Magnetically Confined Fusion:
Transport in the core and in the
Scrape- off Layer
Bogdan Hnat
Joe Dewhurst, David Higgins, Steve Gallagher,
James Robinson and Paula Copil
Fusion Reaction
+ 3H → 4He + n
Lower binding energy for
4He + energy of the
neutron = ~ 17.6 MeV
z Extra neutron in each nuclei increases collision rate
z Electrostatic forces small – one proton per nucleus
z Result: cross section maximum at relatively low
temperature of ~ 300 milion K.
2
3
z H extracted from see water, H can be produced
6Li + n → 4He + T or 7Li +n → 4He + T + n
2H
Lawson’s criterion
Confinement time τE=Energy content / Rate of Loss
z Stationary state: equalise energy loss with input of
thermal energy (keep T constant)
z Define L=n τ , look for T at which fusion reaction
e E
produces enough energy to sustain itself
z
Take 50/50 D + T fusion reaction
Assume cold ions (electron
pressure only)
L = neτ E ≥ 1.5 × 10 [s/m3]
20
Temperature at min. ~ 300 milion K
Realising Lawson’s criterion
Condition L = neτ E ≥ 1.5 × 10
20
can be achieved by:
Large values of τE – magnetic confinement, or
z Large density n – inertial confinement
e
z
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 B 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
z Ergodic surface: magnetic
field line never closes on itself,
thus covering densly the
surface
z Stochastic regions (volums):
magnetic field has radial
component which allows for
fast transport between different
flux surfaces (can not support
pressure gradient)
z
Tokamak
Primary coil indices
toroidal magnetic field
z Pure toroidal field
configuration is unstable
z Plasma current is driven
in toroidal direction,
inducing poloidal field
z
Magnetic field: 0.6 T
Density: 2x1019 [1/m^3]
Plasma current: 1-2 MA
∇ r p = ( j × B)r
MAST – Spherical tokamak
Electricity cost ~ β-0.4
More compact in size
Less prone to MHD instabilities
Lower B – 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 B ,
atomic physics
effects
Particle Transport: classical estimates
∂ t n + ∇ iΓ = S part
∂ t (3 / 2 nT ) + ∇ iQ = S heat
Normally we assume that: Γ = − D∇ ⊥ n Q = −κ∇ ⊥T
Γ r ≈ (nr − nr +Δr )vr = ⎡⎣ nr − ( nr + ∂ r nΔr + …) ⎤⎦ vr = −(vr Δr )∇ r n
Δr
e
2
vr ≈
and we assume that Δr ≈ ρe thus Dr = ν ei ρ e
τ ei
ρi = ( mi / me )
1/ 2
ρ e and ν ie = ( me / mi )ν ei thus Γ = Γ
e
r
i
r
Ions and electrons contribute equally to particle transport
z Same specie collisions do not contribute (momentum
conservation)
z
Heat Transport: classical estimates
∂ t n + ∇ iΓ = S part
∂ t (3 / 2 nT ) + ∇ iQ = S heat
Assume temperature gradient in radial direction
ni ρ
1
i
= ni χ i
Qr ≈ mn [ (vth ) r − (vth ) r +Δr ) ] vr = −κ r ∇ rT ; κ r =
2τ ii
2
In parallel direction the step size in the mean free path
2
i
κ ≈ nTτ ee / m ∝ T
5/ 2
Same specie collisions important for heat flux
z Electrons dominate parallel heat transport
z Collision increase χ
┴ and decrease χ║
z
Transport: neoclassical estimates
Assume particle is moving on a flux surface with
magnetic field magnitude between Bmin and Bmax. Define:
2 μ B0 v B
=
λ≡
2
mv
v B
A particle can never enter the region where μ B > E − ZeΦ
2
⊥ 0
2
All particles must then satisfy 0 ≤ λ ≤ (B0/Bmin).
B0
B0
≤λ≤
Bmax
Bmin
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
ρp ε
eB p
vth
a
ρp ≡
, where Ω p =
and ε ≡
Ωp
m
R
The width of the banana orbit is given by: Δr
Taking ft as fraction of trapped particles heat diffusivity χ is
χ
ban
i
ν ii
2
ε)
= 2ε ρ piν ii
ε
= ft (Δr ) ν eff = (2ε ) ( ρ pi
2
1/ 2
2
Comparing classical and neoclassical heat diffusivities
χ
χ
neoc
i
c
i
⎛ B
= 2ε ⎜
⎜B
⎝ p
2
⎞
⎟⎟ ∼ 10 − 50
⎠
Turbulent transport estimate
Turbulence driven by micro scale instabilities (drift wave
instability, Ion Temperature Gradient-ITG, etc…)
z Gradients are sources of free energy – large fluctuations
at plasma edge are expected
z Often modelled as purely electrostatic
z
mdt v = −∇p + q ( E + v × B) multiply B ×
z
Neglect inertial terms, take velocity perpendicular to B
v⊥ = vE + vdiam
E × B B × ∇(nT )
=
+
2
2
B
B
Turbulent transport estimate
Often modelled as purely electrostatic
z Neglect diamagnetic part
z
E×B
∇φ × B
∇φ
v⊥ ≈ vE =
=−
⇒ v⊥ ≈
2
2
B
B
B
2
Δr )
(
φ
≈ vE Δr =
D=
τ
B
turb
i
neoc
i
D
D
∼ 10
turb
e
neoc
e
D
D
∼ 1000
Sources of turbulence – drift waves
• Ions dominate perp dynamics, electron parallel dir.
• Quasineutrality: ne = ni , cold ions: grad(pi)=0
• No e-i collisions: δφ is in phase with δn
vE × B
E×B
1 dE⊥
=
, vP = −
2
B
ω g B dt
ω
Te ∇ x no
vd =
=
k y eB n
δ n = n0 (1 − e
eφ / kT
)
∇ n0