Nonlinear interactions between micro-turbulence and macro-scale MHD

1
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
MHD activities are observed in reversed shear
plasmas with a transport barrier related to zonal
flows and micro-turbulence.
Time evolution of local electron temperature
Takeji, et.al., Nuclear Fusion (2002)
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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.

Magnetic surfaces
tearing m2
kink m1
KBM mgt10
Electric potential
ITG mgt10
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Basic equations
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Initial equilibrium and linear growth rate
We examine the multi-scale nonlinear interaction
among instabilities in a reversed shear plasma.
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Linear instabilitiesmicro-instabilities and
double-tearing mode
n1
n9
n14
Macro-scale MHD Double tearing mode
Micro-instability
Micro-instability

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.

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Zonal flow
t63
Zonal flow
total
n1
t63
Twisted micro-instability
Twisted double-tearing mode
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.
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Nonlinear evolution of electric potential
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Details of nonlinear evolution I
Zonal flow
Magnetic energy
1
total
Toroidal mode number n
n1
t63
t63
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
m3 double tearing mode appears.
1
2
3
Tearing mode dominates
Alteration of equilibrium magnetic field
Nonlinear growth of tearing mode
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Details of nonlinear evolution II
The tearing mode affects the micro-turbulence by
breaking the magnetic surfaces
2
Magnetic energy
total
n1
Toroidal mode number n
t99
t99
Since the tearing mode breaks the magnetic
surfaces through magnetic reconnections, the
dominance of n1 tearing mode results in an
alteration of the equilibrium magnetic field.
1
2
3
Tearing mode dominates
Alteration of equilibrium magnetic field
Nonlinear growth of tearing mode
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Details of nonlinear evolution III
3
total
n1
Magnetic energy
Toroidal mode number n
t108
t108
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 ngt1.
1
2
3
Tearing mode dominates
Alteration of equilibrium magnetic field
Nonlinear growth of tearing mode
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Initial equilibrium and linear growth
ratebeta1.5
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Linear analysis beta1.5Double-tearing modes
and micro-instabilities
n10
n14
n1
Macro-scale MHD Double tearing mode
Micro-instability
Micro-instability
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Nonlinear evolution of electric potential
Beta1.5, 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.

Macro-MHD n1 (tearing modes)
Linear instabilities
Micro-turbulence ngtgt1
3. destabilize
2.stabilize
1.destabilize
2. stabilize
n toroidal mode number
Zonal flow n0
4. destabilize
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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
Kinetic energy
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Electro-static (beta0) and back ground profiles
are fixed
Time evolution of zonal flow
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Single helicity double tearing background
profile relax
t96
n1
/home/ishizawa/fivefieldVer8/reversed/singleHelici
ty