eB∞

= 1.6DBohm

Vekshtein, Doklady, 271, 1983 & JETP, 84, 1983.

β∞ = 2nTB2 / 8π

>>1,

4

On-going experiments with laser heated plasmas inside liners with embedded axial magnetic fields

ICF on Z K-shell rad

sources photoionization on Z Beamlet

Slutz, et al., PoP, 17, 056393, 2010. Sefkow, et al., PoP, 21, 072711, 2014. Ryutov, et al., PoP, 19, 062706, 2012.

Kemp, LLNL LDRD Kemp, et al., PoP, 23, 101204, 2016. Kemp, et al., APS-DPP, 2017.

Carpenter, et al., APS-DPP, 2017. Mancini, private comm.

split solenoid

cylindrical target, gas or metal foam fill

MagLIF(preheat)

MagLIF(imploded)

radsource photo-ioniza:on

ne(cm-3) 3e20 1e23 1e21(Z~10) 2e20

Te(eV) 300 8000 1000 600

B(T) 30 13,000 10 8.5

β 80 4.6 4.4e3 1.1e3

χe=ωeτe 4.2 608 0.24 4.5

Lm 760 8.9e5 2.4e4 1.5e4

D2+Ar

CUNEO et al.: MDIs FOR INERTIAL CONFINEMENT FUSION AT SANDIA NATIONAL LABORATORIES 3231

to achieve higher ρR’s than in cylindrical geometry at Cr of12–24:1, without pulse shaping or cryogenic fuel, and withimplosion velocities (14–30 cm/µs) that were suggested to behigh enough for adequate self-heating of the hot spot. However,the designs use high aspect ratio targets (AR = 20 − 120) toachieve high implosion velocities, and thus would be moresusceptible to the MRT instability than implosions of lower AR

liners.VanDevender et al. proposed [83] quasi-spherical targets

with cryogenic layers imploded in 40 ns. Extensive simulationsof these targets indicated they could achieve high ρRs andhigh implosion velocities (> 35 cm/µs) for rapid plasma self-heating, ignition and fusion gain. These targets required an AR

of 21 and a Cr of about 13. However, this approach requireslarger dI/dt(≥ 1.5 MA/ns) than existing multi-MA pulsed-power technology has demonstrated (∼0.3MA/ns). Equation(11) suggests that at dI/dt’s of > 1 MA/ns, this AR and Crresult in implosion velocities of > 40 cm/µs. Several technolo-gies have been proposed [84], [85] to achieve such current riserates. A significant research program to develop this capabilitywould be required before progress could be made with thistarget approach.

Slutz and Herrmann et al. show [30] that the use of fuelpre-heating and fuel magnetization can allow relatively slowliners (10 cm/µs) with target designs possibly resilient to MRT(AR = 6) to achieve significant fusion yields using existingpulsed-power technology (≤ 0.3 MA/ns) on the 100-ns driveZ facility. Indeed, (11) suggests that under these conditions, thepeak implosion velocities of magnetically driven liners on Z areno more than 20 cm/µs. The next section discusses our plansto evaluate MagLIF. As we make progress in our understandingof the MRT and its dependence on liner and driver parameters(AR, Cr, and dI/dt), other target designs can be proposed andevaluated.

B. MagLIF

The MagLIF concept and phases are shown in Fig. 9[30]. The liner is initially filled with gaseous D2 or DT fuel[Fig. 9(a)]. The system is magnetized with an axial magneticfield Bz [Fig. 9(a)] produced by an independent coil system (seeFig. 12). The field is established slowly enough over severalms to allow it to fully penetrate all the hardware, including thetarget. Current from the Z generator is delivered, producinga Bθ field to implode the target [Fig. 9(a)]. To pre-heat thefuel, the multi-TW, multi-kJ, 527-nm Z-Beamlet laser [108] co-located with the Z facility can be used (see Fig. 10). After thestart of the implosion, a 10 ns pulse from the Z-Beamlet laseris directed into the liner and pre-heats the fuel to 250–500 eV[Fig. 9(b)]. The implosion continues to convergence ratios of20 : 1, adiabatically heating the fuel, compressing the Bz to >50 MGauss fields, which hinders electron thermal conductionloss to the liner wall.

Integrated MagLIF target designs predict that DT fusionyields of order 100 kJ may be possible on Z at peak drivecurrents of about 27 MA, with an initial axial magnetic fieldof 30 Tesla, and with about 6 kJ of Z-Beamlet laser pre-heatenergy. This yield would represent scientific breakeven wherethe fusion energy output (Efusion) equals the energy invested in

Fig. 9. Phases of the MagLIF concept, showing (a) axial pre-magnetizationphase, (b) laser pre-heat phase, and (c) magnetically driven liner implosion andflux compression phase, described more fully in the text.

heating the fuel (Efuel). This is sometimes represented by thecondition QHS = 1, where QHS = Efusion/Efuel is the gain ofthe hot spot. Scientific breakeven, to our knowledge, has notbeen achieved in any fusion system to date.

