Direct calculation of using eigenvalue perturbation theory S.R.Hudson

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From local to global ray tracing
with grid spacing h
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Alternatively, the eigenvalue derivatives can be
determined directly using perturbation theory
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The direct calculation of the derivatives is
beneficial because . . .

• The rays may be integrated directly
• the data-cube need not be constructed
• the eigenvalue derivatives may be given directly
  to an o.d.e. integrator
• this may be useful if only a few ray trajectories
  are required
• simple to locally refine ray trajectories using
  higher numerical accuracy
• The calculation of the derivatives is consistent
  with the calculation of the eigenvalue
• The derivatives enable a higher order
  interpolation of the data-cube.
• Consider a 2 point interpolation in 1 dimension,

without derivative 2 point interpolation with derivative 2 point interpolation
linear error O(h2) cubic error O(h4)
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For example, consider a tokamak

• A circular cross section tokamak is simple
• there is no ? dependence, minimal Fourier
  harmonics
• note that the ballooning code, interpolation, ray
  tracing etc. is fully 3D
• Shown below are unstable ballooning contours

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In 3D, 4th order interpolation is easily obtained
eigenvalue interpolation error
derivative interpolation error
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The use of the derivatives enables a crude-grid
to give good interpolation
solid exact calculated at 100 radial points
dashed 2-point interpolation
ballooning profile
X grid points
X grid points
radial (VMEC) coordinate
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Construction of data-cube allows eigenvalue
iso-sufaces to be visualized
Another example LHD variant studied by Nakajima
et al. ISW 2005
as eigenvalue is increased, iso-sufaces become
more localized
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Future work possibly includes . . .

• compare results of ray-tracing to global
  stability results
• investigate discrepancy between local and global
  stability limits
• appropriate mass normalization for comparison
  with CAS3D / TERPSICHORE
• include FLR effects / chaotic ray-dynamics as
  studied by MacMillan Dewar