Chromosphere Model: Heating, Structures, and Circulation

P. Song1, and V. M. Vasyliūnas1,2

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. Song1, and V. M. Vasyliūnas1,2

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 loss4

Photosphere

Chromosphere

Corona

Required Heating (for Quiet Sun): Radiative Losses & Temperature Rise

• Total radiation loss in chromosphere: 106~7 erg cm-2 s-1 .• Radiation rate:

– Lower chromosphere: 10-1 erg cm-3 s-1 – Upper chromosphere: 10-2 erg cm-3 s-1

• Power to launch solar wind or to heat the corona to 2~3 MK: 3x105 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 ~ 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

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

0 B j

( )i i int

V

j B V U

( )n n nit

U

V U

t

B

E

0 ( ) ee e

mN e

e E V B j B j

2( ) | |in iq j E V B V U

Ohmic/Joule Frictional 9[Vasyliūnas and Song,JGR, 110, A02301, 2005]

Total Heating Rate from a Power-Law Source1-D Stratified Without Vertical Flow or Current

strong background field: B << B, low frequency: << ni

0

1 220 01

0 21 1

1

21 0

2

0 0

00

0 0

1 3 ,

2 (1 ) 2

where ,

1 '

( ) (1 )

[1 (1 ) ]

/

/

( )

( ) ( 1)

ni

n

ni At

y x

a

z

At ni

e in e i

i n

n

n H nQ z F

V

x a e y dy

Hdz

z V

H

N N

F S d

FS n

10

Total Heating Rate from a Power-Law Source1-D Stratified Without Vertical Flow or Current

strong background field: B << B, low frequency: << ni

0

1 220 01

0 21 1

1

21 0

2

0 0

00

0 0

1 3 ,

2 (1 ) 2

where ,

1 '

( ) (1 )

[1 (1 ) ]

/

/

( )

( ) ( 1)

ni

n

ni At

y x

a

z

At ni

e in e i

i n

n

n H nQ z F

V

x a e y dy

Hdz

z V

H

N N

F S d

FS n

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.

11

Heating Rate Per Particle

Logarithm of heating rate per particle Q/Ntot 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

5/3

3log

2

d pQ R T p

dt

13

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.

Thermal 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

2 5 4 16 -3 -17/ ~ 10 erg cm sconductQ T L

5 -1 -1 -110 erg s K cm

5/3

3log

2

d pQ R T p

dt

14

Importance of ConvectionEnergy Equation

Lower chromosphere: density is high, optical depth is significant ~ black-body radiationR~ 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 radiationQ/NNi~~ 10-26 erg cm3 s-1

Convection, r.h.s./NNi, ~ 10-28 erg cm3 s-1

(for N~Ni~1011 cm-3, p~10-1 dyn/cm2)

Convection is negligible in the chromosphere to the 0th order: Q/N=Ni

Temperature ,T, increases with increasing heating rate per particle Q/N

5/3

3log ,

2

( )

i i

i

p d pQ R Q NN Q NN

dt

Q R NN T

5/3

3log

2

d pQ R T NkT

dt

0( ) ~ 10 ~R b

FR F F Q

z

5/3

3log

2

pQ R p T

V

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/Ntot in eV s-1, with constant velocity perturbation at the lower boundary. For B=20 G, F0 is 107 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

(ω0: lower cutoff frequency)

(0, ) nP

0

00 0

0

0

1

0

nF

n

S

2 201 0

0 0

0

( 1) exp /

0

( , )

n

z

Fn

S z

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

t

B

E

0 2

1

c t

E

B j

( )( ) ( )n en e i in e en en in n

N mm N m m

t e

UV U j F

0 ( )( ) ( )e ei e i in e en en in

N mm N m m

t e

V

j B V U j F

0 0( ) ( )( ) ( )e e e ee e e en in in en ei e i

i i

m m m mN e N m

e t e m m

jj B E V B V U j F F

2 2 25/3

2 2 25/3

2 2 21 2

3 1[log ] ' [ ] [( ) ( )]

2 2

3 1[log ] [( ) ( )]

2 2

( ) /[ ( ) ] ~ 1

, : thermal speeds; ' = .

p pn n

nn n pn n

n

n n

n q

d Pq P w w

dt c

Pdq P w w

dt

c w w c w w

w w

VJ E B U V

U V

V U

J J V

[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)

( )W

t

S E j

20

2

/ , or

( )(1 )

z t At

ZAt ni

S V V

Hq S

V

S E B

j E V B

1

1 /ni ni

i

i

V U V

2

2 2

2 2

( ) | |

(1 )(1 / )

in i

t

ni ni

q

H V

j E V B V U

E j 2 2where 1 (1 ) ( / ) , e inni

e i

H

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)

21

20 /exp)(),( SzS z

2

2

1

(1 )z

z t At ni At

S q H

S z V V V

12

0

1 '

( ) (1 )

z

At ni

Hdz

z V

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

2 22 2

0 1

2 20 1

( , ) ( , ) ( ) exp /(1 ) (1 )

( , ) ( ) exp /

zni At ni At

H Hq z S z S

V V

q z Ss

e

in

23

e in

e i

Total Heating Rate• Total heating rate integrated over frequency

• Total input wave energy flux

• Total heating integrated over height

0

2 20 10

( ) ( , ) ( )

( ) ( ) exp /

dQ z q z d F z

dz

F z d S

00 0 ( )

ni

F S d

0 0 .

zQ z dz F F z 24

Spectrum Dependence on

Height and Field Strength

B=100G

B=10G n=5/3

n=5/3

25

Damping as function of frequency and altitude

Reardon et al., 2008

200 km

1000 km

2

2 2

22 2

0 12 2

( , ) ( , )(1 )(1 / )

( ) exp /(1 )(1 / )

zni At ni

ni At ni

Hq z S z

V

HS

V

26

Observation Range

1000 km

200 km

27

26 September, 1999

The SunStructure of the Sun

• Core• Radiative Zone• Interface Region• Convection Zone• Photosphere• Chromosphere• Transition Region• Corona• Solar Wind

28

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 chromosphereImages 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 ~ 10-300 sec.

• 150-200 km in diameter• 20-150 km/s upward speed, • shoot up to 3-10 Mm• Fresh denser gas flowing from

chromosphere into corona• Rooted in strong field ~ 1kG• 6000-7000 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 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) 34