Reconstruction of the equilibrium of the plasma in a Tokamak and - PowerPoint PPT Presentation

Reconstruction of the equilibrium of the plasma in a Tokamak and identification of the current density profile in real time. EQUINOX Blaise Faugeras Jacques Blum et C´ edric Boulbe GT Plasma, F´ evrier 2012 BF () EQUINOX GT Plasma, F´ evrier 2012 1 / 38

Outline Introduction 1 Mathematical modelling of axisymmetric equilibrium 2 Input measurements 3 Inverse problem 4 Verification and Validation 5 BF () EQUINOX GT Plasma, F´ evrier 2012 2 / 38

JET : vacuum vessel and plasma BF () EQUINOX GT Plasma, F´ evrier 2012 3 / 38

Tokamak BF () EQUINOX GT Plasma, F´ evrier 2012 4 / 38

Introduction Equilibrium of a plasma : a free boundary problem Equilibrium equation inside the plasma, in axisymmetric configuration : Grad-Shafranov equation Right-hand side of this equation is a non-linear source : the toroidal component of the plasma current density Goal Identification of this non-linearity from experimental measurements. Perform the reconstruction of 2D equilibrium and the identification of the current density in real-time. BF () EQUINOX GT Plasma, F´ evrier 2012 5 / 38

Mathematical modelling of the equilibrium Momentum equation : ρ ( ∂ u ∂ t + u . ∇ u ) + ∇ p = j × B At the slow resistive diffusion time scale ρ ( ∂ u ∂ t + u . ∇ u ) ≪ ∇ p Equilibrium equations   ∇ p = j × B (Conservation of momentum) ∇ . B = 0 (Conservation of B )  ∇ × B = µ j (Ampere’s law) + axisymmetric assumption = > Grad-Shafranov equation BF () EQUINOX GT Plasma, F´ evrier 2012 6 / 38

Model 2D problem. Cylindrical coordinates ( r , φ, z ) State variable ψ ( r , z ) poloidal magnetic flux B p = 1 r ∇ ψ ⊥ In the plasma : Grad-Shafranov equation − ∆ ∗ ψ := ∂ ∂ r ( 1 ∂ψ ∂ r ) + ∂ ∂ z ( 1 ∂ψ 1 ∂ z ) = rp ′ ( ψ ) + µ 0 r ( ff ′ )( ψ ) µ 0 r µ 0 r In the vacuum − ∆ ∗ ψ = 0  1  − ∆ ∗ ψ = [ rp ′ ( ψ ) + µ 0 r ff ′ ( ψ )]1 Ω p ( ψ ) in Ω  ψ = g on Γ BF () EQUINOX GT Plasma, F´ evrier 2012 7 / 38

Definition of the free plasma boundary Two cases outermost flux line inside the limiter (left) magnetic separatrix : hyperbolic line with an X-point (right) BF () EQUINOX GT Plasma, F´ evrier 2012 8 / 38

z − ∆ ∗ ψ = 0 Ω C i Γ L ⊗ ⊗ Γ ⊗ ⊗ − ∆ ∗ ψ = j ( r, ψ ) b C 0 r Ω p − ∆ ∗ ψ = j i ⊗ ⊗ flux loop Ω ⊗ ⊗ Ω 0 B probe BF () EQUINOX GT Plasma, F´ evrier 2012 9 / 38

From discrete magnetic measurements to Cauchy conditions on a fixed contour Magnetic measurements Flux loops : ψ ( M i ) B probes : B p ( N i ) . d i Cauchy conditions ( ψ , ∂ n ψ ) on Γ = ∂ Ω Dirichlet BC : direct problem Neumann BC : inverse problem Numerical methods Direct Interpolation (TCV, ToreSupra) Reconstruction of ψ in the vacuum - plasma boundary identification ◮ JET - Xloc : ∆ ∗ ψ = 0, ψ piecewise polynomial ◮ ToreSupra Apolo : ψ ( ρ, θ ), spline interp. in θ on ref. contour, Taylor expansion in ρ ◮ Toroidal harmonics + PFcoils current filaments model BF () EQUINOX GT Plasma, F´ evrier 2012 10 / 38

