SEMI-EMPIRICAL 2-D MHD MODEL OF THE SOLAR CORONA ANDSOLAR WIND: ENERGY FLOW IN THE CORONA

ED SITTLER1, MADHULLIKA GUHATHAKURTA1 and RUTH SKOUG2

1NASA/Goddard Space Flight Center, Greenbelt, MD 20771, U.S.A.2Los Alamos National Laboratory, Los Alamos, NM, 87545, U.S.A.

Abstract. We have developed a semi-empirical 2-D MHD model of the solar corona and solar windfor which the major data inputs are white light coronagraph data and plasma and magnetic fielddata from the Ulysses spacecraft. With regard to the white light coronagraph data we have used datafrom Spartan 201–05 to construct our empirical models of the electron density and magnetic field.We then use conservations laws of mass, momentum and energy to compute estimates of the flowvelocity, effective temperature and effective heat flux as a function of radial distance and latitude. Wewill then compare our empirical model estimates with that of other theoretical models. An example,is the WKB contribution of Alfven waves to the effective temperature and effective heat flux. Wehave also investigated the importance of electron heat conduction in the context as past theoreticaland empirical models and present a preliminary description of a semi-empirical model of electronheat conduction.

1. Introduction

Using the semi-empirical 2-D MHD model originally introduced by Sittler andGuhathakurta (1999a) using Skylab and Ulysses data under solar minimum condi-tions we have now applied this model for the time period when the Sun was makingits transition toward solar maximum using Spartan 201–05 data in 1998. Themodel presented here is an expansion upon the work by Sittler and Guhathakurta(1999b) which introduced the notion of multiple current sheets with just a singleequatorial streamer under solar minimum conditions to a model using three stream-ers which were observed during the transition to solar maximum when multiplestreamer belts are evident in the white light coronagraph data; and then used amagnetic field model with current sheets co-located with each of the three streamerbelts. The details of this model will be discussed in more detail in the accompany-ing paper by Guhathakurta et al. (2001). Since the coronagraph data are observedat a specific interval in time, while Ulysses latitude scans can take more than ayear to do, we have used a hybrid boundary condition which applies for the 1998period at low latitudes, while at high latitudes it is more like solar minimum (seeGuhathakurta et al., this issue). The Ulysses data provides estimates of ne, Vr , andBr as a function of latitude to give us an effective ‘mass flux’ α = ρVr/Br for eachopen field line. Here we note that the mass density at the Alfvén critical point is

Space Science Reviews 97: 39–44, 2001.© 2001 Kluwer Academic Publishers. Printed in the Netherlands.

40 ED SITTLER ET AL.

ρA = 4πα2. We will now use these results to concentrate our discussions on theradial profiles of Teff and qeff for polar (θ = 1◦) and equatorial field lines (θ = 71◦)for which we indicate the field line co-latitudes at the base of the corona and applythem to such models of WKB Alfvén waves and electron heat flux.

2. Comparison of model with predictions of wave pressure and energy fluxterms due to Alfvén waves using WKB approximation

Here we will attempt to compare the wave pressure and heat flux due to Alfvénwaves within the context of our semi-empirical model of the solar wind. To beginwe display the expression for the wave amplitude of Alfvén waves as a function ofradial distance from the Sun 〈δV 2〉 = 〈δV 2

A〉(4χ/(1+χ)2) where χ = √ρ/ρA. We

then use the following definition of the wave pressure term and the wave energyflux PA = 〈δB2〉/8π and qA = (1/2V + 2.0VA)PA, where VA is the Alfvén speed,the brackets indicate time averages and we assume circularly polarized waves.The latter equation can be derived from the energy equation (see Belcher, 1971,who assumed linearly polarized waves). For this comparison we have assumeda δV = 20 km s−1 at base of corona. In Figure 1 we display our comparisonsfor polar and equatorial field lines. Near the Sun all cases are consistent with themodel calculations, while further out the polar wave predictions definitely violateobservations and model predictions. While equatorial solutions give better results.In polar regions the Twave solution tends to peak at the Alfvén critical point ∼ 10solar radii, while Teff tends to peak at 5 RS . If Alfvén waves are important in theheating and acceleration of the solar wind then they must be damped inside 10 RS

in order not to violate observations (see Roberts, 1989). The mid-latitude solutioncompares fairly well with observations but at large r is too high in both T and q.The equatorial solution compares well close to the Sun and far from the Sun, butthe wave pressure at about 10 RS is too high.

3. Heat Flux Calculations and Comparisons for Electrons

We will now attempt to see whether electron heat flux can account for our modelcalculations. We will begin by considering the empirical model by Feldman et al.(1975) based on solar wind electron observations at 1 AU. This model makes agood test case as to whether heat flux by suprathermal tails can produce the re-quired heat flux for all r. The empirical expression for the heat flux as described byFeldman et al. (1975) is shown as follows: qe = (nH�VH)(5/2)kTC(TH/TC − 1)where �VH = 700 km s−1, TH/TC = 6 and nH = 5% where C stands for coreelectrons and H for halo electrons. Assuming Teff = Te we used this expressionto compute radial profiles of the electron heat flux using the model results andis shown in Figure 2. As can be seen the model clearly shows that electrons canproduce a significant heat flux with a relatively small suprathermal component.

