Ideal MHD Equilibria

Transcription

1 CapSel Equil - 01 Ideal MHD Equilibria keppens@rijnh.nl steady state ( t = 0) smoothly varying solutions to MHD equations solutions without discontinuities conservative or non-conservative formulation equivalent stationary MHD equations governed by (ρv) = 0 ρ (v ) v = p + ( B) B + ρg (v ) p = γp v (v B) = 0 and B = 0 given g and parameter γ, solve for ρ, v, p, B

2 CapSel Equil - 02 dramatic simplication when considering STATIC equilibria v = 0 leaves only p + ( B) B + ρg = 0 and B = 0 governing equations for magnetostatic equilibria without external gravity g = 0 p }{{} pressure gradient = ( B) B } {{ } Lorentz force B = 0

3 CapSel Equil - 03a magnetostatic case without gravity: choose dimensionality and geometry 1D cartesian geometry p(x), B x (x), B y (x), B z (x) constant (or zero) B x from B = 0 d p + B2 y + B 2 z dx 2 } {{ } total pressure = 0 x z B y

4 CapSel 1D cylindrical geometry p(r), B R (R), B ϕ (R), B Z (R) (constant RB R hence) zero B R from B = 0 d p + B2 ϕ + B 2 Z = B2 ϕ dr 2 Z ϕ } {{ } total pressure R }{{} radially inward tension Equil - 03b R B ODE governs 1D static equilibria: 4 functions ρ, p, B 2, B 3 3 free profiles, namely ρ and 2 from p, B 2, B 3

5 CapSel Equil D cylindrical case: dimensions from cylinder radius a, axial field strength B 0, axial density ρ 0 2 essential dimensionless parameters β 0 = 2p 0 B 2 0 and µ 0 = ab ϕ RB Z R=0 = aj Z0 2B 0 axial plasma beta β 0 inverse pitch µ = ab ϕ /RB Z Latter uses axial current J Z0 = J Z (R = 0) = 1 R d dr (RB ϕ) R=0

6 CapSel Equil - 05 introduce dimensionless quantities f and unit profiles f R R a R p p B 2 0 ρ ρ ρ 0 ρ = β 0 2 p p 0 β 0 2 p B Z B Z B 0 B Z B ϕ B ϕ B 0 µ 0 µ R B Z

7 CapSel Equil D cylindrical equilibrium fully specified by β 0, µ 0 and two (three with ρ) unit profiles p( R) and µ = µ/µ 0 determine B Z from equilibrium relation d d R β 0 B Z 2 2 p + 2 ( R 2 µ 2 0 µ ) find B ϕ R µ B Z and B ϕ = µ 0 Bϕ BZ 2 = Rµ 2 0 µ 2 core problem, freedom in parameter values and unit profile variations for cylinder of finite length L 2πR 0 : alternatively introduce ɛ = a/r 0 and work with q 0 = ɛ/µ 0 instead q-profile from q = RB Z /R 0 B ϕ

8 CapSel Grad-Shafranov equation Equil - 07a consider axisymmetric ( ϕ = 0), 2D static equilibria assume right-handed coordinate system (R, Z, ϕ) to solve for 5 two-dimensional functions ρ(r, Z), p(r, Z), B R (R, Z), B Z (R, Z), B ϕ (R, Z) density profile arbitrary, not in equilibrium p = ( B) B from B = 0 or 1 R R (RB R) + B Z Z = 0 flux function ψ(r, Z) with B = 1 R eˆ ϕ ψ + B ϕ eˆ ϕ such that B R = 1 R ψ Z and B Z = 1 ψ R R

9 CapSel Equil - 07b note that B ψ = 0 constant ψ surfaces (or contours in (R, Z)-plane) are flux surfaces field lines lie on these flux surfaces 3 orthogonal directions from p = ( B) B Poloidal R-Z plane ϕ J J p B

10 CapSel Equil - 08 since J = B we have J = 0 analogously define current stream function I(R, Z) J = 1 R eˆ ϕ I + J ϕ eˆ ϕ this yields J R = 1 I R Z and J Z = 1 I R R from J = B we have as well ϕ component of p = ( B) B J R = B ϕ Z I = RB ϕ 0 = J R B Z J Z B R = 1 I R 2 Z since commutator vanishes, find I(ψ) ψ R I ψ R Z

11 CapSel Equil - 09 R and Z component of force balance yield p R = J ZB ϕ J ϕ B Z = p Z = J ϕb R J R B ϕ = again find p(ψ) and conclude II R J ϕ 2 R II p = II R J ϕ 2 R R J ϕ 2 R ψ R ψ Z

