The Physical Processes in the Chromosphere: From a View of Partially Ionized Plasma Physics

1. The Physical Processes in the Chromosphere: From a View of Partially Ionized Plasma Physics Paul Song and V. M. Vasyliūnas Space Science Laboratory and Department of Physics, University of Massachusetts Lowell Fundamental processes in a partially ionized plasma Ionization and recombination Inter-species collisions Heating and radiation Collisional MHD waves Chromosphere Alfven wave propagation and damping Heating and cooling Circulation Summary Acknolwledgments: Jiannan Tu

2. Ionization and Recombination • Mass exchange among species photo, collisional • Determining ionization ratio: Ionization equilibrium at chromosphere • Momentum transfer (to plasma) Chromosphere

3. Interspecies Collisions Generalized Ohm’s law: Magnetic diffusion, ohmic dissipation Momentum equation: Frictional, momentum transfer Momentum Conservation: effects depend on ionization fraction Chromospheric conditions

4. Heating and Radiation Heating Wave heating Poynting theorem (EM energy flux conservation) Radiation Dissipation equation Imbalance may result in convection Joule heating Frictional heating Collisional heat exchange

5. Left-hand mode Collisional MHD Dispersion Relation(parallel propagation, incompressible) Right-hand mode [Song, Vasyliūnas, and Ma, 2005]

6. Propagation velocity: Decreases from VAtoNeutral inertia-loading process Neutral collision and neutral motion effect collisionless Left-hand mode Infinite neutral collisionless Infinite neutral Right-hand mode [Song, Vasyliūnas, and Ma, 2005]

7. Attenuation Left-hand mode Attenuation (penetration) depth Shorter for higher frequencies Longer for lower frequencies Low frequencies (PC frequencies) can survive from damping Right-hand mode [Song, Vasyliūnas, and Ma, 2005]

8. Snell’s Law and Generalized Law of Reflectionfor an Ideal Alfven Wave Incident onto Transition Region For parallel Alfven incidence Reflection angle is not equal to incident angle [Song and Vasyliūnas, 2013]

9. Fresnel Conditions: Amplitudes of Reflection and Refraction for an Incident Alfven Wave • Boundary Conditions (for a slow varying contact discontinuity) • Velocity continuous • Magnetic field continuous • Pressure continuous [Song and Vasyliūnas, 2013]

10. Summary • When an Alfven wave is incident onto a discontinuity, such as the transition region, • All 3 MHD modes can be generated (in order to satisfy BC) • All 3 modes can be reflected and transmitted • A total of 7 (including the incidence) waves are present • If the incidence perturbations is in the same plane containing B and normal n • Reflection and penetration both are slow and fast modes • In low beta plasma, slow modes will dominate • In high beta plasma, fast modes dominate • If the incidence perturbations is normal to the plane containing B and normal n • Reflection and penetration both are Alfven modes

11. Conditions in the Chromosphere General Comments: • Partially ionized • Neutrals produce emission • Motion is driven from below Objectives: self-consistently explaining • T profile, • T minimum at 650 km • Sharp Transition Region (TR) • Spicules: rooted from strong field regions • Wine-glass shaped magnetic field geometry Avrett and Loeser, 2008

12. [Song and Vasyliūnas, 2011] Plasma-neutral Interaction • Plasma (red dots) is driven with the magnetic field (solid line) perturbation from below • Neutrals do not directly feel the perturbation while plasma moves • Plasma-neutral collisions accelerate neutrals (open circles) • Longer than the neutral-ion collision time, the plasma and neutrals move nearly together with a small slippage. Weak friction/heating • On very long time scales, the plasma and neutrals move together: no collision/no heating • Similar interaction/coupling occurs between ions and electrons in frequencies below the ion collision frequency, resulting in Ohmic heating

13. Total Heating Rate from a Power-Law Source1-D Stratified Without Vertical Flow or Current strong background field: low frequency . [Song and Vasyliūnas, 2011]

14. Damping as function of frequency and altitude 1000 km 200 km [Song and Vasyliūnas, 2011] [Reardon et al., 2008]

15. 1000 km Observation Range [Song and Vasyliūnas, 2011] 200 km 1000km [Reardon et al., 2008] 200km

16. Perturbation in the Photosphere [Tu and Song, 2013] Random perturbations of the photosphere which assumes a power law spectrum of slop 5/3 and total energy flux of 10-7 erg cm-2 s-1. for B=50G

17. [Tu and Song, 2013] Heating rate as function of height and time for chromospheric density and temperature assumed according to the empirical model [Avrett and Loeser, 2008], B0 = 50 G.

