## 2. Stationary equilibrium state
We consider a plasma configuration consisting of two semi-infinite
homogeneous regions, separated by a nonuniform layer. The
axis in a Cartesian coordinate system is
perpendicular to the boundaries of the nonuniform plasma layer that
are located at and . The
basic state quantities depend on the variable The magnetic field is assumed homogeneous throughout the whole space while the plasma flow with the velocity parallel to the magnetic field, exists for only: The flow speed is discontinuous at while the other physical quantities, such as the density and the temperature , 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: This means that the thermal pressure is uniform due to the assumption const, and that we can freely specify either the plasma density or the temperature . For analytical and numerical reasons, we prescribe the dependence of the cusp speed instead, and express the other basic state quantities in terms of . The square of the cusp speed is defined in terms of squares of the Alfvén speed and of the speed of sound as Here, is the ratio of specific heats while is the ratio of the thermal to the magnetic pressure: with . Clearly, 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 as: 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: The basic state profiles (3) are prescribed by the values of and in Eq. (4) for the cusp speed. However, the speeds and 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 and the temperature ratio of the two uniform regions as follows: In our model, we assume i.e. and . Region 1 () is thus warmer but less dense than region 2 (). © European Southern Observatory (ESO) 1998 Online publication: September 30, 1998 |