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Tutorial Lectures on Stellarator Transport

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16 pages
Tutorial on Stellarator Transport H.E. Mynick (PPPL) April 28, May 12, 2005 1 -Magnetic field Structure: (a) (d) (b) (e) (c)(f) -B-field Model: B(x)=B [1-ε(r)cos θ-δ (r,θ)cosη], (1) 0 h-parameter p≡δ /ε =measure of distance of hstellarator from symmetric limits δ =0, ε=0 h 2 -Neoclassical Transport - Overview: 2~ 1/Eraxiaxihel axi hel Dps DplhelDbn 3 -Radial diffusion: ~ 2νD ≈ F ∆ , with F≡fraction of particles contributing, ~ ν≡ freq of taking step in random walk ∆≡radial step size. -Eg-1: Banana regime, tokamak: 1/2F=F≡ (2ε) =frac of toroidally-trapped particles, t1/2∆=ρ ≡banana width≈ v /(v /qR) ≈qρ/ε , Bt ║bt~ν = ν ≡ toroidal detrapping frequency = ν/(2ε), t with v ≈ ρv/2R = toroidally-induced radial drift Btvelocity. axi 1/2 2 2 2 3/2⇒D ≈ (2ε) ν ρ ≈ ν q ρ /ε bn t bt 4 -Eg-2: Banana regime, straight stellarator: [A.Pytte, A.H. Boozer, Phys. Fluids 24, 88 (1981).] 1/2F=F ≡ (2δ ) =frac of helically-trapped particles, h h 1/2∆=ρ ≡banana width≈v /(v /L ) ≈(q R /r)ρδ , bh Bh ║ h h h~ν = ν ≡ ripple detrapping frequency = ν/(2δ), h with L ≡R/n, v ≈ (ρv/2)(mδ /r) = helically-h Bh hinduced radial drift velocity, q ≡ m/n. h sym 1/2 2 2 2 1/2⇒D ≈ (2δ ) ν ρ ≈ ν(q R/r)ρ δ bn h h bh h h 5 -Ripple-trapped particles: -Well-depth ...
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Tutorial on Stellarator Transport


H.E. Mynick (PPPL)


April 28, May 12, 2005




































1
-Magnetic field Structure:

(a) (d)



(b) (e)


(c)
(f)


-B-field Model:
B(x)=B [1-ε(r)cos θ-δ (r,θ)cosη], (1) 0 h
-parameter p≡δ /ε =measure of distance of h
stellarator from symmetric limits δ =0, ε=0 h
2
-Neoclassical Transport -
Overview:
2~ 1/Er
axi
axi
hel axi hel Dps
Dpl
hel
Dbn











3
-Radial diffusion:

~ 2
νD ≈ F ∆ , with
F≡fraction of particles contributing,
~ ν≡ freq of taking step in random walk
∆≡radial step size.


-Eg-1: Banana regime, tokamak:

1/2F=F≡ (2ε) =frac of toroidally-trapped particles, t
1/2
∆=ρ ≡banana width≈ v /(v /qR) ≈qρ/ε , Bt ║bt
~
ν = ν ≡ toroidal detrapping frequency = ν/(2ε), t
with v ≈ ρv/2R = toroidally-induced radial drift Bt
velocity.
axi 1/2 2 2 2 3/2⇒D ≈ (2ε) ν ρ ≈ ν q ρ /ε bn t bt







4
-Eg-2: Banana regime, straight stellarator:
[A.Pytte, A.H. Boozer, Phys. Fluids 24, 88 (1981).]

