1. VFP simulations of transport in Polar D-D & Effect of IB heating on B-field phenomena R. J. Kingham, C. Ridgers, J. Bissell Plasma Physics Group, Imperial College London A. ThomasP.W. McKenty University of MichiganLLE, University of Rochester DD+FIW, Prague, 3 rd — 6 th May 2009

2. Polar Direct Drive – the concept Method for doing direct-drive ICF on NIF - beam repointing (LLE) Greater heating on equator Temperature gradients in  Density gradients in  [ Skupsky et al., Phys. Plasmas 11, 2763 (2005) ] ‘main’ gradient in r + [ Stamper et. al., PRL 1012 (1971) ]

3. Shock Ignition – Has related transport issues Non-uniform irradiation – on LMJ, use 33 o beam cone for driving shock (?) I ~ 2.5 x 10 15 W/cm 2 I ~ 0.3—1.2 x 10 15 W/cm 2 2x higher peak intensity than in PDD [ Skupsky et al., PoP 11, 2763 (2005) ] PDD Shock ignition [ Ribeyre et al., PPCF 51, 015013 (2009) ]

4. Simulation set up – region from 0.25 n cr < n e < 4 n cr Energy dep. rate 300 140 0 0 8 x 10 -4 x / mfp y / mfp Took a ‘snapshot’ of n e (r,  ), T e (r,  ), dU(r,  )/dt from DRACO (2D-ALE code) Used as initial conditions & heating rate in VFP transport sim - “IMPACT” B-fields, 2D-Cartesian, static density n cr = 10 22 cm -3 (Radius = 1.08mm) 770  m Peak heating rate: ~ 1.5 keV / ns at n cr I ~ 3 x 10 14 W/cm 2 ~ 8 x 10 -4 (n e T eo /  ei ) cr

5. Simulation set up – region from 0.25 n cr < n e < 4 n cr n cr = 10 22 cm -3 T eo = 2.9 keV ei = 5.5  m  ei = 0.17 ps (L n ) r = 65 ei (L n )  ~ 400 ei Density scale length Z = 3.5 300 140 0 x / mfp y / mfp n cr 2 n cr 0.5 n cr 04000 x /  m 24 22 20 log 10 ( n e /cm 3 )

6. Simulation set up – region from 1.9 < T e < 3.7 keV ei = 5.5  m  ei = 0.17 ps (L T ) r = 120 ei (L T )  ~ 1000 ei Temp. scale length Simulation details  x = 2.5 ei  y = 7.5 ei  t = 0.5  ei T e in keV x / mfp y / mfp 0 4000 x /  m 4 2 T e / keV X-BC fixed f o = f m (n eb,T eb ) Y boundaries reflective

7. See B-field grow to  ~ 0.03 by 500ps  at 8.5ps Initial B-field growth consistent with (  n) r  (  T)  Simulation t=0 ; only radial n e & T e gradients  at 500ps See Nernst advection  at 85ps x / mfp y / mfp

8. t = 85ps with B-field no B-field B-field does affect lateral T e profile t = 500ps with B-field no B-field t = 1ns with B-field no B-field y / mfp  T e / eV  T e = T e (y) -  T e  y at n e = 2 n cr B-field modifies T e (  ) via Righi-Leduc heat flow Similar effect in “corona” - 10eV change in T e “Modification” of similar size to intrinsic non-uniformity due to heater beams at n e = 2n cr

9. Flux limiter for q  and q r not generally the same Implied flux limiter for q  larger than that for q r where heating occurs Flux limiter measure: RMS ave. in  q r is classical at 4n cr, where ei small q r is classical at 4n cr, where ei small yields small ei /L n,T t = 85ps frfr ff 0 140 r / ei 10 1 6 E r & E  also show departures from locality “Flux” limiter for E r and E q 3 1 2 q  “diffuses” toward ablation surface - Braginskii underestimates q  here! - Analogous effect to Nernst (?) c.f. [ Rickard, Epperlein & Bell., PRL 62, (1989) ]

