MHD Effects in Laser-Produced Plasmas

Transcription

1 MHD Effcts in Lasr-Producd Plasmas OLEG POLOMAROV and RICCARDO BETTI Fusion Scinc Cntr and Laboratory for Lasr Enrgtics Univrsity of Rochstr

2 Abstract Th implmntation of th magnto-hydrodynamic (MHD) modul in th arbitrary Lagrang-Eulrian (ALE) hydrocod for lasr-plasma simulation DRACO 1 is dscribd. Th MHD block accounts for convction, diffusion, and gnration of th magntic fild by th thrmolctric/magntic ffcts causd by th non-paralll tmpratur and dnsity gradints and th Nrnst trm. Th ffct of th magntic fild on th transport cofficints for MHD quations is xplicitly takn into account and th influnc of th strong magntic fild on hydrodynamics and hating of th lasr-implodd plasma pllts ar studid. This work was supportd by th U.S. Dpartmnt of Enrgy undr Cooprativ Agrmnt Nos. DE-FC-4ER54789 and DE-FC5-8NA83. 1 P. B. Radha t al., Phys. Plasmas 1, 5637 (5).

3 Summary Mgagauss magntic filds ar gnratd in sphrical implosions Isotropic and anisotropic MHD quations ar addd to th ALE hydrocod DRACO. A gnration of th mgagauss magntic fild for sphrical implosions is numrically dmonstratd. Th influnc of th magntic fild on transport cofficints is analyd and th important rol of th Nrst trm is dmonstratd. TC8333

4 Govrning quations for isotropic MHD (no dpndnc of transport cofficints on th magntic fild) 1 A - c t = E Ohm s law: E v # B dp v- v # B = h j - c - n +^ h c j c = d # B = -n, Zn n, 4 ^ v - v h i = ^ v - v h<< r v A t c h = v # d# A- d# d# A+ n c dp 4r B t c h c dp = d# ^v # Bh - d# d# Bo + # 4 d f p r n Equation of motion + c 7 j # BA, Thrmal transport + E: j TC8334

5 A /B rprsntation of th magntic fild in th cylindrical gomtry {r,,} with rotational symmtry B A 1 = r r (-, B, `ra j; Assumption: =, v = A t 1 1 c h r r r r 4r 1 1 r r r r = - : v `ra j+ v `ra jd+ ' : `ra jd+ : `ra jd1 B t Advction: Diffusion: Sourc: c h 1 r r r 4r r r = -: `v B j+ `v B jd + ) < `rb jf + < `rb jf3 + < b l- b lf c h 1 c 1 p 1 p 4r r n r r n d : B / by construction A and B ar volvd indpndntly from ach othr Th slf-gnratd magntic fild is aimuthal as th sourc trm gos only in th quation for B TC8335 Whn implmnting: Th advction, diffusion, and sourc trms ar split from ach othr

6 Th advction part of th quations for aimuthal magntic fild and vctor potntial rprsntd as flow drivativs and solvd on th moving DRACO msh Advction contribution: A t 1 1 d r r r r dt = -: v `ra j+ v `ra jd & `ra j = B t d B r r dt rt = -: `v B j+ `v B jd & d n = DRACO implmntation A for cll nods; B for cll cntrs TC8336 n n+ 1 n yl n+ 1 n n + 1 yl A = A B = B ycnt ycnt n + 1 n + 1 n rho rho n

7 Diffrntial oprators ar discrtid on a non-orthogonal, non-vn msh Symbolic rprsntation: d = lim s 1 # " V S nds Oprators CURL, GRAD, and DIV ar discrtid by th control volum approach. Thy ar rprsntd as fluxs through th boundaris of corrsponding control volums: Nodal (A) to cntrd _ da drd, A di i, j + 1 i + 1, j + 1 Cntrd (B) to nodal _ db drd, B di i, j + 1 i, j i, j i 1, j i, j i + 1, j i, j i + 1, j TC8337 i, j 1

8 Implmntation of th diffusion and sourc trms B Diffusion contribution: =-d# _ D d# Bi t 1. Subroutin rotrotb(i,j,rrb) discrtis th oprator d# _ DBd# Bi on th msh by th control volum approach.. Subroutin Cof(CoB) calculats th diagonal cofficints CoB[i,j] at B i,j. 3. B at th nxt stp ar found from an implicit schm by th hyprsor itrativ approach: B old Bij, = ] 1-~ g : Bij, + ~ :: Bij, -dt : _ rrb- CoBij, ibij, D _ 1 + dtcob : ij, i Sourc contribution: TC8338 B t = c dp d # d n 1. Discrtid by th modifd control volum tchniqu along th contour corrsponding to th cll s boundaris.. Discrtiation numrically satisfis d# d fr _, i = to round off rrors. n

9 Govrning quations for anisotropic MHD (transport cofficints dpnd on th magntic fild) v # B d p RT + R n Ohm s law: E = - c - n + j Friction forc/diffusion: Rj = n 1 $ a h j: h + a B h# j# h - a B h# j _ i = ] g 7 A ^ ] g7 A. h = B B Thrmal forc/ quasi-sourcs : Nrnst trm RT = -b ut ut ut h_ dt: hi -b= ] Bgh# 8dT# hb- b^ ] Bg8h# dtb B v B c d p c RT+ Rj = d# ^ # h + t d# d n n - d # n o TC8339

