Computation of Ideal Perturbed Equilibria and its Applications · 2007. 11. 28. · Jong-kyu Park,...

Jong-kyu Park, PPPL, Princeton University, November 16, 2007

Computation of Computation of Ideal Perturbed Ideal Perturbed EquilibriaEquilibria and its Applicationsand its Applications

Jong-kyu Park,Jonathan E. Menard, Allen H. Boozera, Michael J. Schafferb

Princeton Plasma Physics Laboratory, Princeton, NJ08543aDepartment of Applied Physics and Applied Mathematics, Columbia University,

New York, NY10027bGeneral Atomics, San Diego, CA92186

Supported by Office ofScience


I.I. Computation of ideal perturbed equilibria

• Ideal perturbed force balance equation

( ) 0ξ = =∇δ + δ × + ×δidealf p J B J B

( )∇δ = −ξ ⋅∇ − γ ∇ ⋅ξp P P ( )δ =∇× ξ×B B 0( ) /δ =∇×∇× ξ× μJ B

31 ( )2

δ = − ξ ⋅ξ∫ idealW dx fCan be obtained by the minimization of the perturbed potential in ideal MHD

Three-component equations of plasma displacement ξ

ξ ⋅∇ψ• Euler-Lagrange equation for is solved in DCON' † ''( ) ( ) 0ψ ψ ψ ψ⋅Ξ + ⋅Ξ − ⋅Ξ + ⋅Ξ =F K K G

{ }( )ψψΞ ≡ ξ ψmn

(1) (2) (3)

0ψΞ =ˆ, ( ) 0

(1)

(2)

(3)

δ ⋅ =mnContinuous but B n

ˆ ˆ( ) ( )ξ ⋅ δ ⋅b bGiven n or B n

Boundary conditions are


I.II. Boundary conditions at rational surfaces

• Ideal MHD constraint does not allow islands, so produces a singular current to preserve magnetic topology -

⎡ ⎤⎛ ⎞∂ δ ⋅∇ψΔ = ⎢ ⎥⎜ ⎟∂ψ ⋅∇ϕ⎢ ⎥⎝ ⎠⎣ ⎦

mn

mn

BB ( )

( )

32 20

( )/

θ− ϕΔ= δ ψ −ψμ ∇ψ∫

i m nmn

s mnimej B

n dSB ( )ˆδ ⋅ mnB n

Jump of tangential fields Parallel singular current preventing magnetic islands from opening

Total resonant normal field driving magnetic islands

P-S current

Asymtotic region

( ) 0 /δ ⋅∇ψ = =mn

B at q m n


I.III. Boundary condition on the boundary

( )ˆ( , , ), ( , ), ( , , ) ( ), .......ξ ψ θ ϕ ξ ⋅ θ ϕ δ ψ θ ϕ =∇× ξ×i i bn B B ˆ ˆ,δ ⋅ ×δb bB n n B

ˆδ ⋅ bB n ˆ ×δv

bn B

0 ˆ ˆμ = ×δ − ×δv

b bK n B n B

ˆδ ⋅x

bB n0μ =∇×δx

K Bˆˆ ˆ( ) ( )⎡ ⎤δ ⋅ = δ ⋅⎢ ⎥⎣ ⎦

x

b bB n P B n

ˆˆ( ) ⎡ ⎤δ ⋅ = Λ ⎣ ⎦bB n K

ˆˆ( ) ⎡ ⎤δ ⋅ = ⎣ ⎦x

bB n L K

• The surface current gives the external field

• DCON gives M neighboring perturbed equilibria with the associated total field on the plasma boundary

• The normal field is continuous across the boundary and gives the unique vacuum field outside

• If an wall at infinity is taken as a boundary condition, then the jump of the tangential field on the boundary is precisely the representative surface current on the control surface


I.IV. (1) Perturbed normal displacement

#124800 at t=600ms


I.IV. (2) Perturbed normal field

#124800 at t=600ms


I.IV. (3) Perturbed |B|

#124800 at t=600ms


I.V. 3D equilibrium reconstruction in tokamaks

• IPEC can be used for three-dimensional equilibrium reconstruction in tokamakswhen non-axisymmetric external field and current are specified

