MHD equilibrium calculations for stellarators. Antoine Cerfon, MIT PSFC with F. Parra, J. Freidberg (MIT NSE)

Transcription

1 MHD equilibrium calculations for stellarators Antoine Cerfon, MIT PSFC with F. Parra, J. Freidberg (MIT NSE) March 20, 2012

2 MAGNETIC FIELD LINE HAMILTONIAN Consider a general toroidal coordinate system (r, θ, φ) θ poloidal angle, φ toroidal angle Using the gauge freedom for the vector potential A, it is always possible to construct Ψ and χ such that B = φ Ψ + χ θ Ψ and χ not necessarily poloidal and toroidal flux In (χ, θ, φ) coordinates, the field line trajectories are given by dχ dφ = Ψ θ dθ dφ = Ψ χ Hamilton s equations with H Ψ, x θ and p χ The time φ is periodic, so this is a 1.5 degree of freedom Hamiltonian

3 FUNDAMENTAL COMPLICATIONS IN 3D Pendulum Energy conserved Energy not conserved Magnetic field surfaces Axisymmetry No Symmetry

4 THE WORK HORSE: VMEC (1) Based on an idea by Kruskal & Kulsrud; Numerical scheme first proposed by F. Bauer, O. Betancourt, and P. Garabedian; Implemented and developed as VMEC by S. Hirschman et al. Minimization of the plasma potential energy ( ) B 2 E = + ργ dv = E B + E int 2µ 0 γ 1 Ω Only allowed virtual displacements ξ satisfy δρ = ρ ξ δ B = ( ξ B ) Assuming there is a nested family of flux surfaces s = cst, define B ds = FT (s 0 ) B ds = FP (s 0 ) ρd 3 x = M(s 0 ) s s 0 s s 0 s s 0 F T, F P and M are given functions of s, held fixed during variation

5 THE WORK HORSE: VMEC (2) Turn constrained minimization into unconstrained minimization: Flux and solenoidal constraints incorporated by writing B = s G with G = F T (s)u + F P (s)v + λ(s, u, v) and with λ(s, u, v) a periodic function of u and v Mass constraint obtained from Hölder s inequality M(s) = m(s) = ρ dsdvdu J ρ J 1 γ 1 J γ 1 γ, M (s) m(s) = ( ρ γ dvdu J dvdu ρ (s, u, v) dvdu, J = J (x, y, z) ) 1 γ ( dvdu J ) γ 1 γ Minimum of E int when ρ = ρ(s) = m(s)/( dudv/j ) p = p(s)

6 THE WORK HORSE: VMEC (3) Choose v s.t. v = φ, φ usual toroidal angle Write E = D D D2 3 dsdvdu 2µ 0 D + 1 γ 1 D m(s) γ ( Ddvdu ) γ 1 ds (R, Z) (s, u), D (ψ, R) 1 (u, v), D (ψ, Z) 2 Rψ u, D 3 (u, v) R = R mn (s)cos(mu + nv) Z = Z mn (s)cos(mu + nv) Find R mn and Z mn corresponding to minimum of E using the Steepest Descent Method SIESTA can be used in conjunction with VMEC to compute equilibria with islands

7 A DIRECT SOLVER WITH ISLANDS: PIES Solves the MHD equilibrium equations iteratively 1. B p = 0 2. B p J = B 2 3. B ( ) J = J B 4. B = J Eq.(1) is used to compute magnetic coordinates This step is where most of the action is complications with rational surfaces, stochasticity... Eq.(2)-(3) are then solved analytically, mode by mode, in magnetic coordinates Eq.(4) is then solved for the updated magnetic field, in lab (toroidal) coordinates

8 ASYMPTOTIC APPROACHES: GREENE-JOHNSON (1961) ATF LHD Assumes strong toroidal guide field B 0 Expansion based on the small parameter δ B hel /B 0 1 Parameter Symbol Scaling Beta β β t δ 2 Inverse aspect ratio a/r 0 ɛ δ 2 Number of helical periods N 0 1/δ 2 Poloidal helicity l 1 Rotational transform ι/2π 1 NOTE: β ɛ

9 CONSEQUENCES OF THE G-J ORDERING R = R 0 + aρcosθ Z = aρsinθ φ = φ N 0 To lowest order, B ψ = 0 ɛn 0 B 0 ψ φ Greene-Johnson stellarators are perturbed tokamaks! a = + = ρ eρ + 1 e θ ρ θ 1 = ɛn 0 e φ 1 + ɛρcosθ φ 1 = 0 ψ = ψ(ρ, θ)

10 A NEW ORDERING W7-X HSX NCSX Same expansion parameter as Greene-Johnson: δ B hel /B 0 1 Parameter Symbol G-J Scaling New scaling Beta β β t δ 2 δ Inverse aspect ratio ɛ δ 2 δ Number of helical periods N 0 1/δ 2 1 Poloidal helicity l 1 1 Rotational transform ι/2π 1 1 NOTE: In both expansions, β ɛ

