1. Electron transport in the shock ignition regime Tony Bell University of Oxford Rutherford Appleton Laboratory Acknowledgements: Guy Schurtz, Xavier Ribeyre et al (CELIA, Bordeaux) Robert Kingham, Alex Robinson, Mark Sherlock (Imperial/RAL) Michail Tzoufras (Oxford/RAL) Key papers: Betti et al PRL 98 155001 (2007) Theobald et al Phys Plasmas15 056306 (2008) Ribeyre et al PPCF (in press)

2. Shock ignition Compress target on low isentrope Final laser spike launches ignition shock Figures from: Betti et al (2008) JPhys conf series 112 022024 Pressure (Gbar)

3. Starting point: work at CELIA on off-axis drive Does electron transport increase symmetry? Benefits of going to higher laser intensity (‘fast shock ignition’) Ribeyre et al PPCF 51 015013 (2009)

4. Gitomer et al Phys Fluids 29 2679 (1986) I 2 =10 16 Wcm -2  m 2 : T~10-30keV I 2 =10 17 Wcm -2  m 2 : T~10-100keV Fast electrons produced by ignition pulse Beg et al 1997: T hot = 100 (I 2 /10 17 Wcm -2 ) 1/3 keV Can heat with 100keV electrons without excessive preheat Pressure at critical: 0.32 (T/10keV) (n e /10 22 cm -3 ) Gbar need strong shock convergence high T at critical Pressure in core: 800 (T/5keV) (n e /5x10 25 cm -3 ) Gbar Fast electron range Betti et al PRL 98 155001 (2007): Ignition shock pressure ~ 1Gbar Laser spike: ~ 6x10 15 Wcm -2, 47kJ, 540TW, 100-300psec, 3  10% 100keV electrons from instabilities - beneficial

5. Explore shock ignition driven by high energy electrons using KALOS electron transport code

6. Features of non-local transport: Reduced heat flow for scalelengths < 30 x mfp (‘flux limiter’) Increased heat flow at base of heat front Heat flow at angle to  T Magnetic field where  n x  T = 0 mfp of 10keV electron at critical density ~ 80  m ( laser =0.33  m) transport is non-local Non-local electron transport

7. deflected heat flow Non-local mag feld Epperlein et al (1988) Heat flow at angle to -  T Extra heat flow at base of heat front Nishiguchi et al (1992) Kingham & Bell (2002) Reduced heat flow L < 30 mfp Bell et al (1981)

8. Other ‘non-(non-local)’ effects Borghesi et al (1998)  n x  T source of magnetic field Guerin et al PPCF 41 285 (1999) Resistive electric field inhibition with collisions without collisions

9. Electron transport model requirements Kinetic: non-Maxwellian, anisotropic Energy range: 100 eV – 100 keV Density range: less than critical – more than solid Collisional to collisionless Magnetic field Implicit on electron plasma frequency timescale Unified treatment of thermal (0.1-30keV) with hot (10-1000keV) electrons

10. KALOS code Expand velocity dist n in spherical harmonics f(x,y,v, , ,t) =  f nm (x,y,v,t) P n |m| (cos  ) e im  Any degree of anisotropy by expanding to any order Without collisions operates as efficient Vlasov code Collisions and B easily included E calculated implicitly Equations simple – efficient despite small explicit timestep velocity coordinates in 3D Kinetic a Laser-plasma o Simulation PPCF 48 R37 (2006)

11. collisions magnetic field advection electric field

12. 20 grid-points in magnitude of momentum Spherical harmonics up to 10 th order No collisions ExB drift & rotation KALOS as a pure Vlasov code

13. pxpx pypy 20 grid-points in magnitude of momentum Spherical harmonics up to 10 th order No collisions ExB drift & rotation 0 0

14. pxpx pypy 0 0 After nearly one rotation

15. Tests: Collisions Advection Electric field Reproduce Spitzer conductivity KALOS as a Fokker-Planck code Uses an approximate electron-electron collision term

16. Epperlein & Haines Phys Fluids 29 1029 (1986) KALOS time-dependent calculation for  T proportional to sin(kx) x x x x Comparison with Spitzer conductivity x Spitzer applies in limit of: long scalelength small temperature variation steady state (long times)

17. Simulations to test effect of varying hot electron temperature Parameters relevant to possible expts (not fusion targets)

18. n=10 22 cm -3 100 micron T=3keV T=150eV Initial conditions at start of ‘ignition pulse’ density temperature Cylindrical target Polar drive, absorbed intensity = 8x10 16 cos 2  Wcm -2 Absorption at n = 10 22 cm -3 Constant for 32psec T hot =100keV n=3x10 23 cm -3 n=5x10 21 cm -3

19. Electron pressure (Mbar) 800 0 t = 0 psec

20. Electron pressure (Mbar) 800 0 t = 32 psec P max =640Mbar at edge of high density lower pressure at absorption surface symmetric pressure central preheat (but not for fusion  R) coronal heating

21. n=10 22 cm -3 100 micron T=3keV T=150eV Reduced intensity: initial conditions density temperature Cylindrical target Polar drive I absorbed = 8x10 15 cos 2  Wcm -2 Absorption at critical: n = 10 22 cm -3 Constant for 28psec T hot =10keV n=3x10 23 cm -3 n=5x10 21 cm -3

22. Electron pressure (Mbar) 400 0 t = 28 psec Polar drive I absorbed = 8x10 15 cos 2  Wcm -2 T hot =10keV

23. Electron pressure (Mbar) 400 0 t = 28 psec Polar drive I absorbed = 8x10 15 cos 2  Wcm -2 T hot =10keV Pressure lower by only 50% Less energy into corona Less energy into core: Stronger shock Less symmetric Lack of symmetry compensated for by hydro? (Ribeyre et al)

24. n=3x10 23 cm -3 n=5x10 21 cm -3 n=10 22 cm -3 100 micron T=1keV T=50eV density temperature Cylindrical target Polar drive I absorbed = 1.5x10 15 cos 2  Wcm -2 Absorption at n = 10 22 cm -3 Constant for 32psec T hot =3keV Further reduce intensity & larger scalelength larger scalelength

25. Electron pressure (Mbar) 80 0 t = 32 psec Polar drive I absorbed = 1.5x10 15 cos 2  Wcm -2 T hot =3keV Large pressure asymmetry Much lower pressure in core Max pressure occurs at critical

26. -0.16 to 0.85 MG Magnetic field Electron pressure up to 330 Mbar Electron density 0.5 to 30x10 22 cm -3 Q radial -6.7x10 15 to.3x10 15 Wcm -2 Q theta -2.5x10 15 to 4.5x10 15 Wcm -2 |Q Spitzer | up to 41x10 15 Wcm -2 I absorbed = 8x10 15 cos 2  Wcm -2, T hot =10keV, t=28psec More details of calculation at intermediate intensity

27. Electron densityElectron pressure Heat flow into target I absorbed = 8x10 15 cos 2  Wcm -2, T hot =10keV, t=28psec Planar target 5x10 21 cm -3 3x10 23 cm -3 240 Mbar 120 Mbar (T=250eV) 5x10 15 Wcm -2 3x10 15 Wcm -2 300  m 75  m Magnetic field 480 kG Electric field along surface 3x10 7 Vm -1 Heat flow along surface

28. Conclusions Energetic electrons are useful: Deposit energy at high density - giving high pressure Spread energy around target allowing uneven irradiation Preheat not a problem Crucial parameter: electron range compared with ablation scalelength & target radius Prospect of integrated simulation of transport expts relevant to shock ignition