Transcript Slide 1

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
Heating Rate Per Particle
•
•
•
•
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
1 in
Formation of spicules by stronger heating
strong field region in upper chromosphere
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 KatlenburgLindau, Germany
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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:
Corona
radiation can go to infinity
=> energy radiated from the
chromosphere is lost
Chromosphere
• Corona: nearly fully ionized:
Little radiation is emitted and
little energy is lost via radiation
Photosphere
• Total radiative loss is ~ Tn
• The temperature profile is maintained
by the balance between heating and radiative loss
• Temperature increases where heating rate > radiative loss
4
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
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• More heating at lower altitudes
Conditions in the Chromosphere
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
Avrett and Loeser, 2008
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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 (?)
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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
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ion collision frequency, resulting in Ohmic heating
Simplified Equations for Chromosphere
(Leading terms)
Faraday’s law
B
E  
t
Ampere’s law
 B  0 j
Generalized
Ohm’s law
me
0  N e e( E  V  B )  j  B  n e j
e
V
Plasma momentum equation
i
 j  B  n
i in ( V  U )
t
Neutral momentum equation
Heating rate
[Vasyliūnas and Song,
JGR, 110, A02301, 2005]
n
U
  nn ni (V  U)
t
q  j  (E  V  B) n in i | V  U |2
Ohmic/Joule
Frictional
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Total Heating Rate from a Power-Law Source
1-D Stratified Without Vertical Flow or Current
strong background field: B << B, low frequency: << nni
n 1
 H
Q  z   F0
2 (1   )n niVAt
2
1
 0 
 
 1 
n 1
 3  n 02 

, 2
2
1 


where   x, a    e  y y x 1dy
a
z
1
Hdz '

12 ( z ) 0 (1   )VAtn ni
H  [1   (1   ) 2 ]
  n en in /  e i
  Ni / N n
n ni
F0   S0 ( )d 
0
S0 ( )  ( n  1)
F0   
 
0  0 
n
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Total Heating Rate from a Power-Law Source
1-D Stratified Without Vertical Flow or Current
strong background field: B << B, low frequency: << nni
n 1
 H
Q  z   F0
2 (1   )n niVAt
2
1
 0 
 
 1 
n 1
 3  n 02 

, 2
2
1 


where   x, a    e  y y x 1dy
a
z
1
Hdz '

12 ( z ) 0 (1   )VAtn ni
H  [1   (1   ) 2 ]
  n en in /  e i
  Ni / N n
n ni
F0   S0 ( )d 
0
S0 ( )  ( n  1)
F0   
 
0  0 
n
Logarithm of heating per cm, Q, as function of field
strength over all frequencies in erg cm-3 s-1 assuming11
n=5/3, ω0/2π=1/300 sec and F0 = 107 erg cm-2 s-1.
Heating Rate Per Particle
Heating is stronger at:
•lower altitudes for
weaker field
•higher altitudes for
stronger field
Logarithm of heating rate per particle Q/Ntot in
erg s-1, solid lines are for unity of nin/i
(upper) and ne/e (lower)
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Energy Transfer and Balance
• Heating: Ohmic+frictional
• Radiative loss: electromagnetic
• Thermal conduction: collisional without
flow
• Convection/circulation: gravity/buoyancy
3 d
p 
Q  R    T  p  log 5/ 3 
2 dt 
 
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Importance of Thermal Conduction
Energy Equation
3 d
p 
Q  R    T  p  log 5/ 3 
2 dt 
 
Time scale:~ lifetime of a supergranule:> ~ 1 day~105 sec
Heat Conduction in Chromosphere
– Perpendicular to B: very small
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-1
-1
-1
– Parallel to B: Thermal conductivity:  10 erg s K cm
– Conductive heat transfer: (L~1000 km, T~ 104 K)
Qconduct  T / L2 ~1054167 erg cm-3s-1
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
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Importance of Convection
Energy Equation
Q  R    T 
3
d
p 
NkT  log 5/ 3 
2
dt 
 
Lower chromosphere: density is high, optical depth is significant ~ black-body radiation
R~ 100 erg cm-3 s-1 (Rosseland approximation)
F
R

  R  ( Fb  F ) ~ 100 ~ Q
0
-3
-1
Q~ 10 erg cm s (Song and Vasyliunas, 2011)
z
Convective heat transfer: maybe significant
in small scales
QR 

3
p 
pV   log 5/ 3   T
2
 

Upper chromosphere: density is low, optical depth very small: not black-body radiation
Q/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)
Q  R  Q  NNi  
3p d 
p 
log
 Q, NNi 

