Flow, current, & electric field in omnigenous stellarators

Transcription

1 Flow, current, & electric field in omnigenous stellarators Supported by U.S. D.o.E. Matt Landreman with Peter J Catto MIT Plasma Science & Fusion Center Oral 2O4 Sherwood Fusion Theory Meeting Tuesday May 3, 2011

2 Preview W7-X can be and any reactor must be nearly omnigenous. Omnigenous stellarators: More general than quasisymmetric devices. B has nice properties. Formulae for current & flow simplify dramatically. Universal radial electric field. Landreman and Catto, PPCF 53, (2011).

3 Omnigenity = no unconfined orbits. Tokamak Stellarator Flux surface B B Trapped particle Unconfined α particles can damage plasma-facing components.

4 Omnigenity = no unconfined orbits. Tokamak Stellarator Flux surface B B Trapped particle For a reactor, then, a stellarator must be nearly omnigenous: 0 =Δψ per bounce = ( v ) d ψ dt for all μ. bounce Unconfined α particles can damage plasma-facing components.

5 Omnigenity = no unconfined orbits. Tokamak Stellarator Flux surface For a reactor, then, a stellarator must be nearly omnigenous: 0 =Δψ per bounce = ( v ) d ψ dt for all μ. Equivalent definition: J is a flux function, where J B B Trapped particle = υ d is the longitudinal invariant. bounce Unconfined α particles can damage plasma-facing components.

6 Omnigenity = no unconfined orbits. Tokamak Stellarator Flux surface B B Trapped particle For a reactor, then, a stellarator must be nearly omnigenous: 0 =Δψ per bounce = ( v ) d ψ dt for all μ. bounce Unconfined α particles can damage plasma-facing components. Equivalent definition: J is a flux function, where J = υ d is the longitudinal invariant. J contours for W7-X ( ψ θ) in, polar coordinates

7 If Omnigenity places strong constraints on B. ( d ψ ) v dt = 0, then All B contours link the torus toroidally, poloidally, or both. 2π 2π poloidal angle θ poloidal angle θ θ B 0 0 toroidal angle ζ 2π/ π/5 2π/N ζ toroidal angle ζ 0.9

8 If Omnigenity places strong constraints on B. ( d ψ ) v dt = 0, then All B contours link the torus toroidally, poloidally, or both. 2π 2π poloidal angle θ poloidal angle θ θ Field line B 0 0 toroidal angle ζ 2π/ π/5 2π/N ζ toroidal angle ζ 0.9 Each maximum & minimum of B along a field line is the same: B max B min

9 One way to achieve omnigenity is quasisymmetry. Quasisymmetry: B = B(ψ, Mθ -Nζ). Ignorable coordinate in the guiding-center Lagrangian Tokamak-like trajectories.

10 One way to achieve omnigenity is quasisymmetry. Quasisymmetry: B = B(ψ, Mθ -Nζ). Ignorable coordinate in the guiding-center Lagrangian Tokamak-like trajectories. Neoclassical formulae simplify dramatically. Explicit non-numerical forms possible: i e e i Quasisymmetry: jb 4.8 q dp dp 0.74n dt e 1.17 dt MG+ NI = ε + ne dψ dψ dψ dψ M qn Pytte & Boozer PoF (1981), Boozer PoF (1983) i e e i Tokamak: jb 4.8 q dp dp 0.74n dt e 1.17 dt = ε + ne G dψ dψ dψ dψ where ( ) G ψ = poloidal current outside the flux surface, G ( ψ ) ( ) toroidal current I ψ = inside the flux surface I ( ψ )

11 Omnigenity is more general than quasisymmetry. Cary & Shasharina, PoP (1997), PRL (1997) All toroidal fields Omnigenous Quasisymmetric Axisymmetric LHD TJ-II W7-X HSX NCSX tokamaks Omnigenity: B contours may be curved 2π Quasisymmetry: B contours must be straight 2π 2π poloidal angle θ θ B poloidal angle θ θ θ B π/5 2π/N ζ toroidal angle ζ π/N 0 ζ 2π/5 toroidal angle ζ 0.9

12 Omnigenity is more general than quasisymmetry. Cary & Shasharina, PoP (1997), PRL (1997) All toroidal fields Omnigenous Quasisymmetric Axisymmetric LHD TJ-II W7-X HSX NCSX tokamaks Quasisymmetry is more restrictive than omnigenity and does no more to reduce α-particle loss or neoclassical transport.

13 Omnigenity is more general than quasisymmetry. Cary & Shasharina, PoP (1997), PRL (1997) All toroidal fields Omnigenous Quasisymmetric Axisymmetric LHD TJ-II W7-X HSX NCSX tokamaks Quasisymmetry is more restrictive than omnigenity and does no more to reduce α-particle loss or neoclassical transport. Quasisymmetric fields may have reduced instability & turbulent transport: Flow speeds comparable to υ th,i are only permitted in quasisymmetric fields. (Helander, PoP 2007) Flows are clamped to a weakly sheared neoclassical value except in quasisymmetry, so quasisymmetric fields permit larger flow shear. (Helander & Simakov, PRL 2008)

14 The quasisymmetry helicity (M, N ) can be generalized to omnigenity. Recall: all B contours encircle the torus poloidally, toroidally, or both.

