A Model of the Chromosphere: Heating, Structures, and Convection P. Song 1, and V. M. Vasyliūnas 1,2 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): 10 6~7 erg cm -2 s -1. Radiative loss rate: –Lower chromosphere: erg cm -3 s -1 –Upper chromosphere: erg cm -3 s -1 Power carried by solar wind: 10 5 erg cm -2 s -1 Power to ionize: small compared to radiation Observed wave power: ~ 10 7 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 VAVA

Damping as function of frequency and altitude Reardon et al., km 1000 km

Observation Range

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 n=5/3, ω 0 /2π=1/300 sec and F 0 = 10 7 erg cm -2 s -1.

Heating Rate Per Particle Logarithm of heating rate per particle Q/N tot 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 Condition Energy Equation Time scale: ~ lifetime of a supergranule:> ~ 1 day~10 5 sec Heat flux: negligible (see next page) Lower chromosphere: Optically thick medium R~ erg cm -3 s -1 (Rosseland approximation) Q~ 10 0 erg cm -3 s -1 Convection ~ 10 0 erg cm -3 s -1 (for p~10 5 dyn/cm 2 ) If R<<Q Upper chromosphere: Optically thin medium Q/NN i ~  ~ erg cm 3 s -1 Convection, r.h.s., ~ erg cm 3 s -1 (for N~N i ~10 11 cm -3, p~10 -1 dyn/cm 2 ) Convection is negligible in the upper chromosphere: Q/N=N i  Convection in the lower chromosphere may be important Temperature T increases with increasing heating rate per particle Q/N

Heat Transfer via Thermal Conduction Perpendicular to B: very small Parallel to B: Thermal conductivity: Conductive heat transfer: (L~1000 km, T~ 10 4 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~10 6 K, L~100 km): Q conduct ~ erg cm -3 s -1 : (comparable to greater than the heating rate) important to provide for high rate of radiation

Horizontal Force Balance Momentum equation Force imbalance is mitigated by sound or fast waves Time scales: –Sound wave: 1.5x10 4 km/10 km/s ~ 10 3 sec –Alfven speed in the upper region: 10 4 km/s, ~10 0 sec –Compared with the time scales of the pressure imbalance creation by heating: (from energy equation) 10 5 sec in the lower chromosphere 10 3 sec in upper chromosphere Lower chromosphere: horizontally pressure balanced Upper chromosphere: pressure in higher heating region may be higher

Vertical Force Balance Momentum equation: or: Average over horizontal dimensions (steady state) Vertical acceleration –Upward T>T m –Downward T<T m Vertical flow produces additional pressure imbalance because of the different temperature and density the flow carries

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 km, T m =6200, from radiation function table: N im  m =2x x2x (From our model, Q m /N m ~ since Q m cannot be used quantitatively), assume: N i  /N im  m =N m Q/NQ m Since Q/N~6Q m /N m,  TX=6  m T m X m, where X=N i /N, from radiation function: T= T/T m =1.06, T/T m km T m =6700 N im  m =1.2x x4x10 10, Q/N=6Q m /N m, T=7100, T/T m =1.06 g=274m/s 2 Vz~sqrt(2x274m/s 2 x0.06x10 6 m)=sqrt(33x10 6 )=6 km/s, a number within possible range, but maybe too big if V x is LV z /H=30000/2/200*V z =450km/s, supersonic. C s =10 km/s

Circulation: Connecting the Vertical Flow with Horizontal Flow Continuity equation 2-D Cartesian coordinates Horizontal momentum equation

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 Institution

Lower cell vertical flow speed estimate Energy Equation Vertical velocity