Nonlinear interactions between micro-turbulence and macro-scale MHD

A. Ishizawa, N. Nakajima, M. Okamoto, J. Ramos*

National Institute for Fusion Science

*Massachusetts Institute of Technology

US/Japan JIFT Workshop “Issues in the theoretical analysis of three dimensional configuration” Princeton, March 14-16, 2006

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Our goal• Effects of MHD instabilities and micro-

turbulence on plasma confinement have been investigated separately.

• But these instabilities usually appear in the plasma at the same time.

• Our goal is to understand multi-scale-nonlinear interactions among micro-instabilities, macro-scale-MHD instabilities and zonal flows.

3

Introduction

Time evolution of local electron temperature

Takeji, et.al., Nuclear Fusion (2002)

MHD activities are observed in reversed shear plasmas with a transport barrier related to zonal flows and micro-turbulence.

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Motivation and results• The appearance of the macro-scale MHD

instability, which leads to the disruption, can be affected by micro-turbulence and zonal flows.

• We investigate multi-scale-nonlinear interactions among micro-instabilities, macro-scale tearing instabilities and zonal flows, by solving reduced two-fluid equations numerically.

We find that the nonlinear interactions of these instabilities lead to an alteration of macro-equilibrium magnetic field, then this alteration spreads the micro-turbulence over the plasma.

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Reduced two-fluid equations• We carry out three-dimensional simulations with a

reduced set of two-fluid equations that extends the standard four-field model, by including temperature gradient effects.

• Basic assumptions– Flute approximation– Large aspect ratio– High-beta ordering

• By solving this set of equations, we can describe the nonlinear evolution of tearing modes, interchange modes, ballooning modes and ion-temperature gradient modes.

R

a

k

k//r

Magnetic surfaces

kink m=1 tearing m=2 ITG m>10 KBM m>10 Electric potential

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Basic equations

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7

Initial equilibrium and linear growth rate

ir /

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10i

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n

T

Double tearing mode

Toroidal mode number: n

Gro

wth

rat

e

Micro-instabilities

256128256 rnm NNN80/1~01.0

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We examine the multi-scale nonlinear interaction among instabilities in a reversed shear plasma.

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4-12

104

102

S

m

8

Linear instabilities:micro-instabilities and double-tearing mode

　 n=1

1. In the linear phase, a ballooning structure of micro-instability appears in the bad curvature region. The structure is twisted by ion and electron diamagnetic effects.

2. A double-tearing mode is also unstable, but its growth rate is small compared to that of the micro-instability.

n=9 n=14

Micro-instability Micro-instabilityMacro-scale MHDDouble tearing mode

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Zonal flow t=63Zonal flow total n=1 t=63

linear

The zonal flow induced by the micro-instability has a stabilizing effect on the micro-instability and on the tearing mode by twisting their radial structure.

Twisted micro-instability Twisted double-tearing mode

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Nonlinear evolution of electric potential

11

Details of nonlinear evolution IMagnetic energy

atvti /

Tearing mode dominates

Alteration of equilibrium magnetic field

Nonlinear growth of tearing mode

))(log( nEM

1 2 3

Toroidal mode number: n

t=63

1

Zonal flow

total n=1

The nonlinear mode coupling is so strong that it overcomes the stabilizing effect due to the zonal flow, and the tearing mode growth rate is enhanced by the nonlinear-mode-coupling. Thus an m=3 double tearing mode appears.

t=63

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Details of nonlinear evolution II

Magnetic energy

atvti /

Tearing mode dominates

Alteration of equilibrium magnetic field

Nonlinear growth of tearing mode

))(log( nEM

1 2 3

Toroidal mode number: n

t=99

2

The tearing mode affects the micro-turbulence by breaking the magnetic surfaces

Since the tearing mode breaks the magnetic surfaces through magnetic reconnections, the dominance of n=1 tearing mode results in an alteration of the equilibrium magnetic field.

total n=1 t=99

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Details of nonlinear evolution IIIMagnetic energy

atvti /

Tearing mode dominates

Alteration of equilibrium magnetic field

Nonlinear growth of tearing mode

))(log( nEM

1 2 3

Toroidal mode number: n t=108

3

The alteration spreads the micro-turbulence over the plasma after , and it also increases the energy of the turbulence as indicated by the traces with n>1.

100t

total n=1 t=108

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Initial equilibrium and linear growth ratebeta=1.5%

ir /

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n

T

Double tearing mode

Toroidal mode number: n

Gro

wth

rat

e Micro-instabilities

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412-

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80/1~015.0

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15

Linear analysis: beta=1.5%Double-tearing modes and micro-instabilities

n=10 n=14　 n=1

Macro-scale MHDDouble tearing mode

Micro-instability Micro-instability

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Nonlinear evolution of electric potential

Beta=1.5%, eta3

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Summary• We have found that the multi-scale nonlinear interactions

among micro-turbulence, tearing modes, and zonal flows lead to an alteration of the macro-magnetic field, then this alteration spreads the turbulence over the plasma.– The mechanism of the spreading is as follows. The micro instability

induces zonal flows which attempt to suppress the tearing mode. However, the nonlinear-mode-coupling due to the micro-instability overcomes this suppression and accelerates the growth of the tearing mode. This tearing mode alters the macro-equilibrium magnetic field by breaking the magnetic surfaces, and thus the tearing mode spreads the micro-turbulence over the plasma.

n : toroidal mode number

2. stabilize2.stabilize1.destabilize

3. destabilize

4. destabilize

Zonal flow n=0

Macro-MHD n=1(tearing modes)

Linear instabilities

Micro-turbulence n>>1

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Future plan• Include the effects of a radial electric field

in the initial equilibrium– The present simulation adopted an initial

static equilibrium without radial electric field.

• The importance of the choice of initial perturbation– The present simulation is based on a linear

mode initial condition.

• Solve a set of reduced two-fluid equations derived by Prof. Ramos numerically

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Zonal flow

atvti /

atvti /

Kinetic energy

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Electro-static (beta=0) and back ground profiles are fixed

Time evolution of zonal flow

ar /

tivta /

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Single helicity: double tearing: background profile relax

t=96

n=1

tivta / /home/ishizawa/fivefieldVer8/reversed/singleHelicity