Tokamak Fusion Basics and the MHD Equations

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1 MHD Simulations for Fusion Applications Lecture 1 Tokamak Fusion Basics and the MHD Equations Stephen C. Jardin Princeton Plasma Physics Laboratory CEMRACS 1 Marseille, France July 19, 21 1

2 Fusion Powers the Sun and Stars Can we harness Fusion power on earth?

3 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 1+ 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.)

4 Controlled Fusion uses isotopes of Hydrogen in a High Temperature Ionized Gas (Plasma) Deuterium Helium nuclei (α-particle) sustains reaction Tritium Neutron Lithium α Deuterium exists in nature (.15% abundant in Hydrogen) Tritium has a 12 year half life: produced via 6 Li + n T + 4 He key proton Lithium is naturally abundant neutron T

5 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 <σv>(u α +U n ) at 1 kev, <σv> ~ T 2 P DT ~ (plasma pressure) 2 Operating point ~ 1 kev Need ~ 5 1 kev Note: 1 kev = 1,, deg(k)

6 Toroidal Magnetic Confinement Charged particles have helical orbits in a magnetic field; they describe circular orbits perpendicular to the field and free-stream in the direction of the field. TOKAMAK creates toroidal magnetic fields to confine particles in the 3 rd dimension. Includes an induced toroidal plasma current to heat and confine the plasma TOKAMAK : Russian abbreviation for toroidal chamber

7 ITER is now under construction International Thermonuclear Experimental Reactor: European Union Japan United States Russia Korea China India scale World s largest tokamak 5 MW fusion output all super-conducting coils Cost: $ 5-1 B Originally to begin operation in 215 (now 228 full power)

8 ITER has a site Cadarache, France June 28, 25 Ministerial Level Meeting Moscow, Russia ITER Tore Supra

9 Progress in Magnetic Fusion Research and Next Step to ITER Fusion Power Megawatts Kilowatts Watts Milliwatts 1, 1 1 1, 1 1 1, 1 1 1, 1 1 Data from Tokamak Experiments Worldwide TFTR (U.S.) Years JET (EU) Power (MW) Power (MW) Start of ITER Operations A Big Next Step to ITER Plasma Parameters Duration (Seconds) Plasma Duration (Seconds) TFTR/JET ITER Operation with full power test ITER (Multilateral) Gain Power Gain (Output/Input) 225

10 Simulations are needed in 4 areas How to heat the plasma to thermonuclear temperatures ( ~ 1,, o C) How to reduce the background turbulence How to eliminate device-scale instabilities How to optimize the operation of the whole device

11 These 4 areas address different timescales and are normally studied using different codes ELECTRON TRANSIT ω -1 Ω -1 LH ce SAWTOOTH CRASH ENERGY CONFINEMENT TURBULENCE ISLAND GROWTH CURRENT DIFFUSION Ω -1 τ A ci SEC. (a) RF codes (b) Microturbulence codes (c) Extended- MHD codes (d) Transport Codes

12 Extended MHD Codes solve 3D fluid equations for device-scale stability ELECTRON TRANSIT ω -1 Ω -1 ce LH TURBULENCE SAWTOOTH CRASH ISLAND GROWTH ENERGY CONFINEMENT CURRENT DIFFUSION Ω -1 τ ci A 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

13 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 5 wallclock hours and over 2, CPU-hours

14 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

15 2-Fluid MHD Equations: n + ( nv) = t continuity B = E ib = μj = B t Maxwell V 2 nm i( + V V ) + p = J B iπgv + μ V t momentum 1 E + V B = ηj + ( J B pe) ne Ohm's law 3 pe i pev = pe iv + ηj iqe + QΔ 2 t 2 electron energy 3 pi i piv = pi iv + μ V iqi Q 2 t 2 n number density Ideal MHD Β magnetic field Resistive MHD J current density 2-fluid MHD E electric field nm i ρ mass density e electron charge V p p e i Δ fluid velocity electron pressure ion pressure p p + p e i ion energy μ viscosity η resistivity q,q i Q μ Δ e heat fluxes equipartition permeability 15

16 Ideal MHD Equations: n + ( nv) = t continuity B = E ib = μj = B t Maxwell V nm i( + V V ) + p = J B t momentum E+ V B= Ohm's law 3 p 3 + i pv = p iv 2 t 2 energy Ideal MHD n number density Β magnetic field J current density E electric field nm i ρ mass density V p p e i fluid velocity electron pressure ion pressure p p + p e i μ permeability 16

17 Ideal MHD Equations: ρ + ( ρv) = t continuity B = E ib = μj = B t Maxwell V ρ( + V V ) + p = J B t momentum E+ V B= Ohm's law 3 p 3 + i pv = p iv energy 2 t 2 s + Vi = entropy t 5/3 s pρ s Ideal MHD n number density Β magnetic field J current density E electric field nm i ρ mass density V p p e i fluid velocity electron pressure ion pressure p p + p e i E,J can be eliminated ρ / t is redundant ib is redundant μ permeability 17

18 Ideal MHD Equations: B = ( V B) t V 1 ρ( + V V ) + p = ( B ) B t μ 3 p 3 + i pv = p iv 2 t 2 s + Vi s = t ρ = ( p/ s) 3/5 ρ Β V s p ib μ mass density magnetic field fluid velocity entropy density fluid pressure is redundant permeability Quasi-linear Symmetric Hyperbolic real characteristics 18

