Progressing Performance – Tokamak Core Physics Marco Wischmeier Max-Planck-Institut für Plasmaphysik 85748 Garching marco.wischmeier at ipp.mpg.de Joint ICTP-IAEA College on Advanced Plasma Physics, Triest, Italy, 2016
Specific Fusion Plasma Physics Ø heat insulation (energy transport) Ø magnetohydrodynamic (MHD) stability Ø tokamak operational scenarios Ø exhaust of heat and particles (tomorrow, Wednesday)
Specific Fusion Plasma Physics Ø heat insulation (energy transport) Ø magnetohydrodynamic (MHD) stability Ø tokamak operational scenarios
Reactor energetics: the ‚Lawson‘ criterion for n τ Ε α -heating compensates losses: • radiative losses (Bremsstrahlung) • heat conduction and convection τ E = W plasma /P loss (‘energy confinement time’) leads to which has a minimum for n τ Ε = 2 x 10 20 m -3 s at T = 20 keV
Figure of merit for fusion performance nT τ Power P loss needed to sustain plasma • determined by thermal insulation: τ E = W plasma /P loss (‘energy confinement time’) Fusion power increases with W plasma • P fus ~ n D n T < σ v> ~ n e 2 T 2 ~ W plasma 2 Present day experiments: P loss compensated by external heating • Q = P fus /P ext ≈ P fus /P loss ~ nT τ E Reactor: P loss compensated by α -(self)heating • Q = P fus /P ext =P fus /(P loss -P α ) → ∞ (ignited plasma)
How is heat transported across field lines?
Energy confinement time determined by transport Simplest ansatz for heat transport: B • Diffusion due to binary collisions • • • • χ ≈ r L 2 / τ c ≈ 0.005 m 2 /s τ E ≈ a 2 /(4 χ ) • table top device (R ≈ 0.6 m) collision should ignite! Important transport regime for tokamaks and Transport to the edge stellarators: • Diffusion of trapped particles on banana R orbits due to binary collisions • neo-classical transport (important for impurities) Experimental finding: • ‚Anomalous‘ transport, much larger heat losses • Tokamaks: Ignition expected for R ~ 8 m
Energy confinement: empirical scaling laws In lack of a first principles physics model, ITER has been designed on the basis of an empirical scaling law • very limited predictive capability, need first principles model
From empirical scaling laws to physics understanding P heat First principle based understanding of temperature (density, … ) profiles
Anomalous transport due to turbulence Simplest estimation for heat transport due to turbulence: D ≈ ( Δ r eddy ) 2 / τ tear ≈ 2 m 2 /s
Global turbulence simulations
Energy Transport in Fusion Plasmas T(0.8) T(0.4) Anomalous transport determined by gradient driven turbulence • temperature profiles show a certain ‘stiffness’ • ‘critical gradient’ phenomenon – χ increases with P heat (!) ⇒ increasing machine size will increase central T as well as τ E N.B.: steep gradient region in the edge governed by different physics!
Energy Transport in Fusion Plasmas ASDEX Upgrade Locally, critical gradients can be exceeded (‘Transport Barrier’) • sheared rotation can suppress turbulent eddies • works at the edge (H-mode, see later) and internally (‘ITB’)
Anomalous transport determines machine size Fusion Power [MW] 5 2 4 3 B R Q β = P c N c q 3 . 1 A 3 . 53 = fus 1 2 4 5 1 q A 2 − 95 0 . 1 c H 3 . 23 R 2 . 7 B 3 . 7 β 1 N ITER ( β N =1.8) ITER (Q=10) DEMO ( β N =3) DEMO (ignited) Major radius R 0 [m] Major radius R 0 [m] • ignition (self-heated plasma) predicted at R = 7.5 m • at this machine size, the fusion power will be of the order of 1 GW
Specific Fusion Plasma Physics Ø heat insulation (energy transport) Ø magnetohydrodynamic (MHD) stability Ø tokamak operational scenarios
Plasma discharges can be subject to instabilities Desaster Self-organisation β -limit, disruption sationarity of profiles j(r), p(r)
Plasma discharges can be subject to instabilities linearly stable linearly unstable Equilibrium ∇ p = j x B means force balance, but not necessarily stability Stability against perturbation has to be evaluated by stability analysis Mathematically: solve time dependent MHD equations • linear stability: small perturbation, equilibrium unperturbed, exponentially growing eigenmodes • nonlinear stability: finite peturbation, back reaction on equilibrium, final state can also be saturated instability
Free energies to drive MHD modes current driven instabilities pressure driven instabilities Ex.: kink mode Ex.: interchange mode (only tokamaks) (tokamak and stellarator) N.B.: also fast particle pressure (usually kinetic effects)!
