Chromosphere Model: Heating, Structures, and Circulation P. Song 1, and V. M. Vasyliūnas 1,2 1.Center for Atmospheric Research, University of Massachusetts Lowell 2.Max-Planck-Institut für Sonnensystemforschung,-Lindau, Germany 1 Heating by strong Alfven wave damping Damping is heavier at high frequencies Heating is stronger at lower altitudes for weaker field Heating is stronger at higher altitudes for stronger field Temperature profile is determined by heating rate per particle Temperature minimum at 600 km: transition from Ohmic heating to frictional heating Formation of wine-glass shaped field geometry by circulation Formation of spicules by stronger heating in strong field region in upper chromosphere Heating Rate Per Particle

A Model of the Chromosphere: Heating, Structures, and Circulation P. Song 1, and V. M. Vasyliūnas 1,2 1.Center for Atmospheric Research and Department of Physics, University of Massachusetts Lowell 2. Max-Planck-Institut für Sonnensystemforschung,37191 Katlenburg- Lindau, Germany 2

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 3

The Atmospheric Heating Problem, cont. (Radiative Losses) Due to emissions radiated from regions above the photosphere Photosphere: optical depth=1: radiation mostly absorbed and reemitted => No energy loss below the photosphere Chromosphere: optical depth<1: radiation can go to infinity => energy radiated from the chromosphere is lost Corona: nearly fully ionized: Little radiation is emitted and little energy is lost via radiation Total radiative loss is ~ T The temperature profile is maintained by the balance between heating and radiative loss Temperature increases where heating rate > radiative loss 4 Photosphere Chromosphere Corona

Required Heating (for Quiet Sun): Radiative Losses & Temperature Rise Total radiation loss in chromosphere: 10 6~7 erg cm -2 s -1. Radiation rate: –Lower chromosphere: erg cm -3 s -1 –Upper chromosphere: erg cm -3 s -1 Power to launch solar wind or to heat the corona to 2~3 MK: 3x10 5 erg cm -2 s -1 –focus of most coronal heating models Power to ionize: small compared to radiation The bulk of atmospheric heating occurs in the chromosphere (not in the corona where the temperature rises) Upper limit of available wave power ~ 10 8~9 erg cm -2 s -1 Observed wave power: ~ 10 7 erg cm -2 s -1 Efficiency of the energy conversion mechanisms More heating at lower altitudes 5

Conditions in the Chromosphere Avrett and Loeser, 2008 General Comments: Partially ionized Similar to thermosphere -ionosphere Motion is driven from below Heating can be via collisions between plasma and neutrals Objectives: to explain Temperature profile, especially a minimum at 600 km Sharp changes in density and temperature at the Transition Region (TR) Spicules: rooted from strong field regions Funnel-canopy-shaped magnetic field geometry 6

Heating by Horizontal Perturbations (previous theories) Single fluid MHD: heating is due to internal “Joule” heating (evaluated correctly?) Single wave: at the peak power frequency, 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 (?) 7

Plasma-neutral Interaction 8 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

Simplified Equations for Chromosphere (Leading terms) Faraday ’ s law Ampere ’ s law Generalized Ohm ’ s law Plasma momentum equation Neutral momentum equation Heating rate Ohmic/Joule Frictional 9 [Vasyliūnas and Song, JGR, 110, A02301, 2005]

Total Heating Rate from a Power-Law Source 1-D Stratified Without Vertical Flow or Current strong background field:  B << B, low frequency:  << ni 10

Total Heating Rate from a Power-Law Source 1-D Stratified Without Vertical Flow or Current strong background field:  B << B, low frequency:  << ni Logarithm of heating per cm, Q, as function of field strength over all frequencies in erg cm -3 s -1 assuming n=5/3, ω 0 /2π=1/300 sec and F 0 = 10 7 erg cm -2 s

Heating Rate Per Particle Logarithm of heating rate per particle Q/N tot in erg s -1, solid lines are for unity of in /  i (upper) and e /  e (lower) Heating is stronger at: lower altitudes for weaker field higher altitudes for stronger field 12

Energy Transfer and Balance Heating: Ohmic+frictional Radiative loss: electromagnetic Thermal conduction: collisional without flow Convection/circulation: gravity/buoyancy 13

Importance of Thermal Conduction Energy Equation Time scale: ~ lifetime of a supergranule:> ~ 1 day~10 5 sec Heat Conduction in Chromosphere –Perpendicular to B: very small –Parallel to B: Thermal conductivity: –Conductive heat transfer: (L~1000 km, T~ 10 4 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. Thermal Conduction at the Transition Region (T~10 6 K, L~100 km): Q conduct ~ erg cm -3 s -1 : (comparable to or greater than the heating rate) important to provide for high rate of radiation 14

Importance of Convection Energy Equation Lower chromosphere: density is high, optical depth is significant ~ black-body radiation R~ 10 0 erg cm -3 s -1 (Rosseland approximation) Q~ 10 0 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 i ~  ~ erg cm 3 s -1 Convection, r.h.s./NN i, ~ erg cm 3 s -1 (for N~N i ~10 11 cm -3, p~10 -1 dyn/cm 2 ) Convection is negligible in the chromosphere to the 0 th order: Q/N=N i  Temperature,T, increases with increasing heating rate per particle Q/N 15

Heating Rate Per Particle for Constant Velocity Perturbation Poynting flux is stronger: in strong B field region where damping is weaker Heating is higher near the top boundary for stronger field constant near the lower boundary 16 Logarithm of heating rate per particle Q/N tot in eV s -1, with constant velocity perturbation at the lower boundary. For B=20 G, F 0 is 10 7 erg cm -2 s -1. B hyperbolic-tangentially changes to 20 G in the height from 600 km to 1200 km.

