Zonal magnetic field generation and its effect on the stability of the Neoclassical Tearing Mode

1
Zonal magnetic field generation and its effect on
the stability of theNeoclassical Tearing Mode
F. Militello, M. Romanelli, R.J. Hastie, N.F.
Loureiroa EURATOM/UKAEA Fusion Association,
Culham Science Centre, United Kingdom a
Associação Euratom/IST, Centro de Fusão Nuclear,,
Portugal
Work jointly founded by UK EPSRC and EURATOM
2
Introduction and Motivation

• When b is finite the Zonal Flows are associated
  with Zonal Fields1,2.
• The elusive Zonal Fields have been recently
  observed for the first time3 in the CHS
  experiment in Japan.
• Current Corrugations are a consequence of the
  Zonal Fields (through Amperes law).

1 Chen et al. PoP 2001, Guzdar et al. PoP 2001 2
Thyagaraja et al. PoP 2005, Waltz et al. PoP
2006 3 Fujisawa et al. PRL 2007
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Introduction and Motivation

• It is known that small modifications of the
  current profile in the neighborhood of the
  resonant surface can significantly change D' 1
• Experiments in several machines have reported
  NTMs growing without strong MHD precursors
  (sawteeth, fishbones, or edge localized modes)
  which could generate seed islands.
• Whether this could happen also in ITER is a
  relevant problem.
• Current corrugations induced by zonal fields
  could explain the seedless NTMs.

From Gude et al.
1 Zakharov Subbotin 1989, Westerhof NF 1990,
Connor et al. 1992 2Gude et al. NF 1999,
Fredrickson PoP 2002, Buttery et al. NF 2003
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Outline

• How are the Current Corrugations generated ? i.e.
  an interpretation in terms of Parametric
  Instability.
• What happens to the stability of the NTM if the
  current profile is corrugated?
  i.e. effect of the Current Corrugations
  induced by the Zonal fields on D'.

5

• How are the Current Corrugations generated ?

6
Model for the Parametric Instability
Equations

• Normalized three-field reduced MHD model, slab,
  with constant Te and cold ions.
• No dissipation, electron inertia and FLR effects
  present.
• 2 adimensional parameters.
• Valid for small b and large aspect ratio.

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Parametric Instability - Solution

• We linearize the equations and we find the
  dispersion relation of the waves (pump) present
  in the system.
• The pump is assumed to grow to finite size and to
  interact with other waves (background
  turbulence).
• The pump couples with two sidebands in a four
  wave interaction that can destabilize zonal
  perturbations.

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Pump

• Equilibrium constant density gradient, no
  magnetic shear, no flows.
• We take perturbations in eikonal form and we
  linearize the equations
• The dispersion relation becomes

w0/wA
Pump Drift wave
w/wA b½
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Secondary instability

• We assume that the pump reaches a finite
  amplitude and couples with other waves.
• These waves are in the form

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Parametric Instability - Solution

• The problem is solved by finding the eigenvalues
  of a linear system
• A(S)(0)
• where A is a 9x9 matrix that contains the
  amplitude of the pump.
• The result is a dispersion relation
• w/w W(rk -, de/r, w/wA f0/ w, rKx,
  kx/Kx)
• Guzdar et al. (PRL 2001) made a similar
  calculation, with de/r kx/Kx0 (but omitted
  several relevant terms).

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Dispersion Relation and zonal fields
ef0/Te25r/Ln

• New destabilization for b 0.019 (in this
  case)
• dF1V gt dY 10-4 Gcm, dJ/Jeq 0.1 , dn/neq
  0.1

• When b grows, the zonal perturbation is first
  damped and then it grows again
• The new destabilization is associated with strong
  a magnetic component.

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Dispersion Relation and zonal fields
W W(rk -0.25, de/r0, w/wA f0/ w1, rKx,
kx/Kx0)

• New destabilization for b 0.019 (in this
  case)
• dF1V gt dY 10-4 Gcm, dJ/Jeq 0.1 , dn/neq
  0.1

• When b grows, the zonal perturbation is first
  damped and then it grows again
• The new destabilization is associated with strong
  a magnetic component.