19 v 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 BC) can be given on any curve that each characteristic curve intersects only once: s s + u = t x Cannot be tangent to characteristic curve To calculate characteristics in 3D, we suppose that the boundary conditions are given on a 3D surface φ(,) r t = φ and ask under what conditions this is insufficient to determine the solution away from this surface. If so, φ is a characteristic surface. ( ) Perform a coordinate transformation: (,) r t φ, χστ,, and look for power series solution away from the boundary surface φ = φ v v v v φχστ,,, = v χστ,, + φ φ + χ χ + σ σ + τ τ φ χ σ τ ( ) ( ) ( ) ( ) ( ) ( ) If this cannot be constructed, then φ is a characteristic surface φ φ φ φ These can all be calculated since they are surface derivatives within φ = φ

20 Ideal MHD characteristics-2: Introduce a characteristic surface φ( r, t) = φ spatial normal nˆ = φ / φ characteristic speed: u ( φt + Vi φ) / φ ( ) = ( ) φ Ideal MHD All terms containing derivatives involving φ AX= i B = nˆ = u nv z A nv x A nxcs u nv z A u nzc S nv z A u A = nv z A u nv x A u nc x S nc z S u u (,, B) ( n,, n ) x All known quantities if det A = φ is characteristic surface z B is in z ˆ direction propagation in (x,z) V c A S B/ 5 3 ρv x ρv y ρv z ρ μ B x X = ρ μ B y ρ μ B z 1 cs p s μ ρ p/ ρ 2

21 Ideal MHD wave speeds: u = u = 2 2 u = u = V 2 2 A ( ) ( ) An det A = D= u u VAn u VA cs u VAnc + + S = entropy disturbance Alfven wave ( ) ( ) = s = 4 2 A + S 2 A + S An S u u V c V c V c ( ) ( ) = f = 4 2 A + S + 2 A + S An S u u V c V c V c 1/2 1/2 slow wave fast wave B = (,, B) ( n n ) nˆ =,, V c V A S x B/ 5 3 n V An Z A z μ ρ p/ ρ In normal magnetically confined plasmas, we take 2 2 the low-β limit c V S A u = u = V u u 2 2 A = u s z S 2 2 f An nc = u V + n c A x S Alfven wave slow wave fast wave 21

22 Ideal MHD surface diagrams Reciprocal normal surface diagram u = u = V u u 2 2 A = u s z S 2 2 f An nc = u V + n c A x S Alfven wave slow wave fast wave Ray surface diagram B = (,, B) ( n n ) nˆ =,, V c V A S x B/ 5 3 n V An Z A z μ ρ p/ ρ 22

23 ρv x ρv y ρv z ρ μ B x X = = ρ μ B y ρ μ B z 1 cs p s Ideal MHD eigenvectors entropy Alfven fast n x = 1 n z = fast n x = n z = 1 slow n x = n z = ± 1 ± 1 1 cs / V A ± 1 ± 1 B = (,, B) ( n n ) nˆ =,, V c V A S x B/ 5 3 n V An Z A z μ ρ p/ ρ The Alfven wave only propagates parallel to the magnetic field, and does so by bending the field. It is purely transverse (incompressible) Only the fast wave can propagate perpendicular to the background field, and does so by compressing and expanding the field The slow wave does not perturb the magnetic field, only the pressure

24 Background magnetic field direction Slow Wave Alfven Wave Fast Wave propagation only propagates parallel to B only compresses fluid in parallel direction does not perturb magnetic field propagation only propagates parallel to B incompressible only bends the field, does not compress it propagation can propagate perpendicular to B only compresses fluid in direction compresses the magnetic field This is the troublesome wave!

25 Tokamaks have Magnetic Surfaces, or Flux Surfaces φ Magnetic field is primarily into the screen, however it has a twist to it. After many transits, it forms 2D surfaces in 3D space. Because the particles are free to stream along the field, the temperatures and densities are nearly uniform on these surfaces. Only the Fast Wave can propagate across these surfaces, but it will have a very small amplitude compared to the other waves.

26 Must deal with Fast Wave Tokamak schematic Tokamak cross section The field lines in a tokamak are dominantly in the toroidal direction. The magnetic field forms flux surfaces. Only the fast wave can propagate across these surfaces. Since the gradients across surfaces are large (requiring high resolution), the time-scales associated with the fast wave are very short However, the amplitude will always be small because it compresses the field. The presence of the fast wave makes explicit time integration not practical

27 Summary Nuclear fusion is a promising energy source that will be demonstrated in the coming decades by way of the tokamak (ITER) Global dynamics of the plasma in the tokamak are described by a set of fluid like equations called the MHD equations A subset of the full-mhd equations with the dissipative terms removed are called the ideal-mhd equations These have wave solutions that illustrate that there are 3 fundamentally different types of waves. Unstable plasma motions are always associated with the slow wave and Alfven wave. The fast wave is a major source of trouble computationally because it is the fastest and the only one that propagates across the surfaces Largely because of the fast wave, implicit methods are essential

28 28

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