A Model of the Chromosphere: Heating, Structures, and Convection

1
A Model of the Chromosphere Heating, Structures,
and Convection

• P. Song1, and V. M. Vasyliunas1,2
• Center for Atmospheric Research and Department of
  Environmental, Earth Atmospheric Sciences,
  University of Massachusetts Lowell
• 2. Max-Planck-Institut für Sonnensystemforschung,3
  7191 Katlenburg-Lindau, Germany

2
Abstract

• We propose a model of local convection in the
  chromosphere, with scale size of supergranules.
  The strong heating required in order to balance
  the radiative losses in the chromosphere is
  provided by strong damping, through
  plasma-neutral collisions, of Alfvén waves that
  are driven by motions below the photosphere. On
  the basis of a self-consistent plasma-neutral-elec
  tromagnetic one-dimensional model, we derive the
  vertical profile of wave spectrum and power by a
  novel method, including the damping effect
  neglected in previous treatments. The
  high-frequency portion of the source power
  spectrum is strongly damped at lower altitudes,
  whereas the lower-frequency perturbations are
  nearly undamped and can be observed in the corona
  and above. As a result, the waves observed above
  the corona constitute only a fraction of those at
  the photosphere and, contrary to supposition in
  some earlier Alfvén-wave-damping models, their
  power does not represent the energy input.
  Calculated from parameters of a semi-empirical
  model for quiet-Sun conditions, the mechanism can
  generate sufficient heat to account for the
  radiative losses in the atmosphere, with most of
  the heat deposited at lower altitudes. When the
  magnetic field strength varies horizontally, the
  heating is likewise horizontally nonuniform.
  Since radiative loss is a strong function of
  temperature, the equilibrium temperature
  corresponding to local thermal balance between
  heating and radiation can be reached rapidly.
  Regions of stronger heating thus maintain higher
  temperatures and vice versa. The resulting uneven
  distribution of temperature drives chromospheric
  convection and circulation, which produces a
  temperature minimum in the chromosphere near 600
  km altitude and distorts the magnetic field to
  create a funnel-canopy-shaped magnetic geometry,
  with a strong field highly concentrated into
  small areas in the lower chromosphere and a
  relatively uniform field in the upper
  chromosphere. The formation of the transition
  region, corona, and spicules will be discussed.

3
Conditions in the Chromosphere

• General Comments
• Partially ionized
• Strong magnetic field
• 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
4
Radiative Losses/Required Heating

• Total radiation loss from chromosphere (not
  including photosphere) 1067 erg cm-2 s-1 .
• Radiative loss rate
• Lower chromosphere 10-1 erg cm-3 s-1
• Upper chromosphere 10-2 erg cm-3 s-1
• Power carried by solar wind 105 erg cm-2 s-1
• Power to ionize small compared to radiation
• Observed wave power 107 erg cm-2 s-1 .

5
Plasma-neutral Interaction
VA

• 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
• In very long time scales, the plasma and neutrals
  move together no collision/no heating

6
Damping as function of frequency and altitude
1000 km
200 km
Reardon et al., 2008
7
Observation Range
8
Total Heating Rate from a Power-Law Source
Logarithm of heating per km, Q, as function of
field strength over all frequencies in erg cm-3
s-1 assuming n5/3, ?0/2p1/300 sec and F0 107
erg cm-2 s-1.
9
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 ?in/?i
(upper) and ?e/?e (lower)
10
Local Thermal Equilibrium Condition

• Energy Equation
• Time scale lifetime of a supergranulegt 1
  day105 sec
• Heat flux negligible (see next page)
• Lower chromosphere Optically thick medium
• R 10-1 erg cm-3 s-1 (Rosseland approximation)
• Q 100 erg cm-3 s-1
• Convection 100 erg cm-3 s-1 (for p105 dyn/cm2)
• If RltltQ
• Upper chromosphere Optically thin medium
• Q/NNi? 10-26 erg cm3 s-1
• Convection, r.h.s., 10-28 erg cm3 s-1
• (for NNi1011 cm-3, p10-1 dyn/cm2)
• Convection is negligible in the upper
  chromosphere Q/NNi?
• Convection in the lower chromosphere may be
  important
• Temperature T increases with increasing heating
  rate per particle Q/N

11
Heat Transfer via Thermal Conduction

• Perpendicular to B very small
• Parallel to B
• Thermal conductivity
•  
• Conductive heat transfer (L1000 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
  (T106 K, L100 km)
• Qconduct 10-6 erg cm-3 s-1 (comparable to
  greater than the heating rate) important to
  provide for high rate of radiation

12
Horizontal Force Balance

• Momentum equation
• Force imbalance is mitigated by sound or fast
  waves
• Time scales
• Sound wave 1.5x104 km/10 km/s 103 sec
• Alfven speed in the upper region 104 km/s, 100
  sec
• Compared with the time scales of the pressure
  imbalance creation by heating (from energy
  equation)
• 105 sec in the lower chromosphere
• 103 sec in upper chromosphere
• Lower chromosphere horizontally pressure
  balanced
• Upper chromosphere pressure in higher heating
  region may be higher

13
Vertical Force Balance

• Momentum equation
• or
• Average over horizontal dimensions (steady state)
• Vertical acceleration
• Upward TgtTm
• Downward TltTm
• Vertical flow produces additional pressure
  imbalance because of the different temperature
  and density the flow carries

14
Upper cell vertical flow speed estimate

• Steady state momentum equation
• Estimate the upflow in upper cell strong B region
• With sub m measured and corresponding values
• without sub strong field upper cell.
• _at_1000 km, Tm6200, from radiation function table
  Nim?m2x10-27x2x1011. (From our model, Qm/Nm
  10-18 since Qm cannot be used quantitatively),
  assume Ni?/Nim?mNmQ/NQm
• Since Q/N6Qm/Nm, ?TX6?mTmXm, where XNi/N, from
  radiation function T 6600. T/Tm1.06,
  T/Tm-10.06
• _at_2000 km Tm6700
• Nim?m1.2x10-26x4x1010, Q/N6Qm/Nm, T7100,
  T/Tm1.06
• g274m/s2
• Vzsqrt(2x274m/s2x0.06x106m)sqrt(33x106)6 km/s,
  a number within possible range, but maybe too big
  if Vx is LVz/H30000/2/200Vz450km/s,
  supersonic. Cs10 km/s

15
Circulation Connecting the Vertical Flow with
Horizontal Flow

• Continuity equation
• 2-D Cartesian coordinates
• Horizontal momentum equation

16
Inhomogeneous Heating Chromosphere Circulation
17
Chromospheric Circulation Distortion of
Magnetic Field
18
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 2-D when the magnetic field strength is
  horizontally nonuniform
• The temperature is higher in higher heating rate
  regions.
• The nonuniform heating drives chromospheric
  convection/circulation
• The observed temperature profile, including the
  temperature minimum at 600 km, is consistent with
  the convection/circulation without invoking
  thermal conduction
• Temperature minimum occurs in the place where
  there is a change in heating mechanism electron
  Ohmic heating below and ion frictional heating
  above.

19
Preprints

• Name email Institution

20
Lower cell vertical flow speed estimate

• Energy Equation
• Vertical velocity