Neoclassical transport Dr Ben Dudson Department of Physics, University of York Heslington, York YO10 5DD, UK 28 th January 2013 Dr Ben Dudson Magnetic Confinement Fusion (1 of 19)
Last time Toroidal devices such as the Stellarator and Tokamak Need for a rotational transform to short out the vertical electric field caused by the B drift This can either be created using shaped coils (Stellarators) or by running a current in the plasma (Tokamaks) Dr Ben Dudson Magnetic Confinement Fusion (2 of 19)
Last time Toroidal devices such as the Stellarator and Tokamak Need for a rotational transform to short out the vertical electric field caused by the B drift This can either be created using shaped coils (Stellarators) or by running a current in the plasma (Tokamaks) Calculated the classical transport of particles and energy out of a tokamak. This gave a wildly unrealistic confinement This lecture we ll look at one reason why... Dr Ben Dudson Magnetic Confinement Fusion (2 of 19)
Particle trapping In a tokamak, the magnetic field varies with the major radius Z B φ R B φ 1/R r R Dr Ben Dudson Magnetic Confinement Fusion (3 of 19)
Particle trapping In a tokamak, the magnetic field varies with the major radius Z B φ R B φ 1/R There is now a minimum in the B field at the outboard (large R) side of the tokamak Trapped particles. The study of these particles and their effect is called Neoclassical theory. R Dr Ben Dudson Magnetic Confinement Fusion (3 of 19)
Large aspect-ratio approximation A useful approximation is that the variation in major radius R is small, and that the toroidal field is much bigger than the poloidal field. For a circular cross-section of radius r, the major radius varies Z like r Hence For small ɛ 1 θ R B R = R 0 +r cos θ = R 0 (1 + ɛ cos θ) ɛ = r/r 0 ratio B 0 1 + ɛ cos θ B B 0 (1 ɛ cos θ) is inverse aspect Dr Ben Dudson Magnetic Confinement Fusion (4 of 19)
Particle trapping (redux) If there is no electrostatic potential φ, Kinetic energy 1 2 mv 2 is conserved (i.e. speed is conserved) Magnetic magnetic moment also conserved µ = mv 2 /2B Dr Ben Dudson Magnetic Confinement Fusion (5 of 19)
Particle trapping (redux) If there is no electrostatic potential φ, Kinetic energy 1 2 mv 2 is conserved (i.e. speed is conserved) Magnetic magnetic moment also conserved µ = mv 2 /2B Define velocity at the outboard side (θ = 0): v 0 and v 0. K.E. conserved ( v 2 = v 2 + v 2 = v 0 2 + v 0 2 v 2 = v 2 1 v 2 ) v 2 Conservation of µ v 2 ( v 2 = v 2 1 v 0 2 1 ɛ cos θ v 2 1 ɛ v 2 0 B 0 (1 ɛ cos θ) = B 0 (1 ɛ) ) ( = v 2 1 v 0 2 v 2 [1 + ɛ (1 cos θ)] ) Dr Ben Dudson Magnetic Confinement Fusion (5 of 19)
Particle trapping (redux) If there is no electrostatic potential φ, Kinetic energy 1 2 mv 2 is conserved (i.e. speed is conserved) Magnetic magnetic moment also conserved µ = mv 2 /2B Define velocity at the outboard side (θ = 0): v 0 and v 0. K.E. conserved ( v 2 = v 2 + v 2 = v 0 2 + v 0 2 v 2 = v 2 1 v 2 ) v 2 Conservation of µ v 2 ( v 2 = v 2 1 v 0 2 1 ɛ cos θ v 2 1 ɛ v 2 0 B 0 (1 ɛ cos θ) = B 0 (1 ɛ) ) ( = v 2 1 v 0 2 v 2 [1 + ɛ (1 cos θ)] ) v 2 = v 2 ( 1 v 2 0 v 2 [ 1 + 2ɛ sin 2 (θ/2) ] ) Dr Ben Dudson Magnetic Confinement Fusion (5 of 19)
Particle trapping (redux) ( v 2 = v 2 1 v 0 2 [ 1 + 2ɛ sin 2 v 2 (θ/2) ] ) If v 2 < 0 for any θ then a particle is trapped. Therefore, if v 2 (θ = π) 0 Particle is trapped v 0 2 v 2 [1 + 2ɛ] 1 v 0 v 1 ɛ For trapped particles v 0 Slope 1 2ɛ v 0 v 2 v 2 0 1+2ɛ v 2 0 + v 2 0 v 2 0 1+2ɛ Untrapped v 0 v 0 2ɛ Dr Ben Dudson Magnetic Confinement Fusion (6 of 19)