10. VFP predicts 5x larger B-field than with Classical sim B z t = 510ps Used an equivalent non-kinetic transport simulation Solves 1) Elec. energy equation 2) Ohm’s law 3) heat-flow eqn 4) Ampere-Maxwell 5) Faraday’s law Transport coeffs.  [ Epperlein & Haines, Phys. Fluids 29, 1029 (1986) ] No flux limiter used in classical simulation -->  T e (y) smaller --> less B-field Collapse of  T e (y) outweighs tendancy for Braginskii to overestimate E ? VFP Classical

11. f1 min / max = -2.1e-3 / 1.7e-3 Anisotropic pressure --> makes a difference to B IMPACT IMPACTA f1 min / max = -4.3e-3 / 1.5e-3 f1+ f2 min / max = -6.4e-3 / 4.6e-3 t = 85ps ( Thomas et al., NJP 11, 033001 (Mar. 2009) ) B z IB - heating f1+ f2 min / max = -2.1 / +1.5 kG f1 min / max = -1.4 / +0.49 kG Preliminary ! EyEyEyEy f1 min / max = -0.69 / +0.56 kG 2--3x larger

12. Anisotropic pressure --> makes a difference to B f1 min / max = -9.0e-3 / 4.3e-3 f1 min / max ~ -5e-3 / 2e-3 f1+ f2 min / max = -2e-2 / 1.7e-2 Preliminary ! t = 340 ps ( Thomas et al., NJP 11, 033001 (Mar. 2009) ) IMPACT IMPACTA B z IB - heating f1 min / max = -3.0 / +1.4 kG f1 min / max ~ -1.5 / 0.7 kG f1+ f2 min / max = -6.6 / +5.6 kG ~4x larger

13. Anisotropic pressure – Suppresses B-field advection? y / mfp IMPACT IMPACTA IMPACTA + f 2 B z (y) at n cr IMPACT IMPACTA IMPACTA + f 2 B z (y) at 2 n cr t = 340 ps Units: “0.002” --> 0.7 kG

14. PART 2 — Effect of Inverse-Bremsstrahlung heating on transport & B-field phenomena Theoretical: Better understanding of B-fields and transport Practical: Inertial Confinement Fusion and other experiments D.H. Froula et al., PRL 98, 085001 (2007) 1  m, 100J, 1ns laser 20 < T e < 800eV n e ~ 1.5 x 10 19 cm -3 B applied up to 120 kG (12 T) I ~ 4x10 14 W/cm 2  ~ 150  m

15. Super-Gaussian electron distribution function Breakdown of Maxwellian Assumption A. B. Langdon, PRL 44, 9 (1980): EDF f 0 (r,v,t) tends to Super-Gaussian due to I.B. Super-Gaussian fit ( m=3.3 ) Langdon ( m=5 ) Maxwellian ( m=2 ) General m Involved in transport

16. Where IB heating distortion is important Langdon parameter ‘  ’ & Matte’s fit for ‘m’ PDD  ~ 0.02, m ~ 2.1 Te = 3keV, I ~ 3x10 14,  0.33  m, Z=3.5 Froula’s N 2 gas jet expt Te = 200 eV, I ~ 4x10 14,  1  m, Z = (4) — 7  ~ 6, m ~ 4 shock ign. I ~ 3x10 15,  1  m  ~ 0.15 — 1.5 m ~ 2.4 — 3.3 Non-local transport

17. Transport Relations Extension to Super-Gaussian EDF Dum (1978) & Ridgers (2008): transport theory for 2  m  5 Braginskii: valid m=2 ( f 0 = f M ) New coefficients, old ones changed, Onsager symmetry broken

18. Components of transport coefficients (tensors)

19. Extended Transport Theory: Ridgers’ Ohm’s Law Functions of Hall parameter and m : C. P. Ridgers, PoP 15, 092311 (2008)

20. Extended Transport Theory: Ridgers’ Ohm’s Law Functions of Hall parameter and m : C. P. Ridgers, PoP 15, 092311 (2008)