Chromospheric Circulation: Distortion of Magnetic Field 17

Conclusions Based on the 1-D analytical model that can explain the chromospheric heating –The model invokes heavily damped Alfvén waves via frictional and Ohmic heating –The damping of higher frequency waves is heavy at lower altitudes for weaker field –Only the undamped low-frequency waves can be observed above the corona (the chromosphere behaves as a low-pass filter) –More heating (per particle) occurs at lower altitudes when the field is weak and at higher altitudes when the field is strong Extend to horizontally nonuniform magnetic field strength –The temperature is higher in higher heating rate regions –Temperature is determined by the balance between heating and radiation in most regions –Heat conduction from the coronal heating determines the temperature profile near TR –The nonuniform heating drives chromospheric convection/circulation The observed temperature profile, including the temperature minimum at 600 km, is consistent with the convection/circulation –Temperature minimum occurs in the place where there is a change in heating mechanism: electron Ohmic heating below and ion frictional heating above. The circulation distorts the field lines into a funnel-canopy shaped geometry 18

Power-Law Spectrum of Perturbations Assume the perturbations at the source below photosphere can be described by a power law (turbulence theory). At the surface of the photosphere At height z, due to damping (ω 0 : lower cutoff frequency) n=5/3 19

Three-fluid Equations (neglecting photo-ionization, horizontally uniform) Faraday ’ s law Ampere ’ s law Generalized Ohm ’ s law Plasma momentum equation Neutral momentum equation Energy equations [Vasyliūnas and Song, JGR, 110, A02301, 2005] 20

1-D Stratified Without Vertical Flow or Current strong background field:  B << B, low frequency:  << ni Flow slippage Heating rate (for oscillations) Note E·j is frame dependent and heating q is not. Heating is the same as “ Ohmic ” heating in rest frame (to the sun) Heating rate depends on frequnecy! For strong damping, wave energy flux decreases with height From Poynting theorem (general) Poynting flux For Alfven mode (for low frequencies) 21

1-D Stratified Without Vertical Flow and Current, cont. 1-D Poynting Theorem For strong damping, amplitude of wave decreases with height Upper cutoff frequency (a function of height)  1 /2  Observed peak frequency 22

Heating rate at a given height Heating/damping rate is –higher at higher frequencies –higher at lower altitudes –higher for weaker magnetic field –Ohmic heating (when  >1) is dominant in lower altitudes –Frictional heating is dominant above 600 km Heating Rate ni e in 23

Total Heating Rate Total heating rate integrated over frequency Total input wave energy flux Total heating integrated over height 24

Spectrum Dependence on Height and Field Strength B=100G B=10G n=5/3 25

Damping as function of frequency and altitude Reardon et al., km 1000 km 26

Observation Range 1000 km 200 km 27

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The Solar Photosphere White light images of the Sun: granules, networks, sunspots, The photosphere reveals interior convective motions & complex magnetic fields: β << 1 β ~ 1 β > 1 29

The solar chromosphere Images through H-alpha filter: red light (lower frequency than peak visible band) Filaments, plage, prominences 30

1990s: SOHO’s View of the Corona 31

Type II Spicules Thin straw-like structures, lifetime ~ sec km in diameter km/s upward speed, shoot up to 3-10 Mm Fresh denser gas flowing from chromosphere into corona Rooted in strong field ~ 1kG of them at a give time on the sun A sample DOT Ca II H image obtained on November 4, 2003 showing numerous jet–like structures (spicules, active region fibrils, superpenumbral fibrils) clearly visible on the limb in addition to a large surge. The dark elongated structures near the limb are sunspots. At the bottom of the image thin bright structures, called straws, are emanating, from the chromospheric network (which is hardly visible in this image), while around the active regions several dynamic fibrils and penumbral fibrils are visible (from Tziotziou et al. 2005) 32

Lower quiet Sun atmosphere (dimensions not to scale): The solid lines: magnetic field lines that form the magnetic network in the lower layers and a large-scale (“canopy”) field above the internetwork regions, which “separates” the atmosphere in a canopy domain and a sub-canopy domain. The network is found in the lanes of the supergranulation, which is due to large-scale convective flows (large arrows at the bottom). Field lines with footpoints in the internetwork are plotted as thin dashed lines. The flows on smaller spatial scales (small arrows) produce the granulation at the bottom of the photosphere (z = 0 km) and, in connection with convective overshooting, the weak-field “small-scale canopies”. Another result is the formation of the reversed granulation pattern in the middle photosphere (red areas). The mostly weak field in the internetwork can emerge as small magnetic loops, even within a granule (point B). It furthermore partially connects to the magnetic field of the upper layers in a complex manner. Upward propagating and interacting shock waves (arches), which are excited in the layers below the classical temperature minimum, build up the “fluctosphere” in the internetwork sub-canopy domain. The red dot-dashed line marks a hypothetical surface, where sound and Alfvén are equal. The labels D-F indicate special situations of wave-canopy interaction, while location D is relevant for the generation of type-II spicules (see text for details). 33

Chromospheric Heating by Vertical Perturbations Vertically propagating acoustic waves conserve flux (in a static atmosphere) Amplitude eventually reaches V ph 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) 34