Particle drifts As particles move around the torus, their orbits drift. We have already come across the B drift: v B = v 2 B B 2Ω B 2 but there is also the curvature drift R C Dr Ben Dudson Magnetic Confinement Fusion (7 of 19)
Particle drifts As particles move around the torus, their orbits drift. We have already come across the B drift: but there is also the curvature drift v B = v 2 B B 2Ω B 2 In the frame of the particle, there is a centrifugal force F R = mv 2 R C R 2 C R C This therefore causes a drift ( ) v R = 1 mv 2R C /RC 2 B q B 2 = m qb v 2R C B RC 2 B = v 2 Ω Dr Ben Dudson Magnetic Confinement Fusion (7 of 19) R C B R 2 C B
Particle drifts So how big are these drifts, and what direction are they in? v B = v 2 B B 2Ω B 2 B B 0 e φ B 0 1 R B B 0 R R B B B 2 e φ R R v B = v 2 2ΩR e Z = e Z R Dr Ben Dudson Magnetic Confinement Fusion (8 of 19)
Particle drifts So how big are these drifts, and what direction are they in? v B = v 2 B B v R = v 2 R C B 2Ω B 2 Ω RC 2 B B B 0 e φ B 0 1 R B B 0 R R B B B 2 e φ R R v B = v 2 2ΩR e Z = e Z R R C = R R v R = v 2 R e φ Ω R = v 2 e Z Ω R Dr Ben Dudson Magnetic Confinement Fusion (8 of 19)
Particle drifts So how big are these drifts, and what direction are they in? v B = v 2 B B v R = v 2 R C B 2Ω B 2 Ω RC 2 B B B 0 e φ B 0 1 R B B 0 R R B B B 2 e φ R R v B = v 2 2ΩR e Z Total drift is therefore = e Z R v B + v R = ( v 2 + v 2 /2 ) RΩ R C = R R v R = v 2 R e φ Ω R = v 2 e Z Ω R Note that the drift is in the vertical direction and opposite for electrons and ions e Z Dr Ben Dudson Magnetic Confinement Fusion (8 of 19)
Banana orbits Z Without drifts Z With drifts R R δr b Characteristic shape of the orbits in the poloidal plane gives this the name banana orbit. The width of this orbit is the banana width, often denoted δr bj or ρ bj with j indicating electrons or ions. Dr Ben Dudson Magnetic Confinement Fusion (9 of 19)
Banana orbit characteristics Let us calculate the banana width for a barely trapped particle i.e. one with a bounce point at θ = π, on the inboard side Time for half an orbit: Velocity along the field-line v 2ɛv is small. Total speed is therefore approximately v v. This will be approximately the thermal speed v v th. v 2ɛv th For trapped particles π θ 0 π Bounce point Bounce point 2πR φ Distance travelled = 2πRq so time for a trapped particle to execute half an orbit t b = 2πRq v th 2ɛ Dr Ben Dudson Magnetic Confinement Fusion (10 of 19)
Banana orbit characteristics ) During this time t b = 2πRq/ (v th 2ɛ, the particle drifts to a new flux surface, a distance δr b from the original one. δr b = (V B + V R ) 2πRq v th 2ɛ Dr Ben Dudson Magnetic Confinement Fusion (11 of 19)
Banana orbit characteristics ) During this time t b = 2πRq/ (v th 2ɛ, the particle drifts to a new flux surface, a distance δr b from the original one. δr b = (V B + V R ) 2πRq v th 2ɛ 1 RΩ [ 2ɛv th 2 }{{} v 2 + v th 2 ] }{{} 2 v 2 /2 2πRq v th 2ɛ Dr Ben Dudson Magnetic Confinement Fusion (11 of 19)
Banana orbit characteristics ) During this time t b = 2πRq/ (v th 2ɛ, the particle drifts to a new flux surface, a distance δr b from the original one. δr b = (V B + V R ) 2πRq v th 2ɛ 1 RΩ [ 2ɛv th 2 }{{} = π v th (4ɛ + 1) q π r L q 2 }{{} Ω ɛ 2 ɛ r L v 2 + v th 2 ] }{{} 2 v 2 /2 Note that this is much larger than the Larmor radius r L does this provide another transport mechanism? 2πRq v th 2ɛ Dr Ben Dudson Magnetic Confinement Fusion (11 of 19)