21. B-field Evolution with Super Gaussian Effects Induction equation Fourth termThird term

22. B-field Evolution with Super Gaussian Effects Modifies Nersnt advection other terms Nernst Velocity Nersnt effect Advection of B-field by heat flow J. Bissell — PhD research

23. B-field Evolution with Super Gaussian Effects Suppression of Nernst advection 80% Suppression J. Bissell — PhD research Classical transport Extended transport Extended / Classical

24. B-field Evolution with Super Gaussian Effects Density gradient effects Fourth term

25. B-field Evolution with Super Gaussian Effects Suppression of 30% Suppression J. Bissell — PhD research

26. B-field Evolution with Super Gaussian Effects A New Effect — Advection up density gradients positive  advection of B-field up density gradients As with Nernst write + other terms Low magnetization limit (  << 1) …. m = 2.5 m = 5

27. Conclusions Polar D-D ; get  ~ 0.03 after 500ps of heating B-field strong enough to modify Te(  ) At 2 n cr similar in size to intrinsic variation due to laser non-uniformity Implied flux limiter varies in space (and time) + limiter for q  > q r Inclusion of anistropic pressure (f 2 ) stronger B-field Calc. prone to instabilities + need moving plasma + radial BCs tricky ! IB heating new & mod. B-field dynamics via new tranport coeffs Suppression of Nernst &  n x  T + advection up density gradient PDD relevant to shock ignition IBhohlraum walls ( + shock ignition ? )

29. Implicit finite-differencing  very robust + large  t (e.g. ~ps for  x~1  m vs 3fs) Solves Vlasov-FP + Maxwell’s equations for f o, f 1, E & B z IMPACT – Parallel Implicit VFP code First 2-D FP code for LPI with self consistent B-fields IMPLICT LAGGED EXPLICIT Kingham & Bell, J. Comput. Phys. 194, 1 (2004) f o can be non-Maxwellian  get non-local effects

30. Higher resolution run with shell – 0.1 n cr < n e < 84 n cr 0 4000 x /  m 4 2 T e / keV 4000 x /  m 24 22 20 log 10 ( n e /cm 3 ) 0 Low res  x = 2.5 ei  y = 7.5 ei  t = 0.5  ei Hi. res  x = 1.9 ei  y = 1.1 ei  t = 0.2  ei nx = 160 ny = 280 nv = 80  v = 0.1 v t0

31. min / max = -1.1e-2 / 5.4e-3 min / max = -3.6 / +1.8 kG B z t = 85ps min / max = -0.97e-2 / 3.4e-3 Low resHi. res Higher resolution agrees well early on min / max = -3.2 / +1.1 kG x / mfp y / mfp x / mfp y / mfp

32. min / max = -0.13 / 0.4 min / max = -1.8e-2 / 5.4e-3 Low resHi. res B z t = 255ps … but an instability grows later on min / max = -5.9 / +1.8 kG min / max = -33 / +33 kG x / mfp y / mfp

33. Braginskii’s transport relations VFP heat flow profile more diffuse than Braginskii t = 85ps Braginskii heat flow VFP heat flow x / mfp y / mfp qxqxqxqx qyqyqyqy

34. Larger  when initial n e & T e not  -averaged  substantially larger; ~5x at 2n cr PLANAR n e & T e (t=0) y / mfp x / mfp EXACT n e & T e (t=0) t = 85ps y / mfp 0 300 0 0.001 -0.002 -0.001  (  ) at n e ~ 2 n cr Exact Planar Some problems with simulation - Numerical instability happen early on - Need better spatial resolution Due mainly to (  n)   (  T) r

35. Heating mechanism (Maxwellian or IB) -- Same to 10% min / max = -10e-3 / 6.6e-3 min / max = -9.4e-3 / 5.5e-3 Maxwellian Heating Inverse Bremsstrahlung B z t = 500ps

36. Extended Transport Theory: Modified heat-flux Extended theory has been shown to predict Heat-flux better than classical transport theory (magnetized plasma) C. P. Ridgers, PoP 15, 092311 (2008) m = 3.3