Direct calculation of using eigenvalue perturbation theory SR by malj

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									   Direct calculation of , and ,    k (where    2 is the ballooning growth rate), using eigenvalue perturbation theory, and the use of the derivatives to improve interpolation on a grid.

S.R.Hudson Stellarator Theory Teleconference 9/8/05

From local to global : ray tracing
 Global ballooning analysis requires the quantization of the ray trajectories,      ,    k , k  k   k  k    However, most ballooning codes calculate  , and not the derivatives.     To calculate , , , a data cube i , j , k   ( i ,  j , kk ) is constructed,    k  The derivatives are then approximated:  This has the drawbacks :  i 1, j , k  i , j ,k  , ..  h
with grid spacing h

- significant computational overhead before any ray tracing can proceed - loss of accuracy on taking finite-difference approximation to derivatives

Alternatively, the eigenvalue derivatives can be determined directly using perturbation theory
   The ballooning equation is P( ,  , k )   Q( ,  , k )   R( , , k ) .    Consider a small change    + ,    + ,  k   k + k ,  The ballooning coefficients change  P   P    P    P k  k ,  Q  . . .     the change in the eigenvalue is given by       P     Q      R   d     R   d  similar to the variational refinement performed by COBRA  the perturbed functions  P,  Q,  R are obtained by differentiating P, Q and R  a increase  33% in cpu is involved due to the extra Fourier summations required
( note : most of the cpu in ballooning calculations is consumed by Fourier summations )

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 (I need to work on this : eigenvalue and derivative are both extrapolated; are they still consistent ?) • The derivatives enable a higher order interpolation of the data-cube.
– Consider a 2 point interpolation in 1 dimension,
without derivative linear 2 point interpolation error O(h2) with derivative cubic 2 point interpolation error O(h4)

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

In 3D, 4th order interpolation is easily obtained
eigenvalue interpolation error derivative interpolation error

 tri-cubic interpolation  


i 0 j 0 k 0

3

3

3

ai , j ,k xi y j z k , C1 continuity, error ~ O(h 4 )

 higher order interpolation schemes are possible

the use of the derivatives allows the order of the error to be increased .

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

radial (VMEC) coordinate

Future work possibly includes . . .
• the eigenvalue and derivatives are calculated using Richardson’s extrapolation;
– extrapolation wrt #grid points along field line for each ballooning calculation

– need to check the consistency of extrapolated-eigenvalue with extrapolated-derivatives

• higher order interpolation on data-cube grid
– eg: extend 23 point interpolation to 43 interpolation: O(h4)O(h?)

• higher order derivatives can be calculated using perturbation theory
– higher order derivatives can further improve the data-cube interpolation – is it worthwhile to calculate the higher order derivatives ?

• study some configurations of interest
– – – – need to understand the theory in more detail !!! probably start with axisymmetric approximation, slowly add non-axisymmetry towards NCSX appropriate mass normalization for comparison with CAS3D / TERPSICHORE include FLR effects


								
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