Ideal and resistive MHD instabilities Resistive MHD: η ≠ 0 Ideal MHD: η = 0 • reconnection of field lines • flux conservation • topology changes • topology unchanged
Magnetic islands impact tokamak discharges coupling between island chains (possibly stochastic regions) ⇒ sudden loss of heat insulation ('disruptive instability')
Disruptive instability limits achievable density High density clamps current profile and leads to island chains excessive cooling, current can no longer be sustained disruptions lead to high thermal and mechanical loads!
Removal of magnetic islands by microwaves n ν ECR = ν wave – k || v || Electron Cyclotron Resonance at ν = n 28 GHz B [T] Plasma is optically thick at ECR frequency Deposition controlled by local B-field ⇒ very good localisation
Ideal MHD instabilities limit achievable pressure Optimising nT means high pressure and, for given magnetic field, high dimensionless pressure β = 2 µ 0 <p> / B 2 This quantity is ultimately limited by ideal instabilities ‘Ideal’ MHD limit (ultimate limit, plasma β N = β /(I/aB)=3.5 unstable on Alfvén time scale ~ 10 µ s, β only limited by inertia) [%] • ‘Troyon’ limit β max ~ I p /(aB) , leads to definition of β N = β /(I p /(aB)) • at fixed aB , shaping of plasma cross- section allows higher I p → higher β
Specific Fusion Plasma Physics Ø heat insulation (energy transport) Ø magnetohydrodynamic (MHD) stability Ø tokamak operational scenarios
What is a ‚tokamak scenario‘? A tokamak (operational) scenario is a recipe to run a tokamak discharge Plasma discharge characterised by • external control parameters: B t , R 0 , a, κ , δ , P heat , Φ D … • integral plasma parameters: β = 2 µ 0 <p>/B 2 , I p = 2 π ∫ j(r) r dr … • plasma profiles: pressure p(r) = n(r)*T(r) , current density j(r) β p = 1 β p = 1 I p = 800 kA current density (a.u.) I p = 800 kA current density (a.u.) f NI = 14% f NI = 37% total j(r) total j(r) noninductive j(r) noninductive j(r) → operational scenario best characterised by shape of p(r), j(r)
Control of the profiles j(r)and p(r) is limited Pressure profile determined by combination of heating / fuelling profile and radial transport coefficients • ohmic heating coupled to temperature profile via σ ~T 3/2 • external heating methods allow for some variation – ICRH/ECRH deposition determined by B -field, NBI has usually broad profile • gas puff is peripheral source of particles, pellets further inside but: under reactor-like conditions, dominant α -heating ~ (nT) 2
The (low confinement) L-mode scenario Standard scenario without special tailoring of geometry or profiles • central current density usually limited by sawteeth • temperature gradient sits at critical value over most of profile • extrapolates to very large ( R > 10 m, I p > 30 MA) pulsed reactor
The (high confinement) H-mode scenario With hot (low collisionality) conditions, edge transport barrier develops • gives higher boundary condition for ‘stiff’ temperature profiles • global confinement τ E roughly factor 2 better than L-mode • extrapolates to more attractive ( R ~ 8 m, I p ~ 20 MA) pulsed reactor
Quality of heat insulation Turbulent transport limits (on a logarithmic scale) the gradient of the temperature profile Analogy of a sand pile: limited gradient But total height is variable by barriers
Turbulent transport strongly increases with logarithmic temperature gradient Existence of a critical logarithmic temperature gradient (nearly independent on heating power) “ stiff ” temperature profiles d ln T ∇ T 1 ⎝ T(a) = T(b) exp b - a ⎛ = - = ⎛ L T,cr ⎝ dr T L T,cr
Core temperature determined by temperature at the edge … … nearly independent of heating power Transport barrier at the edge ( “ high ” confinement mode) in divertor geometry
Energy Transport in Fusion Plasmas Anomalous transport determined by gradient driven turbulence • linear: main microinstabilities giving rise to turbulence identified • nonlinear: turbulence generates ‘zonal flow’ acting back on eddy size • (eddy size) 2 / (eddy lifetime) is of the order of experimental χ - values
Sheared flows – the most important saturation mechanism Macroscopic sheared rotation deforms eddies and tears them Radial transport increases with eddy size
Stationary H-modes usually accompanied by ELMs Edge Localised Modes (ELMs) regulate edge plasma pressure • without ELMs, particle confinement ‚too good‘ – impurity accumulation
Cross section of the spherical tokamak MAST MAST, CCFE, UK
Plasma discharges can be subject to instabilities MAST, CCFE, UK
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