12 CapSel Equil - 10a use J = B to find J ϕ = B Z R B R Z in terms of ψ we get RJ ϕ = R R Grad-Shafranov equation thus reads R R ( 1 R ψ R ) + 2 ψ Z 2 1 ψ + 2 ψ R R Z = 2 II p R 2 to solve for ψ(r, Z) under given profiles p(ψ) and I(ψ)

13 CapSel Equil - 10b GS: 2nd order PDE character from aψ RR + 2bψ RZ + cψ ZZ +... find b 2 ac = 1 < 0: elliptic Boundary value problem well-posed for elliptic Laplace equation: solution reaches extremum on boundary here we have (different from Laplace ψ = 0): R 2 1 R 2 ψ = II p R 2 normally have ψ [0, ψ 1 ]: zero at magnetic axis, fixed BV ψ 1

14 CapSel Equil - 11 Grad-Shafranov for tokamak equilibria tokamak = toroidal vessel for achieving controlled nuclear fusion task to achieve steady state magnetically confined plasma within poloidal cross-section of vessel: ideal MHD equilibrium static plasma and axisymmetry: solve GS within given boundary overall configuration: donut with nested flux surfaces Z δ ϕ R R 0 2a

15 CapSel Equil - 12 Scaling for Grad-Shafranov natural to set units by a (length) and B 0 (magnetic and pressure) from 2a horizontal diameter of vessel geometric center of vessel at R 0 ɛ a/r 0 inverse aspect ratio use strength of vacuum field at R 0 for B 0 value of flux on outer boundary ψ 1 use to define a unit flux label ψ = ψ/ψ1 define dimensionless inverse flux α a 2 B 0 /ψ 1 change to dimensionless poloidal coordinates (x, y) R Z x = R R 0 a = R 1 ɛ y = Z = Z a

16 CapSel Equil - 13 LHS of GS changes to R R 1 ψ + 2 ψ ψ 1 R R Z 2 a 2 2 ψ x 2 ɛ ψ 1 + ɛx x + 2 ψ y 2 RHS of GS becomes I di dψ R2 dp dψ a2 B 2 0 ψ 1 Ĩ dĩ 2 d p + R d ψ d ψ using Ĩ RB ϕ/ab 0 and p p/b 2 0

17 CapSel Equil - 14 introduce dimensionless profiles scaled pressure P = P ( ψ) α2 ɛ p Q = Q( ψ) ɛα2 G = G( ψ) 1 ɛ 2a 2 B 2 0 [ I 2 (ψ) R 2 0B 2 0 ] [ Q( ψ) P ( ψ) ] under these definitions, GS becomes 2 ψ x 2 ɛ ψ 1 + ɛx x + 2 ψ y = dg dp x(2 + ɛx) 2 d ψ d ψ two arbitrary profiles G dg/d ψ and P dp/d ψ

18 CapSel Equil - 15 core problem now identified, separate freedom in G and P magnitude and shape: amplitude and unit profile G AΓ( ψ) with Γ(0) = 1 P 1 2 ABΠ( ψ) with Π(0) = 1 roughly related to current profile and pressure gradient profile Final form of GS is then 2 ψ x 2 boundary conditions ψ = 1 on boundary ɛ ψ 1 + ɛx x + 2 ψ y = A 2 ψ = ψ x = ψ y = 0 at magnetic axis (δ, 0) Γ( ψ) + Bx(1 + ɛx 2 2nd order PDE with 4 BCs: overdetermined!!! )Π( ψ)

19 CapSel Equil - 16 A (and B if input δ) to be determined with solution ( eigenvalues ) once solution ψ, A, and B identified reconstruct pressure from p = ɛ α 2 ψ magnetic field components from 1 2 ABΠ d ψ + P (0) B R = 1 ɛ ψ α 1 + ɛx y B Z = 1 ɛ ψ α1 + ɛx x B ϕ = ɛ ψ A(ɛΓ 1 1/2 BΠ)d ψ 1 + ɛx α 2 2 different realization of same core problem solution for different α!

