Astron. Astrophys. 330, 1070-1076 (1998)

Appendix A

Eqs. (9) and (10) determine a boundary layer problem (cf. Nayfeh 1981). It can be solved by expanding the linearized equations for the outer region and a thin (resistive) region inside the current layer and by asymptotically matching the solutions of the expanded equations (cf. Janicke 1982; Otto 1991). In order to solve Eqs. (9) and (10) in the outer region these equations are expanded in a scaling with . In the following the tilde is omitted for the perturbed quantities denoted by the index 1. Assuming and Eq. (9) for the outer region reduces to

with the solution

In order to solve Eq. (9) and (10) in the inner region one applies a scaling with and and Taylor expansions for symmetric (with respect to ) quantities (like and ) of the form and for of the form . With these expansions Eqs. (9) and (10) can be combined to give (for more details see Otto 1991):

where is an integration constant. Eq. (A3) can be related to the differential equation for the hypergeometric functions which provides us with the solution (cf. again Otto 1991):

where is a complicated combination of Kummer functions and the constant is given by

with

Expansion for of the inner solution gives

and expansion of the outer solution for small arguments of y leads to

The dispersion relation is obtained by a matching of the expanded solutions in significant order

which gives Eq. (12).

© European Southern Observatory (ESO) 1998

Online publication: January 27, 1998
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