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Astron. Astrophys. 332, 786-794 (1998)

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5. The numerical procedure

In the two outer regions of the sharp interface model, i.e. the two semi-infinite homogeneous regions, the solutions for [FORMULA] and P can be found analytically. However in the nonuniform intermediate region, the Eqs. (7) have to be integrated numerically. Application of the continuity condition for [FORMULA] and P at [FORMULA] and [FORMULA] finally yields the dispersion relation.

Since we are primarily interested in localized surface type eigenmodes, with evanescent amplitudes in both homogeneous regions to the left and the right of the interface, the related oscillation frequencies should lie in the intervals :

[EQUATION]

and

[EQUATION]

The analytical evanescent solution of (7) valid for [FORMULA] is simply

[EQUATION]

where [FORMULA] and A is an arbitrarily complex factor. Analogous, the analytical solution for [FORMULA] is given by

[EQUATION]

where [FORMULA] and B is an arbitrarily complex factor.

Thus the procedure for solving the eigenvalue problem for the MHD surface modes on the sharp interface is a shooting method from [FORMULA] to [FORMULA]. Starting at [FORMULA] with the evanescent analytical solution for the homogeneous region to the left of the interface, we numerically integrate the ideal MHD Eqs. (7). If a resonance is encountered during the calculations, then the dissipative solutions (11) or (13) are applied continuously between the endpoints [FORMULA] of the corresponding dissipative layer. After having passed through the dissipative layer the computations return to the ideal Eqs. (7) until the final point [FORMULA] is reached. Application of the continuity condition for [FORMULA] and P at [FORMULA] yields thus the dispersion relation which has to be solved for [FORMULA] with prescribed [FORMULA], [FORMULA] and V.

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© European Southern Observatory (ESO) 1998

Online publication: March 23, 1998
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