Appendix A: the temperature profiles, source terms and area factor
The wave dissipation and plasma heating (perpendicular temperature) are described by the source term (see McKenzie et al. 1997)
with (for the case we use) r, H, and C, C, where the values of C and C are in SI units. The second term in Q represents extended heating. The area factor for the flow tube is based on the magnetic field model of Banaszkiewicz et al. (1997) which consists of a dipole, quadrupole and current sheet, so
where a and q, c if r is in units of .
T, T, T and T denote respectively electron temperature, `wave temperature' (defined by p) and the perpendicular and parallel plasma temperatures as obtained from single-ion model. In our calculation T, T are unchanged while the ion temperature profiles we assume to be given by
(m is the effective particle mass in the one-ion model, where 5% alpha admixture was assumed). The constant factor K allows us to consider the case when T is larger than the mass ratio m. The resulting temperature profiles are shown in Fig. 1.
To specify the source terms Q in the wave force (Eq. (3)) we would need to know the parameters in the energy equations corresponding to the above temperature profiles. Lacking this, we assume that
where , and Q is given by Eq. (A1). This is approximately consistent with constant T ratio if the heating equation is of the form
Appendix B: the critical solution
After eliminating the electric field from Eq. (1) by using the electron equation the i momentum equations can be written in the form (i)
We use c, c, c and write the wave force as
The explicit form of the coefficients g can be obtained from Eq. (3).
The system Eq. (B1) is singular at det unless the numerators vanish. The singularity appears when the condition for existence of stationary waves is met for one of the compound modes of the system (McKenzie et al. 1993). For the stationary flow equations under consideration this singularity (and the critical point structure) depends on the formulation: the manifold det would be different if, instead of temperature, we would specify the entropy profiles because the expressions for the sound speeds would be modified.
In the space the det surface consists of two parts corresponding to the two eigenvalues (slow and fast modes) of the matrix a. Acceleration from subsonic to supersonic speeds requires the solution to cross through each. This is possible provided that the numerators on the right hand side of the Eq. (B1) vanish which defines the critical curves on the respective det surface sheets. In our approach the critical solution is constructed by first finding (numerically) the two critical lines and then searching for the pair of points on them that can be linked by a solution trajectory. We use the dummy time variable such that to re-formulate the problem as the 3-dimensional autonomous system with the critical line singularities becoming the lines of fixed points. The relevant segments of the critical lines consist of the saddle point type critical points and the solution crosses them along the separatrix directions, so that integration away from the critical lines is stable.
© European Southern Observatory (ESO) 1998
Online publication: June 12, 1998