18. Tu and Song, [2013] Collisional MHD Simulation [Song and Vasyliūnas, 2011] Analytical • Stronger heating: • weaker B in lower region • stronger B in upper region

19. Total Heating Rate Dependence on Bat the photosphere [Song and Vasyliūnas, 2014] Logarithm of heating per cm, Q, as function of field strength over all frequencies in erg cm-3 s-1 assuming n=5/3, ω0/2π=1/300 sec and F0 = 107 erg cm-2 s-1.

20. Heating Rate Per Particle • Heating and radiative loss balance R≈Q • Radiative loss is R ~ N(T) • Temperature is T ~(Q/N) Expected T profile • Tmin near 600 km • T is higher in upper region with strong B • T is higher in lower altitudes with weaker B pn/i e/e B (Gauss) Logarithm of heating rate per particle Q/Ntot in erg s-1 [Song and Vasyliunas, 2014]

21. Chromospheric Circulation [Song and Vasyliūnas, 2014] • Two (neutral) convection cells • Upper cell: driven by expansion of hotter region in strong field (networks), sunk in weaker field (internetworks) region of colder gas, and completed by continuity requirement • Lower cell: downdraft in strong field regions (consistent with Parker [1970] • B-field: wine-glass shaped • expanding in the upper region to become more uniform by convection in addition to total pressure balance • B-field: more concentrated in the lower cell as pushed by the flow

22. Summary • Since the chromosphere is a weakly ionized plasma, horizontal oscillations at the photosphere can propagate upward as Alfven waves. • Because of the collisions between plasma and neutrals, the wave energy is damped to become thermal energy. • The thermal energy is radiated and maintains the chromosphere at an equilibrium temperature profile. • The energy from the weak field region of the photosphere is nearly completely damped and radiated in the lower chromosphere. • The energy from the strong field region of the photosphere is nearly undamped in the lower region and partially damped to support the upper chromosphere structure, such as the formation of the canopy. • The uneven heating produces circulation cells. • The circulation plays an important role in the formation of the wine-glass shaped magnetic geometry.

23. Importance of Thermal Conduction Energy Equation Time scale:~ lifetime of a supergranule:> ~ 1 day~105 sec Heat Conduction in Chromosphere • Perpendicular to B: very small • Parallel to B: Thermal conductivity: • Conductive heat transfer: (L~1000 km, T~ 104 K) Thermal conduction is negligible within the chromosphere: the smallness of the temperature gradient within the chromosphere and sharp change at the TR basically rule out the significance of heat conduction in maintaining the temperature profile within the chromosphere. Heat Conduction at the Transition Region (T~106 K, L~100 km): Qconduct ~ 10-6 erg cm-3 s-1: (comparable to or greater than the heating rate) important to provide for high rate of radiation

24. Importance of Convection Energy Equation Lower chromosphere: density is high, optical depth is significant ~ black-body radiation R~ 100 erg cm-3 s-1 (Rosseland approximation) Q~ 100 erg cm-3 s-1 (Song and Vasyliunas, 2011) Convective heat transfer: maybe significant in small scales Upper chromosphere: density is low, optical depth very small: not black-body radiation Q/Nn~~ 10-16 erg s-1 Convection, r.h.s./Nn, ~ 10-17 erg s-1 (for N~Ni~1011 cm-3, p~10-1 dyn/cm2) Convection is negligible in the chromosphere to the 0th order: Q/Nn = Temperature, T, increases with increasing Q/Nn

25. Chromospheric Circulation: Consequences

26. Figure 10. Energy flux spectra of transmitted waves calculated at z=3100 km for the ambient magnetic field B0 =10, 50, 100, 500 G. High frequency waves strongly damped and completely damped above a cutoff frequency which depends on the magnetic field (~0.014 Hz, 0.1 Hz, 0.4 Hz, and 0.7 Hz for B0 =10, 50, 100, 500 G). [Tu and Song, 2013]

27. Chromospheric Circulation • Two (neutral) convection cells • Upper cell: driven by expansion of hotter region in strong field (networks), sunk in weaker field (internetworks) region of colder gas, and completed by continuity requirement • Lower cell: downdraft in strong field regions (consistent with Parker [1970]) • B-field: wine-glass shaped • expanding in the upper region to become more uniform by convection in addition to total pressure balance • B-field: more concentrated in the lower cell as pushed by the flow

28. M-I Coupling via Waves (Perturbations) • The interface between magnetosphere and ionosphere is idealized as a contact discontinuity with possible small deformation as the wave oscillates • Magnetospheric (Alfvenic) perturbation incident onto the ionospheric interface • Reflected waves feedback to the magnetosphere • B  k (2 possible B directions) • Polarizations: (noon-midnight meridian) • Alfven mode (toroidal mode) B,u  k-B0 plane • Fast/slow modes (poloidal mode) B,u in k-B0 plane • Antisunward ionospheric motion =>fast/slow modes (poloidal) => NOT Alfven mode(toroidal) Magnetosphere Ionosphere

