KINETIC SIMULATION OF LASER-PLASMA INTERACTIONS

Transcription

1 KINETIC SIMULATION OF LASER-PLASMA INTERACTIONS D. J. Strozzi 1, A. K. Ram, A. Bers, M. M. Shoucri Plasma Science and Fusion Center Massachusetts Institute of Technology Cambridge, MA 139 Institut de Recherche de l Hydro Québec Varennes, Canada MIT work supported by DoE Contract DE-FG-91ER-5419 The work of M. M. Shoucri is supported by IREQ 1 dstrozzi@mit.edu. Poster available at dstrozzi 43 rd DPP Meeting, 3 Albuquerque, NM, USA Poster BP October 3

2 Abstract We use a one-dimensional Eulerian Vlasov Code to study laser-plasma interactions. The excitation of parametric instabilities, such as stimulated Raman scattering (SRS) and stimulated Brillouin scattering (SBS), is an important issue for the efficiency of inertial confinement fusion. Our aim is to understand the growth, saturation, and coupling of these instabilities, in particular the kinetic aspects of these phenomena. The code evolves both species with the fully relativistic Vlasov equation in the direction of laser propagation. It also incorporates a cold fluid transverse velocity for each species, which is needed to couple the transverse and longitudinal dynamics. The plasma is finite and not periodic. On the boundaries we place absorbing plates that collect the particles flowing onto them. This charge enters Poisson s equation as a boundary condition. We run the code for much longer than it takes the laser to cross the computational domain. Once the laser passes a region of plasma, it modifies the equilibrium. This modification is sufficient for SRS to grow from; we do not seed SRS by imposing perturbations or a secondary laser. For parameters similar to the recent Trident single hot spot experiments, SRS is not limited by the Langmuir Decay Instability (LDI) or other ion processes, but by kinetic effects. Large vortices form in phase-space due to electron trapping in the field of the SRS electron plasma wave. At the observed amplitudes of the electrostatic field, the process of wave-breaking may be relevant. Although we have not yet observed stimulated electron acoustic scattering (SEAS), beam-like structures in the electron distribution develop off of which SEAS may occur. We also see SBS starting to develop, but we have not run for long enough times for the ions to move through several ion-acoustic wave periods. Work is underway to extend the code to 1 1 dimensions and to parallelize it. D. J. Strozzi, A. K. Ram, A. Bers, M. M. Shoucri: DPP 3 1

3 Single Hot Spot Experiments [D. S. Montgomery, J. A. Cobble, et al. Phys. Plasmas, 9(5):311 3,.] target (parylene foil) heater beam interaction beam "hot spot" plasma flow One-Dimensional Hot Spot Model Geometry E y,y laser n(x) reflected light E x,k,x x= hot spot plasma x x=l B z,z D. J. Strozzi, A. K. Ram, A. Bers, M. M. Shoucri: DPP 3

4 1-Dimensional Kinetic Model Spatial variation only in x: / x Fields: E = (Ex, E y, ) B = (,, Bz ) Transverse cold beam: F s (x, p x, P y, t) = δ(p y )f s (x, p x, t), P y = m s v ys + q s A y = y canonical mom. y momentum equation becomes [Gabor, Proc. IRE 33:79, 1945] v ys t = q s m s E y Longitudinal dynamics: Relativistic 1-D Vlasov Eqn. for f s (x, p x, t) f s t + v f s x x + q s (E x + v ys B z ) f s = m s p x Longitudinal field: E x (electrostatic perturbations) Poisson E x x = e ɛ (n i n e ) Transverse fields: laser, electromagnetic perturbations Ampère E ± E y ± cb z E y t + c B z x = 1 ɛ J y ( t ± c ) E ± = 1 x ɛ J y D. J. Strozzi, A. K. Ram, A. Bers, M. M. Shoucri: DPP 3 3