15 The quasisymmetry helicity (M, N ) can be generalized to omnigenity. Recall: all B contours encircle the torus poloidally, toroidally, or both. Define M and N: B contours close after linking the torus M times toroidally and N times poloidally.

16 The quasisymmetry helicity (M, N ) can be generalized to omnigenity. Recall: all B contours encircle the torus poloidally, toroidally, or both. Define M and N: B contours close after linking the torus M times toroidally and N times poloidally. In the quasisymmetric limit, then B = B(ψ, Mθ Nζ).

17 The quasisymmetry helicity (M, N ) can be generalized to omnigenity. Recall: all B contours encircle the torus poloidally, toroidally, or both. Define M and N: B contours close after linking the torus M times toroidally and N times poloidally. In the quasisymmetric limit, then B = B(ψ, Mθ Nζ). e.g. HSX: M = 1, N = 4 2π θ B (T) 0 0 ζ 2π 0.9

18 The quasisymmetry helicity (M, N ) can be generalized to omnigenity. Recall: all B contours encircle the torus poloidally, toroidally, or both. Define M and N: B contours close after linking the torus M times toroidally and N times poloidally. In the quasisymmetric limit, then B = B(ψ, Mθ Nζ). e.g. HSX: M = 1, N = 4 2π θ B (T) New geometric consequence of omnigenity: By applying Ampère s Law to a B contour on a flux surface, 0 0 ζ 2π 4π B d r = c MG+ NI ( enclosed current) 0.9

19 The quasisymmetry helicity (M, N ) can be generalized to omnigenity. Recall: all B contours encircle the torus poloidally, toroidally, or both. Define M and N: B contours close after linking the torus M times toroidally and N times poloidally. In the quasisymmetric limit, then B = B(ψ, Mθ Nζ). e.g. HSX: M = 1, N = 4 2π θ B (T) New geometric consequence of omnigenity: 4π d = ( enclosed current) By applying Ampère s Law to a B contour on a flux surface, 0 0 ζ 2π B r c MG+ NI B ψ B 2q MG+ NI + H = where H = 0. B B c M qn 0.9

20 Current in an omnigenous plasma is described by a concise, explicit, analytical formula. fqb t NI + MG dpe dpi dte dti j = ne ne B qn M dψ dψ dψ dψ 2 2q dpe dpi B + 1 ( NI MG) + W + + B( qn M) dψ dψ 2 B ( ) ( ) Tokamak result with G NI + MG / qn M W 2 2B = qg + I q ( ) ζ dζ B B N M B θ ζ I ( ψ ) G( ψ ) and are the toroidal & poloidal currents. G ( ψ ) W = 0 I ( ψ )

21 Bootstrap j can vanish if M = 0. Subbotin et al, NF 46, 921 (2006), Helander & Nührenberg, PPCF 51, (2009).

22 Bootstrap j can vanish if M = 0. Subbotin et al, NF 46, 921 (2006), Helander & Nührenberg, PPCF 51, (2009). Toroidal current inside a flux surface ψ 2 ( ψ ) = d = I j r.

23 Bootstrap j can vanish if M = 0. Subbotin et al, NF 46, 921 (2006), Helander & Nührenberg, PPCF 51, (2009). Toroidal current ψ 2 = I( ψ ) =. inside a flux surface j d r di 4π I dp 2π dψ = 2 d 2 B ψ + B jb

24 Bootstrap j can vanish if M = 0. Subbotin et al, NF 46, 921 (2006), Helander & Nührenberg, PPCF 51, (2009). Toroidal current ψ 2 = I( ψ ) =. inside a flux surface j d r di 4π I dp 2π dψ = 2 d 2 B ψ + B ( NI MG) From last page: j B +. jb

25 Bootstrap j can vanish if M = 0. Subbotin et al, NF 46, 921 (2006), Helander & Nührenberg, PPCF 51, (2009). So if B contours close poloidally ( M = 0) rather than toroidally or helically, di = dψ Toroidal current ψ 2 = I( ψ ) =. inside a flux surface j d r di 4π I dp 2π dψ = 2 d 2 B ψ + B ( )... I. ( NI MG) From last page: j B +. jb

26 Bootstrap j can vanish if M = 0. Subbotin et al, NF 46, 921 (2006), Helander & Nührenberg, PPCF 51, (2009). Toroidal current ψ 2 = I( ψ ) =. inside a flux surface j d r di 4π I dp 2π dψ = 2 d 2 B ψ + B ( NI MG) From last page: j B +. So if B contours close poloidally ( M di = (...) I. dψ = jb 0) rather than toroidally or helically, Boundary condition: I ( ψ = 0) = 0.