1/2F=F ≡ (2δ ) =frac of helically-trapped particles, h h
1/2
∆=ρ ≡banana width≈v /(v /L ) ≈(q R /r)ρδ , bh Bh ║ h h h
~
ν = ν ≡ ripple detrapping frequency = ν/(2δ), h
with L ≡R/n, v ≈ (ρv/2)(mδ /r) = helically-h Bh h
induced radial drift velocity, q ≡ m/n. h

sym 1/2 2 2 2 1/2⇒D ≈ (2δ ) ν ρ ≈ ν(q R/r)ρ δ bn h h bh h h















5
-Ripple-trapped particles:

-Well-depth parameter
y= 0, deeply ripple-trapped particle
1, marginally-trapped particle
>1, non-ripple-trapped particle.

-For model B-field (1), have
y =[K/µB -1+ε cos θ+δ ]/(2δ), (2) 0 h h
with η≡nζ-mθ=ripple phase, K≡ (E- eΦ)=
kin.energy

-Diffusion in y due to pitch-angle scattering:
2<(δy) > ≈ ν t, with ν ≡ν/(2δ ). h h h

⇒time τ to detrap from ripple-well for δy≈1: h
τ ≈ 1/ν . h h








6
- “1/ν-regime” (ν /Ω > 1 ): h E
[ Galeev, Sagdeev, Furth, Zh.Prikl.Mekh. i Tekhn.Fiz., 3 (1968),
Gibson, Mason, Plasma Phys. 11, 121 (1969),
Stringer, Nucl. Fusion 12, 689 (1972),
Connor, R.J. Hastie, Nucl. Fusion 13, 221 (1973)]
∆ ≈ v /ν , with v ≈ ρv/2R = toroidally-induced Bt h Bt
radial drift velocity,
1/2F ≈ (2δ ) = frac of ripple-trapped particles, h
~
ν ≈ ν = detrapping frequency. h
1/2 2 3/2 2⇒ D ≈ (2δ ) ν (v /ν ) ≈ (2δ ) v /ν -1 h h Bt h h Bt


z
R



7/2D has strong energy dependence, ~ K , -1
and is indep of Ω ~ E = -∂Φ. E r r

1/2D /D ~ (M /M ) >> 1. -1i -1e i e

7
1 1/2- “ν , ν superbanana regimes” (ν /Ω < 1 ) : h E

-Collisions perturb orbits from ν=0 superbananas,
having sb width ∆ =v /Ω . 0 Bt E

ν≠0:


- sb’s within a distance ∆y =1/p of y=1 detrap 0
collisionlessly, making sb excursion δr(y)
continuous. (p≡δ /ε): h
-2For ν /Ω > p , a collisional boundary layer h E
½
∆ is formed, of width ∆y =(ν /Ω ) , swamping
ν h E0

∆y 0.
δr
1 0 y

∆y
ν


8
1/2 -2- “ν sb-regime” (p <ν /Ω <1 ): h E
[Galeev, Sagdeev, Sov.Phys.Usp. 14, 810 (1969),
Galeev, Sagdeev, Furth, Rosenbluth, Phys. Rev. Letters 22, 511 (1969).]

∆ = ∆ , 0
1/2 1/2F ≈ F ≡ (2δ ) ∆y = (ν/Ω ) ,
ν h ν E
~ 2
ν
≈ ν /(∆y ) =Ω h ν E
1/2 2 3/2⇒ D ≈ν v /Ω 1/2 Bt E


1 -2- “ν sb-regime” (ν /Ω < p ): h E
[Galeev, Sagdeev, Sov. Phys. Usp. 12, 810 (1970)]

∆ = ∆ , 0
1/2F ≈ F ≡ (2δ ) ∆y , 0 h 0
~ 2
ν
≈ ν /(∆y ) , h 0
-1/2 2 2⇒ D ≈νp(2δ ) v /Ω 1 h Bt E









9
-Banana-drift branch:

- “stochastic regime”
[Goldston, White and Boozer, Phys.Rev. Lett. 47, 647 (1981).]

1 -1- “ν ,ν bd-regimes”
[Linsker, Boozer, Phys. Fluids 25, 143 (1982).]

- “banana-plateau regime”
[Boozer, Phys. Fluids 23, 2283 (1983).]
















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