5 The Time-Stepping Algorithm Operator splitting with cubic spline for Vlasov; leapfrog for electromagnetic fields [Ghizzo, Bertrand, Shoucri et al., Journ. Comp. Phys. 9(431), 199] t/dt (leapfrog) f f ( t +v x x )f= f 1 ( t +F 1/ px )f=, F =q(e x +v y B z ) f 1 + ( t +v x x )f= f 1 (what we do) f f ( t +v x x )f= ( t +F 1/ px )f= f 1 f 1 n n n 1 = 1 (n + n 1 ) n 1 [f 1 ] N p N p N p 1 = 1 (N p + N p1 ) N p1 [n 1 ] dn w dn w dn w 1 = 1 (dn w + dn w1 ) dn w1 [f 1 ] N w N w N w 1 = N w + 1 dn w 1 N w1 = N w + dn w1 E x E x 1 [n 1, N p 1, N 1 w ] E ± E ± + ( t ± x )E ± = E ± 1 = 1 (E± + E± 1 ) E ± ( t ± x )E ± = ɛ 1 1 J y1 E ± 1 + v 1 y m t v y v y v =qey 1 y = 1 (v y + v y1 ) v y1 D. J. Strozzi, A. K. Ram, A. Bers, M. M. Shoucri: DPP 3 4

6 Nonperiodic grids in x and p x ; plasma is finite. Boundary Conditions Absorbing plates (transparent to electromagnetic fields) at x = and x = L. Particles that flow out to the plates stay there and no longer evolve. N sl, N sp, N sr = # of particles of species s on left plate, in plasma, on right plate X(p) = dt v(p) v(p) = p γm N L = dt p min dp X(p) dxf(x, p) p min dp v f(, p) left plate p min X(p) N L p dx x Charges on plates enter E x as a boundary condition. E x x = ɛ 1 ρ Q L + Q P + Q R = E x (x < left plate) = E(x = ) = 1 (Q L Q P Q R ) ɛ E(x) = E(x = ) + ɛ 1 x ρ(x )dx Q L N el N il left plate plasma Q P N ep N ip L Q R N er N ir right plate x D. J. Strozzi, A. K. Ram, A. Bers, M. M. Shoucri: DPP 3 5

7 Outflow to plates Change in total number of particles 3 x 1 3 (N s,tot N s,tot ())/N s,tot () 1 1 N e,tot N i,tot ω p t Electrons on left, right walls 1 N e,left N e,right 1 4 Ions on left, right walls N i,left N i,right N e /N ep 1 4 N i /N ip ω t p ω p t D. J. Strozzi, A. K. Ram, A. Bers, M. M. Shoucri: DPP 3 6

8 Run Parameters Scales used in code: time: t = ω 1 p length: d e = c/ω p speed: c momentum: p = m e c E field: v quiv (E x ) = c Plasma Parameters: Plasma length: L = 75 µm = 16 d e Plasma density: n e = cm 3 =.3 n crit initial electron Temperature: T e = 35 ev initial ion Temperature: T i = 1 ev electron thermal speed: v T e =.6 c ion thermal speed: v T i = c Plasma frequency: ω p = rad/s t = s, d e = 47 nm Debye length: λ De = 1.3 nm =.6 d e Laser Parameters: Laser frequency: ω = rad/s = 5.59 ω p Laser instensity: I = 1 16 W/cm electron quiver velocity: v quiv =.45 c Numerical parameters: free-space wavelength: λ = 57 nm = 1.1 d e Timestep: dt = c dx =.1/ω p Total runtime: T = 1.56 ps = 1 t Spatial gridpoints: 161 dx =.1 d e electron momentum gridpoints: 31 dp e =.7 m e c =.1 p T e ion momentum gridpoints: 151 dp i =.69 m e c =.187 p T i D. J. Strozzi, A. K. Ram, A. Bers, M. M. Shoucri: DPP 3 7

