chromosphere model: heating, structures, and circulation p. song 1, and v. m. vasyliūnas 1,2...
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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
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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
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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.
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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
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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
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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
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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
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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
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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.
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
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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
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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
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
26 September, 1999
The SunStructure of the Sun
• Core• Radiative Zone• Interface Region• Convection Zone• Photosphere• Chromosphere• Transition Region• Corona• Solar Wind
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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
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The solar chromosphereImages through H-alpha filter: red light (lower frequency than peak visible band)
Filaments, plage, prominences
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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)
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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).
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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