*Astron. Astrophys. 335, 303-308 (1998)*
## 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
helpdesk.link@springer.de |