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).

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

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

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

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.

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

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

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

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

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

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

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

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

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.

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

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

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

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

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

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)

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)

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

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

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?

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