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Tokamak Transport • Transport Coefficients (∆X)2 Γ = −D∇N : Fick’S Law D = : Diffusion Coefficient (M2/S) 2Τ Q = −Κ∇T : Fourier’S Law

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Tokamak Transport • Transport Coefficients (∆X)2 Γ = −D∇N : Fick’S Law D = : Diffusion Coefficient (M2/S) 2Τ Q = −Κ∇T : Fourier’S Law Introduction to Nuclear Fusion Prof. Dr. Yong-Su Na Force balance in a tokamak 2 Tokamak Equilibrium • The Grad-Shafranov Equation dp dF ∆*ψ = −µ R2 − F 0 dψ dψ JΦxBp: ▽p: JpxBΦ: strength plasma load strength of Bp on a magnetic of BΦ flux surface What kind of forces does a plasma have regarding equilibrium? 3 Tokamak Equilibrium • Basic Forces Acting on Tokamak Plasmas - Radial force balance Bp Jt x Bp X Jt Magnetic pressure, Tension force - Toroidal force balance: Tire tube force FNET ~ −eR ( pAI − pAII ) 4 J.P. Freidberg, “Ideal Magneto-Hydro-Dynamics”, lecture note Tokamak Equilibrium • Basic Forces Acting on Tokamak Plasmas - Toroidal force balance: Hoop force φI = φII BI > BII , AI < AII 2 2 BI AI > BII AII 2 2 FNET ~ eR (BI AI − BII AII ) / 2µ0 - Toroidal force balance: 1/R force φI = φII BI > BII , AI < AII 2 2 BI AI > BII AII net pressure B2 F = 2π 2a2 NET 2µ 0 5 J.P. Freidberg, “Ideal Magneto-Hydro-Dynamics”, lecture note Tokamak Equilibrium • Basic Forces Acting on Tokamak Plasmas - External coils required to provide the force balance I p × Bv Fv = BIL = 2πR0 I p Bv How about vertical movement? Bφ > Bθ > Bv 6 Plasma transport in a Tokamak 7 Energy Confinement Time Temperature Time 8 Energy Confinement Time Temperature 1/e Time τE • τE is a measure of how fast the plasma loses its energy. • The loss rate is smallest, τE largest if the fusion plasma is big and well insulated. 9 Tokamak Transport • Transport Coefficients (∆x)2 Γ = −D∇n : Fick’s law D = : diffusion coefficient (m2/s) 2τ q = −κ∇T : Fourier’s law Thermal diffusivity κ (∆x)2 a2 a2 χ ≡ ≈ D ≈ ≈ → τ E ≈ n τ τ E χ - Particle transport in fully ionised plasmas η⊥n∑ kT D⊥ = with magnetic field B2 10 Tokamak Transport • Classical Transport - Classical thermal conductivity (expectation): χi ~ 40χe - Typical numbers expected: ~10-4 m2/s 2 - Experimentally found: ~1 m /s, χi ~ χe 1 kTe Bohm diffusion (1946): D⊥ = 16 eB David Bohm (1917-1992) 11 Tokamak Transport • Classical Transport 1 kTe Bohm diffusion: D⊥ = 16 eB τE in various types of discharges in the Model C Stellarator F. F. Chen, “Introduction to Plasma Physics and Controlled Fusion” (2006) 12 Tokamak Transport • Neoclassical Transport - Major changes arise from toroidal effects characterised by inverse aspect ratio, ε = a/R0 13 Tokamak Transport • Particle Trapping ∇ ⋅ B = 0 HW. Derive this. θ 1 1 ∂ 1 ∂ 1 ∂Bφ ⇒ (rBr ) + [(1+ ε cosθ )Bθ ]+ = 0 1+ ε cosθ r ∂r r ∂θ R0 ∂φ 0 Bθ (θ = 0) ⇒ Bθ (r,θ ) = 1+ ε cosθ ˆ ˆ B0 B(r,θ ) = Bθ (r,θ )θ + Bφ (r,θ )φ = 1+ ε cosθ - Particle trapping by magnetic mirrors trapped particles with banana orbits Discuss the particle motion untrapped particles with circular orbits −ε ε - Trapped fraction: 1 Bmin 1 2 ftrap = 1− = 1− = 1− = Rm Bmax 1+ ε 1+ ε for a typical tokamak, ε ~ 1/3 → ftrap ~ 70% 14 1 B×∇B v ∇ = ± v⊥r Tokamak Transport D, B 2 L B2 • Particle Trapping mv2 R ×B = || 0 0 v D,R 2 2 qB0 R trapped particles passing particles 15 Tokamak Transport • Particle Trapping 16 Tokamak Transport • Neoclassical Bootstrap current Toroidal direction Projection of poloidally trapped ion trajectory Poloidal direction B Ion gyro-motion R 17 Tim Hender, “Neoclassical Tearing Modes in Tokamaks”, KPS/DPP, Daejun, Korea, 24 April 2009 http://tfy.tkk.fi/fusion/research/ Tokamak Transport • Neoclassical Bootstrap current J ∝ ∇p Currents due to BS neighbouring bananas largely cancel orbits tighter where field stronger - More & faster particles on orbits nearer the core (green .vs. blue) lead to a net “banana current”. - This is transferred to a helical bootstrap current via collisions. 