1. TEC Short introduction to plasma fluid theory Dominik Schega

2. TEC Outlook 1.Why a fluid theory? 2.On the way to the fluid theory 3.Braginskii fluid equations 4.Fundamental orderings – description of various plasmas 5.Example of an application of the fluid theory 6.Overwiev

3. TEC General problem definiton Using the statistics we are able to describe the plasma via simple equations With normalization condition Phase-space conservation requiers With the acceleration The total distribution function F s compose essentially of Dirac’s delta distribution functions each following the detailed trajectory of a simple particle. Finally we end up with a classical electromagnetic many-body problem

4. TEC Kinetic equation – ensemble averaging Tractable equation – ensemble averaging Correlations between acceleration and distribution function – collision operator Ensemble average of the phase-space conservation equation – kinetic equation Vlasov equation

5. TEC On the way... Classical electromagnetic many-body problem Unsolvable Kinetic equation Partly solvable Unintuitive 6D Ensemble average

6. TEC On the way... Classical electromagnetic many-body problem Unsolvable Kinetic equation Partly solvable Unintuitive 6D Ensemble average Why a fluid theory? We expect it to be simpler to understand and simpler to apply in computation - 3D Simple idea – compute the moment of kinetic equation before solving it

7. TEC Moments of the distribution function Particle density Particle flux density Stress tensor Energy flux density

8. TEC Moments of the distribution function In the rest-frame of the species under consideration Pressure tensor and ”scalar” presure Heat flux density Relations between moments in different frames Note that

9. TEC Moments of collision operator Particle conservation Momentum conservation (friction force)‏ Collisional energy conservation

10. TEC Moments of kinetic equation Continuity equation Momentum conservation equation Energy conservation equation

11. TEC Typical form of plasma fluid equations Electron fluid equations Ion fluid equations Resistive force Thermal force Work done by the resistive force Work done by the thermal force Convective derivative

12. TEC On the way... Classical electromagnetic many-body problem Unsolvable Kinetic equation Partly solvable Unintuitive 6D Ensemble average Braginskii Fluid Equations Solvable Intuitive 3D Moments of kinetic equation Applicability? Collisional plasma only! S.I. Braginskii Transport processes in a plasma Revievs of Plasma Physics 1 (1965)‏

13. TEC Fundamental orderings cold plasma equations Electronfluid equations Ion fluid equations The fluid velocities are much greater than the thermal velocities

14. TEC Fundamental orderings – MHD equations Electronfluid equations Ion fluid equations The fluid velocities are of order the thermal velocities

15. TEC Fundamental orderings – drift equations Electronfluid equations Ion fluid equations The fluid velocities are of order the drift velocities

16. TEC On the way... Classical electromagnetic many-body problem Unsolvable Kinetic equation Partly solvable Unintuitive 6D Ensemble average Braginski fluid equations Solvable Intuitive 3D Moments of kinetic equation Fluid equations in a chosen limit Solvable Intuitive 3D Choose a limit A. Cole and R. Fitzpatrick Drift-hydrodynamical model of error-field penetration in tokamak plasmas Physics of plasmas 13 (2006)‏

17. TEC Choose a coordinate system and describe the equilibrium Coordinate system: - toroidal, intrinsic (non-orthogonal); - Intrinsic angle The model equilibrium field:

18. TEC Apply the perturbation theory Each function compose of equilibrium part and perturbed part We predict periodic solutions Fuorier analysis allows us to find a set of equations for 1D mode evolution

19. TEC Edge structure difference between vacuum error field and plasma response case

20. TEC Describe the equilibrium and apply coordinate system That was the way Classical electromagnetic many-body problem Unsolvable Kinetic equation Partly solvable Unintuitive 6D Ensemble average Braginski fluid equations Solvable Intuitive 3D Moments of kinetic equation Fluid equations in a chosen limit Solvable Intuitive 3D Choose a limit Model equations to be solved Solvable Intuitive 3D