max-planck-institut für plasmaphysik, euratom association different numerical approaches to 3d...
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Max-Planck-Institut für Plasmaphysik, EURATOM Association
Different numerical approaches to 3D transport modelling of
fusion devices
Alexander Kalentyev
Max-Planck Institut für Plasmaphysik, EURATOM Association
Stellarator Theory Division
Max-Planck-Institut für Plasmaphysik, EURATOM Association
Introduction
3D effects:
•In tokamaks near divertor plates
•stellarators are intrinsically 3D
Ergodicity:
•Perturbation coils in tokamaks (TEXTOR-DED, DIII-D)
•In stellarators ergodic region always present
Max-Planck-Institut für Plasmaphysik, EURATOM Association
Transport equations
Max-Planck-Institut für Plasmaphysik, EURATOM Association
Finite volume approach (BoRiS)
plasma core (non-ergodic)
ergodic region
island (non-ergodic)
Divertors
Generalized Newton solver
Special application - W7-X using Boozer coordinates for 7 separate domains
Max-Planck-Institut für Plasmaphysik, EURATOM Association
Flexibility of BoRiS
Solution of the Navier-Stokes equationsfor a flow in a square cavity
Predicted streamlines Influence of the under-relaxation parameters on convergence rate
Convergence regionPeric et al. 1988
Scrape Off Layer
Plasma core
Wall
Parallel direction
Rad
ial d
irec
tio
n
Ergodic region
|| flr D
Enhancement of radial transport due to contribution from
parallel transport
Rechester Rosenbluth, Physical Review Letters, 1978
Electron temperature
r
Max-Planck-Institut für Plasmaphysik, EURATOM Association
Transport in an ergodic region
Kolmogorov length LK is a measure of field line ergodicity
0
1log
SLK
10
S
exponential divergence
Typical value in W7-X : LK = 10 – 30 m
Max-Planck-Institut für Plasmaphysik, EURATOM Association
Kolmogorov length
central cut
backward cut
forward cut
x1
x2
x3
333231
232221
131211
ggg
ggg
ggg
g ij
One coordinate aligned with the magnetic field to minimize numerical diffusion
Area is conserved
Use a full metric tensor
Local system shorter than Kolmogorov length to handle ergodicity
Max-Planck-Institut für Plasmaphysik, EURATOM Association
Local magnetic coordinates
Max-Planck-Institut für Plasmaphysik, EURATOM Association
Interface problem
1) Optimized mesh (finite-difference scheme)
/10 1
,100 ,2 ,
24
||||
||
smmm
NRNLL
numerics
Problem: numerical diffusion induced
by interpolation on the interface
Max-Planck-Institut für Plasmaphysik, EURATOM Association
Monte-Carlo 1st Order Algorithm
smnumerical /10 28
Random process random step
Realization
Diffusion Convection
Monte-Carlo combined with Interpolated Cell Mapping
High accuracy transformation of the perpendicular coordinates of a particle(mapping between cuts) needed!
Max-Planck-Institut für Plasmaphysik, EURATOM Association
Finite Difference Approach
Fieldline tracing
Triangulation
Metric coefficients
333231
232221
131211
ggg
ggg
ggg
g ij
Transport code
GridNeighborhoods
Temperature solution
Magnetic field
Linearization matrix
Mesh optimization
Max-Planck-Institut für Plasmaphysik, EURATOM Association
“Semi-implicit” scheme
RTA
nnn RTA
RTA
RTA
222
111
Implicit scheme
„Semi-implicit“ scheme
Memory usage: 7 times less
Solver: 50 times faster
Max-Planck-Institut für Plasmaphysik, EURATOM Association
Results
Max-Planck-Institut für Plasmaphysik, EURATOM Association
Conclusion and Future Work
Conclusion
•Comparisons between three different codes for a W7-X geometry were done.
Future Work
•To complete the physics (including all transport equations).
•To compare results in more realistic cases (including target plates, finite beta).
Max-Planck-Institut für Plasmaphysik, EURATOM Association
Conduction-convection
Convection-conduction equation for a „fluid quantity“ f:
fSfhhDDfDfVhVt
f||||
B/Bh,hhDDgDD
fSfVx
fDg
xgt
f
ji||
ijij
ij
iji
ere wh
1
or
0 21 hh x1 =constx 2=const x3
B
reference cut
„Magnetic“ coordinate system:
- contribution from D|| in D33 only
Metric tensor: determined by field line tracing
Max-Planck-Institut für Plasmaphysik, EURATOM Association
Monte-Carlo 1st Order Algorithm
Fokker-Planck Eq. for pseudoscalar density of test particles,
Random process
Requirement
Realization
diffusion, convection sink, source
random step
independent random numbers
physics: diffusion and convection of the “fluid quantity”
Higher order schemes in 3D get much too complex
Interpretation as probabilistic approximationof Green functions possible
gfN