1. Overview of equations and assumptions Elena Khomenko, Manuel Collados, Antonio Díaz Departamento de Astrofísica, Universidad de La Laguna and Instituto de Astrofísica de Canarias (IAC), La Laguna, Tenerife (Spain). http://www.iac.es/proyecto/spia/

2. Elena Khomenko Degree of Ionization in VAL-C model Chromosphere Photosphere

3. Elena Khomenko Degree of Ionization in VAL-C model photosphere chromosphere nini nNnN

4. Elena Khomenko + + + + + + Important for magnetized photosphere and chromosphere B Hall effectAmbipolar diffusion Relative motions between electrons, ions and neutrals (Hall effect and Ambipolar diffusion) produce additional important sources of heating. Reynolds number around 1-10 Additional diffusitivy due to collisions between ions and neutrals is 5-6 orders of magnitude larger than the usual Coulomb diffusivity (Reynolds number around 1-10). 10 km1 sec Characteristic scales in a partially ionized plasma depend on the degree of ionization reaching 10 km and 1 sec for the solar atmosphere. Multi-fluid approximation

5. Elena Khomenko How detailed? MHD Quasi-MHD: single fluid 2- or 3-fluid MHD Kinetic equations; motion of individual particles Depending on scales of phenomena to study, and number of particles. 10 km1 sec Events on scales below 10 km & 1 sec and chromospheric density & neutral fraction probably require multi-fluid approach.

6. Elena Khomenko Equations for individual species Mass conservation Momentum conservation Energy conservation  =e electrons  =i ions  =n neutral hydrogen

7. Elena Khomenko Combined quasi-MHD equations Mass conservation Momentum conservation Energy conservation electrons +ions +neutral hydrogen

8. Elena Khomenko Definitions Center of mass velocity Kinetic pressure tensor Diffusion velocity see Bittencourt (1986)

9. Elena Khomenko Combined quasi-MHD equations Mass conservation Momentum conservation Energy conservation electrons +ions +neutral hydrogen

10. Elena Khomenko Combined quasi-MHD equations Mass conservation Momentum conservation Energy conservation electrons +ions +neutral hydrogen

11. Elena Khomenko Combined quasi-MHD equations Mass conservation Momentum conservation Energy conservation electrons +ions +neutral hydrogen We need Ohm’s law

12. Elena Khomenko Generalized Ohm’s law: all terms Ohmic term Hall term Ambipolar term Biermann battery term …

13. Elena Khomenko Generalized Ohm’s law: all terms Definitions: - partial pressures term neutral fraction ion fraction unity vector in the direction of mag. field; collisions

14. Elena Khomenko Generalized Ohm’s law: all terms Assumptions: Neglect electron inertia and momentum terms; Quasi-neutrality; Same temperatures of all species; Time variations of diffusion velocity are neglected; Pressure tensor has to be assumed scalar.

15. Elena Khomenko Generalized Ohm’s law Ohmic term Hall term Ambipolar term Depends on neutron fraction, collision frequency and B. VALC +

16. Elena Khomenko Generalized Ohm’s law spatial and temporal scales VALC +

17. Elena Khomenko Generalized Ohm’s law: all terms Photospheric magneto-convection snapshot from Vögler et al (2005) Z=600 km ‹B›=180 G

18. Elena Khomenko Generalized Ohm’s law: all terms Inversion of chromospheric sunspot observations by Socas-Navarro (2005) Z=1400 km B=2500 G

19. Elena Khomenko Center of mass velocity of charges Ion-electron kinetic pressure tensor Diffusion velocities of charges see Bittencourt (1986) Two-fluid equations: definitions

20. Elena Khomenko Two-fluid equations see Bittencourt (1986) Mass conservation for charged species Momentum conservation for charged species Energy conservation for charged species c = (e + i ) = (electrons + ions)

21. Elena Khomenko Two-fluid equations see Bittencourt (1986) Mass conservation for charged species Momentum conservation for charged species Energy conservation for charged species c = (e + i ) = (electrons + ions)

22. Elena Khomenko Two-fluid equations Mass conservation for charged species Momentum conservation for charged species Energy conservation for charged species

23. Elena Khomenko Transport theory in partially ionized plasmas Collisional frequencies; Elements of the pressure tensor; Heat flux vector; Ionization-recombination terms; Radiative transfer terms ? How can we compute

24. Elena Khomenko Numerical solution Special treatment for collisional terms? How can we solve the 2-fluid equations numerically? How much new physics do we gain in 2-fluid compared to quasi-MHD?

25. Elena Khomenko Where can it be important? Wave propagation Force balance Magnetic reconnection Flux emergence Chromospheric heating Interstellar medium Shock fronts Turbulence…etc.