Equilibrium and transport in Tokamaks

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1 Equilibrium and transport in Tokamaks Jacques Blum Laboratoire J.-A. Dieudonné, Université de Nice Sophia-Antipolis Parc Valrose Nice Cedex 02, France 08 septembre 2008 Jacques Blum (Université de Nice) Equilibrium and transport in Tokamaks 08 septembre / 47

2 Outline 1 MHD equations 2 Mathematical modelling of axisymmetric equilibrium 3 The inverse equilibrium problem 4 The transport system Jacques Blum (Université de Nice) Equilibrium and transport in Tokamaks 08 septembre / 47

3 Magnetohydrodynamic (MHD) equations Derivation of the fluid equations from the kinetic equations f α t + (v. x)f α + F α m α. v f α = C α fα (x, v, t) : distribution function (α = e : electrons, α = i : ions) Fα : force on the particle m α : mass of the particle C α : collision operator Density of particles : n α (x, t) = f α (x, v, t)dv Fluid velocity : u α (x, t) = 1 n α f α (x, v, t)vdv Jacques Blum (Université de Nice) Equilibrium and transport in Tokamaks 08 septembre / 47

4 Pressure tensor P α (x, t) = m α f α (x, v, t)(v u α )(v u α )dv Isotropic case p α (x, t) = m α 3 f α (x, v, t)(v u α ) 2 dv First moment n α t +. f α vdv 1 Fα m α v f αdv = 0 F α v = 0 for the electromagnetic forces n α t +.(n αu α ) = 0 Jacques Blum (Université de Nice) Equilibrium and transport in Tokamaks 08 septembre / 47

5 Second moment m α t (n αu α ) + m α x. f α vvdv v.(f α.v)f α dv = m α vc α dv v = (v u α ) + u α with the equation of conservation of density, we obtain m α n α ( u α t + u α. u α ) =.P α + n α F α + R α with F α = Ze(E + u α B) Third moment : Energy equation + closure by some assumption on the heat flux vector (transport model) Jacques Blum (Université de Nice) Equilibrium and transport in Tokamaks 08 septembre / 47

6 Magnetohydrodynamic equations - One fluid model Mass density ρ = m e n e + m i n i = m e Zn i + m i n i m i n i Flow velocity Current density u = m en e u e + m i n i u i ρ u i j = en e u e + Zen i u i = en e (u i u e ) Scalar pressure p = n e kt e + n i kt i where k is the Boltzmann constant Jacques Blum (Université de Nice) Equilibrium and transport in Tokamaks 08 septembre / 47

7 Single fluid resistive magnetohydrodynamic equations ρ t +.(ρu) = s ( Conservation of particles) ρ( u + u. u) + p = j B (Conservation of momentum) t 3 2 ( p t + u. p) + 5 p.u + Q = s (Conservation of particle energy) 2 E = B (Faraday s law) t.b = 0 (Conservation of B) E + u B = ηj (Ohm s law) H = j (Ampere s law) B = µh (Magnetic permeability) p = nkt (Law of perfect gases) T : temperature, Q : heat flux, η : resistivity tensor, s and s : source terms and k : Boltzmann constant Jacques Blum (Université de Nice) Equilibrium and transport in Tokamaks 08 septembre / 47

8 Characteristic time constants of the plasma Alfven time constant : τ A = a(µ 0m i n i ) 1/2 B 0 Diffusion time constant of the particle density n τ n = a2 D where D is the particle diffusion coefficient. Time constants for diffusion of heat of the electrons and of the ions τ e = n ea 2 K e τ i = n ia 2 K i Jacques Blum (Université de Nice) Equilibrium and transport in Tokamaks 08 septembre / 47

9 Characteristic time constants of the plasma Resistive time constant for the diffusion of the current density and magnetic field in the plasma τ r = µ 0a 2 η Global time constant for plasma diffusion On the diffusion time-scale τ p τ p = inf(τ n, τ e, τ i, τ r ) ρ( u t τ A τ p. + u u) p Jacques Blum (Université de Nice) Equilibrium and transport in Tokamaks 08 septembre / 47

10 Mathematical modelling of the equilibrium Momentum equation : ρ du dt = j B p At the slow resistive diffusion time scale ρ du can be neglected dt Equilibrium equations : p = j B consequence : B = 0 B = µj p B = 0 p j = 0 => Field lines and current lines are on isobaric surfaces (p = cst) = magnetic surfaces Jacques Blum (Université de Nice) Equilibrium and transport in Tokamaks 08 septembre / 47