Such fusion yields were never envisioned in designs for anyindirect-drive concept on Z. The 100 kJ predicted DT fusionenergy yield is equivalent to 3.5 × 1016 neutrons. Equivalentyields from D2 MagLIF implosions would be of order 4.4 ×1014 neutrons (80X lower). Such an output could be considereda “scaled scientific breakeven” condition. For comparison,the maximum experimentally measured D2 yields from theindirectly driven ZPDH were of order 2 − 4 × 1011 [17],[19], a factor of 103 lower. The maximum predicted neutronyields for the DEH were of order 108 − 109. Although both theZPDH and DEH designs could be improved, clearly the relative

Be liner Bz

Bφ

laser preheated D2

laser

a) Magnetized, hot, β>> 1 plasma bounded by a cold wall: b) Magnetized, hot, arbitrary β plasma bounded by a cold wall:

5

Outline

Self-similar analyses provide “exact” solutions to a simplified problem – multi-physics codes provide approximate solutions to an “exact” problem. Similarity solutions provide verification tests for multi-physics codes: e.g., LASNEX, HYDRA, PERSEUS

∂∂t(3nT )+ ∂

∂x(3nTux )+ 2nT

∂ux∂x

= ∂∂x

⎡⎣ (κ⊥(e) +κ⊥

(i ) ) ∂T∂x

thermal conduction

⎤⎦

6

1D MHD, planar MHD eqns at large plasma beta includes the Nernst electro-thermal term

ni = ne = n Te = Ti = T

continuity:

momentum:

thermal:

induction:

∂n∂t

+ ∂∂x(nu) = 0

∂Bz∂t

= −c∂Ey

∂x and Jy = − c

4π∂Bz∂x

Ey =uxcBz +η⊥Jy −

β∧uT

en∂T∂x

Maxwell:

2nT = constant

∂Bz∂t

+ ∂∂x(uxBz ) =

∂∂x

⎡⎣c2η⊥

4π∂Bz∂x

magnetic diffusion

+ cβ∧uT

en∂kBT∂x

Nernst

⎤⎦

E field:

∞ β

7

Self-similar solution for heat & magnetic flux loss to a cold liner from a planar, isobaric, plasma of large

boundary conditions

H plasma

B∞

n∞ T∞ non-conducting,

insulating wall

η = x / D⊥∞(T )t

Self-similar ansatz

Θ = Η = V = 0

Θ = Η = Ν = 1 at η→∞

Velikovich, Giuliani, & Zalesak, PoP, 22, 042702, 2015.

thermal boundary layer

magnetic boundary layer

η ≡ xD⊥∞

(T )t V(η) = 2ux (x,t)

D⊥∞(T ) / t

Ν(η) = n(x,t)n∞

Θ(η) = T (x,t)T∞

Η(η) = Bz (x,t)B∞

D⊥∞(T ) = κ⊥∞

(e) +κ⊥∞(i )

3n∞

∞ β

8

Magnetic & thermal losses to an unmagnetized, cold liner at large

Q(T ) ~ D⊥eff(T ) t

diffu

sivi

ties

norm

aliz

ed t

o

Bohm

(=cT∞/1

6eB ∞

)

χe ∞

~32DBohm

~8DBohm

Heat loss is dominated by thermal conduction & advection, Magnetic flux loss dominated by advection & Nernst effect.

∞ β

D⊥eff(T ) ~ 2cT∞

eB∞

= 32DBohm

D⊥eff(B) ~ cT∞

2eB∞

= 8DBohm

D⊥(T ) = κ⊥

(e) +κ⊥(i )

3n= Tτ e3me

f1(χe, lnΛ)

D⊥(B) = c

2η⊥

4π= c2me

4πe2nτ ef2 (χe, lnΛ)

El-Nadi & Rosenbluth, PoF, 16, 2036, 1973

(Same eqn for drift instability driven, turbulent thermal diffusion in a high β collisionless plasma

∂∂t(3nT )+ ∂

∂x(3nTux )+ 2nT

∂ux∂x

= ∂∂x

⎡⎣ (κ⊥(e) +κ⊥

(i ) ) ∂T∂x

thermal conduction

− β∧uT

enTJy

Ettingshausen

⎤⎦ + η⊥Jy2

Joule heating

− β∧uT

en∂T∂x

JyThomson

9

1D MHD, planar eqns have three thermo-electric transport terms for arbitrary

ni = ne = n Te = Ti = T

continuity:

momentum:

thermal:

induction:

