MHD Simulations for Fusion Applications Lecture 1 Tokamak Fusion Basics and the MHD Equations Stephen C. Jardin Princeton Plasma Physics Laboratory CEMRACS ‘10 Marseille, France July 19, 2010 1
Fusion Powers the Sun and Stars Can we harness Fusion power on earth?
The Case for Fusion Energy • Worldwide demand for energy continues to increase – Due to population increases and economic development – Most population growth and energy demand is in urban areas • Implies need for large, centralized power generation • Worldwide oil and gas production is near or past peak – Need for alternative source: coal, fission, fusion • Increasing evidence that release of greenhouse gases is causing global climate change . . . “Global warming” – Historical data and 100+ year detailed climate projections – This makes nuclear (fission or fusion) preferable to fossil (coal) • Fusion has some advantages over fission that could become critical: – Inherent safety (no China syndrome) – No weapons proliferation considerations (security) – Greatly reduced waste disposal problems (no Yucca Mt.)
Controlled Fusion uses isotopes of Hydrogen in a High Temperature Ionized Gas (Plasma) Deuterium Helium nuclei ( α -particle) … sustains reaction Tritium Neutron α Lithium Deuterium exists in nature (0.015% abundant in Hydrogen) T Tritium has a 12 year half life: produced via 6 Li + n � T + 4 He key proton Lithium is naturally abundant neutron
Controlled Fusion Basics Create a mixture of D and T (plasma), heat it to high temperature, and the D and T will fuse to produce energy. P DT = n D n T < σ v>(U α +U n ) at 10 keV, < σ v> ~ T 2 Operating point ~ 10 keV P DT ~ (plasma pressure) 2 Need ~ 5 atmosphere @ 10 keV Note: 1 keV = 10,000,000 deg(K)
Toroidal Magnetic Confinement TOKAMAK creates toroidal magnetic fields to confine particles in the 3 rd Charged particles have helical orbits dimension. Includes an induced in a magnetic field; they describe toroidal plasma current to heat and circular orbits perpendicular to the confine the plasma field and free-stream in the direction of the field. “TOKAMAK”: Russian abbreviation for “toroidal chamber”
ITER is now under construction I nternational T hermonuclear E xperimental R eactor: • European Union • Japan • United States • Russia • Korea • China • India • World’s largest tokamak scale • all super-conducting coils • 500 MW fusion output • Cost: $ 5-10 B • Originally to begin operation in 2015 (now 2028 full power)
June 28, 2005 ITER has a site… Ministerial Level Meeting Cadarache, France Moscow, Russia ITER Tore Supra
Progress in Magnetic Fusion Research and Next Step to ITER Operation with Fusion full power test Power 1,000 Start of Megawatts ITER 100 JET ITER Operations (EU) (Multilateral) 10 Data from Tokamak TFTR Experiments Worldwide (U.S.) 1,000 A Big Next Step to ITER Kilowatts Plasma Parameters 100 500 10 450 9 10 400 8 350 7 300 6 1,000 250 5 200 4 Watts 150 3 100 100 2 50 1 10 0 0 Power Duration Power Gain Power Gain Power (MW) Plasma Duration (Seconds) (MW) (Seconds) (Output/Input) 1,000 Milliwatts TFTR/JET ITER 100 10 2025 1975 1985 1995 2005 2015 Years
Simulations are needed in 4 areas • How to heat the plasma to thermonuclear temperatures ( ~ 100,000,000 o C) • How to reduce the background turbulence • How to eliminate device-scale instabilities • How to optimize the operation of the whole device
These 4 areas address different timescales and are normally studied using different codes SAWTOOTH CRASH ENERGY CONFINEMENT ELECTRON TRANSIT TURBULENCE ω LH τ A ISLAND GROWTH CURRENT DIFFUSION -1 Ω ce Ω ci -1 -1 10 -10 10 -8 10 -6 10 -4 10 -2 10 0 10 2 10 4 SEC. (b) Micro- (c) Extended- (d) Transport Codes turbulence codes (a) RF codes MHD codes
