Kein Folientitel

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Max-Planck-Institut für Plasmaphysik, EURATOM
Association Introduction
The Scrape Off Layer, and ergodic regions.
Why do we want to model them.
How can we model them (numerical methods).
How exactly do we apply the Finite
Difference method.
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Max-Planck-Institut für Plasmaphysik, EURATOM
Association The Scrape Off Layer
pump
Scrape Off Layer or plasma edge
Plasma core
LCFS
islands
pump
Divertor plates
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Max-Planck-Institut für Plasmaphysik, EURATOM
Association Ergodic Regions
Strong anisotropy
Plasma core (non-ergodic)
ergodic region
electron
Enhances radial transport
Flattening of temperature profile
Last closed flux surface
T
Scrape Off Layer
r0
rLCFS
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Max-Planck-Institut für Plasmaphysik, EURATOM
Association Comparison of Numerical Methods
Finite Volume method
Finite Difference method
Finite Element method
Points are generated along magnetic field lines
Volumes formed by magnetic field lines
Clearer picture of the topology of the transport
Temperature at each point is a function of the
temperature at the neighboring points
Flux balances are evaluated across surfaces of
volumes
Physical quantities are interpolated in each
triangle, but errors accumulate
Ergodicity is not a constraint
Ergodicity makes volume generation complicated
Difficult to separate parallel and perpendicular
transport
Exact parallel transport
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Max-Planck-Institut für Plasmaphysik, EURATOM
Association Applying the Finite Difference Method
To model plasma transport using the Finite
Difference method we need
Discretized transport equations
A set of points filling the plasma volume
Neighborhood information for each point
Initial and boundary conditions
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Max-Planck-Institut für Plasmaphysik, EURATOM
Association Electron Heat Conduction Equation
?
High anisotropy
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Max-Planck-Institut für Plasmaphysik, EURATOM
Association Metric Coefficient
Vector form of heat conduction equation
Conduction tensor
3-D metric coefficient
Generalized spatial coordinate
hy
hz
hx
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Max-Planck-Institut für Plasmaphysik, EURATOM
Association General Ansatz
Use a system of closed field lines or field lines
hitting the wall
?1 and ?2 are the local perpendicular directions
?1
1 neighbor behind
1 neighbor in front
?2
Parallel direction is along the field lines
Parallel direction is coming out of the page
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Max-Planck-Institut für Plasmaphysik, EURATOM
Association Unstructured 3-D Mesh
A set of Poincaré plots showing field line
intersection points
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Max-Planck-Institut für Plasmaphysik, EURATOM
Association Local Magnetic Coordinates
?2

?1
Local perp1-perp2 surface. In general, the
surface can be curved (mixed ?1 ?2 derivatives)
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Max-Planck-Institut für Plasmaphysik, EURATOM
Association Unstructured 2-D Mesh (1)
Generated points
hx
Points found by interpolation
Perp1-perp2 surface
hx
Derivative of metric coefficient is zero because
hx is constant with radial position.
2-D metric coefficient
Calculation of divergence of fluxes (central
differences) variation of metrics in other
directions enters.
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Max-Planck-Institut für Plasmaphysik, EURATOM
Association Unstructured 2-D Mesh (2)
Generated points
Points found by interpolation
hx
Perp1-perp2 surface
hx
2-D metric coefficient
Derivative of metric coefficient is no longer
zero because hx changes with radial position.
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Max-Planck-Institut für Plasmaphysik, EURATOM
Association Neighborhood Array
Points generated by field line trace
Perp1-perp2 surface containing radial neighbors
Interpolated ghost points
Non-parallel field lines
Parallel neighbor in front
Parallel neighbor behind
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Max-Planck-Institut für Plasmaphysik, EURATOM
Association Curved Magnetic Surface
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Max-Planck-Institut für Plasmaphysik, EURATOM
Association Temperature Contours from Findif
ASDEX-Upgrade
chamber wall
plasma core
separatrix
plasma core
X-point
divertor plates
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Max-Planck-Institut für Plasmaphysik, EURATOM
Association Temperature Contours from Findif
0.02 m
50 eV
3 eV
Rectilinear mesh heat flux 100 kW Sheath BC ?T
0.02 m
0.006 m
4.2 m
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Max-Planck-Institut für Plasmaphysik, EURATOM
Association Status
Solves electron heat conduction equation
Implicit and explicit methods
Flexible boundary conditions and geometries
Structured and unstructured meshes
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Max-Planck-Institut für Plasmaphysik, EURATOM
Association Inconsistent Local Coordinates
Generated points
Points found by interpolation
Discretization is not using a consistent local
orthogonal coordinate system
stencil
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Max-Planck-Institut für Plasmaphysik, EURATOM
Association Consistent Local Coordinates
Consistent local coordinates for the whole
stencil, from A. Runov, including metrics
Field lines
x1
x2
x3
Area is conserved
n-1
n
n1
Price mixed derivatives
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Max-Planck-Institut für Plasmaphysik, EURATOM
Association Outlook
Algorithm for finding good field line start
points
Discretization using the consistent local
coordinates
Benchmarking with Finite Volume and Monte Carlo
code
Applying the code to different devices and
different regimes.
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Max-Planck-Institut für Plasmaphysik, EURATOM
Association IMPRS, April 2002
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Max-Planck-Institut für Plasmaphysik, EURATOM
Association IMPRS, April 2002
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Max-Planck-Institut für Plasmaphysik, EURATOM
Association IMPRS, April 2002
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Max-Planck-Institut für Plasmaphysik, EURATOM
Association Next
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Max-Planck-Institut für Plasmaphysik, EURATOM
Association Unstructured 2-D Mesh (1)
hx
Generated points
Points found by interpolation
Perp1-perp2 surface
hx
Derivative of metric coefficient is zero because
hx is constant with radial position.
2-D metric coefficient
Parallel derivative
Perpendicular derivative
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Max-Planck-Institut für Plasmaphysik, EURATOM
Association Unstructured 2-D Mesh (2)
Generated points
Points found by interpolation
hx
hx
Derivative of metric coefficient is no longer
zero because hx changes with radial position.
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Max-Planck-Institut für Plasmaphysik, EURATOM
Association Wendelstein 7-X plasma
Use a system of closed field lines or field lines
hitting the wall
Finite Difference modelling of electron heat
conduction in the plasma edge
? 0
? -18
? 18
Conduction tensor