11 Magnetic surfaces Jacques Blum (Université de Nice) Equilibrium and transport in Tokamaks 08 septembre / 47

12 Axisymmetric configuration Cylindrical coordinates (r, z, φ) B independent of φ B = B p + B t B p = (B r, B z ) = ( 1 r B t = f r e φ ψ z, 1 ψ r r ) = 1 r Ψ ψ(r, z) : poloidal flux f = f (ψ) : diamagnetic function From Ampere s theorem : j = j p + j φ j p = 1 r [ ( f µ ) e φ] j φ = ( ψ)e φ Jacques Blum (Université de Nice) Equilibrium and transport in Tokamaks 08 septembre / 47

13 Tokamak Jacques Blum (Université de Nice) Equilibrium and transport in Tokamaks 08 septembre / 47

14 Grad-Shafranov Equation Equilibrium equation : with In the plasma : In the vacuum : p = ψ ψ f r µ 0 r 2 f = r ( 1 µ 0 r r ) + z ( 1 µ 0 r z ) ψ = rp (ψ) + 1 µ 0 r (ff )(ψ) ψ = 0 Jacques Blum (Université de Nice) Equilibrium and transport in Tokamaks 08 septembre / 47

15 Jacques Blum (Université de Nice) Equilibrium and transport in Tokamaks 08 septembre / 47

16 Definition of the free plasma boundary Two cases : magnetic separatrix : hyperbolic line with an X-point (left) outermost flux line inside the limiter (right) Jacques Blum (Université de Nice) Equilibrium and transport in Tokamaks 08 septembre / 47

17 Equation for Ψ(r, z) inside the vacuum vessel where { ψ = λ[ra( ψ) + 1 r B( ψ)]χ Ωp in Ω ψ = h on Ω p ( ψ) = λa( ψ) 1 (ff )( ψ) = λb( ψ), µ 0 ψ max ψ ψ Ω = ψ b max ψ [0, 1] in Ω p Ω χ Ωp is the characteristic function of Ω p Jacques Blum (Université de Nice) Equilibrium and transport in Tokamaks 08 septembre / 47

18 The inverse equilibrium problem : experimental measurements magnetic measurements interferometry and polarimetry ψ(m i ) = g i on Γ 1 ψ r n (N j) = h j n e (ψ)dl = α m C m n e (ψ) ψ C m r n dl = β m kinetic pressure p(r, 0) = p d (r) Jacques Blum (Université de Nice) Equilibrium and transport in Tokamaks 08 septembre / 47

19 Jacques Blum (Université de Nice) Equilibrium and transport in Tokamaks 08 septembre / 47

20 Jacques Blum (Université de Nice) Equilibrium and transport in Tokamaks 08 septembre / 47

21 Statement of the inverse problem State equation ψ = g on Γ Least square minimization ψ = λ[ r R 0 A( ψ) + R 0 r B( ψ)]1 Ωp J(A, B, n e ) = J 0 + K 1 J 1 + K 2 J 2 + J ɛ with J 0 = j (1 r J 1 = i ( J 2 = i ( ψ n (N j) h j ) 2 C i n e r ψ n dl α i) 2 C i n e dl β i ) 2 Jacques Blum (Université de Nice) Equilibrium and transport in Tokamaks 08 septembre / 47

22 Tikhonoff regularization Find A 0, B 0, n e0 such that : J ɛ = ɛ ( 2 A ψ 2 )2 d ψ 1 +ɛ 2 ( 2 B ψ 2 )2 d ψ 0 1 +ɛ 3 ( 2 n e ψ 2 )2 d ψ 0 J(A 0, B 0, n e0 ) = inf J(A, B, n e ) Jacques Blum (Université de Nice) Equilibrium and transport in Tokamaks 08 septembre / 47

23 Numerical identification Finite element resolution 1 ψ vdx = λ[ r A( Ω µ 0 r Ω p R ψ) + R 0 B( ψ)]vdx 0 r with Fixed point A(ψ) = i a iφ i (ψ), u = (a i, b i ) where K stiffness matrix u coefficients of j P g Dirichlet boundary condition Kψ = D(ψ)u + g ψ = K 1 [D(ψ n )u + g] B(ψ) = i b iφ i (ψ) Jacques Blum (Université de Nice) Equilibrium and transport in Tokamaks 08 septembre / 47

