Theoretical Modelling of

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Theoretical Modelling of The Neoclassical
Tearing Mode in Tokamaks (Threshold
Mechanisms) Howard Wilson Department of
Physics, University of York, Heslington, York,
YO10 5DD
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Outline

• Schematic overview of physics
• Drive mechanisms
• Bootstrap current and the neoclassical tearing
  mode
• Threshold mechanisms
• Transport effect
• Polarisation current effect
• Key unresolved issues in understanding
• Analytic calculation
• Existing theoretical formalism, highlighting the
  challenges
• Identification of new physics
• The requirements of numerical simulations
• Summary

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Tearing Mode Theory Ampères Law
d2y/dr2m0J (via Ampères law)
Integrate Ampères law across current layer
Obtained by matching to solution of ideal MHD
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(Modified) Rutherford Equation the basic NTM
drive
?

• Contributions to J
• Ohmic current
• Bootstrap current
• Combine components
• Many islands should grow at all rational
  surfaces
• Theory suggests tokamak confinement should be
  appalling!

Bootstrap term dominates
w
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Cross-field transport provides a threshold for
growth

• In the absence of sources in the vicinity of the
  island, a model transport equation is
• For wider islands, c?gtgtc???? p flattened
• For thinner islands such that c?c???
• pressure gradient sustained
• bootstrap current not perturbed

Thin islands, field lines along symmetry
dn...??0
Wider islands, field lines see radial variations
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Lets put some numbers in (JET-like)
Ls10m c?3m2s1 kq3m1
c1012m2s1
3mm

• There are complications which theory must
  address
• (1) This width is comparable to the orbit width
  of the ions
• (2) It assumes diffusive transport across the
  island, yet the length scales are comparable to
  the diffusion step size
• (3) It assumes a turbulent perpendicular heat
  conductivity, but takes no account of the
  interactions between the island and turbulence
• To understand the threshold, the above three
  issues must be addressed
• a challenging problem, involving interacting
  scales.

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Electrons and ions respond differently to the
island Localised electrostatic potential is
associated with the island

• Electrons are highly mobile, and move rapidly
  along field lines
• electron density is constant on a flux surface
  (neglecting c?)
• For small islands, the E?B velocity dominates
  the ion thermal velocity
• For small islands, the ion flow is provided by
  an electrostatic potential
• this must be constant on a flux surface
  (approximately) to provide quasi-neutrality
• Thus, there is always an electrostatic potential
  associated with a magnetic island (near
  threshold)
• This is required for quasi-neutrality
• It must be determined self-consistently
• resulting E?B flow influences ion profiles
  across small islands

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An additional complication the polarisation
current

• For islands with width ion orbit (banana)
  width
• electrons experience the local electrostatic
  potential
• ions experience an orbit averaged electrostatic
  potential
• the effective E?B drifts are different for the
  two species
• a perpendicular current flows the polarisation
  current
• The polarisation current is not divergence-free,
  and drives a current along the magnetic field
  lines via the electrons
• Thus, the polarisation current influences the
  island evolution
• a quantitative model remains elusive
• if stabilising, provides a threshold island
  width ion banana width (1cm)

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Combining the effects further modifies the
Rutherford Equation

• The modified Rutherford equations becomes

Equilibrium current gradients
Bootstrap current
Inductive current
polarisation current
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The Modified Rutherford Equation summary

• Need to generate seed island
• additional MHD event
• poorly understood

• Stable solution
• saturated island width
• well understood?

w

• Unstable solution
• Threshold
• poorly understood
• needs improved transport model
• need improved polarisation current

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Let us explore the theory on which this is
based Identify weaknesses, and roles for
modelling
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An analytic calculation the essential ingredients

• The drift-kinetic equation
• neglects finite Larmor radius, but retains full
  trapped particle orbits
• We write the ion distribution function in the
  form
• where gi satisfies the equation
• Solved by identifying two small parameters

Lines of constant W
x
q
c
Self-consistent electrostatic potential
Vector potential associated with dB
rbjparticle banana width wisland width rminor
radius
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An analytic calculation the essential ingredients

• The ion drift-kinetic equation

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Order D0 solution

• To O(D0), we have
• The free functions introduce the effect of the
  island geometry, and are determined from
  constraint equations on the O(D) equations

No orbit info, no island info
Orbit info, no island info
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Order D solution

• To O(Dd0), we have
• Average over q coordinate (orbit-averagea bit
  subtle due to trapped ptcles)
• leading order density is a function of perturbed
  flux
• undefined as we have no information on
  cross-field transport
• introduce perturbatively, and average along
  perturbed flux surfaces

