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# Tutorial Lectures on Stellarator Transport

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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 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 ...

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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
2 -Particle motion: -Magnetic field: B=∇ψt×∇θ+∇ζ×∇ψp=∇ψt×∇αp (1) withαpθ-ιζ,ψ≡ψtB0r2toroidal flux. -Drift eqns: vD=vB+vE=MB×V, with V≡µB+eΦ. (2)&=∇ψ.vD=ecαpV,α&p=∇αp.vD=ecψV (3) -Bounce average, using bounce actionJ(ψ,αp|µ,E)=(2π)-1ds Mv||(s): (4) v ,|| ,/ ψ,αpJ=2dsπψ,αpM= −2dπsv||ψ αpV= −∂ψ αpVb, (5) with1/b2dπsv||= ∂EJ bounce time/(2π). = J
cψ= ψ&=ceαEpJ=ceαpH,α&p=eEJJceH ψ, with E=H(x,ρ||,µ)=21Mρ|2|2+V(x,µ) the Hamiltonian.
(6)
3 -Early ideas(assumeΦ=0 for simplicity):-Isodynamic(Palumbo) condition: [Palumbo, Nuovo CimentoX53B, 507 (1968).] -If can create config with B=B(ψ), (7) then&=ecαpV=0, ie,&=∇ψ.vB~∇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)=B0(ζ)[1-κ(ζ)x]+ O(x2), (8) where x-r.(ζ)=ψ1/2x1(ζ)cos(θ-α1(ζ)), κaxis curvature=κ(ζ),ψ~O(x2). So can have B=B(ψ) only if (a)B0(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).& Thenψ=ecαpH∝ −∂αpJ=0.
(9)
4 -Meyer-Schmidt (MS) configurations: [Meyer, Schmidt, Z. Naturforsch.13A, 1005 (1958).] -Sought to improve high-βequilibrium properties by reducing Pfirsch-Schlueter currents JPS. Achieved by localizing ripple to the inside of the torus in such a way that toroidal term et(ψ,θ)=εtcos(θ strength) in field B(x)=B0[1-et(ψ,θ)-eh(ψ,θ)cosη],(10) is approximately eliminated.
5 -QH (& helically-symmetric) configurations:[Boozer, Phys.Fluids26, 496 (1983), Nuehrenberg, Zille, Phys. Lett. A129, 113 (1988).]B=B(ψ,η), whereη≡nζ-mθ. ds..dηLh.., Lh(ds/dη)R/n, J=J(ψ), andψ&=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..dθLt.., Lt(ds/dθ)qR, J=J(ψ), andψ&=0 Create QAs by modulating κ(ζ) hatκ(ζ) x1(ζ)=const so t asζvaries:-All QS systems [QA,QH, QP?s] approximate omnigenity.
6 -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)].
Bmax ridge, on high
Bmin valley, in low - However, axis-expansion solnsB(ψ,θ,ζ)=B0(ζ)[1-κ(ζ)x]+ O(x2) 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 Bmin-valley is quite flat, there is appreciable variation along the Bmax-ridge, making QPS a low-A member of the QO family.
7 -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 requirementψ&=0 forallparticles to just ψ&0 formost troublesome particles. -Makes use of contribution toψ&from modulation of eh(ψ,θ). Take model /2|~| eh(ψ,θ)=εh(ψ)(1-σcosθ),εh(ψ) ~ψ|mrm|. (11)
θ
=1
σ=0
σ=1
σ=-1
=2σ=.5
=3σ=1
=σ=1
-LHD, Rax=3.53 m:
a
b
θ
b
0
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a -From (3) or (6), find r&=(dr/dψ)ψ&=vB0sinθ(εt-σεh<cosη>), & θ= vB0(et+meh<cosη>)/r, (12) where vB0≡µB0/(Mr), 1/2 1/2
<cosη>
1/2y Can maker&=0for 1=σp<cosη>, where p≡εh/εt. σ=-1
Since D~&2 ~ 1- 1.5.
(13)
σ=1
, can reduce D by factors 10-50 by usingσp
9 -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 JJ(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|
Bmax
Bmin
η/2π-Theθ-dependence makes the banana-width vary as the s.b. precesses:
-No concrete implementations have yet been attempted. -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).]-Widenrangeoftransport-optimizedconfigsbyimposing 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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