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Dispersion Relation and zonal fields
W W(rk -0.25, de/r0, w/wA f0/ w1, rKx,
kx/Kx0)

• Electron inertia increases dy/df at large b,
• Finite kx/Kx shifts the contourplot to smaller
  rKx,
• With magnetic shear the threshold is crossed more
  easily as w/wAk//-1 and k// can be small at
  resonant surfaces

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• What happens to the stability of the NTM if the
  current profile is corrugated?

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General picture

• Questions
• 1) How does dD depend to the features of the
  corrugation?
• 2) What is the effect of these small current
  perturbations on Tearing Mode stability?

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Stability of the Tearing Mode

• D' measures the stability of the tearing mode
• From the linear theory of the tearing mode, D'
  is calculated by integarting JxB0
• D' is a function of (m,n), the global boundary
  conditions and Jeq.

m2,n1
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Modification of D'

• With current corrugations, Newcombs equation
  can be written as

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Calculation of dD'

• An expression for dD' is obtained with simple
  algebra.
• In the next slides, the expression is applied to
  some cases of practical interest.

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Slab model

• We apply the formula obtained to this simple slab
  equilibrium
• We assume that the constant-y approximation is
  valid
• for the matching theory
  to be valid (L is a
  macroscopic scale and l the width of the
  resistive layer)

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Cylindrical case

• We study the same problem in cylindrical
  geometry with numerical tools (with a shooting
  code).

• dD' scales linearly with A
• dD' scales like Kx3
• Maximum amplitude when Rrs

A 5x10-5, Kx 40 A 2.5x10-5, Kx 40 A
5x10-5, Kx 30
m3, n2
m2, n1
rs
rs
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Linear Behavior

• Linear growth rate as a function of the
  corrugation wavelength, for a fixed amplitude of
  the symmetric perturbation (A5x10-5, 1x10-4,
  2x10-4).
• Marks numerical solution of linear problem,
  given a symmetric corrugation of wavelength Kx.
• Solid line Analytical prediction with the
  current curvature contribution and the
  corrugation effect.

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NTM trigger

• We fix all the constants to realistic values and
  we change dD'max and ztzf/th..
• We find three different responses to the
  corrugations
• No island,
• Small island,
• Full NTM.
• In standard operations zltzcrit, but possible
  trigger in peculiar shots. Realistic cases more
  unstable.

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Summary and Implications

• We have described a possible generation mechanism
  for the Zonal Fields as magnetic components of
  the zonal flows and evaluated their features.

• In the standard paradigm turbulence can release
  its energy through the generation of Zonal Flows.

• In the new paradigm turbulence can release
  energy also by generating zonal fields and
  destabilizing Tearing Modes.

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Summary and Implications

• We have described a possible generation mechanism
  for the Zonal Fields as magnetic components of
  the zonal flows and evaluated their features.

• In the standard paradigm turbulence can release
  its energy through the generation of Zonal Flows.

• In the new paradigm turbulence can release
  energy also by generating zonal fields and
  destabilizing Tearing Modes.

25
Summary and Implications

• We have described a possible generation mechanism
  for the Zonal Fields as magnetic components of
  the zonal flows and evaluated their features.

• In the standard paradigm turbulence can release
  its energy through the generation of Zonal Flows.

• In the new paradigm turbulence can release
  energy also by generating zonal fields and
  destabilizing Tearing Modes.

26
Summary and Implications

• We have described a possible generation mechanism
  for the Zonal Fields as magnetic components of
  the zonal flows and evaluated their features.

• In the standard paradigm turbulence can release
  its energy through the generation of Zonal Flows.

• In the new paradigm turbulence can release
  energy also by generating zonal fields and
  destabilizing Tearing Modes.

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Summary and Implications

• We have found a scaling for the maximum change of
  D' as a function of the current corrugation
  features and shown that it is significant
  Militello et al. PoP 2009
• This work shows that the experimental evaluation
  of D' can be extremely difficult and might lead
  to misinterpretation of the observed phenomena.
• Our model provides a viable explanation to the
  puzzling experimental observation of the
  appearance of NTMs in relatively quiescent
  plasmas, where the instability is not preceded by
  a precursor (i.e. the NTM is not triggered by any
  MHD activity).