5/ 3 
2 dt 
 
Q  R  NNi (T )
Convection is negligible in the chromosphere to the 0th order: Q/N=Ni
Temperature ,T, increases with increasing heating rate per particle Q/N
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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
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
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from 600 km to 1200 km.
Chromospheric Circulation:
Distortion of Magnetic Field
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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).
P(0, )    n
n
  n1 F0   
At the surface of the photosphere
 0
0  0 

S0     
0
 0
At height z, due to damping

(ω0: lower cutoff frequency)
n
 ( n1) F0    exp  2 /  2 
1
0  0 

S z ( z,  )  
0
 0

n=5/3
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 0
Three-fluid Equations
(neglecting photo-ionization, horizontally uniform)
Faraday’s law
Ampere’s law
B
t
1 E
  B  0 j  2
c t
E  
Generalized Ohm’s law

me j
m m
m
 j  B0  Ne e(E  V  B0 )  Ne me (n en n in )(V  U)  e ( e n in n en n ei ) j  Fe  e Fi
e t
e mi
mi
Plasma momentum equation
mi
Neutral momentum equation
Energy equations
[Vasyliūnas and Song,
JGR, 110, A02301, 2005]
Ne V
m
 j  B0  Ne (min in  men en )(V  U)  e (n en n in ) j  F
t
e
mn
N n U
m
 N e (min in  men en )(V  U)  e (n en n in ) j  Fn
t
e
3 d
P
V
1
P [log 5 / 3 ]  J ' [E   B]  n pn [( U  V ) 2   ( wn2  w2 )]
2 dt

c
2
P
3 d
1
qn  Pn [log 5n/ 3 ]  n pn  [(U  V ) 2   ( wn2  w2 )]
2 dt
n
2
qp 
  (V  U) 2 /[c1 ( wn2  w2 )  c2 wn w] ~ 1
wn , w : thermal speeds; J' = J   q V.
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1-D Stratified Without Vertical Flow or Current
strong background field: B << B, low frequency: << nni
i
1
Flow slippage
VU  
V
n ni 1  i /n ni
Heating rate
(for oscillations)
q  j  (E  V  B) n in i | V  U |2

t  2 H V 2
n ni (1   )(1   2 /n ni2 )
 E j
where H  1   (1   )2  ( /n ni ) 2  ,  
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
W
From Poynting theorem (general)
t
Poynting flux
For Alfven mode
   S  (E  j)
S  E  B / 0 , or S z  tVAt  V 2 
2H
 q 
S Z  j  (E  V  B )
VAtn ni (1   )
(for low frequencies)
21
n en in
e i
1-D Stratified Without Vertical Flow and Current, cont.
1-D Poynting Theorem
1 S z
q
2H


2
S z z
tVAt  V 
n niVAt (1   )
For strong damping, amplitude of wave decreases with height

Sz ( z, )  S0 () exp   2 / 12
Upper cutoff frequency
(a function of height)

Observed
peak
frequency
z
1
Hdz '

12 ( z ) 0 VAtn ni (1   )
1/2
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Heating Rate
• Heating rate at a given height
q( z,  ) 
2H
n niVAt (1   )
q( z,  )  
S z ( z,  ) 
2H
n niVAt (1   )

S0 ( ) exp  2 / 12
 
2
2

S
(

)
exp


/

0
1

s 




n en in
 e i
• 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
nin
ne
nni
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Total Heating Rate
• Total heating rate integrated over frequency
Q( z )  

0
d
q( z ,  )d    F ( z )
dz


F ( z )   d  S0 ( ) exp  2 / 12
0

• Total input wave energy flux
n ni
F0   S0 ( )d
0
• Total heating integrated over height
 Q  zdz  F  F  z  .
z
0
0
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Spectrum
Dependence on
Height and
Field Strength
B=10G
n=5/3
n=5/3
B=100G
25
Damping as function of
frequency and altitude
2H
q( z,  ) 
S z ( z,  )
n niVAt (1   )(1   2 /n ni2 )
2H

S0 ( ) exp   2 / 12 
2
2
n niVAt (1   )(1   /n ni )
1000 km
200 km
Reardon et al., 2008
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Observation Range
1000 km
200 km
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The Sun
Structure of the Sun
•
•
•
•
•
•
•
•
•
Core
Radiative Zone
Interface Region
Convection Zone
Photosphere
Chromosphere
Transition Region
Corona
Solar Wind
26 Se ptember, 1999
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
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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 ~
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 32
fibrils are
visible (from Tziotziou et al. 2005)
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
33 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).
Chromospheric Heating by Vertical Perturbations
• Vertically propagating acoustic waves
Bird (1964)
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…
~
• 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