2. The model
We consider the flow along a tube of infinitesimal cross section A starting at the base of the corona. The magnetic field is parallel to the flow and the flow tube is directed radially outward. By n, p, p, m, Z and u we denote the number density, partial pressure (perpendicular or parallel to the field), particle mass, electric charge number and the flow speed of the j-th (j) ion species; the subscript e denotes the electron component. , are the mass and the radius of the Sun. The magnetic flux and the particle flux through the tube are conserved: BA, n (j), n. For electrons we assume isotropic temperature T and take the zero m limit in the momentum equation which then determines the electric field E. The momentum equation for the i-th ion component is (i):
in which E is given by
In addition there are charge neutrality and zero electric current constraints. The system is closed by assuming that the temperature profiles are prescribed as above and the flow tube area factor is given by the magnetic field model of Banaszkiewicz et al. (1997).
In Eq. (1) denote the characteristic collision frequencies associated with Coulomb friction between the ion components and are the correction factors as given in Leer et al. (1992). The wave force F acting on the i-th ion species (i) is given by
where p is the wave pressure. The first term represents the non-resonant interactions (McKenzie). The second (see Isenberg & Hollweg 1982, 1983, Isenberg 1984) is induced by presence of wave dissipation, including the resonant contribution (McKenzie & Marsch, 1982). The source terms Q determine heating of the ion components and the wave dissipation. The latter can be expressed as an equation for the decay of the wave action
Here u and V denotes the phase speed of the waves, given by the dispersion formula
We take into account only the outward propagating waves. Eq. (5) corresponds to the cold plasma case. In hot anisotropic plasma the terms proportional to p would appear in the dispersion formula. We decided to keep the cold plasma expression for consistency with McKenzie et al. (1997). The induced error should not be large, because the Alfven speed much exceeds the thermal speeds.
The dispersion formula we use is not applicable in the high frequency region, in particular near the ion-cyclotron frequency. This implies an inconsistency in our model, because the main heating process is assumed to be due to ion-cyclotron resonance. However, in distinction to Isenberg (1984) and Isenberg and Hollweg (1983), the main term in the heat source (Eq. (A1)) is in our model concentrated within from the solar surface, so that the description of the outside region should be less affected by the approximation. A consistent treatment of the resonant interaction would require not only a kinetic description of ion-cyclotron damping but also the evolution of the wave spectrum, which is beyond the scope of this model.
When the evolution of wave action (Eq. (4)) is taken into account, the wave force diverges as u: this is related to the streaming instability. When Q there is another apparent divergence at V. This should not appear in a fully consistent treatment, because the waves which are at rest relative to the i-th component should not contribute to its heating (Q should become zero). We simulate this by taking Q with a given function of heliospheric distance. Similar form of Q (with power of V dependent on the wave spectrum) was used by Isenberg and Hollweg (1983). The resonant interaction is expected to switch off when relative flow speed u exceeds a critical value which is lower than the Alfven speed (McKenzie and Marsch, 1982) but we do not include this feature in our approximate expression for the source. However, at the distances where the relative speed becomes comparable with the Alfven speed, the resonant effects are in our model not dominant.
© European Southern Observatory (ESO) 1998
Online publication: June 12, 1998