Extension of collisionless discharge models for 1/47 application to - PowerPoint PPT Presentation

Extension of collisionless discharge models . . . Extension of collisionless discharge models for 1/47 application to fusion-relevant and general Introduction plasmas Overview of existing models Analytic- numerical Leon Kos method ( ε = 0) Extension of University of Ljubljana the theoretical Faculty of Mechanical Engineering model ( ε > 0) LECAD laboratory Results Conclusion October 21, 2009 / PhD Thesis Defense

What is plasma? Fourth state of the matter (fire). Extension of collisionless discharge models . . . 2/47 Introduction Plasma Applications Motivation and thesis aims Solids - have very strong intermolecular bonds. Overview of existing models Liquid - molecules are tied together by loose strings. Analytic- Gas - atoms bounce around freely in space. numerical method Plasma - ionized gas, electrons and ions are separately free ( ε = 0) Extension of Temperature is the average amount of kinetic energy per atom. the theoretical model ( ε > 0) Results Conclusion

Plasma properties Extension of collisionless discharge models . . . 3/47 Introduction Quasi-neutral ( n i = n e ). Plasma Applications Thin sheath is observed at the wall ( λ D ≪ L ). Motivation and thesis aims Exhibit collective motion - collisionless. Overview of existing models Very conductive - can be shaped and confined by Analytic- electro-magnetic forces. numerical method ( ε = 0) Extension of the theoretical model ( ε > 0) Results Conclusion

Laboratory plasmas Aparatus used for producing a plane symmetric positive column in argon showing the position of the probes. [from Harrison-Thompson, 1959] Extension of collisionless discharge models . . . 4/47 Introduction Plasma Applications Motivation and thesis aims Overview of existing models Analytic- numerical method ( ε = 0) Extension of the theoretical model ( ε > 0) Results Conclusion

Fusion Tokamak - Joint European Torus Extension of collisionless discharge models . . . 5/47 Introduction Plasma Applications Motivation and thesis aims Overview of existing models Analytic- numerical method ( ε = 0) Extension of the theoretical model ( ε > 0) Results Conclusion

Particle tracing developed in LECAD Toroidal and poloidal magnets Extension of collisionless discharge models . . . 6/47 Introduction Plasma Applications Motivation and thesis aims Overview of existing models Analytic- numerical method ( ε = 0) Extension of the theoretical model ( ε > 0) Results Figure: Particle trajectories by Erˇ zen et al. Conclusion

Plasma diagnostics Extension of collisionless discharge models . . . 7/47 Introduction Plasma Applications Motivation and thesis aims Overview of existing models Analytic- numerical method ( ε = 0) Extension of the theoretical model ( ε > 0) Results Conclusion

Geometry One dimensional model Extension of collisionless discharge Φ( x ) models . . . ( x ′ , v ′ ) ( x, v ) x 8/47 Φ( x ′ ) Φ( x ) Introduction Plasma Applications Φ w Motivation and thesis aims Overview of x = − L x = 0 x = L existing models Figure: The geometry and coordinate system. Analytic- numerical method ( ε = 0) Extension of Plane–parallel geometry the theoretical model Symmetric - we observe only one half ( ε > 0) Results We normalize problem to L = 1. Conclusion

Motivation Extension of collisionless discharge models . . . Provide precise treatment of the sheath region to fluid 9/47 codes (SOLPS, EDGE2D). Introduction Plasma No analytic-numeric kinetic code available for T n > 0. Applications Motivation and Particle In Cell (PIC) methods are not enough precise and thesis aims Overview of can’t simulate ε = 0 case. existing models Existing ε = 0 models are limited in temperature range. Analytic- numerical No solution to ε > 0 kinetic model available. method ( ε = 0) Can velocity distribution function be obtained from Extension of potential curves? the theoretical model ( ε > 0) Results Conclusion

Thesis statement Extension of collisionless discharge models . . . 10/47 Introduction The problem of a special integro-differential equations Plasma Applications should be solved numerically without approximations Motivation and thesis aims to achieve an extended solution range applicable to Overview of existing fusion-relevant and general plasmas for an arbitrary models ion temperature and arbitrary finite ε . Analytic- numerical method ( ε = 0) Extension of the theoretical model ( ε > 0) Results Conclusion

Methodology In this thesis the author presents investigations and results with the following assumptions Extension of collisionless discharge models . . . 11/47 The Poisson equation is employed in the whole discharge Introduction region. Plasma Applications A two-scale approximation is obtained just within the limit Motivation and thesis aims of the infinitely small Debye length in comparison with the Overview of existing system length. models The ion-source temperature can take an arbitrary value. Analytic- numerical method The electron-neutral impact is considered as a ionization ( ε = 0) mechanism. Extension of the theoretical model ( ε > 0) Results Conclusion

