UNIVERSITY OF SCIENCE AND TECHNOLOGY OF CHINA

Simulation of Resistive Instabilities in the presence of Runaway Current

Huishan Cai1, Guoyong Fu2

1University of Science and Technology of China 2Princeton Plasma Physics Laboratory

The IEA Workshop on the "Theory and Simulation of Disruptions", July 9-11, 2014, Princeton, NJ

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Outline

u Motivation

u Physical Model

u Simulation Results Ø Linear results Ø Nonlinear results

u Conclusion and discussion

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Motivation

u During the disruption:

Ø Damage of runaway electrons to the first wall

Ø During the thermal quench, temperature drops quickly, resistivity increases correspondingly, plasma current starts to decay.

Ø During the current quench, the current is contributed partly or completely by runaway current.

Ø Runaway current profile is usually more peaked than the pre-disruption current, may induce resistive instabilities.

Ø Resistive instabilities may influence the confinement of runaway electrons.

Motivation

u The generation of runaway current (H. Smith, et.al.,PoP,2006) Ø  Two of the generation mechanisms: Dreicer and Avalanche

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)3(,1 //0

//

tJ

rEr

rr ∂

∂=⎟

⎠

⎞⎜⎝

⎛∂

∂

∂

∂µ

)2(,////// cenEJ re+=σ

)1(,tn

tn

tn II

reIrere

∂

∂+

∂

∂=

∂

∂

Ø Ohm’s Law and Faraday’s Law

Motivation

u The initial and final current profile

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calculated from the above model H. Smith, et.al.,PoP,2006

Physical Model

u Parallel Ohm’s Law

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u Momentum equation

)5(BJPv×+⋅∇−−∇= recpdt

dρ

)4()( //// reJJE −=η

u Runaway density equation

)6(02//

// =⎟⎟⎠

⎞⎜⎜⎝

⎛ ×−×−⋅∇+

∂

∂

Bn

eBvmnvn

tn

ree

rerere Ebκbb

Eq.(6) is derived from the drift kinetic equation, similar to that of Helander,et. al., PoP, 2007

cv |~| //,, 2//// vmnPvenJ erererere =−= is assumed. Due to this assumption,

u Implement the above extended-MHD model in the M3D-k code.

Simulation results

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u As a first step, we consider only the effects of runaway current. The

effects of runaway pressure tensor are neglected.

u For simplicity, the current is assumed to be completely carried by runaway electrons.

u Given the profile of safety factor as

93.0,83.00 =q•  Two cases:

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Linear results

u Case1: q0=0.83 Ø Without runaway current, the growth rate of 1/1 mode is about

2.14e-2, the mode structures are shown below

•  It is the resistive kink mode.

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Linear results

u Case 1:q0=0.83 Ø With runaway current, the growth rate of 1/1 mode is about

2.56e-2, the mode structures are shown below

•  The growth rate is enhanced slightly, and the mode structures keep almost the same.

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Linear results

u The physics: the linearized equations in cylindrical geometry are

( ) ( ) )7(,ˆˆˆ/ 2RJRinqim δδψηγδψφδ −∇=+−− ⊥

)8(,ˆˆ 20//2 δψδψφδγ ⊥⊥ ∇⎟

⎠

⎞⎜⎝

⎛ −+−=∇ qnm

qin

drJd

rimR

( ) )9(,0ˆ

/ =−− δψδdrJd

rmJnqm R

R

where runaway current is a flux function approximately, due to .|~| // cv

RlRl JJwOJwOJ δδδψδδψδ >→ //2

// )/(~),/(~Order analysis: The perturbation of runaway current is of higher order. The characteristic functions in the resistive layer without runaway are

( ) ( ) ( )( )[ ]2/exp//2/ lll wxwwxxerfcb −−= πδψ

( )( ) ( )ll wxerfcwbi /2/2/=δφ

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Linear results

,ˆ

xdrJd

sqJ Rs

Rδψ

δ −= ,0ˆ<

drJd R

//// JJJJ Rohm δδδδ >>−>=<<

The perturbation of runaway current tends to enhance the perturbation of Ohm’ current, so that it will increase the growth rate of the resistive kink mode.

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Linear results

Ø With runaway current, without electric drift contribution in runaway equation, the growth rate is 2.45e-2, mode structures are

•  The contribution of electric drift in runaway equation is small,

.1/ >>Avcdue to

Linear results

Ø With runaway current, without magnetic drift contribution in runaway equation, the growth rate is 2.6e-2, mode structures are

•  The effect of magnetic drift velocity is small due to .1/ >>ΔΔ layerb

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Linear results

u Case2: q0=0.93 Ø Without runaway current, the growth rate of 1/1 mode is about

9.3e-3, the mode structures are

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Linear results

u Case 2:q0=0.93 Ø With runaway current, the growth rate of 1/1 mode is about

1.72e-2, the mode structures are

•  The growth rate is enhanced, and the mode structures keep almost the same.

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Nonlinear results

u The usual sawteeth are observed in the simulation.

u Case1:q0=0.83, without runaway current

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Nonlinear results

u Resistive kink mode results a single sawtooth crash only. u A steady state with axi-symmetric equilibrium is reached after the crash. u The plasma current and runaway current become flatten within q=1 surface, q0 greater than 1. Physics: sawtooth crash tends to flatten current within q=1 surface. After the crash, the runaway current, acting as a current source, stays flatten.

u Case1:q0=0.83, with runaway current

Simulation results

•  The evolution of current

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Nonlinear results

u Case1:q0=0.83,without electric drift,

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Nonlinear results

u Case1:q0=0.83, without magnetic drift

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Nonlinear results

u Case2:q0=0.93.

Ø The same trend with q0=0.83.

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Conclusion and discussion

u The linear growth rate of 1/1 resistive kink mode is enhanced by the effects of runaway current.

u After a single sawtooth crash, plasma reaches a new steady state with axi-symmetric equilibrium.

u The profiles of plasma current and runaway current are flatten within q=1 surface.

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Thank you for your attention!