Rep

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Multi-Scale Interaction of Magnetic Islands and
Micro-Turbulence in Tokamaks
M. Muraglia, O. Agullo, S. Benkadda, P. BeyerLIA
FJ-Magnetic Fusion Lab/LPIIM CNRS - Université de
Provence, Marseille, France X.
Garbet Association Euratom-CEA, CEA/DSM/DRFCCEA
Cadarache, France
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Posters

• Nonlinear visco-resistive dynamics of the Harris
  current sheet
• K. Takeda, O. Agullo et al
• Nonlinear dynamics of multiple NTMs in tokamaks
• D. Chandra et al
• Nonlinear dynamics of magnetic islands embedded
• in micro-turbulence
• M. Muraglia et al

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Outline
1. Introduction and motivation 2. Model
derivation p, ?, ? 3. Destabilisation of the
Tearing saturated state 4. Effect of a large
viscosity 5. Conclusion
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1. Introduction and Motivation

• The effects of MHD instabilities and
  micro-turbulence on plasma
• confinement have been investigated separately.
• However these instabilities usually appear in
  the plasma
• at the same time.
• - Micro-turbulence is observed in Large Helical
  Device plasmas
• that usually exhibit MHD activities.
• K. Tanaka, et al., Nuclear Fusion (2006)
• - MHD activities are observed in reversed shear
  plasmas with a
• transport barrier related to zonal flows and
  micro-turbulence.
• Takeji, et al., Nuclear Fusion (2002)
• In the present work, we study the interaction
  between Tearing
• Modes and a pressure gradient instability
  (Interchanges like instability).

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1. Introduction and Motivation
produce
- Large scale flows - Zonal flows
Micro-turbulence
stabilize
?
Island poloidal rotation ?
Macro-MHD
Goal Focus on the nonlinear multi-scale
interaction of the fields and get some insight on
the origin of island poloidal rotation .
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2. Model derivation - Equilibrium

• The equilibrium pressure field is given by
• The equilibrium magnetic field is given by

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2. Model derivation - Equations

• Reduced MHD equations for electrostatic
  potential ?, pressure p, and magnetic flux ???
• Model takes into account both Tearing Mode and
  Interchange in slab geometry (2D).

Tearing Normalisation
Parameters
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2. Model derivation - Assumptions
1. gt Diamagnetic effect
is suppressed gt C4 0. 2. C0 acts only on the
instability level of the interchange, we suppose
C0 0.
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2. Model derivation - Assumptions
1. gt Diamagnetic effect
is suppressed gt C4 0. 2. C0 acts only on the
instability level of the interchange, we suppose
C0 0.
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3. Destabilisation of the Tearing saturated state

• Parameters Values

• Energies

• Fields

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3. Destabilisation of the Tearing saturated state

• Parameters Values

• Energies

• Fields

Linear regime
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3. Destabilisation of the Tearing saturated state

• Parameters Values

• Energies

• Fields

First Saturated State
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3. Destabilisation of the Tearing saturated state

• Parameters Values

• Energies

• Fields

Transition to Turbulence
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3. Destabilisation of the Tearing saturated state

• Parameters Values

• Energies

• Fields

Second Saturated State
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Nonlinear Simulation
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3. Destabilisation of the Tearing saturated
state Linear regime

• Linear formation of a magnetic island.
• Linear growth rate
• for ky 1.
• However
• Pressure energy higher than
• the kinetic energy.
• Quadrupole structure appears on pressure.
• Tearing instability driven
• by a pressure gradient

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3. Destabilisation of the Tearing saturated
state First saturated state

• Growth of the magnetic island stopped gt
  classical saturated state.
• Rutherfords like regime
• E p gtgt E??
• ???An interplay between p, ?,
• and not ?, ?, controls the
• dynamics of the saturated
• state.

Variation of the derivative of the magnetic
island size
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3. Destabilisation of the Tearing saturated
state Transition to turbulence

• Just before the transition, large-scales Tearing
  structure (ky 1) contains most of the energy.

K-0.7
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3. Destabilisation of the Tearing saturated
state Transition to turbulence

• Persistence of large-scales tearing structures
  (ky 1) .
• After the transition generation of turbulent
  small-scales with strong interaction between
  pressure and magnetic flux.
• gt Transition to a new
• saturated state.
• Magnetic island is strongly
• destabilised and a poloidal
• flow appears.

