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ICF Ignition, the Lawson Criterion and Comparison with MCF FSC Deuterium–Tritium Plasmas 100 Ignition

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ICF Ignition, the Lawson Criterion and Comparison with MCF FSC Deuterium–Tritium Plasmas 100 Ignition ICF Ignition, the Lawson Criterion and Comparison with MCF FSC Deuterium–Tritium Plasmas 100 Ignition E Q ~ 10 x i OMEGA (2009) T i 10 Q ~ 1 n Q = Tokamaks 1993–1999 W /W ) Fusion Input Q ~ 0.1 1 W = energy Laser direct kev s kev Q ~ 0.01 3 drive (1996) – 0.1 m Q ~ 0.001 Laser direct 20 0.01 drive (1986) 10 ( Q ~ 0.0001 Laser indirect 0.001 drive (1986) Q ~ 0.00001 Lawson fusion parameter, Lawson fusion parameter, 0.0001 0.1 1 10 100 “Review of Burning Plasma Physics,” Central ion temperature (keV) FESAC Report (September 2001). 51st Annual Meeting of the R. Betti American Physical Society University of Rochester Division of Plasma Physics Fusion Science Center and Atlanta, GA Laboratory for Laser Energetics 2–6 November 2009 Collaborators FSC K. S. Anderson (UO5.00004) P. Chang (TO5.00004) R. Nora C. D. Zhou University of Rochester Laboratory for Laser Energetics B. Spears (UO5.00013) J. Edwards S. Haan J. Lindl Lawrence Livermore National Laboratory Summary The measurable Lawson criterion and hydro-equivalent curves determine the requirements for an early hydro-equivalent demonstration of ignition on OMEGA and on the NIF (THD) FSC • Cryogenic implosions on OMEGA have achieved a Lawson parameter PxE á 1 atm-s comparable to large tokamaks • Performance requirements for hydroequivalent ignition on OMEGA and NIF (THD)* GtRH GT H YOC Px Hydro-equivalent ignition n i n E g/cm2 keV (atm s) 15% OMEGA (25 kJ) ~0.30 ~3.4 (~3 × 1013 2.6 neutrons) NIF (THD) ~1.8 ~4.7 ~40% 20 * NIF will begin the cryogenic implosion campaign using a surrogate Tritium– Hydrogen–Deuterium (THD) target. Only a very small fraction of Deuterium (<5%) is used to prevent fusion reactions from affecting the hydrodynamics. TC8563a See B. Spears (UO5.00013). Outline FSC • A 1-D measurable Lawson criterion for ICF • Hydro-equivalent curves and hydro-equivalent ignition • The 3-D extension of the Lawson criterion • Comparison with magnetic confinement TC8633 Outline FSC • A 1-D measurable Lawson criterion for ICF • Hydro-equivalent curves and hydro-equivalent ignition • The 3-D extension of the Lawson criterion • Comparison with magnetic confinement TC8633a “Classic” work on ICF ignition has focused on static models of the hot spot and has neglected the dense shell FSC • Lawson criterion applied to the hot spot 15 –3 nTixE > 3 × 10 cm keV s n, x cannot be measured • Static models of the ignition condition use the hot-spot areal density and ion temperature 2 tRhot spot á 0.3 g/cm Ti á 5 to 10 keV tRhot spot cannot be measured • J. D. Lawson, Proc. Phys. Soc. London • R. Kishony et al., Phys. Plasmas 4, 1385 (1997). B70, 6 (1957). • J. D. Lindl, Inertial Confinement Fusion: The Quest • S. Yu. Gus’kov et al., Nucl. Fusion for Ignition and Energy Gain Using Indirect Drive 16, 957 (1976). (Springer-Verlag, New York, 1998). • S. Atzeni and A. Caruso, Phys. Lett. A • S. Atzeni and J. Meyer-ter-Vehn, The Physics of Inertial 85, 345 (1981). Fusion: Beam Plasma Interaction, Hydrodynamics, Hot • S. Atzeni and A. Caruso, Nuovo Cimento B Dense Matter, International Series of Monographs on 80, 71 (1984). Physics (Clarendon Press, Oxford, 2004). TC8532a Ion temperatures, areal densities, and neutron yields are parameters of the fuel assembly that can be measured with existing diagnostics FSC dNneutron • Neutron yields and neutron rates Nneutron are measured with scintillators dt • Ion temperature (neutron averaged) is Ti measured with the neutron time-of-flight neutron detectors 3 tR / tdr • Total areal density (neutron averaged) neutron #0 is measured with magnetic recoil neutron spectrometer measuring the Shell downscattered neutron fraction.* Hot spot Total tR . shell (tD)stagnation Dstagnation TC8533a See J. Frenje (NI2.00004). A dynamic model of ignition relates the hot-spot stagnation properties to those of the shell FSC Dynamic model of hot-spot formation and ignition Alpha Deceleration deposition Peak implosion Stagnation Ignition velocity V = Vi The initial conditions are those at peak Thermal implosion velocity instability • R. Betti et al., Phys. Plasmas 8, 5257 (2001). • R. Betti et al., Phys. Plasmas 9, 2277 (2002). • J. Sanz et al., Phys. Plasmas 12, 112702 (2005). • Y. Saillard, Nucl. Fusion 46, 1017 (2006). • J. Garnier and C. Cherfils-Clérouin, Phys. Plasmas 15, 102702 (2008). • C. D. Zhou and R. Betti, Phys. Plasmas 15, 102707 (2008). TC8534a The heat and radiation energy lost by the hot spot does not propagate through the dense cold shell; no heat or radiation flux at the hot-spot boundary FSC mo < Dshell 5 2 mo qheat =-l] Tg dTT lSp ] g .l0 T T t shell 3 2 nm TeV hokeV 1000 m = c m o 200 5 etg/ cco Heat flux Rad. flux Radiation flux . 