Diffusion and Transport in Axisymmetric Geometry Stephen C. Jardin - PowerPoint PPT Presentation

MHD Simulations for Fusion Applications Lecture 2 Diffusion and Transport in Axisymmetric Geometry Stephen C. Jardin Princeton Plasma Physics Laboratory CEMRACS ‘10 Marseille, France July 20, 2010 1

These 4 areas address different timescales and are normally studied using different codes SAWTOOTH CRASH ENERGY CONFINEMENT ELECTRON TRANSIT TURBULENCE ω LH τ A ISLAND GROWTH -1 CURRENT DIFFUSION Ω ce Ω ci -1 -1 10 -10 10 -8 10 -6 10 -4 10 -2 10 0 10 2 10 4 SEC. (b) Micro- (c) Extended- (d) Transport Codes turbulence codes (a) RF codes MHD codes

Transport codes solve the same equations as the Extended MHD codes, but in 2D rather than 3D Z ∂ = (d) Transport Codes 0 � axisymmetry ∂ φ First we will consider a model R φ problem to better understand the timescales in 2D, then will proceed to a general approach 3

Tokamak Equilibrium Basics: Need for a vertical field Z Plasma Cross Section (toroidal current in) R φ A tokamak needs an externally generated “vertical field” for equilibrium. A purely vertical field will produce a nearly circular cross-section plasma. We first consider a very crude “rigid plasma” model to better understand where the slow resistive time scale comes from 4

In actual tokamak experiments, external poloidal field is not purely straight but has some curvature to it Z Z R R φ φ Good Curvature Bad Curvature • Stable to vertical mode • Unstable to vertical mode • Oblate plasma • Elongated plasma • low beta limits • higher beta limits 5

In actual tokamak experiments, external field is not purely straight but has some curvature to it Z Z R R φ φ Good Curvature Bad Curvature • Stable to vertical mode • Unstable to vertical mode • Oblate plasma • Elongated plasma • low beta limits • higher beta limits 6

External poloidal field with curvature can be thought of as a superposition of vertical and radial field. Z Z R R φ φ If plasma column is displaced upward, the force Vertical J x B = I P x B Rext Instability will accelerate it further upward. Same for downward. 7 Alfven wave time scale: very fast!

Describe the plasma as a rigid body A nearby conductor will produce of mass m with Z position Z P . eddy currents which act to stabilize Assume time dependence e i ω t Z Conducting wall Equation of motion: with dipolar current I C - inertia external field conductor Circuit equation for wall: Z P R φ resistance plasma coupling inductance + Plasma with current I P 8

Introduce plasma velocity V P = i ω Z P to get a 3x3 matrix eigenvalue equation for ω of standard form Z Conducting wall with dipolar current I C - Z P Three roots: R φ + Plasma with current I P 9

With only passive conductor, still an unstable root but much smaller. Not on Alfven wave time scale but on L/R timescale of conductor. These are high frequency (~10 -7 sec) stable oscillations Three roots: that are slowly damped by the wall resistivity This is the unstable mode. Very slow (~ 10 -1 sec), and independent of plasma mass. 10

Total stability is obtained by adding an active feedback system which only needs to act on this slower timescale. Z This “rigid” mode is easily stabilized by Conducting Wall adding a pair of feedback coils of with current I C opposite sign, and applying a voltage proportional to the plasma displacement -or its time integral or time derivative (PID) Z P Three roots: R φ Plasma with current I P 11

To model this “vertical instability” in realistic geometry, and take the non- rigid motion of the plasma into account, we take advantage of the fact that the unstable mode does not depend on the plasma mass (or inertia), and the stable modes are very high frequency and very low amplitude. We start with the basic MHD + circuit equations and apply a ∂ + ∇ n = i ( n V ) 0 “resistive timescale ordering” ∂ t ∂ B ε � 1 Introduce small parameter = −∇× E ∂ ∂ t η ε ∼ V ∼ E ~ V ~ ~ R ~ ∂ V i ∂ t + •∇ + ∇ = × nM ( V V ) p J B i ∂ t + × = η E V B J ∂ ⎛ ⎞ 3 p 3 + ∇ + = − ∇ + η 2 i i q p V p V J ⎜ ⎟ ∂ 2 t ⎝ 2 ⎠ = ∇× J B ⎡ ⎤ d ∑ ∫ + + + = L I M I J G R R dR ( , ) R I V ⎢ ⎥ φ i i ij j i i i i dt ⎣ ⎦ 12 ≠ i j P

