1. A Model of the Solar Chromosphere: Structure and Internal Circulation
Paul Song
Space Science Laboratory and Department of Physics
University of Massachusetts Lowell
Acknowledgments: V. M. Vasyliūnas
(Song, 2016, ApJ in press)

2. The “Coronal” Heating Problem (since Edlen 1943)
Explain how the temperature of the corona can reach 2~3 MK from 6000K on the surface: against 2nd Law?
Explain the energy for radiation from regions above the photosphere

3. The “Coronal” Heating Problem (since Edlen 1943)
Explain how the temperature of the corona can reach 2~3 MK from 6000K on the surface: against 2nd Law?
power required: ~3x105 erg cm-2 s-1
heating occurs in the corona
Explain the energy for radiation from regions above the photosphere
power required: ~106~7 erg cm-2 s-1
heating occurs in the chromosphere
Observed wave power during quiet time: ~107 erg cm-2 s-1
Power to launch solar wind: ~ 3x104 erg cm-2 s-1

4. Conditions in the Chromosphere
Statistic empirical model ==
Time and horizontal average condition
Objectives: self-consistently explaining
T profile,
Sharp Transition Region (TR)
Spicules: rooted from strong field regions
Wine-glass shaped magnetic field geometry
Radiative cooling
(Avrett and Loeser, 2008) R
220km
Modeled total wave energy flux
of ~107 ergs cm-2 s-1
Radiation cooling rate R=NeNΛ(T), blue line, in ergs cm-3 s-1, and downward integrated radiation over height starting from 2024km, W, red line, in 105 ergs cm-2 s-1.

5. Heating Mechanism and Heat Rate
Mechanical energy source: photospheric horizontal perturbations, propagating along B
Heating mechanism: strong collisional damping [Song and Vasyliūnas, 2011]
Weaker field (~5 G):
Heating rate sufficient to heat
the lower region
Almost all wave energy flux is
damped in the lower region
Strong field (~750G at z=220km):
dominates heating in the
upper region

6. Network Field Expansion
Horizontal Average
Force balance at lower boundary:
mag pressure in network area =
= thermal pressure in internetwork area
Thermal pressure decreases exponentially with height
Strong field expansion to reduce mag Pressure
Uneven horizontal pressure => circulation
A supergranule is formed with:
600 km wide network at
the lower boundary, and
30,000 km wide internetwork

7. Chromospheric Circulation
A supergranule size: 30,000km
Internetwork: B~5G, locally closed field lines; heating from both feet; all wave energy is damped, supporting most radiation from chromosphere
Network: B ~750G at 220km, 600 km wide, open field lines =>connecting to corona and/or other supergranules; waves are weakly damped in lower region, expand with field and heat upper region

8. 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])
The convection further strengthens the wine-glass-shaped field geometry

9. Chromospheric Circulation
Transition region is formed when collisional heating becomes weaker due to high ionization. It is highly thermal stable, allowing ionization/recombination at interface
Larger wave power in strong B region, in particular in intersections of the networks, may produce enough heating to make TR unstable, observed as sporadic upward beams of chromospheric material: type-II spicules
Additional coronal heating is required in the height where Tmax is located.
However, required heating ~5% of the total heating => a tractable problem.

10. Summary
Since the chromosphere is a weakly ionized magnetized 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 with an equilibrium temperature profile.
The energy from the weak field region of the photosphere is nearly completely damped and radiated from the lower chromosphere.
The field in the network expands to form the canopy.
The energy from the strong field region of the photosphere is weakly damped in the lower region and partially damped to support the upper chromosphere structure.
The uneven horizontal neutral pressure produces circulation cells.
The circulation plays an important role in the formation of the wine-glass shaped magnetic geometry.

11. Below surface
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 Internetwork: 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. No upper circulation cells, the lower cells are below surface, Critical height: Ne-min, not T-min

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

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

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

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

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

17. Attenuation Attenuation (penetration) depth
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]

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

19. 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]

20. 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

21. Plasma-neutral Interaction
[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

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

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

24. 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

25. [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.

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

27. Total Heating Rate Dependence on B at 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.

28. 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]

29. 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

30. 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

31. 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~~ erg s-1
Convection, r.h.s./Nn, ~ 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

32. 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]

33. 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

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

35. 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

36. 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

37. Evolution of Chromospheric Models
deposition of mechanical energy - H-ionization - start of outward temperature increase
C. Hydrostatic models (e.g. VAL, etc.) are not sufficient
A. Transition through beta = 1
D.Thus richly structured - Ca II K filtergrams do not (easily)
reveal the chromospheric structure

38. 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

39. 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

40. 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

41. Chromospheric Heating by Vertical Perturbations
Vertically propagating acoustic waves conserve flux (in a static atmosphere)
Amplitude eventually reaches Vph and 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)

42. 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 (?)

43. 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

44. 1-D Empirical Chromospheric Models
Vernazza, Avrett, & Loeser, 1981