24 Least-square minimization k : experimental measurements Λ : regularization terms J(u) = C(ψ)ψ k 2 + u T Λu Approximation J(u) = C(ψ n )ψ k 2 + u T Λu, with ψ = K 1 [D(ψ n )u + g] J(u) = C(ψ n )K 1 D(ψ n )u + C(ψ n )K 1 g k 2 + u T Λu = E n u F n 2 + u T Λu Normal equation (E nt E n + Λ)u = E nt F n Jacques Blum (Université de Nice) Equilibrium and transport in Tokamaks 08 septembre / 47

25 Available basis for A, B and n e Piecewise linear Cubic B-splines Scaling functions (Average Interpolating wavelets) Fig.: Bsplines (left), Scaling functions (right) Jacques Blum (Université de Nice) Equilibrium and transport in Tokamaks 08 septembre / 47

26 Numerical Results : Tore Supra and JET characteristics ToreSupra Finite element mesh JET Number of triangles Number of nodes functions A and B Basis type Bspline Bspline Number of basis func. 7 7 Jacques Blum (Université de Nice) Equilibrium and transport in Tokamaks 08 septembre / 47

27 References : IDENT : J. Blum, E. Lazzaro and al, Problems and methods of self-consistent reconstruction of tokamak equilibrium profiles from magnetic and polarimetric measurements, Nuclear fusion, 30, 1990, p 1475 EQUINOX : J. Blum, C. Boulbe and B. Faugeras, Real-time equilibrium reconstruction in a Tokamak, Burning Plasma Diagnostics, AIP Conference Proceedings, 988, 2008, p 420 Jacques Blum (Université de Nice) Equilibrium and transport in Tokamaks 08 septembre / 47

Fig.: Jet (left), Tore Supra (right) Jacques Blum

28 Examples on Jet and Tore Supra Jet with magnetics. Tore Supra with magnetics, polarimetry and interferometry. Fig.: Jet (left), Tore Supra (right) Jacques Blum (Université de Nice) Equilibrium and transport in Tokamaks 08 septembre / 47

29 The transport system At the resistive diffusion time scale τ r = µ 0a 2, the plasma follows a η sequence of equilibrium states, linked by the transport processes. Evolution of p(ψ, t), f (Ψ, t) : p = n e kt e + n i kt i diffusion velocity parallel to the magnetic surfaces perpendicular diffusion velocity n e = n e (Ψ, t) T e = T e (Ψ, t) n i = n i (Ψ, t) T i = T i (Ψ, t) Jacques Blum (Université de Nice) Equilibrium and transport in Tokamaks 08 septembre / 47

30 The averaging technique (H. Grad and al) χ : arbitrary coordinate that labels the magnetic surface S (Example : χ = Ψ) A = AdV = 1 AdS V V χ, with V V = V χ = S S ds χ. where V is the volume enclosed inside the magnetic surface S. We are now going to average the transport equations, so as to obtain a set of 1-D diffusion equations with respect to the space variable χ. Jacques Blum (Université de Nice) Equilibrium and transport in Tokamaks 08 septembre / 47

31 Properties of the average.w = W. V, W, V t (V A ) = V Ȧ + χ Au χ. V, A where Ȧ denotes the time-derivative at a fixed point (r, z), whereas t denotes the time-derivative at fixed χ. The vector u χ is the velocity of the constant-χ surface, defined by It can be deduced that χ + u χ. χ = 0. V t = χ u χ. V. Jacques Blum (Université de Nice) Equilibrium and transport in Tokamaks 08 septembre / 47

32 The equation of conservation of electrons n e +.(n e u e ) = S 1, where S 1 is a source term. After averaging and multiplying by V t ( n e V ) + χ n e(u e u χ ). V = S 1 V. Γ e : flux particle relative to a constant-χ surface : Neutrality of the plasma n i Γ e = n e (u e u χ ). χ t (n ev ) + χ (Γ ev ) = S 1 V Jacques Blum (Université de Nice) Equilibrium and transport in Tokamaks 08 septembre / 47

