Tutorial Lectures on Stellarator Transport

Hmynick

Description

Tutorial on Stellarator Transport-3 Transport Optimization H.E. Mynick (PPPL) May 12, 2005 -Thanks to: A. Boozer, S. Gerhardt, L.-P. Ku, D. Mikkelsen, M. Redi, G. Rewoldt, D.Spong 1 -Transport Optimization: -Approaches: -Neoclassical: -Quasi-Helical (QH) (HSX) -Quasi-Axisymmetric (QA) (NCSX) -Quasi-Poloidal (QPS) -Quasi-Omnigenous (QO)/ Quasi-Isodynamic (QI) (W7-X, inward- shifted LHD) -Isometric/ Approximately Omnigenous -Pseudo-Symmetric (PS) -Turbulent optimization: -Internal transport barriers via root-jumping -Turbulence modifications from shaping 2 -Particle motion: -Magnetic field: B=∇ψ×∇θ+∇ζ×∇ψ = ∇ψ×∇α (1) t p t p2 with α ≡ θ-ιζ, ψ≡ψ≡B r ≡toroidal flux. p t 0 -Drift eqns: ˆB×∇Vv = v +v = , with V≡µ B+eΦ. (2) D B E MΩc c& ∂ V & − ∂ Vαψ⇒ =∇ψ.v = α , =∇α .v = ψ (3) ppD p De e-Bounce average, using bounce action -1J(ψ,α |µ,E)=(2π) ds Mv(s): (4) p ||∫ ds ds∂ J = ∂ Mv =− ∂ V =−∂ V /Ω ,ψ ,α ψ ,α || ψ ,α ψ ,α b⇒ ∫ ∫ (5) p p p p2π 2πv||ds1/Ω ≡ =∂ Jb E∫with = bounce time/(2π). 2πv|| ∂ J∂ Jc α c c cp ψ− = ∂ H =− ∂ H& α &ψα ψ⇒ = , = , (6) ppe ∂ J e e∂ J eE E2 21with E=H(x, ρ ,µ)= Mρ Ω +V(x, µ) the || ||2Hamiltonian. 3 -Early ideas (assume Φ=0 for simplicity) : -Isodynamic (Palumbo) condition: [Palumbo, Nuovo Cimento X53B, 507 (1968).] -If can create config with B=B(ψ), (7) c& ∂ V &ψ ...

Informations

• Publié par: Ermey
• Nombre de lectures: 24
• Langue: English

Extrait

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   Tutorial on Stellarator Transport-3 Transport Optimization  

H.E. Mynick (PPPL)  

May 12, 2005                         -Thanks to: A. Boozer, S. Gerhardt, L.-P. Ku, D. Mikkelsen, M. Redi, G. Rewoldt, D.Spong   

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-Transport Optimization:  -Approaches: -Neoclassical:  -Quasi-Helical (QH) (HSX) -Quasi-Axisymmetric (QA) (NCSX) -Quasi-Poloidal (QPS)

-Quasi-Omnigenous (QO)/  Quasi-Isodynamic (QI) (W7-X, inward-  shifted LHD) -Isometric/ Approximately Omnigenous -Pseudo-Symmetric (PS)  -Turbulent optimization: -Internal transport barriers via root-jumping -Turbulence modifications from shaping       

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-Particle motion:  -Magnetic field: B=∇ψ t ×∇θ+∇ζ×∇ψ p =∇ψ t ×∇α p  (1) withα p ≡θ-ιζ,ψ≡ψ t ≡B 0 r 2 ≡toroidal flux.  -Drift eqns:  v D =v B +v E = M B ∇× Ω V , with V≡µB+eΦ. (2)  ⇒ & =∇ψ.v D = e c ∂ α p V , α & p =∇α p .v D = − e c ∂ ψ V  (3) -Bounce average , using bounce action  J(ψ,α p |µ,E)=(2π) -1 ∫ ds Mv || (s): (4)  v  ,|| , / ⇒ ∂ ψ,α p J= ∫ 2 ds ∂ π ψ,α p M= − ∫ 2 d π s v || ∂ ψ α p V= −∂ ψ α p VΩ b ,  (5) with 1/Ω b ≡ ∫ 2 d π s v || = ∂ E J   bounce time/(2π). =  ∂J

c ∂ ψ = ⇒ ψ & = − c e∂ α E p J = c e ∂ α p H , α & p = e∂ E J J − c e ∂H ψ , with E=H(x,ρ || ,µ)= 2 1 Mρ | 2 | Ω 2 +V(x,µ) the Hamiltonian.  

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(6)

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-Early ideas (assumeΦ=0 for simplicity) :   -Isodynamic(Palumbo) condition: [Palumbo, Nuovo CimentoX53B, 507 (1968).] -If can create config with B=B(ψ), (7) then & = e c ∂ α p V =0, ie, & =∇ψ.v B ~∇ψ.B×∇B=0. -However, an expansion soln of the equilibrium eqs around the magnetic axis yields [eg, Garren, Boozer, Phys. Fluids B3, 2805 (1991)]  B(ψ,θ,ζ)=B 0 (ζ)[1-κ(ζ)x]+ O(x 2 ), (8) where x≡-r.  (ζ)=ψ 1/2 x 1 (ζ)cos(θ-α 1 (ζ)), κ≡axis curvature=  κ(ζ),ψ~O(x 2 ). So can have B=B(ψ) only if (a)B 0 (ζ)=0 (uninteresting for confinement), or (b)κ(ζ have for all)=0 (cantζfor toroidal config).  -Omnigenouscondition: (originally considered for mirror machines, in  [Hall, McNamara, Phys.Fluids18, 552 (1975).] ) -Create fields with J=J(ψ). & Then  ψ = e c ∂ α p H∝ −∂ α p J=0 . 