10 Th thrmal transport quation for anisotropic MHD dt 1 dv m 1,,, C P C T T Q q q Q dt V dt m ion LRFCP Magn t = - - t x _ - i + - d : _ T+ j i + i Thrmal hat flux: Frictional hat flux: Joul hating: TC834 q = -l h_ dt : hi -l ] Bgh# 8dT # hb - l ] Bg8h# d T B = T = 1 Tu Tu Tu q j = - $ n b h_ j: hi + b= ] Bgh# 7j# ha+ b ] ^ Bg7h# ja. = Q -l ] BgdT -l ] Bg8h # dt Magn = 1 - n % b Tu = Tu ^ ] Bgj+ b ] Bg7h# ja + ` b p RT Rj E : j - d + = = n : j+ n : j ^ B -^ l -l Tu ^ = hh_ d T - b Tu = : hi jh_ j: hi/ Thrm conductivity For B", l = " l ; / l, and l^" For B"3, l =, l^", but l^ > l ; ~t B ~ = mc, x lctron/ion collision tim

11 Th quation for aimuthal magntic fild B slf-gnratd by gradn # gradt and Nrst trms for anisotropic MHD If A (t = ) = & A (t) =, and only B is gnratd B t v B c dp n DB B DB h 1 r r rb r DB h 1 -d# ^ # h - d# o = -d# 7 ] g= d# A + ' ; ] g - r rb + ^ ` je ; ] g ^ ` je1 + d# < n c b] Bg T n c = d F + d# < b] Bg ^ _ h# dtif ut ut Rsistivity a = a ; a ; a ; a^ a ; Diffusion cofficint: D =, ^ ] Bg= c a =, ^ 4r n For B slf-gnration, only D = (B) and b] Bg ut ar ssntial ^ Nrnst trm TC ~t B ~ = mc, x lctron/ion collision tim

12 Th hat-flux limitrs for transvrs hat conductivitis for th anisotropic cas ar a gnraliation of th limitr for th isotropic cas Isotropic cas: q nt nt T nt T n T ~ l dt ~ x x x m dt ~ m ~ m v ~ m T x m mfp 3 Anisotropic cas: q ~ l] Bg d T =, ^ =, ^ lim 1 1 mt q=, ~ f# l] Bg=, T + = f# l] Bg ^ ^, 1 = ^ m r x + x ~ mfp B B TC834 lim q = min8q, q B, f ~ 6.

13 Initial input data for DRACO/MHD simulation of th sphrical targt implosion drivn by th squar lasr puls DRACO msh t = r (i,j) Lasr Imploding shll Implosion Lasr Initiali_grid layr 1 = 8 i_clls of DD xlay(1) =.395 cm (i,j) CH DD layr = 5 i_clls of CH xlay() =.18 cm j_clls = 1 clls.1 i: along targt radius j: along targt circumfrnc Lasr Lasr Simulation_input Initial_lasr_uniformity = Lgndr mod mod_num = 4 Lasr_ampl_prturb = 1 # 1 TC8343 t1 = s powr1 = t = 1 # 1 1 powr = 5 TW t3 = 1 # 1 9 powr3 = 5 TW t4 = 1.1 # 1 9 powr4 =

14 Vorticity in a hydrodynamic flow of a conducting fluid srvs as a good indicator of th prsnc of a magntic fild B t c dp ptot = d# ^v # Bh + d# f p n ^ d # vh= d # 6 v# ^ d # vh@ - d # d o t t B ~ c m - d Z ion # v B [gauss] Implosion Curl V [gauss] 5, 1 5, 1 5, 5 j 5, 5 5 i 1 Lasr 5 1 TC8344

15 Dynamics of slf-gnratd by (gradn # gradt) aimuthal magntic fild B for isotropic MHD B = 5 G B = 1 Implosion t = 1.34 # 1 1 t = 5.7 # G B i 1 5 j 1 Lasr 1, 1, B = 1 4 G t = 8 # 1 1 B = # 1 5 G t = 1 # 1 9 1, 1, 5 1, 1, 1,, TC

16 Dynamics of slf-gnratd (by gradn#gradt and Nrnst trms) aimuthal magntic fild B for anisotropic MHD B = 5 # 1 4 G t = 5.7 # 1 1 Implosion B = 1 5 G t = 8 # 1 1 B 5, 5, 5 j 1 1, 5, 5, 1, 5 i 1 Lasr B = 4 # 1 5 G t = 9.8 # 1 1 B = 1.5 # 1 6 G t = 1.3 # 1 9 4,,, 4, # # TC

17 Summary/Conclusions Mgagauss magntic filds ar gnratd in sphrical implosions Isotropic and anisotropic MHD quations ar addd to th ALE hydrocod DRACO. A gnration of th mgagauss magntic fild for sphrical implosions is numrically dmonstratd. Th influnc of th magntic fild on transport cofficints is analyd and th important rol of th Nrst trm is dmonstratd. TC8333