33 The energy equations Electrons : Neglecting the viscosity terms 3 2 p e +.(Q e p eu e ) = j.e Q u i. p i + S 2, where p e and p i are, respectively, the electron and ion pressure, Q e the electron heat flux, Q = 3m e m i n e t e (T e T i ), u i the ion flow velocity and S 2 a source term. After averaging 3 2(V ) 2/3 t [ p e (V ) 5/3 ] + χ [(q e kt eγ e )V ] = [ j.e u χ. p e u i. p i Q + S 2 ]V, with q e = Q e. χ. Jacques Blum (Université de Nice) Equilibrium and transport in Tokamaks 08 septembre / 47

34 Ions : Ion energy balance equation : 3 2 p i +.(Q i p iu i ) = Q + u i. p i + S 3, where Q i is the ion heat flux and S 3 a source term. After averaging 3 2(V ) 2/3 t (p i(v ) 5/3 ) + χ [(q i with q i = Q i. χ. kt i Γ e Z )V ] Γ e p i n e χ V = Q V + S 3 V, Jacques Blum (Université de Nice) Equilibrium and transport in Tokamaks 08 septembre / 47

35 Flux equations Poloidal flux Ψ Velocity u Ψ of a constant-ψ surface : Ψ + u Ψ. Ψ = 0. Velocity of the constant-χ surface : We have χ + u χ. χ = 0. Ψ = Ψ t + Ψ χ χ. hence the evolution equation for the poloidal flux Ψ is Ψ t = (u χ u Ψ ). Ψ. Jacques Blum (Université de Nice) Equilibrium and transport in Tokamaks 08 septembre / 47

36 Toroidal flux Φ : Φ = Safety factor q V B t 2πr dv q = 1 Φ 2π Ψ. (number of toroidal turns necessary to make one poloidal turn) Hence q = 1 V 4π 2 Ψ r 2 f. Moreover, we have From Faraday s law Φ t = Φ t Ψ 2πq Ψ t. Φ t Ψ = 1 V 2π Ψ E.B. Evolution equation for the toroidal flux Φ, Φ t = 1 V 2π Ψ [f r 2 (u χ u Ψ ). Ψ + E.B ]. Jacques Blum (Université de Nice) Equilibrium and transport in Tokamaks 08 septembre / 47

37 Choice of the flux surface label χ As Φ t = 0, one has χ = ( Φ πb 0 ) 1/2 (u χ u Ψ ). Ψ = E.B f r 2. Hence From Ohm s law, it can be deduced that where η is Spitzer s resistivity. Ψ t = E.B f r 2. E + u B = ηj E.B = η j.b, Jacques Blum (Université de Nice) Equilibrium and transport in Tokamaks 08 septembre / 47

38 Moreover, one can deduce with j.b = f 2 µ 0 V χ (C 2 Ψ f χ ), C 2 = V r 2 2 χ. Therefore the diffusion resistive equation becomes Ψ t ηf µ 0 C 3 χ (C 2 f with C 3 = V r 2. By differentiation with respect to χ Ψ t 1 µ 0 χ [ηf C 3 χ (C 2Ψ )] = 0. f Ψ ) = 0, (4.1) χ Jacques Blum (Université de Nice) Equilibrium and transport in Tokamaks 08 septembre / 47

39 From the definitions of Φ and χ, the following relation for f can be obtained f = 4π2 B 0 χ C 3. Equation for Ψ can be written Ψ t 1 µ 0 χ [ ηχ C3 2 χ (C 2C 3 χ Ψ )] = 0. It is a parabolic 1-D equation which characterizes the resistive diffusion of the poloidal flux with respect to the toroidal one. Ψ is related to the safety factor q by Ψ = B 0χ q. Jacques Blum (Université de Nice) Equilibrium and transport in Tokamaks 08 septembre / 47

40 Electron energy equation One has j.e = j φ/r E.B f r 2 + u χ. p, with j φ /r = 1 (C 2 Ψ ) µ 0 V χ. Let us define σ e by σ e = p e (V ) 5/3 3 σ e 2(V ) 2/3 t + χ [(q e kt eγ e )V ] = Γ ev n e p i χ + ηχ µ 2 0 C 3 2 Q V + S 2 V. χ (C 2Ψ ) χ (C 2C 3 χ Ψ ) Jacques Blum (Université de Nice) Equilibrium and transport in Tokamaks 08 septembre / 47

