1. Recent developments in Vlasov-Fokker-Planck transport simulations relevant to IFE capsule compression R. J. Kingham, C. Ridgers Plasma Physics Group, Imperial College London 9 th Fast Ignition Workshop, Boston, 3 rd —5 th Nov 2006

2. Outline We are coupling our electron transport code, IMPACT, to an MHD code (previously, IMPACT used static density) Example of enhanced code in use  Froula (LLNL) & Tynan’s (USD) expt. effect of B-fields on non-local transport in hohlraum gas-fill context We are starting to investigate transport & B-field generation on outside wall of cone, during implosion Preliminary results  B-field of  > 1 in 0.5ns  affects lateral T e profile next to cone (beneficial?)  lateral heat flow non-local

3. Interested in departures from Braginskii transport… …even in classical transport, B-fields add complexity Braginskii’s transport relations (stationary plasma) Nernst effect Convection of B-field with heat flow Righi-Leduc heat flow  T  T q RL

4. 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

5. VFP equation for isotropic component f 0 Moving with ion fluid - include bulk convection. Compressional heating - from bulk plasma compression/rarefaction Fictitious forces - we are no longer in an inertial frame c f(v) f(w)

6. VFP equation for “flux” component f 1 Moving with ion fluid Bulk flow terms - Bulk momentum flow Fictitious Forces (4) [ Chris Ridgers’ PhD project ] I st velocity moment of this yields

7. Using IMPACT with MHD to model magnetized transport experiments N 2 gas jet 1 , 1J, 1ns laser beam 2 , 1J, 200ps probe beam - Thompson scattering Te (eV) Radius (  m) B=0 nonlocal heat wave B=12T local heat wave LASER Experiment of D. Froula (LLNL), G. Tynan (UCSD) and co. Effect of B-fields on non-local transport in hohlraum gas-fill context [ Tynan et al. submitted to PRL ] [ Divol et al. APS2006 Z01.0014 ] No B-field: k mfp > 0.03  non-local Strong B-field  expected to “localize” D ~  mfp /  D  ~ r 2 ge /   means kr ge << k mfp mfp  r ge

8. “Bottling up” of T e for  >1 seen in VFP simulation too Simulations start at T e =100eV + heating via inverse bremsstrahlung No B-field 12T B-field 1D problem with cylindrical symmetry  code 2D Cartesian so do 2D calc See “bottling up” of temperature in VFP sims with B-field 200  m

9. VFP suggests heat flow is marginally non-local at 12T Radial heat flow

10. VFP code successfully moving plasma & B-field Magnetic Reynold’s # large  resistive diffusion small … Nernst covection responsible for majority of central B-field reduction B-field convecting with plasma… electron pressure blowing out plasma 

11. Allowing for plasma motion affects evolution T e (r) Heat flow - |q| (r) e B z (r) / m e  eio with hydro w/o hydro Simulations starts at T e = 20eV B = 12T

12. “What does the gold cone do to thermal transport in the vicinity of n cr in the adjacent shell?” Focusing on critical surface 0.25 n cr < n e <4 n cr r n, T r crit Could be susceptible to  n x  T B-fields? Radial T e & Lateral n e gradients ? r rr r T r T q RL  n  n B  (  T) r  (  n)   Lateral T e & Radial n e gradients ? r rr r n r n q RL  T  T B  (  T)   (  n) r

13. Simulation set up – region from 0.25 n cr < n e < 4 n cr 04000 r /  m 24 22 20 log 10 ( n e /cm 3 ) 0 4000 r /  m 4 2 T e / keV Radial dens profile Radial T e profile DRACO ‘snapshot’ of n e (r,  ), T e (r,  ), dU(r,  )/dt used as init. cond. for IMPACT [ … as used in APS talk on PDD. DRACO data courtesy Radha & McKenty ] Heating Rate y / mfp x / mfp Peak heating: ~8 keV / ns I ~1.5 x 10 14 W/cm 2 ~ 4 x10 -4 (n e T eo /  ei ) cr nene nini Z y /  m Gold cone: L ni ~ 80  m Z ~ 50 T e ~ 3 keV !!! log 10 ( n/n cr, Z) ei = 5.5  m  ei = 0.17 ps

14. B-fields strong enough to magnetize plasma develops via  n x  T   t = 85ps  ~ 1.3   t = 500ps x / mfp y / mfp log 10 (n e ) (  n)   (  T) r (  n) r  (  T)  Simulation details  x = 2.5 ei (nx = 56) fixed x-bc  y = 7.5 ei (nx = 40) refl. y-bc  t = 0.5  ei ei = 5.5  m  ei = 0.17 ps

15. B-field does affect lateral T e profile  T e = T e (y) -  T e  y at n e = 2 n cr with B-field no B-field t = 8.5ps t = 85ps with B-field no B-field t = 1ns with B-field no B-field  T e / eV (  n)   (  T) r (  n) r  (  T)  Lowering due to Righi-Leduc heat flow from B-field (?) Flattening due to Righi-Leduc heat flow from B-field (?) y / mfp Virtually no change in Te in cone  Low thermal cond. T 5/2  c    ~~ (Z ln   Large heat capacity

16. Classical heat flow into cone up to 4x too large qxqxqxqx qyqyqyqy t = 0.5ns VFP heat flow Braginskii heat flow x / mfp y / mfp Units q fso = n eo m e v To 3

17. B-field alters lateral heat flow in VFP sims q y – B=0 t = 500ps q y – with B-field

18. Conclusions IMPACT (VFP code) + MHD  moving plasma + B-field in 2D Fielded on Froula & Tynan’s experiment; B-field suppr. of non-local effects  still some non-locality at 12 Tesla  B-field cavity, primarily due to Nernst advection Transport & B-field generation on outside wall of cone during CGFI implosion Preliminary results  B-field of  > 1 in 0.5ns  flattens lateral T e profile next to cone (beneficial?)  lateral heat flow non-local Future: use enhanced code + working on adding f 2 + f 3  no radiation transp., ionization (yet) + Au too hot L n to large?

21. Simulation: T eo = 100eV (Au), 500eV (shell) L ni ~ 20  m Radial dens. gradient ~ 3x shorter than before T = 17ps

23. 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