18 Tim Hender, “Neoclassical Tearing Modes in Tokamaks”, KPS/DPP, Daejun, Korea, 24 April 2009 http://tfy.tkk.fi/fusion/research/ Tokamak Transport • Particle Trapping - Collisional excursion across flux surfaces untrapped particles: 2rg (2rLi) Magnetic field lines Magnetic flux surfaces 19 Tokamak Transport • Particle Trapping - Collisional excursion across flux surfaces untrapped particles: 2rg (2rLi) Magnetic field lines Magnetic flux surfaces 20 Tokamak Transport • Particle Trapping - Collisional excursion across flux surfaces untrapped particles: 2rg (2rLi) trapped particles: Δrtrap >> 2rg – enhanced radial diffusion across the confining magnetic field - If the fraction of trapped particle is large, this leakage enhancement constitutes a substantial problem in tokamak confinement. 21 Tokamak Transport • Particle Trapping - Collisional excursion across flux surfaces untrapped particles: 2rg (2rLi) trapped particles: Δrtrap >> 2rg – enhanced radial diffusion across the confining magnetic field - If the fraction of trapped particle is large, this leakage enhancement constitutes a substantial problem in tokamak confinement. 22 Tokamak Transport • Neoclassical Transports - May increase D, χ up to two orders of magnitude: - χi 'only' wrong by factor 3-5 - D, χe still wrong by up to two orders of magnitude! 1 kTe D⊥ = Bohm diffusion 16 eB η⊥n∑ kT D⊥ = B2 J. Wesson, Tokamaks (2004) 23 Tokamak Transport • Transport in fusion plasmas is ‘anomalous’ - Normal (water) flow: Hydrodynamic equations can develop nonlinear turbulent solutions (Reynolds, 1883) ρ: density of the fluid (kg/m³) inertial forces ρvL vL v: mean velocity of the object relative Re = = = to the fluid (m/s) viscous forces µ ν L: a characteristic linear dimension, (travelled length of the fluid; hydraulic diameter when dealing with river systems) (m) μ: dynamic viscosity of the fluid (Pa·s or N·s/m² or kg/(m·s)) ν: kinematic viscosity (μ/ρ) (m²/s) A vortex street around a cylinder. This occurs around cylinders, for any fluid, cylinder size and fluid speed, provided that there is a Reynolds number of between ~40 and 103 24 Tokamak Transport • Transport in fusion plasmas is ‘anomalous’ - Normal (water) flow: Hydrodynamic equations can develop nonlinear turbulent solutions (Reynolds, 1883) - Transport mainly governed by turbulence: radial extent of turbulent eddy: 1 - 2 cm typical lifetime of turbulent eddy: 0.5 - 1 ms - Anomalous transport coefficients are of the order 1 m2/s (∆x)2 D ~ : diffusion coefficient (m2/s) 25 τ http://scidacreview.org/0801/html/fusion.html Tokamak Transport • Microinstabilities - often associated with non-Maxwellian velocity distributions: deviation from thermodynamic equilibrium (nonuniformity, anisotropy of distributions) → free energy source which can drive instabilities - kinetic approach required: limited MHD approach • Two-stream or beam-plasma instability - Particle bunching → E perturbation → bunching↑ → unstable • Drift (or Universal) instability - driven by p (or n) in magnetic field - excited by drift