Measurement

Axisymmetricfield

Non-axisymmetricfield

Intrinsic field +Control field

Equilibrium

Perturbed equilibrium

Island drivingTotal resonant fieldShielding current

Additional control

Perturbed field and displacement Partical orbit

Enhanced transportToroidal torque

LM

RWM

ELM


II.I. LM: Error field correction scheme

• The ith important external field can be defined by ith singular eigenvector of coupling matrix

( ) ( )ˆ ( , ) Re ( ) θ− ϕ⎛ ⎞δ ⋅ θ ϕ = Φ θ⎜ ⎟⎝ ⎠∑

x x i m nb mn

m

B n w e

( )ˆδ ⋅mn

B n

Β = ⋅Φx

ITER

( )ˆ ( )cos ( )sinδ ⋅ = θ φ + θ φx bB n A B


II.II. LM: Error field correction in ITER

• When plasma evolves between possible scenarios, then combined coupling matrix can be used to identify ith important external field

Β = ⋅Φcombined combined x

ITER

• Tolerances guided by the combined coupling matrix are:

( ) 18 3ˆ ( ) (10 )−δ ⋅ ≅critical emnB n Gauss n musing


III.I. RWM: Plasma response to external field

• Plasma response to external field is generally determined by energy and torque required to produce perturbations

( )† †3 14δ = ⋅ = ⋅Φ +Φ ⋅∫W j Adx I I( ) ( )† †3 2ϕ

∂τ = × ⋅ = ⋅Φ −Φ ⋅

∂ϕ∫x nj B dx i I I

Φ = ⋅x

L I

Φ = ⋅Φx

P 1Φ ≈ − Φ− α

x

s i

• Torque parameter can be estimated by measuring amplification of applied field in experiment and by calculating in IPEC, assuming

α1α


III.II. RWM: Torque by neoclassical theory

( ) ( ) ( ) ( )( )δ = ⋅δ = ⋅δ + ξ ⋅∇ δ = ⋅δ δ = ⋅x xL LB B B B B B B B B B B B B B B b

• Torque parameter can be calculated by

( )2 212 21 2, 0 , 0

10

23 /

ϕ φ φ≠ ≠

⎛ ⎞⎜ ⎟μ⎜ ⎟τ = ψ Ω δ + Ω δ⎜ ⎟ψ νπ ν + μ −⎜ ⎟⎝ ⎠

∑ ∑∫ psi ieq nm eq nm nmn m n mTi ii

psTi

dV p pd R C q n B C n B Wd V

m nqV R q

• The variation of field strength is calculated on perturbed flux surfaces

• Torque can not be larger than the maximum, ˆ ˆ( )( )δ ⋅ δ ⋅∫p x

b bR B n B n da

max/α α• Theory and experiment gives different#124801


III.III. RWM: Self-shielding effect

• Anti-correlation between theory and experiment in high density regime indicates plasma self-shielding effect, that is, shielding of external perturbation byamplified viscous force

( )( )

maxexp max 2

max

//

1 /α α

α α =+ α α

the

the

ab


Summary

I. IPEC (Ideal Perturbed Equilibrium Code) has been developed modifying DCON and VACUUM, and constructing interface between -external and total field.This code can be used to reconstruct perturbed equilibria in tokamaks

II. Locked Mode (LM) in tokamaks can be effectively mitigated by resolving ith important mode minimizing total resonant field on rational surfaces.The new approach is applied to ITER

III. Plasma response to external perturbation can be calculated precisely in ideal MHD, and is important to describe Resistive Wall Mode (RWM).Calculation of perturbed energy and torque in theory and experiment shows self-shielding effect of plasma by amplified viscous force

Computation of �Ideal Perturbed Equilibria and its Applications

Computation of Ideal Perturbed Equilibria and its Applications · 2007. 11. 28. · Jong-kyu Park,...

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