29. Phase Velocity Wavelength = Vphase/: High frequencies (> 1 Hz):  << L (gradient scale ~101~3 km) Low frequencies (~1.3 mHz):  ~ 1700 km

30. Reflection and Transmission • B, u  k-B0 plane (Alfven mode) (toroidal mode) • Reflection is significant • B, u in k-B0 plane (fast and slow) (poloidal mode) • Fast mode dominates CA/Cs=3=C’A/C’s Fast mode transmittance CA/Cs=3=C’A/C’s

31. M-I Coupling via Waves (Perturbations) • The interface between magnetosphere and ionosphere is idealized as a contact discontinuity with possible small deformation as the wave oscillates • Magnetospheric (Alfvenic) perturbation incident onto the ionospheric interface • For a field-aligned Alfvenic incidence (for example on cusp ionosphere) B  k, B0: B in a plane normal to k (2 possible components) • Polarizations (reflected and transmitted) (noon-midnight meridian) • Alfven mode (toroidal mode) B,u  k-B0 plane • Fast/slow modes (poloidal mode) B,u in k-B0 plane • Antisunward ionospheric motion =>fast/slow modes (poloidal) => NOT Alfven mode (toroidal) Magnetosphere Ionosphere

32. Evolution of Chromospheric Models

33. Lower quiet Sun atmosphere (dimensions not to scale): Wine-glass or canopy-funnel shaped B field geometry: “network” and “internetwork”; “canopy” and “sub-canopy”. Network: lanes of the supergranulation, large-scale convective flows Smaller spatial scales convection: the granulation, weak-field Upward propagating and interacting shock waves, from the layers below the classical temperature minimum, Type-II spicules: above strong B regions.

34. Something is Wrong: • With increased complexity, the fundamental problems are not resolved (not even addressed)! • Mutual assurance between simulations (with parameters that are 1000 times different from observations) and interpretations • Heating rate is 100 times too small (or waves need 100 times stronger • What forms field geometry? • What forms the temperature profile? • What forms the transition region? • What produces spicules? • Not self-consistent physical processes

35. The Solar Atmospheric Heating Problem(since Edlen 1943) • Explain how the temperature of the corona can reach 2~3 MK from 6000K on the surface • Explain the energy for radiation from regions above the photosphere Solar surface temperature

36. The Atmospheric Heating Problem, cont. • The problem is more of radiative cooling at the photosphere than heating corona (Böhm-Vitense, 1984) • Corona: • not radiative (no cooling) • transparent to radiation (no radiative absorption/heating) • Chromosphere: • radiative absorption/heating is weak • EM and/or mechanical energy input from the photosphere • heat flux from corona (small) • radiative cooling, R~N*Ne*Tn, is strong • T profile is maintained by heating at the balance temperature where radiative loss: R(T)~Q • T increases where heating rate Q/N increases Corona Chromosphere Photosphere

37. Chromospheric Heating by Vertical Perturbations • Vertically propagating acoustic waves conserve flux (in a static atmosphere) • Amplitude eventually reaches Vphand wave-train steepens into a shock-train. • Shock entropy losses go into heat; only works for periods < 1–2 minutes… Bird (1964) ~ • Carlsson & Stein (1992, 1994, 1997, 2002, etc.) produced 1D time-dependent radiation-hydrodynamics simulations of vertical shock propagation and transient chromospheric heating. Wedemeyer et al. (2004) continued to 3D... (Steven Cranmer, 2009)

38. Heating by Horizontal Perturbations(previous theories) Single fluid MHD: heating is due to internal “Joule” heating (evaluated correctly?) Single wave: at the frequencies of peak power, not a spectrum Weak damping: “Born approximation”, the energy flux of the perturbation is constant with height Insufficient heating (a factor of 50 too small): a result of weak damping approximation Less heating at lower altitudes Stronger heating for stronger magnetic field (?)

39. Required Heating (for Quiet Sun): Radiative Losses & Temperature Rise • Power required: • Lower chromosphere: 10-1 erg cm-3 s-1 • Upper chromosphere: 10-2 erg cm-3 s-1 • Power to heat the corona to 2~3 MK: 3x105 erg cm-2 s-1 (focus of most coronal heating models) • Power to launch solar wind 3x104 erg cm-2 s-1 • Power to ionize: small compared to radiation • The bulk of atmospheric heating occurs in the chromosphere (not in the corona where the temperature rises) • Total radiation loss in chromosphere: 106~7 erg cm-2 s-1 . • Upper limit of available wave power ~ 108~9 erg cm-2 s-1 • Observed wave power: ~ 107 erg cm-2 s-1 • Efficiency of the energy conversion mechanisms • More heating at lower altitudes

40. 1-D Empirical Chromospheric Models Vernazza, Avrett, & Loeser, 1981