Overview of existing models Two-scale approximation Extension of collisionless discharge models . . . Φ 12/47 sheath solution exact solution Introduction Overview of plasma solution existing models Tonks- X Langmuir model Bissell-Johnson model Figure: Symbolic picture illustrating the two-scale approximation. Scheuer- Emmert model Analytic- numerical method Plasma solution - Tonks–Langmuir model ( ε = 0) Extension of Sheath solution - Bohm model the theoretical model Exact solution - plasma + sheath (Our extended model) ( ε > 0) Results Conclusion

Plasma parameters Extension of collisionless 1 The macroscopic neutrality n e = n i discharge models . . . 2 Strong electric field is localized to distance λ D with 13/47 Introduction λ D ≪ L , ε ≡ λ D / L ( ≪ 1) , (1) Overview of existing where models Tonks- Langmuir model � ǫ 0 kT e Bissell-Johnson λ D = n 0 e 2 , (2) model Scheuer- Emmert model Analytic- is the Debye radius. numerical method 3 The number of the particles in the Debye sphere is high ( ε = 0) Extension of n λ 3 the theoretical D ≫ 1 . (3) model ( ε > 0) Results Conclusion

Tonks-Langmuir (T&L) model Extension of 0.0 collisionless discharge models . . . -0.2 T =0 n 14/47 (T&L m odel) -0.4 Introduction Overview of -0.6 existing models Tonks- Langmuir -0.8 = -0.85403 model s Bissell-Johnson model -1.0 Scheuer- 0.0 0.2 0.4 0.6 0.8 1.0 Emmert x model Analytic- numerical Ions are born at rest (cold ion-source case). method ( ε = 0) Analytic kinetic solution for ε = 0. Extension of Beakdown of quasi-neutrality at Φ s = − 0 . 85403. the theoretical model ( ε > 0) Lewi Tonks and Irving Langmuir. A general theory of the plasma of an arc. Results Phys. Rev. , 34(6):876–922, Sep 1929. Conclusion

Bissell-Johnson (B&J) model ( ε = 0) 8 8 F ( θ ) F ( θ ) Extension of τ = 0 . 25 τ = 0 . 1 τ = 0 . 33 collisionless 6 6 τ = 0 . 5 discharge models . . . 4 4 15/47 2 2 τ = 1 . 0 τ = 2 . 0 τ = 4 . 0 θ θ τ = 8 . 0 Introduction - 1.0 0.0 1.0 - 1.0 0.0 1.0 - 0.5 0.5 - 0.5 0.5 Overview of existing Figure: Kernel F ( θ ) B&J equation (left, dashed), our approximation models (right, dashed) and the exact kernel (solid). Tonks- Langmuir model Realistic Maxwellian ion-source velocity distribution. Bissell-Johnson model Scheuer- The Bohm criterion is used as the boundary condition to Emmert model the quasi-neutrality equation. Analytic- numerical Kernel approximation with 8-th order Chebyshev method ( ε = 0) polynomial and sinh( . ) switch function. Extension of Plasma Eq. with 9-th order polynomial. the theoretical model ( ε > 0) R. C. Bissell and P. C. Johnson. The solution of the plasma equation in plane parallel geometry with a Maxwellian source. Results Physics of Fluids , 30(2):779–786, 3 1987. Conclusion

Scheuer-Emmert (S&E) model ( ε = 0) Extension of collisionless discharge models . . . Better kernel approximation. 16/47 Did not apply any kind of Bohm criterion in advance. Introduction Dense grid at endpoint singularity. Overview of existing Analytic approximation to sub-integrals with a series models Tonks- expansion. Langmuir model Bissell-Johnson Different normalization than B&J. model Scheuer- Emmert Ion source temperature range is still limited to non-fusion model temperatures. Analytic- numerical method ( ε = 0) J. T. Scheuer and G. A. Emmert. Sheath and presheath in a collisionless plasma with a maxwellian source. Extension of Physics of Fluids , 31(12):3645–3648, 1988. the theoretical model ( ε > 0) Results Conclusion

S&E and our results What ion-source they employed? Extension of collisionless discharge models . . . 17/47 Introduction Overview of existing models Tonks- Langmuir model Bissell-Johnson model Scheuer- Emmert model Analytic- numerical Figure: Comparison of the potential profile with S&E for T i = T e . method ( ε = 0) The original scan is overlayed with our potential profile and axis box. Extension of the theoretical model ( ε > 0) Results Conclusion