9 Parametric Instabilities: Coupling of Modes EMW (electromagnetic) ω = ω pe + c k e/m E k EPW (e plasma) ω = ω pe + 3v T e k e/s E k IAW (ion acoustic) ω = c ak e/s E k 6 4 EMW EPW IAW SBS SRS laser Matching: ω 1 = ω + ω 3 k 1 = k + k 3 E j = a j (x, t) ωj ɛ ˆɛ j e (k jx ω j t) ω/ω p LDI SRS,LDI SBS LDI 1 1 kc/ω p ω las ω SBS = 5.59 k las k SBS = 5.5 ω SRS = 4.49 k SRS = 4.38 ω EP W = 1.1 k EP W = 1.1 ω LDI =.16 k LDI = 19.4 ω SBS IAW =.915 k SBS IAW = 11 a j = ɛ ω j E j = action density Da 1 D 1 t Da D t Da 3 D 3 t = Ka a 3 = K a 1 a 3 = K a 1 a D D j t = t + v gj x + ν j D. J. Strozzi, A. K. Ram, A. Bers, M. M. Shoucri: DPP 3 8

10 Laser-Modified Equilibrium: Electromagnetic Fields v ye at ω p t = 1 E at ω p t = v ye E xω /c p v ye (k) at ω p t = 1, xω p /c = xω /c p E (k) at ω p t = 1, xω p /c = k=1 k las k=1 k las v ye E kc/ω p kc/ω p D. J. Strozzi, A. K. Ram, A. Bers, M. M. Shoucri: DPP 3 9

11 Laser-Modified Equilibrium: Electrostatic Fields n e n e at ω p t = 1 E x at ω p t = 1 6 x 1 3 x n e n e E x xω p /c n e (k) at ω p t = 1, xω p /c = xω p /c E x (k) at ω p t = 1, xω p /c = k=1 k las 1 7 k=1 k las k las k las n e n e E x kc/ω p kc/ω p D. J. Strozzi, A. K. Ram, A. Bers, M. M. Shoucri: DPP 3 1

12 Reflected Light at x = Instantaneous Reflectivity (Avg.=3.1%).5 E (ω) at x =, tω p = 1 1 <(E ) > t /<(E + ) > t E (ω) ω SRS ω SBS ω t p E (ω) at x =, tω p = ω/ω p E (ω) at x =, tω p = ω SRS ω SBS ω SRS ω SBS E (ω) 1 6 E (ω) ω/ω p ω/ω p D. J. Strozzi, A. K. Ram, A. Bers, M. M. Shoucri: DPP 3 11

13 Frequency of reflected light E (x = ) vs. time 7.5 E (ω), log scale ω/ω p ω SBS ω SRS ω p t.5 1 Upshift in SRS frequency at later times. Related to downshift in EPW frequency due to trapping? SBS develops at later times. Not present in run with stationary ions (see below). D. J. Strozzi, A. K. Ram, A. Bers, M. M. Shoucri: DPP 3 1

14 Reflected Light at x =, Stationary Ions Instantaneous Reflectivity (Avg.: 3.4%).5 <(E ) > t /<(E + ) > t ω t p E (ω) at x =, tω p = 4 1 E (ω) at x =, tω p = ω SRS ω SBS ω SRS ω SBS E (ω) 1 6 E (ω) ω/ω p ω/ω p D. J. Strozzi, A. K. Ram, A. Bers, M. M. Shoucri: DPP 3 13

15 Longitudinal Electric Field E x vs. t.4. E x at xω p /c = 4 RMS E x (t) at xω p /c = 4.3 E =.55 = wave breaking threshold. E x E x ω t p E x (ω) at xω p /c = 4, ω p t = ω t p E x (ω) at xω p /c = 4, ω p t = ω EPW E x (ω) 1 8 ω EPW E x (ω) ω/ω p ω/ω p D. J. Strozzi, A. K. Ram, A. Bers, M. M. Shoucri: DPP 3 14

