1. Mikhail Z. Tokar and Mikhail Koltunov
IC-MSQUARE 2013 , Prague,
„Shell“ approach to modeling of impurity spreading from localized sources in plasma
Mikhail Z. Tokar and Mikhail Koltunov
Institut für Energie- und Klimaforschung – Plasmaphysik, Forschungszentrum Jülich GmbH, Association EURATOM-FZJ, Trilateral Euregio Cluster, D Jülich, Germany

2. Outline
Tokamak and impurity injection in fusion devices
Fluid transport equations for charged impurity species
“Shell” model for spreading of impurity ions
Verification of “shell” approach
Conclusions and outlook

3. D+ +T+  He2+(3.5 MeV) + n (14.1 MeV)
Tokamak
Fusion reactions:
Confinement of fusion reagents, D+, T+, by magnetic field in a tokamak:
D+ +T+  He2+(3.5 MeV) + n (14.1 MeV)
JET
Injection of Impurities in tokamaks
Cooling down of plasma edge and weakening of plasma-wall interaction
Investigation and modification of plasma transport properties
Softening of harmful consequences of large instabilities

4. 3-D fluid transport equations for impurity ions
Coordinate system
Particle continuity equations for densities nj of impurity species
Momentum balances along magnetic field for parallel flux components jz
Heat balances for density of impurity ion thermal energy Wj=1.5njTj
Force balance of electrons for parallel electric field
Plasma quasi-neutrality for electron density
Diffusion flux components across magnetic field

5. Shell model for impurity spreadingz
Decay region
Source region
Shell structure:
Source region : j -ions are produced by ionization of j -1 -ions
Decay region : j -ions disappear by ionization into j +1 -ions

6. Approximate solutions
Variation on magnetic surfaces 
Approximate analytical solutions with error  20% (see below):
Recurrent relationships: j-1-shell is the source region of j-shell

7. How to find nj0, j0, jd, ljd,  as functions of t and r ?
New “shell” variables:
total number N and momentum  of j-ions in j -shell and its sub-regions:
y-region
total j-shell
z-region
source
source
source
source
Relations to original variables:

8. Invert interrelations to “shell” variables
1-D equations for “shell” variables 
from integrals of transport equations over shell and its sub-regions:

9. Difference between two solutions does not exceed 20%
Verification of shell approach
1-D flow along magnetic field:
Numerical solution of 1-D equations:
Numerical solution of 0-D shell equations: nj nj
Difference between two solutions does not exceed 20%

10. Conclusion & Outlook
Shell approach is formulated and 1-D equations for the time variation of radial profiles of shell variables are derived
These variables are uniquely related to characteristics of physical ion density, i.e. its maximum value, characteristic dimensions of localization regions
Shell approach is verified by comparing numerical solutions of one-dimensional transport equations found directly and in shell approximation
Compared to 3-D calculations shell approach allows to reduce calculation time by a factor >104 without a significant error: from years to hours for situations of our interest!
Future developments: coupling with particle, momentum and heat balances for main plasma components to describe self-consistently the spreading process and impact of impurity on the plasma

11. Difference between two solutions does not exceed 20%
Verification of shell approach
1-D diffusion equation for perpendicular transport:
Numerical solution of 1-D equation:
Numerical solution of 0-D shell equations: nj nj
Difference between two solutions does not exceed 20%

12. Application of shell approach
Ar-puffing into JET: time evolution of the density radial profiles
averaged over magnetic surface
maximum on magnetic surface
Ar1+
Ar1+
n10 > ni  electron density is significantly perturbed
Ar4+
Ar4+
Very strong localization even for ions of significantly high charges
 Consequences for heat loads on machine wall