27 Bootstrap j can vanish if M = 0. Subbotin et al, NF 46, 921 (2006), Helander & Nührenberg, PPCF 51, (2009). Toroidal current ψ 2 = I( ψ ) =. inside a flux surface j d r di 4π I dp 2π dψ = 2 d 2 B ψ + B ( NI MG) From last page: j B +. So if B contours close poloidally ( M di = (...) I. dψ = jb 0) rather than toroidally or helically, Boundary condition: I ( ψ = 0) = 0. ( ψ ) Self-consistent current profile is I = 0 with j B = 0.

28 Bootstrap j can vanish if M = 0. Subbotin et al, NF 46, 921 (2006), Helander & Nührenberg, PPCF 51, (2009). Toroidal current ψ 2 = I( ψ ) =. inside a flux surface j d r di 4π I dp 2π dψ = 2 d 2 B ψ + B ( NI MG) From last page: j B +. So if B contours close poloidally ( M di = (...) I. dψ = jb 0) rather than toroidally or helically, Boundary condition: I ( ψ = 0) = 0. ( ψ ) Self-consistent current profile is I = 0 with j B = 0. If bootstrap current isn t needed to make rotational transform, minimize it: Reduce drive for instabilities Maintain optimization as pressure is varied.

29 Bootstrap j can vanish if M = 0. Subbotin et al, NF 46, 921 (2006), Helander & Nührenberg, PPCF 51, (2009). Toroidal current ψ 2 = I( ψ ) =. inside a flux surface j d r di 4π I dp 2π dψ = 2 d 2 B ψ + B ( NI MG) From last page: j B +. So if B contours close poloidally ( M di = (...) I. dψ = jb 0) rather than toroidally or helically, Boundary condition: I ( ψ = 0) = 0. ( ψ ) Self-consistent current profile is I = 0 with j B = 0. If bootstrap current isn t needed to make rotational transform, minimize it: Reduce drive for instabilities Maintain optimization as pressure is varied. ( = ) B contours should close poloidally M 0 to minimize j B.

30 Flow in an omnigenous plasma is described by a concise, explicit, analytical formula. V i 2 ( ) ( ) ( + W ) ( ) 2qB dti NI + MG 2q dφ 1 dpi NI + MG = e B dψ qn M B dψ en dψ qn M ( ) ( ) Tokamak result with G NI + MG / qn M W 2 2B = qg + I q ( ) ζ dζ B B N M B θ ζ I ( ψ ) G( ψ ) and are the toroidal & poloidal currents. G ( ψ ) W = 0 I ( ψ )

31 E r is determined by ambipolarity. Non-quasisymmetric stellarators: Neoclassical radial current depends on E r. j neoclassical ψ j turbulence ψ. (Helander & Simakov, Contrib. Plasma Phys. 2010) You can solve for E r using j neoclassical ψ = 0.

32 E r is determined by ambipolarity. Non-quasisymmetric stellarators: Neoclassical radial current depends on E r. j neoclassical ψ j turbulence ψ. (Helander & Simakov, Contrib. Plasma Phys. 2010) You can solve for E r using j neoclassical ψ = 0. Tokamaks & quasisymmetric stellarators: Radial fluxes of ions and electrons are always equal, regardless of E r ( intrinsic ambipolarity ) (Helander & Simakov, PRL 2008) You cannot solve for E r.

33 E r is determined by ambipolarity. Non-quasisymmetric stellarators: Neoclassical radial current depends on E r. j neoclassical ψ j turbulence ψ. (Helander & Simakov, Contrib. Plasma Phys. 2010) You can solve for E r using j neoclassical ψ = 0. Omnigenous stellarators: dφ dni dti j ψ = Zeni + Ti 0.17ni departure from quasisymmetry dψ dψ dψ ( ) 2

34 E r is determined by ambipolarity. Non-quasisymmetric stellarators: Neoclassical radial current depends on E r. j neoclassical ψ j turbulence ψ. (Helander & Simakov, Contrib. Plasma Phys. 2010) You can solve for E r using j neoclassical ψ = 0. Omnigenous stellarators: dφ dni dti j ψ = Zeni + Ti 0.17ni departure from quasisymmetry dψ dψ dψ Universal result: dφ 1 Ti dni dti = dψ Ze ni dψ dψ ( ) 2 Totally independent of the details of B.

35 Summary: omnigenity is a useful limit. More general than quasisymmetry. Relevant (at least for insight and code benchmarking) to W7-X and to any reactor. Concise, explicit, analytical formulae for j, V, and E r. Poloidally closed B contours zero bootstrap current. For non-quasisymmetric B, E r is determined explicitly: dφ 1 Ti dni dti = independent of the B geometry. dψ Ze ni dψ dψ Landreman and Catto, PPCF 53, (2011).