To model this “vertical instability” in realistic geometry, and taking the non-rigid motion of the plasma into account, we take advantage of the fact that the unstable mode does not depend on the plasma mass (or inertia), and the stable modes are very high frequency and low amplitude. ∂ + ∇ n = i ( n V ) 0 We start with the basic MHD + ∂ t circuit equations and apply a ∂ B “resistive timescale ordering” = −∇× E ∂ t ε � Introduce small parameter 1 ∂ V ε + •∇ + ∇ = × 2 nM ( V V ) p J B ∂ i ∂ t η ε ∼ ∼ V E ~ V ~ ~ R ~ + × = η E V B J i ∂ t ∂ ⎛ ⎞ 3 p 3 + ∇ + = − ∇ + η 2 i i q p V p V J ⎜ ⎟ All equations pick up a ∂ 2 t ⎝ 2 ⎠ ε factor of , in all terms, = ∇× J B which cancels out, except in the momentum equation, where the ⎡ ⎤ d ∑ ∫ inertial terms are + + + = L I M I J G R R dR ( , ) R I V ⎢ ⎥ φ i i ij j i i i i ε 2 dt multiplied by . ⎣ ⎦ 13 ≠ i j P

To model this “vertical instability” in realistic geometry, and taking the non-rigid motion of the plasma into account, we take advantage of the fact that the unstable mode does not depend on the plasma mass (or inertia), and the stable modes are very high frequency and low amplitude. ∂ + ∇ n = i ( n V ) 0 We start with the basic MHD + ∂ t circuit equations and apply a ∂ B “resistive timescale ordering” = −∇× E ∂ t 0 ε � Introduce small parameter 1 ∂ V ε 2 + • ∇ + ∇ = × nM ( V V ) p J B ∂ i ∂ t η ε ∼ V ∼ E ~ V ~ ~ R ~ + × = η i ∂ E V B J t ∂ ⎛ ⎞ 3 p 3 This allows us to drop the + ∇ + = − ∇ + η 2 i i q p V p V J ⎜ ⎟ inertial terms in the ∂ 2 t ⎝ 2 ⎠ momentum equation, and = ∇× J B replace it with the equilibrium equation. ⎡ ⎤ d ∑ ∫ Huge simplification…. + + + = L I M I J G R ( , ) R dR R I V ⎢ ⎥ φ i i ij j i i i i dt removes Alfven timescale ⎣ ⎦ 14 ≠ i j P

To model this “vertical instability” in realistic geometry, and taking the non-rigid motion of the plasma into account, we take advantage of the fact that the unstable mode does not depend on the plasma mass (or inertia), and the stable modes are very high frequency and low amplitude. This is the set of equations we ∂ + ∇ n = i ( n V ) 0 solve to simulate control of the ∂ t plasma position and shape. ∂ B = −∇× E ∂ t There are 3 production codes that solve these nonlinear equations in ∇ = × p J B 2D and are used to design and + × = η E V B J test control strategies. ∂ ⎛ ⎞ 3 p 3 + ∇ + = − ∇ + η 2 i i q p V p V J ⎜ ⎟ • TSC (PPPL) ∂ 2 t ⎝ 2 ⎠ • DINA (Russia) = ∇× J B • CORSICA (LLNL) ⎡ ⎤ d ∑ ∫ + + + = L I M I J G R R dR ( , ) R I V ⎢ ⎥ φ i i ij j i i i i dt ⎣ ⎦ ≠ i j P 15

Z Consider first the vector magnetic field equation ∂ B φ = −∇× (1) E R ∂ t + × = η (2) E V B J The most general form for an equilibrium axisymmetric magnetic field is: ( ) = ∇ ×∇Ψ + φ Ψ ∇ φ (3) B g Ψ is "flux function" Substitute (3) into (1): g is toroidal field function ∂Ψ ∂ g ∇ i B =0 ∇ ×∇ φ + ∇ φ = −∇× (4) E ∂ ∂ t t ∂ = 1 2 ∇ φ = ∇ φ 0, Take dot product of (4) with ∂ φ 2 R ∂ 1 g [ ] = −∇ φ ∇× = ∇ ∇ × φ (5) i E i E 2 ∂ R t ∂Ψ ∂Ψ ∇ ×∇ φ = −∇× ∇ φ Noting that , the remaining part of (4) becomes ∂ ∂ t t ∂Ψ = ∇ + φ 2 (6) R E i C t ( ) Constant can be taken to vanish ∂ t to match boundary condition that 16 Ψ =0 at R=0

Z Recall: + × = η (2) E V B J ∂ 1 g [ ] φ = ∇ ∇ × φ (5) i E R ∂ 2 R t ∂Ψ = ∇ φ 2 (6) R E i ∂ t ( ) = ∇ ×∇Ψ + φ Ψ ∇ φ B g Use (2) to eliminate E from (5) and (6): μ = ∇× J B 0 ∂ 1 g = Δ Ψ∇ + ∇ ×∇ φ φ * g ( ) = ∇ −∇ × ⎡ φ × + ∇ × φ η ⎤ i V B J ⎣ ⎦ ∂ 2 R t ⎡ ⎤ ∂ η g g 1 ( ) = ∇ − + ∇ φ ∇ ×∇Ψ + φ ∇ 2 i i (7) R V V g ⎢ ⎥ ∂ μ 2 2 t R R ⎣ ⎦ 0 ∂Ψ = ( ) − × ∇ + φ η ∇ φ 2 2 R V B i R J i ∂ t ∂Ψ = − η ∇Ψ + Δ Ψ * i (8) V ∂ μ t R − Δ Ψ ≡ ∇ ∇Ψ * 2 2 0 R i