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3. Destabilisation of the Tearing saturated
state Transition to turbulence

• Persistence of large-scales tearing structures
  (ky 1) .
• After the transition generation of turbulent
  small-scales with strong interaction between
  pressure and magnetic flux.
• gt Transition to a new
• saturated state.
• Magnetic island is strongly
• destabilised and a poloidal
• flow appears.

Variation of the derivative of the magnetic
island size
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3. Destabilisation of the Tearing saturated
state Second saturated state

• Strong pressure depletion
• in the current sheet.
• Apparition of small-scales
• structure outside the
• island.

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Second Saturated State
3. Destabilisation of the Tearing saturated
state Second saturated state

• Energy peaks.
• Pressure oscillates inside
• the magnetic structures
• (O point X point).
• Outside the island,
• turbulence is present.
• Pressure Shear

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Poloidal Flow
3. Destabilisation of the Tearing saturated
state Island poloidal rotation
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Nonlinear Simulation
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Origin of the Island Poloidal Rotation
3. Destabilisation of the Tearing saturated state
Island poloidal rotation

• At the beginning of the transition, we can
  observe that a poloidal flow appears with Vmean
  10-2 Va.
• What is the origin of this flow ?

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Origin of the Island Poloidal Rotation
3. Destabilisation of the Tearing saturated state
Origin of the island poloidal rotation
Origin of Vyp nonlinear electrostatic term or
nonlinear magnetic term
Origin of Vy ? nonlinear electrostatic (Renolds
stress) term or nonlinear
magnetic term (Maxwell stress)
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Origin of the Poloidal Flow
3. Destabilisation of the Tearing saturated state
Origin of the poloidal flow
0.06
0.6

• Poloidal flow generated by the pressure (V0p )
  is more important than the one generated by the
  electrostatic potential (V0 ? ).

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Pressure Poloïdal Flow
3. Destabilisation of the Tearing saturated
state Poloidal pressure velocity
610-5
1.510-18

• In the pressure equation, the nonlinear
  magnetic term is more important than the
  nonlinear electrostatic term.
• The poloidal pressure flow is driven by the
  nonlinear magnetic term.

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Electric Poloïdal Flow
3. Destabilisation of the Tearing saturated
state Poloidal electrostatic velocity
Reynolds Stress
Maxwell Stress
110-5
310-4
Maxwell stress higher than Reynolds stress gt NO
STRONG ZONAL FLOW
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Viscosity Effect
4. Effect of a large viscosity
increase.
Four regimes - Linear regime, - First
Saturated state - Transition to a new saturated
state - Satured ocillations regime (predator
prey regime)
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Viscosity Effect Evolution of Magnetic Island
4. Effect of a large viscosity Evolution of
magnetic island size

• Linear formation of magnetic island.
• Linear growth rate
• for ky 1.
• Rutherfords like regime.
• After the transition tilting-like
  instability .
• Violent reconnection phenomena at the peaks.

Variation of the derivative of the magnetic
island size
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Viscosity Effect Turbulent Regime ?
4. Effect of a large viscosity Turbulent regime
?

• Spectra shows that after the transition, the
  small-scales are not generated.
• Pressure and electrostatic potentiel macro
  convective cells.
• gt Turbulent small-scales are not generated

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Viscosity Effect - Stresses
4. Effect of a large viscosity Stresses
Reynolds Stress
Maxwell Stress
710-3
0.02
Reynolds stress higher than Maxwell stress gt
STRONG ZONAL FLOW
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Summary

• We studied the interaction between magnetic
  island generated by Tearing
• instability and an interchange type
  micro-turbulence.
• If the coupling parameter between pressure and
  magnetic flux is strong
• - the Tearing mode is driven by a pressure
  gradient,
• - an interplay between p, ? controls the
  dynamics of the saturated state,
• - after a first saturated state, a second
  instability destabilizes the magnetic
• island and produces an island poloidal
  rotation.
• For the case where
• - the destabilisation of the Tearing steady
  state leeds to the generation of
• turbulent small-scales,
• - the pressure poloidal flow is driven by the
  nonlinear magnetic term,
• - we dont observe a strong Zonal Flow.
• The increase of viscocity stops the generation
  of turbulent small-scales.
• In this case, the Zonal Flow dominates and allows
  an island poloidal rotation.

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Small coupling between pressure and magnetic flux
Small coupling No transition and second
saturated State regimes.
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A Numerical Simulation

• The code has been benchmarked for 256² modes for
  Tearing
• and Interchange.
• And Porcelli s results for visco-resistive
  Tearing have been recovered.
• (K. Takeda et al)
• Aliasing problems have been solved.
• The nonlinear terms are calculated thanks to the
  Arakawas method.