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

P. Song1, and V. M. Vasyliūnas1,2

1. Center for Atmospheric Research and Department of Environmental, Earth & Atmospheric Sciences, University of

Massachusetts Lowell

2. Max-Planck-Institut für Sonnensystemforschung,37191 Katlenburg-

Lindau, Germany

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-electromagnetic 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.

Conditions in the Chromosphere

Avrett and Loeser, 2008

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

Radiative Losses/Required Heating

• Total radiation loss from chromosphere (not including photosphere): 106~7 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 .

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

VA

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 zV

H SV

Observation Range

Total Heating Rate from a Power-Law Source

0

12 2 21 0 0

0 21 1

1

2

21 0

0 0

00

0 0

11 3 ,2 2

where ,

1 '1( )

//

( )

( ) ( 1)

ni

n

ni At

y x

a

z

At ni

e in e i

tot n

n

n nQ z FV

x a e y dy

dz

z V

N N

F S d

FS n

Logarithm of heating per km, 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.

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

Local Thermal Equilibrium ConditionEnergy Equation

Time scale:~ lifetime of a supergranule:> ~ 1 day~105 sec

Heat flux: negligible (see next page)

Lower chromosphere: Optically thick mediumR~ 10-1 erg cm-3 s-1 (Rosseland approximation) Q~ 100 erg cm-3 s-1 Convection ~ 100 erg cm-3 s-1 (for p~105 dyn/cm2)If R<<Q

Upper chromosphere: Optically thin mediumQ/NNi~~ 10-26 erg cm3 s-1

Convection, r.h.s., ~ 10-28 erg cm3 s-1 (for N~Ni~1011 cm-3, p~10-1 dyn/cm2)

Convection is negligible in the upper chromosphere: Q/N=N i

Convection in the lower chromosphere may be importantTemperature T increases with increasing heating rate per particle Q/N

5/3

3 log / ,2

/ ( )

i i

i

Q R Q kT d pN Q N NN N N dtQ N N T

5/3

1 3 log2

Q R d pT kTN N N dt

24

2

16 43 R

R F T T T QN N z N HT z T z

5/3

3 log2

Q pkT TN

V

Heat Transfer via Thermal ConductionPerpendicular to B: very smallParallel 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 greater than the heating rate) important to provide for high rate of radiation

2 5 4 16 7 -3 -1/ ~ 10 erg cm sconductQ T L

5/ 25 -1 -1 -1

5

0.224 4.67 4.18 10 erg s K cmln

10

TZ

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

hh h

dp

dt V

j B

Vertical Force Balance• Momentum equation:

• or:

• Average over horizontal dimensions (steady state)

• Vertical acceleration– Upward T>Tm

– Downward T<Tm

• Vertical flow produces additional pressure imbalance because of the different temperature and density the flow carries

zdV p gdt z

lnz zz

V V T pV gt z m z

ln ln 1 1, where m m

p p mgz z T T T

1z

m

dV Tg

dt T

lnzzV T pV gz m z

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. • @1000 km, Tm=6200, from radiation function table: Nimm=2x10-27x2x1011. (From our

model, Qm/Nm ~10-18 since Qm cannot be used quantitatively), assume: Ni/Nimm=NmQ/NQm

• Since Q/N~6Qm/Nm, TX=6mTmXm, where X=Ni/N, from radiation function: T= 6600. T/Tm=1.06, T/Tm-1=0.06

• @2000 km Tm=6700

• Nimm=1.2x10-26x4x1010, Q/N=6Qm/Nm, T=7100, T/Tm=1.06• g=274m/s2

• Vz~sqrt(2x274m/s2x0.06x106m)=sqrt(33x106)=6 km/s, a number within possible range, but maybe too big if Vx is LVz/H=30000/2/200*Vz=450km/s, supersonic. Cs=10 km/s

2

2

1 12

2 1

z z

m

zm

dV V Tgdt z T

TV g zT

Circulation: Connecting the Vertical Flow

with Horizontal Flow• Continuity equation

• 2-D Cartesian coordinates

• Horizontal momentum equation

ln ln zh z h

VV

z z

V V

0 V

200 or ~ ~15000

x zz x x x

V V HV V V Vx H L

1 1 or x x x z

x zV V V Vp p

V Vx z x H x

Inhomogeneous Heating: Chromosphere Circulation

Chromospheric Circulation: Distortion of Magnetic Field

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.

• The circulation may distort the field lines into a funnel-canopy shaped geometry

Preprints

• Name email Institution

Lower cell vertical flow speed estimate

Energy Equation

Vertical velocity 7 0 5 2

3 ln 5 ln2 2

/ ~ 2 10 10 /10 10 /

zz z

z

Q p VV Vp z z H

V HQ p cm s