Collisions We ve already seen collisions, and come across the collision times mi 1 τ ei < τ ii m e Z 2 < τ ie m i τ ei m e τ jk average times it takes to change the velocity of particles of species j by 90 o, through scattering with species k. To take a step of size δr b, a particle doesn t need to be deflected by 90 o. It just needs to be scattered from a trapped into a passing particle. To do this, the parallel velocity needs to be changed by v ɛv th The effective collision time is therefore τ eff τɛ We can also define a collision frequency, which is just ν 1 τ, and so an effective collision frequency for trapped particles ν eff ν ɛ = 1 τɛ Dr Ben Dudson Magnetic Confinement Fusion (12 of 19)
Collisionality A useful quantity used in MCF is the collisionality ν. This is the average number of times a particle is scattered into a passing particle before completing a banana orbit. ν t b τ eff Dr Ben Dudson Magnetic Confinement Fusion (13 of 19)
Collisionality A useful quantity used in MCF is the collisionality ν. This is the average number of times a particle is scattered into a passing particle before completing a banana orbit. ν t b τ eff = ν ɛ Rq ɛvth Dr Ben Dudson Magnetic Confinement Fusion (13 of 19)
Collisionality A useful quantity used in MCF is the collisionality ν. This is the average number of times a particle is scattered into a passing particle before completing a banana orbit. ν t b τ eff = ν ɛ Rq ɛvth ν = νrq ɛ 3/2 v th Note that if ν > 1 then trapped particles do not complete a full banana orbit before being scattered. Thus trapped particles only exist for ν < 1. Dr Ben Dudson Magnetic Confinement Fusion (13 of 19)
Collisionality For electrons colliding with ions or electrons, τ e m1/2 e Te 3/2 n ν e nrq ɛ 3/2 T 2 e For ions colliding with ions (ion-electron negligible) τ i m1/2 i Te 3/2 n ν i nrq ɛ 3/2 T 2 i Dr Ben Dudson Magnetic Confinement Fusion (14 of 19)
Collisionality For electrons colliding with ions or electrons, τ e m1/2 e Te 3/2 n ν e nrq ɛ 3/2 T 2 e For ions colliding with ions (ion-electron negligible) Note that τ i m1/2 i Te 3/2 n ν i The ratio ν i /ν e m e /m i 1 nrq ɛ 3/2 T 2 i Collisionality is independent of mass ( equal for ions and electrons) ν n/t 2 very low for hot tokamaks so trapped particles become more important Dr Ben Dudson Magnetic Confinement Fusion (14 of 19)
Neoclassical transport At low collisionality When ν 1, trapped particles exist for many banana orbits. This is called the banana regime. After N steps in a random direction, particles or energy will diffuse an average of N steps N t 2ɛ τ eff Note that N is multiplied by the fraction of trapped particles. Dr Ben Dudson Magnetic Confinement Fusion (15 of 19)
Neoclassical transport At low collisionality When ν 1, trapped particles exist for many banana orbits. This is called the banana regime. After N steps in a random direction, particles or energy will diffuse an average of N steps N t 2ɛ τ eff Note that N is multiplied by the fraction of trapped particles. The typical distance energy diffuses in a time t is t 2ɛ t 2 d neo δr b = τ eff τ ɛ δr b π q } 2 1/4 {{ ɛ 3/4 } 10 30 t τ ii r Li }{{} Classical result Classical transport needed minor radius r 14cm Neoclassical transport increases this by 10. ITER (expected Q = 10) has a minor radius of 2m. Most transport in tokamaks is anomalous, due to turbulence Dr Ben Dudson Magnetic Confinement Fusion (15 of 19)
