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

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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 [FORMULA] with [FORMULA]. In the following the tilde is omitted for the perturbed quantities denoted by the index 1. Assuming [FORMULA] and [FORMULA] Eq. (9) for the outer region reduces to

[EQUATION]

with the solution

[EQUATION]

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

[EQUATION]

where [FORMULA] 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):

[EQUATION]

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

[EQUATION]

with

[EQUATION]

Expansion for [FORMULA] of the inner solution gives

[EQUATION]

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

[EQUATION]

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

[EQUATION]

which gives Eq. (12).

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© European Southern Observatory (ESO) 1998

Online publication: January 27, 1998
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