41 Ion energy equation 3 2(V ) 2/3 t σ i + χ [(q i with q i = Q i. χ and σ i = p i (V ) 5/3 Density equation kt i Γ e Z )V ] Γ e p i n e χ V = Q V + S 3 V, t N e + χ (Γ ev ) = S 1 V. with N e = n e V To close the system in the neoclassical theory, Γ e, q e and q i are expressed as quasi-linear combinations of n e / χ, p e / χ, p i / χ, and E.B. The adiabatic case : N e, σ e, σ i and Ψ are kept constant during adiabatic evolutions. Jacques Blum (Université de Nice) Equilibrium and transport in Tokamaks 08 septembre / 47

42 Diagonal model Assume that n e (u e u χ ) = D n e, Q e = K e T e, Q i = K i T i, where D is the electronic diffusion coefficient, and K e and K i are the electronic and ionic thermal conductivities, respectively. Hence Γ e = D 2 χ n e χ, q e = K e 2 χ T e χ, q i = K i 2 χ T i χ Jacques Blum (Université de Nice) Equilibrium and transport in Tokamaks 08 septembre / 47

43 Density, electron and ion energy equations become N e t χ (DC n e 1 χ ) = S 1 V, 3 σ e 2(V ) 2/3 t χ (K T e ec 1 χ ) 5k 2 ηχ χ (DC n e 1T e χ ) χ (C 2Ψ ) χ (C 2C 3 χ Ψ ) + DC 1 n e = µ 2 0 C 3 2 Q V + S 2 V, p i n e χ χ 3 2(V ) 2/3 σ i t χ (K ic 1 T i χ 5k 2 with C 1 = V 2 χ. χ (DC 1T i n e Z χ ) + DC 1 n e p i n e χ χ = Q V + S 3 V Jacques Blum (Université de Nice) Equilibrium and transport in Tokamaks 08 septembre / 47

44 The averaged equilibrium equation The Grad-Shafranov equilibrium equation can be written 1 p.( Ψ) = µ 0 r 2 Ψ + 1 f 2 2µ 0 r 2 Ψ The average of this equation over a magnetic surface leads to Ψ χ (C 2Ψ ) = µ 0 V p χ C f 2 3 χ. As p = σ e + σ i (V, this equation enable us to determine the functions ) 5/3 f (χ), if the profiles Ψ (χ), σ e (χ), σ i (χ) and the geometric coefficients r 2 2 χ and r 2 have been previously calculated. The integration of this averaged equation over the plasma gives I p = C 2(χ max )Ψ (χ max ) 2πµ 0 where I p is the total plasma current and χ max the value of χ at the plasma boundary. Jacques Blum (Université de Nice) Equilibrium and transport in Tokamaks 08 septembre / 47

45 Coupling of the equilibrium and the diffusion system 2 χ, r 2 2 χ, r 2 Equilibrium system for Ψ(r, z) Transport system + averaged G.S equation p(χ), f (χ), Ψ (χ) Finite elements + Newton-Raphson iterations Finite differences + θ-method Jacques Blum (Université de Nice) Equilibrium and transport in Tokamaks 08 septembre / 47

46 Boundary conditions n e (χ max ) = n 0 e, T e (χ max ) = T 0 e, T i (χ max ) = T 0 i, f (χ max ) = R 0 B 0, and I p Ψ (χ max ) = 2πµ 0 C 2 (χ max ) Jacques Blum (Université de Nice) Equilibrium and transport in Tokamaks 08 septembre / 47

47 Conclusion The coupling between the 2D Grad-Shafranov equilibrium equation and the 1D transport equations, that depend on the geometry of the flux lines, is usually called a 1 1/2 D transport model and enables us to follow the quasi-static evolution of the equilibrium at the transport time- scale. The resolution of the transport equations requires the computation of the source terms due to the additional heating (neutral beam injection, RF heating) and of the current generated by the RF system in the modified Ohm s law (example : CHRONOS Software) Jacques Blum (Université de Nice) Equilibrium and transport in Tokamaks 08 septembre / 47

Un problème inverse: l identification du profil de courant dans un Tokamak

Un problème inverse: l identification du profil de courant dans un Tokamak Blaise Faugeras en collaboration avec J. Blum et C. Boulbe Université de Nice Sophia Antipolis Laboratoire J.-A. Dieudonné Nice,