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Simulation of Nonlinear Interactions between n=1 Internal Kink and High-n Pressure-driven MHD Instabilities

Y. Kagei1, Y. Kishimoto2,1, and T. Miyoshi3

1 Naka Fusion Institute, Japan Atomic Energy Agency, Japan2 Graduate School of Energy Science, Kyoto University, Japan

3 Graduate School of Science, Hiroshima University, Japan

10th Active Control of MHD Stability Workshop 31th October – 2nd November, 2005

University of Wisconsin-Madison, Wisconsin USA

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Outline

1. IntroductionDevelopment of fully compressible nonlinear MHD simulation code under the NEXT (Numerical EXperiment of Tokamak) Project in JAEA

2. Objective

3. Modeling

4. Results

5. Summary

Our first simulation results showing that m/n=1/1 internal kink mode have an influence on pressure-driven high-m/n modes

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Introduction

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T. Miyoshi and NEXT Group, JAERI-Research 2000-023 (2000).

Discretization :

Satisfy divergence free condition

Grid : (I)Structured rectangular grids,(II)Orthogonal flux coordinate grids, and (III)Unstructured grids

Basic equations :

Time integration : (I)Explicit (Runge-Kutta) or (II)Semi-implicit scheme

* The development of the unstructured code is still underway.

Fully compressible nonlinear MHD Code development under the NEXT Project in JAEA

Fully-compressible, single-fluid MHD

Finite-volume(Poloidal)/Pseudo-spectral(toroidal) method• • •

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Objective

Understanding nonlinear interactions between various unstable modes which differ in their scales and their sources of instabilities

Importance of understanding saturation mechanisms of their modes in such a plasma

e.g. Current-driven 1/1 kink mode and Pressure-driven high-m/n modes

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•

5Simulation Model (1)

Rectangular plane : Quadrilaterals, (NR, NZ)=(256,256)

Toroidal direction:Pseudo-spectral representation (n)from -32 to +31

Basic equations :

A=3 R

Z

Finite-volume(finite-differential)/spectral discretization of 3-D fully compressible nonlinear MHD equations

Boundary :No-slipping and perfect conducting wall

An axisymetric relaxed state obtained by a 2-D simulation with the resistivity η=0, but with a nonzero viscosity ν

Initial Configuration :

*) The finite volume method on the structured grid, such as this work, corresponds to the finite difference method in the generalized coordinate system.

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∂∂ t

g P =− ∂∂ x i

g P V i −−1P ∂∂ x i

g V i −1 g g ij J i J j

∂∂ t

g B i =− ∂∂ x j

E k−∂

∂ xkE j

g J i = ∂∂ x j

Bk− ∂∂ xk

B j

E i =− g V j Bk−V k B j J i

▪ Momentum conservation

▪ Energy conservation

▪ Faraday's law of induction

Simulation Model (2)

▪ Ampere's law

▪ Ohm's law

▪ Mass conservation

g ≡ g R×gZ ⋅g▪ Control volume

∂∂ t g =− ∂

∂ x i g V i

∂∂ t g V =− ∂

∂ x i g V V i

− ∂∂ x i

g P g i g J jBk− J k B j g i ∂∂ xi g g ij ∂

∂ x jV

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Resistivity

Viscosity

Thermal Conductivity

η =1x10-4

ν =1x10-4

κ =1x10-3

Internal kink mode (m/n=1/1)Pressure-driven mode (high m/n)

Initial state dependence of the linear growth rate

A high beta tokamak plasma with q0<1 linearly unstable to the resistive internal kink mode(n=1) and to the resistive pressure-driven modes(Peaked at n=12)

•

•

0

1

2

3

4

2 3 4

βp=1.0

βp=0.7

βp=0.01

q

R

q0=0.81

Growth rates about comparable to each other at βp~0.7 ⇒ Nonlinear simulation for the parameter value are conducted

0

0.01

2 3 4

βp=1.0

βp=0.7

βp=0.01

P

R

0

0.01

0.02

0 2 4 6 8 10 12 14 16

γ

n

βp=1.0

βp=0.7

βp=0.01

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βp=0.7βp=0.01

Nonlinear evolution of instabilities

The growth of n>1 modes are nonlinearly excited by the n=1 internal kink

βp=0.01 :

βp=0.7 :

