SpringerLink
Forum Springer Astron. Astrophys.
Forum Whats New Search Orders


Astron. Astrophys. 339, 215-224 (1998)

Previous Section Next Section Title Page Table of Contents

2. Stationary equilibrium state

We consider a plasma configuration consisting of two semi-infinite homogeneous regions, separated by a nonuniform layer. The [FORMULA] axis in a Cartesian coordinate system is perpendicular to the boundaries of the nonuniform plasma layer that are located at [FORMULA] and [FORMULA]. The basic state quantities depend on the variable z only.

The magnetic field [FORMULA] is assumed homogeneous throughout the whole space while the plasma flow with the velocity [FORMULA] parallel to the magnetic field, exists for [FORMULA] only:

[EQUATION]

The flow speed is discontinuous at [FORMULA] while the other physical quantities, such as the density [FORMULA] and the temperature [FORMULA], have smooth profiles within the layer.

Since the effect of gravity is ignored in our treatment, the statics of the basic state is simply given by:

[EQUATION]

This means that the thermal pressure [FORMULA] is uniform due to the assumption [FORMULA]const, and that we can freely specify either the plasma density [FORMULA] or the temperature [FORMULA]. For analytical and numerical reasons, we prescribe the [FORMULA]dependence of the cusp speed [FORMULA] instead, and express the other basic state quantities in terms of [FORMULA].

The square of the cusp speed is defined in terms of squares of the Alfvén speed [FORMULA][FORMULA] and of the speed of sound [FORMULA][FORMULA] as

[EQUATION]

Here, [FORMULA] is the ratio of specific heats while [FORMULA] is the ratio of the thermal to the magnetic pressure:

[EQUATION]

with [FORMULA]. Clearly, [FORMULA]const in our model.

The squares of the Alfvén and of sound speed, the plasma density and the plasma temperature can be written in terms of [FORMULA] as:

[EQUATION]

To reduce the mathematical complications as much as possible but still keep the basic physics in our analysis, we consider a simple linear profile for the cusp speed:

[EQUATION]

The basic state profiles (3) are prescribed by the values of [FORMULA] and [FORMULA] in Eq. (4) for the cusp speed. However, the speeds [FORMULA] and [FORMULA] are not convenient from a practical point of view as they cannot be estimated in a straightforward way. For this reason, we express them in terms of [FORMULA] and the temperature ratio [FORMULA] of the two uniform regions as follows:

[EQUATION]

In our model, we assume [FORMULA] i.e. [FORMULA] and [FORMULA]. Region 1 ([FORMULA]) is thus warmer but less dense than region 2 ([FORMULA]).

Previous Section Next Section Title Page Table of Contents

© European Southern Observatory (ESO) 1998

Online publication: September 30, 1998
helpdesk.link@springer.de