11 CONSEQUENCE OF THE NEW ORDERING To lowest order, B ( ψ = 0 ɛn 0 B 0 φ + B p ) ψ = 0 Equilibria can have a non-planar magnetic axis Our expansion: O(1) O(δ) B = B0 eφ + ( ) B φ1 ɛxb eφ 0 + B p1 J = Jφ1 eφ + J p1 p = p 1 (ρ, θ, φ) Greene-Johnson expansion: O(1) O(δ) O(δ 2 ) O(δ 3 ) B = B0 eφ + B 1 (ρ, θ, φ) + (B φ2 ɛxb 0 ) e φ + B p2 + B 3 (ρ, θ, φ) J = J2 (ρ, θ) + J 3 (ρ, θ, φ) p = p 2 (ρ, θ) +p 3 (ρ, θ, φ)

12 THE NEW EXPANSION (1) The B = 0 equation To lowest order, this is B p1 = 0. Introduce A 1 (ρ, θ, φ) s.t. ( B = 1 ɛx + B ) φ1 eφ + A 1 e φ B 0 B 0 The µ 0 J = B equation µ 0 a J 1 B 0 = e φ B φ1 B 0 e φ 2 A 1 The J = ( B p)/b 2 equation ( β1 2 + B ) φ1 = 0 B φ1 = β 1 B 0 B 0 2 θ-pinch pressure balance relation β 1 = 2µ 0p 1 B 2 0

13 THE NEW EXPANSION (2) The B p = 0 equation ( ɛn 0 φ e φ A 1 ) β 1 = 0 The J = 0 equation ( ɛn 0 φ e φ A 1 ) 2 A 1 = ɛ e Z β 1 ( φ + e φ A ) β = 0 ( φ + e φ A ) 2 A = e Z β A = A 1 ɛn 0 β = 1 ɛn 2 0 2µ 0 p 1 B 2 0 = β 1 ɛn 2 0

14 THE NEW EXPANSION: RESULTS ( φ + e φ A ) β = 0 ( φ + e φ A ) J φ = e Z β 2 A = J φ A = A 1 β = 1 ɛn 0 ( B = 1 ɛx β 1 B 0 2 ɛn 2 0 2µ 0 p 1 B 2 0 = β 1 ɛn 2 0 J φ = µ 0aJ 1φ ɛn 0 B 0 ) eφ + A 1 µ 0 a J 1 e φ = 1 eφ β 1 e φ 2 B 0 2 A 1 Equations agree with the Greene-Johnson expansion in the right limit Equations are in a convenient form for iterations

15 ITERATION STEPS Start with given A (0) for k = 1, 2,... ( φ + e φ A (k 1) ) β (k) = 0 ( φ + e φ A (k 1) ) J (k) φ = e Z β (k) 2 A (k) = J (k) φ until a stopping criterion holds

16 HYPERBOLIC EQUATIONS AND INITIAL CONDITIONS ( φ + e φ A ) β = 0 How to choose the I.C. at φ = 0 such that β is periodic in φ?

17 DICK AND JANE CALCULATE 3D EQUILIBRIA ( φ + e φ A ) β = 0 Given A, the field line trajectories are given by the characteristics: dρ dφ = 1 A ρ θ dθ dφ = 1 A ρ ρ Given a starting point (ρ 0, θ 0, φ 0 ), we can easily integrate these equations ρ(φ), θ(φ) Integrate for many turns (very large φ) to sample the whole magnetic surface If starting point is always of the form (ρ 0, 0, 0), ρ 0 is a unique label for each magnetic surface

18 EXTRACTING INFORMATION FROM CHARACTERISTICS Through this process, we have ρ = ρ(ρ 0, θ, φ) ( ρ ez β = dβ dρ 0 ρ 0 ) 1 [ sinθ cosθ ρ ] ρ θ

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22 CLOSING THE ITERATION LOOP ( φ + e φ A ) J φ = e Z β We already know the characteristics for this equation, and the RHS of this equation on the characteristics Solving for J φ is a straightforward 1D (Lagrangian) integral We can then evaluate A/ ρ and A/ θ required for the next iteration: [ A 1 (ρ, θ, φ) = dρ dθ ρ ρ ρ cos(θ θ ] ) ρ 2π ρ 2 + ρ 2 2ρρ cos(θ θ J φ (ρ, θ, φ) ) A 1 (ρ, θ, φ) = dρ dθ ρρ 2 cos(θ θ ) θ 2π ρ 2 + ρ 2 2ρρ cos(θ θ ) J φ(ρ, θ, φ)

23 PRELIMINARY RESULTS LAST ISSUE? Start with vacuum field (in this case, dominant l = 2 component, some l = 3 and l = 4) β(ρ 0 ) = 0.01(1 ρ 2 0 )2

24 PRELIMINARY RESULTS LAST ISSUE? Flux surfaces after 1st iteration Need to calculate the new location of the magnetic axis to evaluate new β profile; is there a fast numerical scheme to do that?

25 REFERENCES Variational methods M.D. Kruskal and R.M. Kulsrud, Phys. Fluids (1958) F. Bauer, O. Betancourt and Paul Garabedian in A Computational Method in Plasma Physics (Springer-Verlag, New York, 1978) S. P. Hirshman and J. C. Whitson, Phys. Fluids 26, 3553 (1983) S. P. Hirschman, R. Sanchez, and C.R. Cook, Phys. Plasmas 18, (2011) PIES A.H. Reiman, H.S. Greenside, Comput. Phys. Commun. 43, 157 (1986) A.H. Reiman, H.S. Greenside, J. Comput. Phys (1988)