10-27

10-24

10-21

10-18

10-15

10-12

10-9

10-6

10-3

0 500 1000 1500

n=1n=2n=3n=4n=5n=6n=7n=8n=9

n=10n=11n=12n=13n=14n=15

mag

netic

ene

rgy

t / τA

10-27

10-24

10-21

10-18

10-15

10-12

10-9

10-6

10-3

0 500 1000 1500 2000 2500

n=1n=2n=3n=4n=5n=6n=7n=8n=9

n=10n=11n=12n=13n=14n=15

mag

netic

ene

rgy

t / τA

n=1 and n>1 modes are linearly unstable

Some of the mode with n>1 are accelerated in the early nonlinear regime

n>1 modes appears to saturate at last

early nonlinearstage

Linear stage

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10-27

10-24

10-21

10-18

10-15

10-12

10-9

10-6

10-3

0 500 1000 1500 2000 2500

n=1n=2n=3n=4n=5

mag

net

ic e

ner

gy

t / τA

Nonlinear evolution of instabilities

Linear

Non-linear

t /τA=1500max=1.7e-10

t /τA=2100max=1.7e-8

t /τA=1500max=7.6e-8

t /τA=2100max=7.7e-6

t /τA=2300max=3.7e-5

t /τA=2300max=6.6e-7

βp=0.7

• On a q=1 rational surface with the same helicity

Pn=1

Pn=5

Pn=1

Pn=5

Pn=1

Pn=5

n>1 mode in the early nonlinear regime (t/τA~2000-2500)

the same growth rate

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10

10-27

10-24

10-21

10-18

10-15

10-12

10-9

10-6

10-3

0 500 1000 1500 2000 2500

n=20n=21n=22n=23n=24

mag

net

ic e

ner

gy

t / τA

t /τA=1500max=8.2e-11 max=2.1e-8

t /τA=2300max=4.8e-8

Nonlinear evolution of instabilities

Linear

t /τA=1500max=9.8e-9

t /τA=2200max=2.0e-5

t /τA=2300max=5.9e-5

Non-linear

10-27

10-24

10-21

10-18

10-15

10-12

10-9

10-6

10-3

0 500 1000 1500 2000 2500

n=10n=11n=12n=13n=14

mag

net

ic e

ner

gy

t / τA

t /τA=2200

Linear

• A Similar behavior to that observed for 2 ≤ n ≤ 5 (as shown in the previous viewgraph) can be observed for other n, excepted around the fastest growing mode n=12.

n=12

Pn=20

Pn=12

Pn=20

Pn=12

Pn=20

∂∂ t g V =− ∂

∂ xi gV V i

− ∂∂ xi

g P g ig J jBk− J k B j g i ∂∂ xi g g ij ∂

∂x jV

11 Nonlinear saturation of high-n instabilities

Fully nonlinear regime (t/τA>2500)

Pn=12

max=7.3e-4t /τA=2700

max=4.1e-4t /τA=2640

Pn=12

10-15

10-12

10-9

10-6

10-3

2200 2300 2400 2500 2600 2700

mag

net

ic e

ner

gy

t / τA

n=1

n=12

n=5

n=20

t /τA=2300max=5.9e-5

n=12P

The first phase reversal event occurs on the q=1 rational surface

The amplitude of reversed phase increases with time, and after that, the next event occurs

(t/τA~2640)

At last, a coupling structure between m and m±1 becomes different from that in early nonlinear regime (t/τA~2700)

Several phase reversal events & Saturation of high-n modes

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m-2/n-1 m/n-1

m-1/n m/n m+1/n

m/n+1 m+1/n+1 m+2/n+1

・・・・・・・・・・

・・・

・・・

Poloidal coupling of the pressure-driven mode

Excited by nonlinear coupling effect

⇒ Early nonlinear regime :An excitation of the high-m/n helical modes by the nonlinear coupling effect (yellow) ⇒

The growth of the internal kink mode (1/1) ⇒

⇒ Fully nonlinear regime :A change of an original coupling structure of pressure-driven high-m/n modes (blue)

・・・・・

1/1

m-1/n-1

Lattice model of nonlinear mode coupling

The change of the coupling structure may be related to the saturation of high-m/n modes

・・・

Linear kink mode

13 Time evolution of pressure profiles

t /τA=2200

t /τA=2500

t /τA=2660

t /τA=2200

t /τA=2500

t /τA=2660

ϕ=0 (ϕ is toroidal angle) ϕ=π

R RThe phase reversal events prevent fingers from growing in the bad curvature region

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14Helical finger structure in saturated state

Isosurface of the pressure at t=2660

• Helically distorted fingers are formed accompanied by m/n=1/1 kink motion

• More detailed is under investigation. Consistency of the results will be checked in the future.

• The phenomena is as if the plasma transfers fingers from bad- to good- curvature region

15Summary

Development of a 3-D fully-compressible nonliner MHD simulation code based on the finite-volume/spectral scheme

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• A nonlinear simulation of MHD instabilities of a high beta plasma unstable to pressure driven modes (high-n) whose growth rates are about comparable to resistive internal kink mode (low-n)

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1) Appearance of a coupled modes with the same helicity around a q=1 rational surface in the early nonlinear regime

2) A change of the poloidal mode coupling structure and a saturation of high-m/n modes in the fully nonlinear regime

Findings

• Consideration using a lattice model

Prevention of fingers from growing at bad curvature regionFormation of a helical finger structure

Consistency will be checked and then more detailed will be investigated in the future.