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Note solution implies multi-scale interactions

• Solution for gi(0,0) has important implications
• flatten density gradient inside island
  stabilises micro-instabilities
• steepen gradient outside could enhance
  micro-instabilities
• however, consistent electrostatic potential
  implies strongly sheared flow shear, which would
  presumably be stabilising
• An important role for numerical modelling would
  be to
• understand self-consistent interactions between
  island and m-turbulence
• model small-scale islands where transport cannot
  be treated perturbatively

unperturbed
across X-pt
across O-pt
c
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Aside Physics of the separatrix multi-layer

• In the separatrix layer, electron parallel
  dynamics and cross-field diffusion balance
• formalism for collisionless solution developed
  by Hazelitine et al
• work in progress to understand influence on
  particle responses, and hence polarisation
  current (work in progress (M James, et al))

Defines transport layer
O-point
ge(0,0)
Far from island
c/w
Connects to g(0,0) far from separatrix
Collisional layer connects to Maxwellian
Transport layer (solution not Maxwellian)
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Work in progress Magnetic islands influence ITG
stability
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Geometry magnetic islands in a sheared slab

• We adopt a sheared slab geometry with a
  magnetic island
• Consider long, thin islands weak variation of
  equilibrium in y-direction
• Work in island rest frame (ie, an equilibrium
  radial electric field exists)

Magnetic field
xr-rs
Flux quantity
z
y
Electrostatic potential has 3 pieces
ITG pertbn (linearised)
equilibrium
island
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Methodology

• We perturb about a Maxwellian with density and
  temperature functions of x
• Two pieces
• one time independent response to island
  perturbation
• a time dependent response associated with ITG
• use gyro-kinetic theory to derive particle
  responses
• For time-dependent pieces, we Fourier transform
  perturbations
• order kxrikyriltlt1 Kywltlt1, ri/wltlt1
• The equilibrium radial electric field can be
  used to define a drift frequency (normalised to
  the island diamagnetic drift frequency)
• We can also define diamagnetic frequency
  associated with ITG instability

A measure of the island rotation frequency in ExB
rest frame (ie an O(1) parameter)
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Ion response time-dependent perturbations

• After much algebra
• perturbatively treat FLR and the parallel
  dynamics for the time-dependent ion response
• impose quasi-neutrality to derive
• Terms in ad represent flow shear, terms in an
  represent flattening of island
• anad1 for the case with the island
• anad0 for the case with no island

Doppler-shifted freq
Shear flow
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Local Stability Analysis (1)

• Parameter set hi10, t2, rs/w0.2, Ls/Ln-15,
  wE-0.5, b0.12
• Density profile flattened across island
• Doppler shift effect slightly destabilising
• Pressure profile effects substantially stabilise
• ITG mode is most unstable in vicinity of X-point
• Modification to mode structure, and effect on
  stability will influence c?

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Return to Calculation
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Order Dd equation provides another constraint
equation, with important physics

• Averaging this equation over q eliminates many
  terms, and provides an important equation for
  gi(1,0)
• We write
• We solve above equation for Hi(W) and
• yields bootstrap and polarisation current

Provides bootstrap contribution
Provides polarisation contribution
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Different solutions in different collisionality
limits

• Eqn for Hi(W) obtained by averaging along lines
  of constant W to eliminate red terms
• recall, bootstrap current requires collisions at
  some level
• bootstrap current is independent of collision
  frequency regime
• Equation for depends on
  collision frequency
• larger polarisation current in collisional limit
  (by a factor q2/e1/2)
• Construct current perturbation Amperes law ?
  Rutherford eqn
• A kinetic model is required to treat these two
  collisionality regimes self-consistently
• must be able to resolve down to collisional
  time-scales
• or can we develop clever closures?

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Summary of the Issues

• What is the role of transport in determining the
  threshold?
• Is a diffusive model of cross-field transport
  appropriate?
• How do the island and turbulence interact?
• How important is the transport layer around
  the island separatrix?
• What is the role of the polarisation current?
• Finite ion orbit width effects need to be
  included (ie wion banana width)
• Influenced by transport layer at separatrix?
• Need to treat the case v?vE?
• An accurate derivation of the electrostatic
  potential is required for the polarisation
  current (ie quasi-neutrality) it likely
  influences the bootstrap current at small island
  width too
• Glasser term physics.important for STs?
• What provides the initial seed island?
• Experimentally, usually associated with another,
  transient, MHD event
• How do we determine the island propagation
  frequency?
• Depends on dissipative processes (viscosity, etc)