SEMI-EMPIRICAL 2-D MHD MODEL OF THE SOLAR CORONA AND SOLAR WIND 41

Figure 1. Plot of temperature and heat flux as a function of radial distance for polar and equatorialregions. The solid lines are from the model and correspond to Teff and qeff. The dashed curvescorrespond to Alfven waves Twave and qwave.

Figure 2. Plot of electron heat flux using empirical formula by Feldman et al. (1975) using modelparameterts.

To further show this point we show in Figure 3 plots of the electron mean freepath over radial distance as a function of r using our model results. As can beseen the thermal (core) electrons become collisionless outside 2 RS over the polesand that one expects significant suprathermal tails to develop for the electrons withcorresponding heat flux to accelerate the plasma to speeds approaching 800 km s−1.In the equatorial regions the electrons remain collisional out to 10 RS and onlymodest deviations from a thermal population are expected near the Sun and the

42 ED SITTLER ET AL.

Figure 3. Plot of Coulomb mean free path for electrons over radial distance as function of r usingmodel parameters.

Spitzer’s conductivity law should apply. When we apply Spitzer’s heat conductionprediction for the equatorial solar wind using radial variation of electron temper-ature empirically determined by Sittler and Scudder (1980) (i.e., TC = r−0.35) forTC ∼ 106 K at base of corona the computed heat flux is orders of magnitude lowerthan that required by our model calculations. Therefore, we feel that electron heatconduction is not important in the equatorial plane.

4. Semi-Empirical Model of Electron Heat Flux

We are in the process of developing a semi-empirical model of the electron heatconduction (see Sittler, 1978) which is based on both theoretical and observationalconsiderations. Most specifically we are most interested in applying this model tothe polar regions where suprathermal tails may form. This model is a ‘local model’which uses the Krook’s anzatz in the Boltzmann equation using the collision term(δf ∗/δt)coll = −(f ∗ − f ∗

0 )/τ where τ = (λw/wC)(u3/(u4 + λw/λcoul)) is the

collision time, λw is a characteristic ‘local’ scattering length due to waves, λcoul isa characteristic collision length ‘local’ due to coulomb collisions, u = (w/wc),wc

is the thermal speed of the core electrons, w is the electron speed in the electronproper frame, and f ∗ = f ∗

0 + f ∗1 where f ∗ is the proper frame electron distrib-

ution function, f ∗0 is the unperturbed electron distribution function and f ∗

1 is theperturbed correction term due to plasma gradients and collisions. We can the writethe following equation for f ∗

1 where we assume f ∗0 is a kappa distribution

SEMI-EMPIRICAL 2-D MHD MODEL OF THE SOLAR CORONA AND SOLAR WIND 43

f ∗1 = fκ(Peven + Podd)τ,

Peven = −(Vr

r

) {3

2αT c − αn − ακ

[h(κ) − κ ln

(1 + u2/κ

)] −

−ηu2 [αT c + ακ + 2v (3v + αn) sin2 θ∗] }

,

Podd = w cos θ∗

r

{3

2αT c − αn − ακ

[h(κ) − κ ln

(1 + u

κ

)]−

−ηu2

[αT c + ακ + 2Arr

w2

]},

where h(κ) = 2κβ(2κ−1)− 12 , Ar = eE∗

r /me, η = ((κ+1)/κ)(1/(1+u2/κ)) andαG = rd/dr(ln G) where β(z) is the beta function, E∗

r is the field aligned properframe electric field, ν = D/Dt (ln(B/n)) is the ‘viscosity’ term and G = TC, n,and κ . We can then impose constraint of particle conservation

∫f ∗e1 d3w = 0 the

zero current condition j ∗‖ = −e

∫f ∗e1w‖ d3w = 0 to give us a relationship for

the interplanetary potential. Using the above relationships and our empirical modelwe can compute the variation of κ as a function of r along B and then use therelationship q∗

e‖ = 12me

∫f ∗e1w

2w‖ d3w to give us the electron heat flux along B.In this model λw is a free parameter to be determined by comparing q∗

e‖ with qeff.At present this model is preliminary and we plan to use SUMER data (David et al.,1998) and Ulysses data (Maksimovic et al., 2000; Issautier et al., 1998) to constrainthe model from the corona to the distant solar wind.

5. Conclusion

In conclusion we have shown that the WKB approximation for Alfvén waves doesnot adequately explain our predicted effective temperatures and effective heat flux,although the equatorial regions gave better results. We determined that electronheat flux is not sufficient in the equatorial plane to explain observations, but in polarregions the electron gas becomes collisionless close to the Sun and electron heatconduction could be important for solar wind acceleration. We have proposed asemi-empirical model for electron heat conduction using the Krook’s anzatz. Usingour model we hope to constrain electron kappa and the ratio of λcoul/λwave.

References

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44 ED SITTLER ET AL.

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