Max-Planck-Institut für Plasmaphysik, EURATOM Association Different numerical approaches to 3D...

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


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


Transport equations


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


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


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


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


Local magnetic coordinates


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


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!


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


“Semi-implicit” scheme

RTA

nnn RTA

RTA

RTA

222

111

Implicit scheme

„Semi-implicit“ scheme

Memory usage: 7 times less

Solver: 50 times faster


Results


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).


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


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

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