waves with a phase velocity of v with a very short wavelength ∇ ∇ De - most unstable, dominant for anomalous transport • Trapped particle modes - Preferably when the perturbation frequency < bounce frequency - drift instability enhanced by trapped particle effects - Trapped Electron Mode (TEM), Trapped Ion Mode (TIM) 26 Tokamak Transport • Profile consistency (or profile resilience or stiffness) G. Tardini et al, NF 42 258 (2002) Dexp = D NC + Danomalous > D NC χ exp = χ NC + χ anomalous > χ NC 27 How to reduce plasma transport? 28 Tokamak Transport • Suppression of Anomalous Transport: H-mode - 1982 IAEA FEC, F. Wagner et al. (ASDEX, Germany) - Transition to H-mode: state with reduced turbulence at the plasma edge - Formation of an edge transport barrier: steep pressure gradient at the edge 29 Tokamak Transport • Suppression of Anomalous Transport: H-mode - 1982 IAEA FEC, F. Wagner et al. (ASDEX, Germany) - Transition to H-mode: state with reduced turbulence at the plasma edge - Formation of an edge transport barrier: steep pressure gradient at the edge a2 τ ≈ E χ 30 Tokamak Transport • Suppression of Anomalous Transport: H-mode - 1982 IAEA FEC, F. Wagner et al. (ASDEX, Germany) - Transition to H-mode: state with reduced turbulence at the plasma edge - Formation of an edge transport barrier: steep pressure gradient at the edge Hoover dam 31 Tokamak Transport • Suppression of Anomalous Transport: H-mode - 1982 IAEA FEC, F. Wagner et al. (ASDEX, Germany) - Transition to H-mode: state with reduced turbulence at the plasma edge - Formation of an edge transport barrier: steep pressure gradient at the edge 32 L-H transition threshold power -1 0.82 0.58 1.0 0.81 Pth = 2.84M Bt n20 R a 33 Tokamak Transport • First H-mode Transition in KSTAR (November 8, 2010) - B0 = 2.0 T, Heating = 1.5 MW (NBI : 1.3 MW, ECH : 0.2 MW) After Boronisation on November 7, 2010 34 Tokamak Transport • First H-mode Transition in KSTAR (November 8, 2010) 35 Tokamak Transport • Suppression of Anomalous Transport: ITBs H-mode Reversed shear mode 36 Tokamak Transport • Suppression of Anomalous Transport 37 Improved confinement suitable for the steady-state operation Non-monotonic current profile Turbulence suppression High pressure gradients Large bootstrap current Non-inductive current drive 38 Pulsed Operation Faraday‘s law Plasma Current d (MA) v = − B ⋅ dS dt ∫S t (s) CS Coil Current Power Supply Limit (kA) t (s) Inherent drawback of Tokamak! 30 Steady-State Operation Plasma Current (MA) t (s) CS Coil Current (kA) t (s) d/dt ~ 0 Steady-state operation by self-generated and externally driven current 30 Turbulence Simulations - XGC1 simulation Courtesy of S. H. Ku (PPPL) 41 Energy Confinement Time W W stored energy In steady conditions, neglecting radiation loss, τ E = ≈ = ∂W P applied heating power Ohmic heating replaced by P − in in ∂t total input power • Since tokamak transport is anomalous, empirical scaling laws for energy confinement are necessary. • To predict the performance of future devices, the energy confinement time is one of the most important parameter. • Empirical scaling laws: regression analysis from available experimental database. fit αI αB αP αn αM αR αε ακ τ th,E = CI B P n M R ε κ in engineering variables 43 Energy Confinement Time • H-mode confinement scaling IPB98( y,2) 0.93 0.15 −0.69 0.41 0.19 1.97 0.58 0.78 τ th,E = 0.0562I B P n M R ε κ a τE in KSTAR and ITER? Why should ITER be large? 44.
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