Extended MHD Codes solve 3D fluid equations for device-scale stability SAWTOOTH CRASH ENERGY CONFINEMENT ELECTRON TRANSIT TURBULENCE ω LH τ A ISLAND GROWTH CURRENT DIFFUSION -1 Ω ce Ω ci -1 -1 10 -10 10 -8 10 -6 10 -4 10 -2 10 0 10 2 10 4 SEC. • Sawtooth cycle is one example of global phenomena that need to be understood • Can cause degradation of confinement, or plasma termination if it couples with other modes • There are several codes in the US and elsewhere that are being used to study this and related phenomena: • NIMROD • M3D
Quicktime Movie shows Poincare plot of magnetic field at one toroidal location • Example of a recent 3D calculation using M3D code • “Internal Kink” mode in a small tokamak (Sawtooth Oscillations) • Good agreement between M3D, NIMROD, and experimental results •500 wallclock hours and over 200,000 CPU-hours
Excellent Agreement between NIMROD and M3D Kinetic energy vs time in lowest toroidal harmonics Flux Surfaces during crash at 2 times M3D NIMROD M3D NIMROD
2-Fluid MHD Equations: ∂ + ∇ • n = V ( ) 0 continuity n ∂ t ∂ B = −∇× ∇ = μ = ∇× E i B J B 0 Maxwell ∂ 0 t ∂ V + •∇ + ∇ = × − ∇ + μ ∇ V V J B i Π 2 V ( ) momentum nM p ∂ i GV t 1 ( ) + × = η + × −∇ E V B J J B Ohm's law p e ne ∂ ⎛ ⎞ 3 3 p + ∇ = − ∇ + η −∇ + i V i V 2 i q e ⎜ ⎟ electron energy p p J Q Δ ∂ ⎝ e ⎠ e e 2 2 t ∂ ⎛ ⎞ 3 3 p 2 + ∇ = − ∇ + μ ∇ −∇ − i V i V i q i ⎜ ⎟ ion energy p p V Q Δ ∂ i i i ⎝ ⎠ 2 2 t μ viscosity V fluid velocity number density n Idea l MHD η resistivity Β electron pressure p magnetic field e Resistive MH D q ,q heat fluxes J ion pressure p current density i e i 2- flui d MHD ≡ + equipartition Q E p p p electric field Δ e i μ permeability ≡ ρ 15 electron charge mass density e nM 0 i
Ideal MHD Equations: ∂ + ∇ • n = V ( n ) 0 continuity ∂ t ∂ B = −∇× ∇ = μ = ∇× E i B J B 0 Maxwell ∂ 0 t ∂ V + •∇ + ∇ = × V V J B ( ) momentum nM p ∂ i t + × = E V B 0 Ohm's law ∂ ⎛ ⎞ 3 3 p + ∇ = − ∇ i V i V ⎜ ⎟ p p energy ∂ ⎝ ⎠ 2 2 t V number density fluid velocity n Ideal MHD Β magnetic field electron pressure p e J current density ion pressure p i ≡ + E electric field p p p e i μ ≡ permeability ρ 16 mass density nM 0 i
Ideal MHD Equations: ∂ ρ + ∇ • ρ = V ( ) 0 continuity ∂ t ∂ B = −∇× ∇ = μ = ∇× E i B J B 0 Maxwell ∂ 0 t ∂ V ρ + •∇ + ∇ = × V V J B ( ) momentum p ∂ t + × = E V B 0 Ohm's law ∂ ⎛ ⎞ 3 p 3 + ∇ = − ∇ i V i V ⎜ ⎟ energy p p ∂ ⎝ ⎠ 2 2 t ∂ s ρ − ≡ ⇒ + ∇ = 5/3 V i 0 entropy s p s ∂ t E,J V can be eliminated number density fluid velocity n Ideal MHD ∂ ρ ∂ Β / t is redundant magnetic field electron pressure p e ∇ i B J is redundant current density ion pressure p i μ ≡ + E permeability electric field p p p 0 e i ≡ ρ 17 mass density nM i
Ideal MHD Equations: ∂ B ( ) = ∇× × V B ∂ ρ t mass density ∂ V 1 ( ) Β magnetic field ρ + •∇ + ∇ = ∇× × V V B B ( ) p ∂ μ t V fluid velocity 0 ∂ ⎛ ⎞ 3 3 p entropy density + ∇ = − ∇ s i V i V ⎜ ⎟ p p ∂ ⎝ ⎠ 2 2 t fluid pressure p ∂ + s ∇ = V i 0 s ∂ t ∇ B i is redundant ( ) 3/5 ρ = / p s μ permeability 0 Quasi-linear Symmetric � real characteristics Hyperbolic 18
Ideal MHD characteristics: The characteristic curves are the surfaces along which the solution is propagated. In 1D, the characteristic curves would be lines in (x,t) Boundary data (normally IC and ∂ ∂ s s + = BC) can be given on any curve 0 u ∂ ∂ t x that each characteristic curve intersects only once: � Cannot be tangent to characteristic curve To calculate characteristics in 3D, we suppose that the boundary φ = φ r conditions are given on a 3D surface and ask under what ( , ) t 0 conditions this is insufficient to determine the solution away from this φ surface. If so, is a characteristic surface. ( ) → φ χ σ τ r Perform a coordinate transformation: and look for ( , ) , , , t φ = φ power series solution away from the boundary surface 0 ∂ ∂ ∂ ∂ v v v v ( ) ( ) ( ) ( ) ( ) ( ) φ χ σ τ = χ σ τ + φ − φ + χ − χ + σ − σ + τ − τ v v , , , , , ∂ φ ∂ χ ∂ σ ∂ τ 0 0 0 0 0 φ φ φ φ 0 0 0 0 These can all be calculated since they If this cannot be constructed, φ = φ φ are surface derivatives within then is a characteristic surface 0
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