16 Longitudinal Electric Field E x vs. x. RMS E x at ω p t = 4 E wavebreak = E x at ω p t = 9 E x E x xω p /c E x (k) at ω p t = 4, xω p /c = xω /c p E x (k) at ω p t = 9, xω p /c = k SRS ksbs 1 4 k SRS ksbs E x E x kc/ω p kc/ω p D. J. Strozzi, A. K. Ram, A. Bers, M. M. Shoucri: DPP 3 15

17 Evolution of Trapped Distribution Contour Plots of f e (log scale) f e contours at ω p t=4 f e contours at ω p t=45 f e contours at ω p t= f e contours at ω p t=6 f e contours at ω p t=7 f e contours at ω p t= f e contours at ω p t=9 f e contours at ω p t= p e /m e c.1 p e /m e c xω /c p D. J. Strozzi, A. K. Ram, A. Bers, M. M. Shoucri: DPP 3 16

18 Trapped f e during first SRS burst Contour Plot of f e at ω p t = 4.3. v p,epw p e /m e c xω p /c Spatially-averaged < f e > x < f e > x Maxwellian fit: beam <f e > x T fit = 366 ev n fit =.75 n <f e > x fit <δ f> x 4 n b =.175 n T b =18 ev v b =.315 c p e /m e c p/m e c D. J. Strozzi, A. K. Ram, A. Bers, M. M. Shoucri: DPP 3 17

19 Wave-Breaking: Possible Limit to SRS? Wave-breaking threshold: Tralleving-wave solutions to electrostatic warm-fluid equations n(kx ωt), u(kx ωt) do not exist with electric field amplitudes E > E max. [ T. P. Coffey, Phys Fluids 14, 14 (1971); W. L. Kruer, The Physics of Laser Plasma Interactions. Addison-Wesley (1988)] Emax = β 8 3 β1/4 + β 1/ β T ( e ω ) T 1 T 3 m e,ev = kev ck E max vs. T e of SRS EPW for run parameters.1 E max / [c(n m e /ε ) 1/ ] T e =35 ev E max =.55 ω EPW =ω bounce T e (ev) D. J. Strozzi, A. K. Ram, A. Bers, M. M. Shoucri: DPP 3 18

20 Scaling of Bounce Frequency with Amplitude of E x ω B eex k EP W m e Bounce Freq. vs. E x.6.51 E x =E x,wavebreak.4 ω B /ω p..56 ω B =γ SRS.1..3 E x D. J. Strozzi, A. K. Ram, A. Bers, M. M. Shoucri: DPP 3 19

21 Ion Density Fluctuations vs. x.1 n i n i at ω p t = 45. n i n i at ω p t = 1.1 n i n i n i n i xω /c p n i (ω) at ω p t = 45, xω p /c = xω /c p n i (ω) at ω p t = 1, xω p /c = n i n i k SBS k LDI n i n i 1 5 k SBS k LDI kc/ω p kc/ω p D. J. Strozzi, A. K. Ram, A. Bers, M. M. Shoucri: DPP 3

22 Ion Density Fluctuations vs. t n i n i at xω p /c = 4 n i n i at xω p /c = n i n i n i n i ω p t ω p t Run too short for ion acoustic wave to go through several cycles, e.g.: c a = k IAW SBS = 11 τ = π c a k SBS = 688 D. J. Strozzi, A. K. Ram, A. Bers, M. M. Shoucri: DPP 3 1

23 Future Work: Parallelize, 1 1 -D Code Preliminary results from 1 1 -D code that treats v y c kinetically: f s (x, p x, v y, t). f s t + v f s x x + q s(e x + v y B z ) f s + p x q s γm s (E y v x B z ) f s v y = n e at ω p t = 8 E y at ω p t = 8 n e E y x ω p /c x ω p /c D. J. Strozzi, A. K. Ram, A. Bers, M. M. Shoucri: DPP 3

24 Comments? Reprint? Please leave address. D. J. Strozzi, A. K. Ram, A. Bers, M. M. Shoucri: DPP 3 3