Diffusion equations Consider a volume of plasma containing a particle density n and energy density nt Flux of particles is Γ Flux of energy is Q Dr Ben Dudson Magnetic Confinement Fusion (16 of 19)
Diffusion equations Consider a volume of plasma containing a particle density n and energy density nt Flux of particles is Γ n t = Γ Continuity gives Flux of energy is Q (nt ) = Q t Dr Ben Dudson Magnetic Confinement Fusion (16 of 19)
Diffusion equations Consider a volume of plasma containing a particle density n and energy density nt Flux of particles is Γ Continuity gives Flux of energy is Q n t = Γ (nt ) = Q t Diffusive process: flux gradient (Fick s law) Γ = D n Q = nχ T n t = (D n) so if D is approximately constant: (nt ) = (nχ T ) t Assuming n constant: n t = D 2 n T t = (χ T ) Dr Ben Dudson Magnetic Confinement Fusion (16 of 19)
Diffusion equations From the units of D and χ (L 2 /T ), they must be the step size squared over the step time. For classical transport, D i = D e r 2 Le /τ ei 3 10 4 m 2 /s χ i r 2 Li /τ ii 2 10 2 m 2 /s χ e r 2 Le /τ ei 3 10 4 m 2 /s Dr Ben Dudson Magnetic Confinement Fusion (17 of 19)
Diffusion equations From the units of D and χ (L 2 /T ), they must be the step size squared over the step time. For classical transport, D i = D e r 2 Le /τ ei 3 10 4 m 2 /s χ i r 2 Li /τ ii 2 10 2 m 2 /s χ e r 2 Le /τ ei 3 10 4 m 2 /s For neoclassical transport, χ e χ i δr 2 be δr 2 bi }{{} m e/m i χ i 2ɛδr 2 bi /τ eff 0.4m 2 /s τ ii τ ei }{{} m i /m e = χ i me m i χ i /60 7 10 3 m 2 /s What about neoclassical particle transport D i,e? Dr Ben Dudson Magnetic Confinement Fusion (17 of 19)
Neoclassical particle transport For classical transport, collisions between particles of the same species didn t contribute to particle transport In this case most of the particles a trapped particle is colliding with are passing particles so this is no longer true χ i 60χ e so does this mean that D i 60D e? Dr Ben Dudson Magnetic Confinement Fusion (18 of 19)
Neoclassical particle transport For classical transport, collisions between particles of the same species didn t contribute to particle transport In this case most of the particles a trapped particle is colliding with are passing particles so this is no longer true χ i 60χ e so does this mean that D i 60D e? If this happened (and it does in some situations), the plasma would start to charge up, creating an electric field which held the ions back: this is called non-ambipolar transport It turns out that due to momentum conservation, ions and electrons actually diffuse at the same rate without an electric field (intrinsically ambipolar) Neoclassical particle diffusivity is comparable to χ e χ e D e D i q2 ɛ 3/2 r 2 Le /τ ei χ i mi m e χ e Dr Ben Dudson Magnetic Confinement Fusion (18 of 19)
Summary In toroidal machines, the variation in magnetic field strength leads to particle trapping The Grad-B and curvature drifts cause trapped particles to follow banana orbits Collisions scatter trapped particles into passing particles, and collisionality ν is the average number effective collision times τ eff τɛ per banana orbit This leads to a diffusion with a step size of the banana width δr b and time scale τ eff : χ δr 2 b /τ eff This neoclassical transport is the minimum possible in a toroidal device Measured diffusivities in tokamaks are typically 10 100 times larger than neoclassical: they are anomalous This is due to turbulence, which we ll study later in the course Dr Ben Dudson Magnetic Confinement Fusion (19 of 19)