Astron. Astrophys. 333, 1069-1081 (1998)

2. Geometry and assumptions

As is suggested e. g. by H -filtergrams, the chromosphere is a highly structured region, and it is highly variable on long time-scales of some days (e. g. chromospheric network) as well as on short timescales of minutes and seconds (e. g. spicules or bright points). Thus the first conclusion would be that a stationary description of the atmosphere is impossible.

Nevertheless, stationary and steady models, like those of Vernazza et al. (1981), contributed a lot to the understanding of the relevant processes in the atmosphere. Even if the chromosphere is in a highly non-stationary state, a stationary model should give us a basic idea of the relevant processes by describing a "mean chromosphere", which probably does not exist in the real solar atmosphere. This philosophy corresponds with the one of climate models: without resolving the weather, these can describe some of the basic mechanisms leading to the global behaviour of e. g. the temperature runs in the earth's atmosphere.

As the main aim of this paper is to study in detail the source region of the solar wind, the application of a stationary model is suggested also by recent ULYSSES results: Barnes et al. (1995) found, that the particle flux density in the fast solar wind (normalized to 1 AU) is nearly constant, regardless of heliocentric longitude, latitude and distance or time in the solar cycle. Thus if the interest is in a mean behaviour, it is justified, given the steadiness of the fast wind, to apply stationary conditions also in the chromosphere.

2.1. Geometry

Concerning the geometry of the source regions of the solar wind, for the fast and the slow wind the following two pictures may be applicable (see Fig. 1).

Fig. 1. Sketch of the geometry in the source regions of the fast and slow wind. (see text).

1. Fast solar wind
Between the super-granulation cells vertical magnetic field emerges, which widens up to form the so called canopy and build up the coronal funnels (Dowdy 1986; Fig. 1a). A possible scenario is that the fast wind leaves the sun through these funnels. At the bottom of the funnels, in the chromosphere, the magnetic field is vertical and a one-dimensional stratification is a good approximation (see Fig. 1c)

The velocity in the bottom region of the funnel can be calculated in the following way: At the earth's orbit, at 1 AU, the particle flux density in the high speed wind is (Schwenn 1990). Mapping this flux back to the chromosphere, by using the geometry factors of the spherical expansion from to 1 AU (1/215²), the over-spherical expansion in the fast wind (1/7 after Kopp & Holzer 1976) and the partial filling of the solar surface by coronal funnels ( % after Athay 1981), leads to a particle flux at the bottom of the funnels, i. e. in the chromosphere, of . Using the density of at 8000 K from the atmosphere model of Vernazza et al. (1981), this flux leads to a velocity of the order of 500 m/s.

2. Slow solar wind
In this case the situation is much more complicated. But one possible scenario may be the following: at the top of large coronal loops material is accumulated because of a continuous flow into the loop at its footpoints (see Fig. 1b). Thus from time to time the loops in the equatorial streamer belt have to open and let the accumulated material go out into interplanetary space, forming the variable slow wind (see the recent SOHO observations of Sheeley et al. 1997). In the chromospheric lower part of the loop, which is small compared to the whole loop, the conditions are locally comparable to those at the base of the coronal funnels.

Even if these are very simplifying scenarios they do account for the basic geometric properties as known today. In both cases a one-dimensional stratified atmospheric layer can be assumed to exist in the chromosphere, if the interest is in its the mean behaviour as the source region of the solar wind.

2.2. Assumptions

Besides the so far discussed assumptions - time stationarity, homogeneous vertical magnetic field and one-dimensional stratification - some more obvious assumptions are made.

The material in the chromosphere is ionized by the UV radiation coming from higher layers. This radiation is (partly) absorbed in the chromosphere (see Sect. 3.2). This is a strong simplification - normally the full problem of the radiative transport has to be considered. But up to now no models are available which can handle the radiative transport and the plasma dynamics self-consistently. As a first step this paper concentrates on the latter aspects, an approach which leads to a simplification in the treatment of the radiation.

In the chromosphere the thermal coupling between the different species is still strong enough to equilibrate temperatures. Thus only one energy equation is used to describe the thermodynamics. Additionally, in this energy equation the effects of heating and radiative cooling are simply parameterized as functions of the temperature and density (see Sect. 3.3).

Finally it should be stated that the plasma is assumed to be quasi-neutral and bear no net current.

© European Southern Observatory (ESO) 1998

Online publication: April 28, 1998

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