∂n∂t

+ ∂∂x(nu) = 0

∂∂t(ρux )+

∂∂x(ρux

2 )+ ∂∂x

2nT +Bz2

8π⎛⎝⎜

⎞⎠⎟= 0

for self-similar solutions

2nT + Bz2 / 8π = constant

∂Bz∂t

+ ∂∂x(uxBz ) =

∂∂x

⎡⎣c2η⊥

4π∂Bz∂x

magnetic diffusion

+ cβ∧uT

en∂kBT∂x

Nernst

⎤⎦

∞ β

10

For arbitrary , one has Ettingshausen, Thomson, & Joule in thermal eqn, in addition to Nernst in induction

continuity:

momentum:

thermal:

induction:

dVdη

= (V− η2) d lnΘdη

+ (V− η2) 2Ηβ∞ + (1−Η

2 )dΗdη

ΝΘ = 1+ (1−Η2 ) / β∞

(1+ 1−Η2

β∞

) (V− η2) d lnΘdη

+ 23dVdη

⎡⎣⎢

⎤⎦⎥= ddη

⎡⎣ Θ5/2 Μ1

Μ4

dΘdη

thermal conduction

+ 23

β∧(ET )

β∞

Θ dΗdη

Ettingshausen

⎤⎦ +

43Lm∞β∞

Μ2Μ4

Θ3/2dΗdη

⎛⎝⎜

⎞⎠⎟

2

Joule heating

+ 23

β∧(ET )

β∞

dΘdη

dΗdη

Thomson

(V− η2) dΗdη

+Η dVdη

= ddη

⎡⎣1

Lm∞Μ2Μ4

Θ3/2dΗdη

magnetic diffusion

+ 12β∧(ET ) dΘ

dηNernst

⎤⎦

Consistent with earlier study, for only Nernst in induction remains.

β∧(ET ) = ζ ΗΘ5/2

1+ (1−Η2 ) / β∞

Μ3

Μ4 VNernst = − 1

2

β∧(ET )

ΗdΘdη

∞ β

electro-thermal:

χe =ω ceτ e =

Lm∞λβ∞

ΗΘ5/2

1+ (1−Η2 ) / β∞

1Μ4

lnΛ =Μ4 (Θ,Η,β∞ )lnΛ∞

Lm!∞ =

D!∞(T )

D!∞(B)

⎛

⎝⎜⎞

⎠⎟~ 1

2β∞χe ∞

2 ∝ T 4

n

e- Hall parameter: Coulomb log:

Giuliani & Velikovich, submitted to IEEE TPS, Feb., 2018.

χe =ωceτe10-1 100 10+1

10-2

10-1

100

M1

M2

M3

Fig.1

11

n, T, & V profiles depend on , weakly on , but B profile varies strongly with and

boundary conditions ~ MagLIF preheat stage

B∞(T)

β∞ χe∞electro-thermal

A 1.4 3.7e4 0.2 no

B 1.4 3.7e4 0.2 yes

C 140 3.7 20. no

D 140 3.7 20. yes

∞ β ~^β(ET)

∞ β ~^β(ET)

n∞ = 3×1020 cm−3, T∞ = 300 eV, ⇒ Lm!∞ = 763

H plasma

B∞

n∞ T∞ non-conducting, insulating

wall

η = x / D∞

(T )tΘ = 1/ 4 Η = 1 V = 0

Θ = Η = Ν = 1 at η→∞

12

Case D as a MHD code verification test: =3.7, χe=20 at 3 ns.

self-similar non-isobaric, MHD code

∞ β

Early studies showed anomalously high thermal transport from a hot, magnetized plasma to a cold wall due to Nernst. Planar, isobaric, self-similar solutions with large and show that heat loss by thermal conduction & advection, magnetic flux loss by Nernst & advection: For low plasmas, the Ettingshausen, Thomson, and Joule terms enter the thermal energy eqn, in addition to Nernst in induction. Four self-similar solutions appropriate to the conditions following laser pre-heat: n, T, & V profiles depend on , weakly on electro-thermal effects; but B varies strongly with both and electro-thermal effects. A verification test for electro-thermal physics in MHD codes without isobaric constraint in self-similar solutions. 13

β ∞

Lm!∞ ≈ 1

2β∞χe ∞

2 D⊥eff(T ) ~ 32DBohm D⊥eff

(B) ~ 8DBohm

H plasma

B∞

n∞ T∞

η = x / D∞

(T )t

β ∞

Summary

β ∞

χ e∞

β ∞ ~ ^ β(ET)