Explicit solutions to ∆ ∗ ψ = 0 Laplacian in cylindrical coordinates If ∆ ∗ ψ ( r , z ) = 0 in D then Ψ( r , z , φ ) = 1 r ψ ( r , z ) cos φ , ∆Ψ = 0 in D ′ Bipolar (Toroidal) coordinates z a 2 r 2 + ( z − a cot η ) 2 = sin 2 η z M τ = log( MA MB ) × O × � η = AMB η a a B A r A B O × × a r a 2 a ( r − a coth τ ) 2 + z 2 = sinh 2 τ Quasi-separable solution � � Ψ( τ, η, φ ) = cosh τ − cosh η A ( τ ) B ( η ) cos φ BF () EQUINOX GT Plasma, F´ evrier 2012 11 / 38

Complete set of solutions � � P 1 � � � � cos( k η ) � � 2 (cosh τ ) a sinh τ k − 1 T P , Q √ cosh τ − cos η k ∈ N = Q 1 c , s , k 2 (cosh τ ) sin( k η ) k − 1 k ∈ N References P.M. Morse and H. Feshbach. Methods of Theoritical Physics . 1953 E. Durand. Magn´ etostatique . 1968 N.N. Lebedev. Special Functions and their Applications . 1972 J. Segura and A. Gil. Evaluation of toroidal harmonics . CPC. 1999 Y. Fischer. PhD. 2011 BF () EQUINOX GT Plasma, F´ evrier 2012 12 / 38

Flux in the vaccum N � � ( β P , Q c , s , k )( T P , Q ψ ( r , z ) = c , s , k ) + ψ f ( r , z ; r k , z k ) k =0 k N � � ( β P , Q B ( r , z ) = c , s , k ) B k ( r , z ) + B f ( r , z ; r k , z k ) k =0 k PF coils modelized by filaments of current Current I k at ( r k , z k ) : √ rr k [(1 − α 2 ψ f ( r , z ; r k , z k ) = µ 0 I k 2 ) J 1 ( α ) − J 2 ( α )] , B f = . . . απ Compute ( β P , Q c , s , k ) k =1: N ) by least-square fit to magnetic measurements ψ in the vacuum Ω 0 \ (Ω p ∪ Ω C i ) Evaluate ( ψ, ∂ n ψ ) on Γ BF () EQUINOX GT Plasma, F´ evrier 2012 13 / 38

Experimental measurements magnetic ”measurements” ψ ( M i ) = g i and 1 ∂ψ ∂ n ( N j ) = h j on Γ r on mesh boundary (experimental measurements if possible, or outputs from other codes : toroidal harmonics, XLOC (JET) and APOLO (ToreSupra)) interferometry and polarimetry on several chords � � n e ( ψ ) ∂ψ n e ( ψ ) dl = α m , ∂ n dl = β m r C m C m motional Stark effect f j ( B r ( M j ) , B z ( M j ) , B φ ( M j )) = γ j BF () EQUINOX GT Plasma, F´ evrier 2012 14 / 38

BF () EQUINOX GT Plasma, F´ evrier 2012 15 / 38

Statement of the inverse problem State equation   − ∆ ∗ ψ = λ [ r ψ ) + R 0 A ( ¯ r B ( ¯ ψ )]1 Ω p ( ψ ) in Ω R 0  ψ = g on Γ Least square minimization J ( A , B , n e ) = J 0 + K 1 J 1 + K 2 J 2 + J ǫ with J 0 = � j (1 ∂ψ ∂ n ( N j ) − h j ) 2 r � J 1 = � ∂ψ n e ∂ n dl − α i ) 2 i ( r C i � J 2 = � n e dl − β i ) 2 i ( C i � 1 � 1 � 1 ( ∂ 2 A ( ∂ 2 B ( ∂ 2 n e ψ 2 ) 2 d ¯ ψ 2 ) 2 d ¯ ψ 2 ) 2 d ¯ J ǫ = ǫ ψ + ǫ ψ + ǫ ne ψ ∂ ¯ ∂ ¯ ∂ ¯ 0 0 0 BF () EQUINOX GT Plasma, F´ evrier 2012 16 / 38