20 CapSel Equil - 17 The Soloviev solution analytic solution to Grad-Shafranov R R 1 ψ + 2 ψ R R Z = 2 II p R 2 for linear profiles I 2 /2 = I 2 0/2 Eψ and p = p 0 F ψ E, F free profile parameters polynomial solution derived by Soloviev ψ = ( C + DR 2) two additional parameters C, D verify by insertion [ E + ( F 8D 2 ) R 2] Z 2

21 CapSel Equil - 18 use Soloviev for creating tokamak equilibria instructive to change to ψ and (x, y) coordinates Soloviev solution rewritten as ɛ ψ = x 2 (1 2 x2 ) + 1 ɛ2 [1 + τɛx(2 + ɛx)] y2 4 σ 2 unit contour through 4 points (x = ±1, y = 0) (0, y = ±σ) ɛ = a/r 0 inverse aspect ratio wrt geometric center τ measures triangularity (ellips-like for τ = 0) ψ = 0 at magnetic axis (δ, 0) with δ = 1 ɛ [ 1 + ɛ2 1 ] outward Shafranov shift of magnetic axis geometric effect required for equilibrium in torus

22 CapSel Equil - 19 did not consider boundary as given here determined as unit contour from solution cases ɛ = 0.333, τ = 0 and σ = (0.5, 1, 1.5) elliptic flux contours

23 CapSel Equil - 20 varying the triangularity parameter τ case τ = 0.5 versus τ = 0 and τ = 1.5 triangular flux contours, sign of τ for direction

24 CapSel Equil - 21 Force-Free equilibria So far: 1D and 2D static ideal MHD equilibria (no gravity, no flow) pressure gradient balancing Lorentz force in low beta plasma β 1 only solve ( B) B = 0 plus solenoidal constraint current J = B must be B force-free magnetic configuration, can be 1D, 2D, or 3D

25 CapSel Equil - 22 simplest case: vanishing current J = 0 or potential fields vacuum magnetic field solution since B = 0 define φ from B = φ since B = 0 solve 2 φ = 0 elliptic Laplace equation, BVP well-posed frequently used for reconstructing coronal magnetic field normal B component from magnetograms of solar photosphere B n photosphere = φ n photosphere given

26 CapSel more general: field-aligned currents J = αb from ( B) = 0 find B α = 0 Equil - 23a α is constant along field line 1D examples are slab: B y (x) = sin(αx) and B z (x) = cos(αx) cylinder: B Z (R) = J 0 (αr) and B ϕ (R) = J 1 (αr)

27 CapSel Equil - 23b example taken from Régnier & Amari three reconstructions: potential, constant α, varying α best agreement with observations for nonlinear force-free case

28 CapSel Equil - 24 Stationary equilibria static (v = 0) case only left momentum balance stationary solutions v 0 involve all equations 1D case still fairly trivial slab: when v x = 0 same as before 1D slab with arbitrary background flow v y (x) and v z (x) obeys d dx p + B2 y + Bz 2 2 } {{ } total pressure = ρg with gravity g = gê x density enters example solution: force-free (B y, B z ) and ρ = 1 Dx on x [0, 1] density contrast D and pressure p = p 0 g(x Dx2 2 )

29 CapSel Equil D cylindrical equilibria take v R = 0 then again only force balance remains assume gravitational line source g = GM d dr p + B2 ϕ + BZ 2 2 } {{ } total pressure = B2 ϕ R }{{} radially inward tension + R 2 ê R ρvϕ 2 } R{{} centrifugal force ρ GM suitable for accretion disks R [R inner, R outer ]: midplane Z ϕ R 2 }{{} gravity M * R for jets: neglect gravity and take R [0, R jet ]

30 CapSel Equil D (axisymmetric) stationary ideal MHD equilibria allow for both toroidal and poloidal flow in tokamak steady-state core problem is again 2nd order PDE for flux ψ(r, Z) BUT additional algebraic equation for M 2 (R, Z) ρvp/b 2 p 2 latter expresses energy conservation along fieldline much more complex problem: character of governing PDE can change from location to location elliptic or hyperbolic nature depends on M 2 turns out: 3 elliptic regimes: subslow, slow, fast elliptic flow regions ε ss ε s ε f 0 1 M 2 slow Alfven fast

31 CapSel Equil - 27 References J.P. Goedbloed, Transactions of Fusion Technology 37, 2T, march 2000 B.C. Low, Lecture series HAO 1994 J.P. Goedbloed & S. Poedts, Advanced Magnetohydrodynamics, Lecture series K.U. Leuven (CPA, Doctoraatsopleiding Wiskunde) J.P. Goedbloed & S. Poedts, Principles of Magnetohydrodynamics, Cambridge University Press, to appear 2003 J.P. Goedbloed, Magnetohydrodynamics of astrophysical plasmas, Lectures at Utrecht University 2001 J.P. Goedbloed & A.E. Lifschitz, Phys. Plasmas 4, 1997, 3544 J.P. Goedbloed, Physica Scripta T98, 2002, 43 S. Régnier web site solar.physics.montana.edu/regnier T. Amari web site