Magnetically Confined Fusion: Transport in the core and in the Scrape- off Layer ogdan Hnat Joe Dewhurst, David Higgins, Steve Gallagher, James Robinson and Paula Copil
Fusion Reaction H + 3 H 4 He + n Lower binding energy for 4 He + energy of the neutron = ~ 17.6 MeV Extra neutron in each nuclei increases collision rate Electrostatic forces small one proton per nucleus Result: cross section maximum at relatively low temperature of ~ 300 milion K. H extracted from see water, 3 H can be produced 6 Li + n 4 He + T or 7 Li +n 4 He + T + n
Lawson s criterion Confinement time τ E =Energy content / Rate of Loss Stationary state: equalise energy loss with input of thermal energy (keep T constant) Define L=n e τ E, look for T at which fusion reaction produces enough energy to sustain itself Take 50/50 D + T fusion reaction Assume cold ions (electron pressure only) L = nτ 1.5 10 e E Temperature at min. ~ 300 milion K 0 [s/m 3 ]
Realising Lawson s criterion Condition L= nτ 1.5 10 can be achieved by: e E Large values of τ E magnetic confinement, or Large density n e inertial confinement 0
Confinement and topology Poincare s theorem (in my own words ) Let S be a smooth, closed surface and C(x) be well behaved vector field such that the component of C tangent to S never vanishes. The surface S must then be a torus.
Confinement and topology Consider the outmost bounding surface of some confined plasma. Particles can stream free along the magnetic field lines An ideal confining magnetic field should have no component normal to the bounding surface Magnetic field must cover the entire surface and the tangent component can not vanish anywhere Conclusion: The bounding surface must be a torus.
Flux surfaces Rational surfaces: magnetic field line on such surface closes on itself after n toroidal and m poloidal turns Ergodic surface: magnetic field line never closes on itself, thus covering densly the surface Stochastic regions (volums): magnetic field has radial component which allows for fast transport between different flux surfaces (can not support pressure gradient)
Tokamak Magnetic field: 0.6 T Density: x10 19 [1/m^3] Plasma current: 1- MA Primary coil indices toroidal magnetic field Pure toroidal field configuration is unstable Plasma current is driven in toroidal direction, inducing poloidal field p= ( j ) r r
MAST Spherical tokamak Electricity cost ~ β -0.4 More compact in size Less prone to MHD instabilities Lower cheaper electricity
Different confinement regions Core: Hot collisionless plasma, β~1, δx/<x>«1 Edge: strong flow shear, steep gradients Scrape-Off Layer: Cooler, low, atomic physics effects
Particle Transport: classical estimates n+ i Γ = S (3 / nt ) + iq = S t part t heat Normally we assume that: Γ = D n Q= κ T Γ ( ) ( ) r nr nr+δ r vr = nr nr + rnδ r+ vr = ( vrδr) rn Δr e vr and we assume that Δr ρe thus Dr = ν eiρe τ ei ρ / 1/ e i = m m ρ and ν = m / m ν thus Γ =Γ ( ) ( ) i i e e ie e i ei Ions and electrons contribute equally to particle transport Same specie collisions do not contribute (momentum conservation) r r
Heat Transport: classical estimates n+ i Γ = S (3 / nt ) + iq = S t part Assume temperature gradient in radial direction 1 Q [( ) ( ) )] r mn vth r vth r r v κ +Δ r = r rt; Same specie collisions important for heat flux Electrons dominate parallel heat transport Collision increase χ and decrease χ t κ n heat ρ i i i r = = ni i τ ii In parallel direction the step size in the mean free path κ ntτ / m T ee 5/ χ
Transport: neoclassical estimates Assume particle is moving on a flux surface with magnetic field magnitude between min and max. Define: μ0 v 0 λ = mv v A particle can never enter the region where μ > E ZeΦ All particles must then satisfy 0 λ ( 0 / min ). 0 0 max λ min Particles trapped by the mirror force, move on the outboard side of the flux surface. This trapped particle orbits are called banana orbits.
Transport: neoclassical estimates Δr ρ ε p v e th p a,whereω p = andε Ω m R The width of the banana orbit is given by: ρ p p Taking f t as fraction of trapped particles heat diffusivity χ is ν χ ban 1/ ii i = ft( Δ r) νeff = ( ε) ( ρ ε pi ) = ερ ν pi ii ε Comparing classical and neoclassical heat diffusivities χ neoc i c i χ = ε 10 50 p
Turbulent transport estimate Turbulence driven by micro scale instabilities (drift wave instability, Ion Temperature Gradient-ITG, etc ) Gradients are sources of free energy large fluctuations at plasma edge are expected Often modelled as purely electrostatic md v = p + q( E + v ) multiply t Neglect inertial terms, take velocity perpendicular to E ( nt) v = v + v = + E diam
Turbulent transport estimate Often modelled as purely electrostatic E φ v v = = v E Neglect diamagnetic part ( Δr) φ D= veδ r = τ φ D D turb i neoc i turb D 10 e 1000 D neoc e
Sources of turbulence drift waves Ions dominate perp dynamics, electron parallel dir. Quasineutrality: n e = n i, cold ions: grad(p i )=0 No e-i collisions: δφ is in phase with δn v v E d E 1 de =, v P= ω dt ω T n = = k e n y δ n= n e φ e x o e / kt 0(1 ) g n 0