Numerical method Finite element resolution  Find ψ ∈ H 1 with ψ = g on Γ such that    � � 1 λ [ r ψ ) + R 0  ∀ v ∈ H 1 A ( ¯ r B ( ¯ 0 , µ 0 r ∇ ψ ∇ vdx = ψ )] vdx   R 0 Ω Ω p with A ( x ) = � B ( ψ ) = � i a i f i ( x ) , i b i f i ( x ) , u = ( a i , b i ) Fixed point K ψ = Y ( ψ ) u + g K modified stiffness matrix, u coefficients of A and B , g Dirichlet BC Direct solver : ( ψ n , u ) → ψ n +1 ψ n +1 = K − 1 [ Y ( ψ n ) u + g ] BF () EQUINOX GT Plasma, F´ evrier 2012 17 / 38

Numerical method Least-square minimization J ( u ) = � C ( ψ ) ψ − d � 2 + u T Au d : experimental measurements A : regularization terms Approximation J ( u ) = � C ( ψ n ) ψ − d � 2 + u T Au , with ψ = K − 1 [ Y ( ψ n ) u + g ] = � C ( ψ n ) K − 1 Y ( ψ n ) u + C ( ψ n ) K − 1 g − d � 2 + u T Au J ( u ) = � E n u − F n � 2 + u T Au Normal equation. Inverse solver : ψ n → u ( E nT E n + A ) u = E nT F n BF () EQUINOX GT Plasma, F´ evrier 2012 18 / 38

Numerical method. EQUINOX One equilibrium reconstruction : Fixed-point iterations : ◮ Inverse solver : ψ n → u n +1 ◮ Direct solver : ( ψ n , u n +1 ) → ψ n +1 ◮ Stopping condition || ψ n +1 − ψ n || < ǫ || ψ n || A pulse in real-time : Quasi-static approach : ◮ first guess at time t = equilibrium at time t − δ t ◮ limited number of iterations Normal equation : ≈ 10 basis func. → small ≈ 20 × 20 linear system Tikhonov regularization parameters unchanged K = LU and K − 1 precomputed and stored once for all Expensive operations : update products C ( ψ ) K − 1 and C ( ψ ) K − 1 Y ( ψ ) BF () EQUINOX GT Plasma, F´ evrier 2012 19 / 38

Numerical Results : Tore Supra and JET characteristics ToreSupra JET Finite element mesh Number of triangles 1382 2871 Number of nodes 722 1470 functions A and B Basis type Bspline Bspline Number of basis func. 8 8 Computation time (1.80GHz) One equilibrium 20 ms 60 ms Real-time requirement : 100 ms BF () EQUINOX GT Plasma, F´ evrier 2012 20 / 38

Tore Supra - Magnetics and polarimetry BF () EQUINOX GT Plasma, F´ evrier 2012 21 / 38

JET - Magnetics and polarimetry BF () EQUINOX GT Plasma, F´ evrier 2012 22 / 38

Algorithm verification : twin experiments Method Functions A and B given. Generate ”measurements” with direct code Test the possibility to recover the functions by solving the inverse problem Noise free experiments. Magnetics only. With a well-chosen regularization parameter ε , A and B are well recovered. Averaged current density and q profiles are not very sensitive to ε . Experiments with noise. Magnetics only and mag+polarimetry. Averaged current density and q profiles are less sensitive to noise than A and B . With polarimetry A and B are better constrained. BF () EQUINOX GT Plasma, F´ evrier 2012 23 / 38

Average over magnetic surfaces � � � � ∂ gdv = 1 gds gdl dl < g > = |∇ ρ | = / ∂ V V ′ B p B p V S C ρ C ρ ρ a coordinate indexing the magnetic surfaces BF () EQUINOX GT Plasma, F´ evrier 2012 24 / 38