0 Shell Shell Heat flux . 0 Hot spot R+ Hot spot R r R + + qheat ` r = Rj . 0 qrad `r= R j . 0 TC8536 Shell The hot-spot global energy balance determines the ignition condition FSC d 3 P] tg Vol^th = compression/decompression work dt ;2 E + a_heating — (conduction + radiation losses) dR Compression/decompression work =-S t P t ]]g g dt fa 2 vv Alpha heating = P # dV fa = 3.5 MeV 16 v T2 2 + SR=4r Conduction losses at R = #S qheat :ndS = 0 4r 3 Radiation losses at R+ = q : ndS = 0 Vol= R #S rad 3 TC8537 The dynamic ignition model requires an equation for the temperature evolution FSC 3 3 Use a T approximation of GvvH & GvvH . CaT • Hot-spot energy equation d 3 2 dR 2R 2 P]] tgR t g =-2P]] tgR t g +C P] tg # Tr dr dt 8 B dt a 0 • Shell’s momentum equation (simple “thin-shell” approximation, Msh / shell mass) 2 d R 2 Needs T(r,t) Msh =4rP]] tgR t g dt2 More realistic compressible “thick shell model” see: TC8538 R. Betti et al., Phys. Plasmas 9, 2277 (2002). The heat lost by the hot spot is deposited on the shell inner surface driving an ablative mass flow into the hot spot (ignore radiation losses*) FSC 5 2 Enthalpy flux lsp ]TT g = l0 Mass T t 5 ablation Pv = -l TTd 2 a sp ] g 2tT T : Pva =va =2 ma mi mi Heat flux : m l Shell i 0 dT m = - T3 2 a 5 dr Hot spot 2/ 5 r2 useTT= 0 ^th f 1Z p R2 R r 5/ 2 : 4mi l0T 0 ma = 25 R^ t h *Radiation losses are ignored in this talk for simplicity. TC8539 They are included in C. D. Zhou and R. Betti, Phys. Plasmas 15, 102707 (2008). Mass conservation in the hot spot determines the temperature evolution FSC 5 2 • Ablation rate from shell : 4mi l0 T0] tg ma = to hot spot 25 R] tg • Hot-spot-mass dM : =4rR2 m conservation dt a 3 m P t 4m P]]tg R t g • Mass–Temp relation M=##t dV = i ] g dV. i 2 T r,t T t v v ^ h 0] g 3 • Temperature equation d PR 4r 5 2 f p . l0 RT0 dt T0 25 TC8541 Hot-spot-ignition model: three ODE’s for pressure, radius, and central temperature FSC • Hot-spot energy or pressure equation d dR P t R t 3 =-2P t R t 2 +CP t2 T t R t 3 dt 7 ]]g g A ]]g g dt a ]]g0 g ] g • Shell momentum or hot-spot-radius equation 2 d R 2 Msh =4rP]] tgR t g dt2 • Hot-spot-mass or temperature equation* 3 d P]] tgR t g 4r 5 2 . l0 R]] tgT0 t g dt > T0] tg H 25 TC8542 *Radiation losses are neglected in this talk for simplicity The initial conditions are related to the time when the imploding shell reaches the maximum implosion velocity FSC t = 0 " Maximum implosion velocity Vi Vi Dense shell R(0) Stagnating core P(0) T(0) Hot spot 0 : PP^0h=0TT 0 ^ 0 h = 0 RR^0h=0 RV ^ 0 h =- i • Expansion parameter d % 1 3 Hot- spot thermal energy ^t=0h 2rPR^0h ^ 0 h ! / = % 1 Shell kinetic energy t=0 2 ^ h ^1 2hMVsh i TC8543 The model has an analytic solution in the absence of a-particle energy deposition FSC • Stagnation pressure PPno-a .!-5 2 (without alphas) st 0 • Stagnation radius no-a 1 2 Rst .! R0 (without alphas) R 3 V2 7 • Stagnation temperature no-a S0. 1 MVsh i W Tst = S W (without alphas) l0 no-a 2 S _Rst i W T X 3 2rPR^0h ^ 0 h ! / %1 Expansion parameter 2 ^1 2hMVsh i TC8544 The no-a stagnation values are used in the derivation of the dimensionless ignition model (with a’s); ignition depends on the one dimensionless parameter ca FSC Dimensionless variables ˆ no-a ˆ no-a ˆ no-a no-a PP= PRst = RRst TT= Tst x= tVRi st Energy Mass Momentum ˆ ˆ 3 2 ˆ d ˆ ˆ 5 ˆ ˆ 5 ˆ d PR 5 2 d R ˆ ˆ 2 _PR i = ca PR T d n =RT =PR dx dx Tˆ dx2 Initial conditions Critical parameter—determines ignition t 5 2 P]0g = ! t -1 2 no--ano a no-a R]0g = ! PRst st Tst d"0 ca= C a to 8Vi R]0g = - 1 t 1 2 T]0g = ! TC8545a Ignition condition for the model equations: find the critical value of ca leading to an explosive solution FSC ) Singularity t ( P Time Condition for singular Singularity ca > 1.2 explosive solution caused by 3 GvoH ~ CaT no--ano a no-a PRst st Tst ca~ C a 8 Vi Cannot be measured TC8546a The ignition parameter ca can be written in terms of the measurable parameters tRtotal and Ti FSC Energy conservation Ignition condition: ca > 1.2 1 2rPR3 = MV2 no --ano a no - a st st 2 sh i PRst st Tst ca~ C a 8Vi Shell mass at stagnation 2 Msh .4rtstDst Rst Rst Temperature-velocity relation 2 7 Dst no-a 0.
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