Chaotic Transport in ReverseShear Tokamaks

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Chaotic Transport in Reverse-Shear Tokamaks

• F. A. Marcus, E. C. Silva, Iberê L. Caldas
  (University of São Paulo)
• T. Kroetz, M. Roberto (Aeronautic Technological
  Institute-CTA)
• R. L. Viana (Federal University of Paraná )
• Workshop of Stochasticity in Fusion Plasmas
• Jülich, 2007

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I - Drift Wave Driven Transport
Chaotic trajectories caused by the particle
drift, due to given poloidal electrostatic waves
and the toroidal magnetic field.
Guiding center drift velocity
Fluctuating potential
Hamiltonian
Marcus, Ph. D. thesis, IFUSP. Horton, PPCF (1985)

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Drift waves in y (poloidal) direction due to
gradient density in x (radial) direction. Uniform
magnetic field, B, in z (toroidal) direction.

Monotonic Flow Profile Two wave Hamiltonian in
dimensionless variables
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Ion Guiding Center Trajectories for Two Waves

A2 /A1 0.1
A2 /A1 0.8
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Radial Diffusion Coefficient
Values comparable to the measured diffusion
coefficient
Diffusion coefficient as a function of
the confinement parameter U.
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Reverse-Shear Flow Profile
Two wave Hamiltonian in dimensionless variables.
Reference frame moving with the phase velocity
Phase difference between two waves
Confinement parameter
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Ion Guiding Center Trajectories for Two Waves r
x ?
Confinement parameter radial profile (U x r)
inside the plasma
A2 /A1 0
A2 /A1 0.2
Highest transport at U 0 r 0.8 (plasma edge)
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• II- ESCAPE PATERN
• In tokamaks, nonmonotonic plasma profile reduces
  transport. However, plasma wall interaction may
  increase with the amplitude of magnetic resonant
  modes and damage the tokamak wall.
• Previous works
• Nontwist maps in fusion Balescu, Misguich,
  Petrisor, Wingen
• Escape pattern Evans, Finken
• Transport Spatchek, Abdullaev

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Procedure and Results

• MHD plasma equilibria with nonmonotonic current
  profile (with a shearless region).
• Resonant perturbations caused by ergodic magnetic
  limiters.
• Nontwist symplectic maps to describe perturbed
  field lines.
• Calculation of escape pattern and footprints of
  magnetic field lines.
• Roberto, Silva, Caldas, Viana, Phys. Plasmas,
  (2004, 2006) Portela, Caldas, Viana, Morrison,
  IJBC (2007)

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Equilibrium Magnetic Field Lines in a Tokamak
Lines describe toroidal (?) and poloidal (?)
angles, on toroidal magnetic surfaces.




Integrable field
Field line equation
Frequency
t (canonical time) ? ? (toroidal angle)
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Nonmonotonic Equilibria
d?\dI dq\dI 0 at Q 4
Ergodic limiter creates two resonances at Q 4
Radial safety factor profile for the MHD
equilibria
Coil current Ih . Dominant m/n resonant
perturbation in toroidal geometry (Silva et
al., IEEE Trans. Plasma Science)
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Poincaré Map m/n 4/1 Ih / Ip 8.57 .
Correlation lengths for a grid of initial
conditions
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• Poincaré Map m/n 5/1
• Ih / Ip 8.57

Correlation Lengths for a Grid of Initial
Conditions
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Footprints Correlation Length vs Scape Angle
m/n 4/1
m/n 5/1
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Conclusions

• Turbulence induced transport, observed in
  tokamaks, can be estimated by considering the
  chaotic trajectories caused by the particle drift
  due to given poloidal electrostatic waves and the
  toroidal magnetic field.
• Perturbed magnetic configuration described by
  nontwist symplectic maps.
• For some high amplitude resonances, magnetic
  lines with long correlation lengths reach the
  wall in concentrated footprints.