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(9)

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-Meyer-Schmidt (MS) configurations: [Meyer, Schmidt, Z. Naturforsch.13A, 1005 (1958).] -Sought to improve high-βequilibrium properties by reducing Pfirsch-Schlueter currents J PS . Achieved by localizing ripple to the inside of the torus in such a way that toroidal term e t (ψ,θ)=ε t cos(θ strength) in field  B(x)=B 0 [1-e t (ψ,θ)-e h (ψ,θ)cosη],(10)  is approximately eliminated.            

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-QH (& helically-symmetric) configurations: [Boozer, Phys.Fluids26, 496 (1983), Nuehrenberg, Zille, Phys. Lett. A129, 113 (1988).]  ⇒B=B(ψ,η), whereη≡nζ-mθ. ⇒∫ds..→∫dηL h .., L h ≡(ds/dη)≈R/n, ⇒J=J(ψ), and ψ & =0 .    -QA (& axisymmetric) configurations: [Nuehrenberg, Lotz, and S. Gori, inTheory of Fusion Plasmas, E. Sindoni, F. Tryon and J. Vaclavik eds., SIF, Bologna, (1994), Garabedian, Phys. Plasmas3, 2483 (1996).]  ⇒B=B(ψ,θ). ⇒∫ds..→∫dθL t .., L t ≡(ds/dθ)≈qR, ⇒J=J(ψ), and ψ & =0   Create QAs by modulating  κ(ζ) hatκ(ζ) x 1 (ζ)=const so t  asζvaries:      -All QS systems [QA,QH, QP?s] approximate omnigenity.

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-QP configurations: B=B(ψ,ζ). -Optimizers like the one (STELLOPT) used to design NCSX, QPS, have achieved excellent confinement aiming at QP symmetry:  [Spong, Phys. Plasmas 12, 056114 (2005)]. 

B max  ridge, on high

B min  valley, in low   - However, axis-expansion solns B(ψ,θ,ζ)=B 0 (ζ)[1-κ(ζ)x]+ O(x 2 ) lead to the conclusion [Mikhailov, Shafranov, Subbotin, Isaev, Nuehrenberg, Zille, Cooper, Nucl.Fusion 42, L23 (2002)] : The third type of symmetry, poloidal symmetry, cannot be satisfied in toroidally closed stellarator configurations, in particular not in a linear approximation with respect to the distance from the magnetic axis.  -While the B min -valley is quite flat, there is appreciable variation along the B max -ridge, making QPS a low-A member of the QO family. 

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-QO/QI configurations: -Discovered while investigating configs related to MS configs. [Mynick, Chu, Boozer, Phys. Rev. Letters48, 322 (1982), Nuehrenberg, Zille, Phys. Lett.114A, 129 (1986)]  -Relaxes requirement ψ & =0 forallparticles to just ψ & ≈0 formost troublesome particles. -Makes use of contribution to ψ &  from modulation of e h (ψ,θ). Take model /2| ~ | e h (ψ,θ)=ε h (ψ)(1-σcosθ),ε h (ψ) ~ψ |m r m| . (11)

θ 

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=1

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σ=0

σ=1

σ=-1

=2σ=.5

=3σ=1

=∞σ=1

 -LHD, R ax =3.53 m: 

 a

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b

θ

b

0

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a

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-From (3) or (6), find r & =(dr/dψ)ψ &  =v B0 sinθ(ε t -σε h <cosη>), & θ = v B0 (e t +me h <cosη>)/r, (12) where v B0 ≡µB 0 /(MΩr), 1/2 1/2

<cosη> 

1/2   y  ⇒Can make r & =0  for 1=σp<cosη>, where p≡ε h /ε t .  σ=-1

Since D~ & 2 ~ 1- 1.5.

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(13)

σ=1 

 , can reduce D by factors 10-50 by usingσp

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-Isometric/ Approximately Omnigenous Configs: [Skovoroda, Shafranov, Plasma Physics Reports21, 886 (1995), Cary, Shasharina, Phys.Rev.Lett.,78, 674 (1997).]  -Set of configurationsproperlycontaining QS ones, satisfying J≈J(ψ), ψ & ≈0 foralmost allparticles, but permitting variation in shape of ripple-well as particle drifts poloidally. -J=J(ψ) results in isometry condition, that lengths alongBbetween any 2 contours with constant B=|B|

B max

B min

η/2π   -Theθ-dependence makes the banana-width vary as the s.b. precesses:

-No concrete implementations have yet been attempted.

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-Pseudo-symmetric Configurations:  [ Shafranov, et al., Proc of Int. Symposium on Plasma Dynamics in Complex Electromagnetic Fields for Comprehension of Physics in Advanced Toroidal Plasma Confinement, Dec.8-11, 1997, Research Report, Inst. of Advanced Energy, Kyoto Univ, March 20 (1998) p.193, Mikhailov, Shafranov, Suender, Plasma Physics Reports24653 (1998).]   -Widenrangeoftransport-optimizedconfigsby imposing less stringent condition of no ripple-wells. ⇒Removes sb mechanism [no ripple-trapped particles (τ=h)], leaving only the banana-drift branch, due to toroidally-trapped ones (τ=t).              

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