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Astron. Astrophys. 331, 392-404 (1998)

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7. Conclusions

We developed a new method in the framework of the full understanding of both Alfvén and slow resonant absorption. This method is a derivative of the numerical scheme, used before to calculate resonant absorption in non-uniform flux-tubes. This scheme included the SG&H prescription to treat both slow and Alfvén resonant absorption and used numerical integration to deal with the non-uniform internal region.

We extended this method to derive the spectrum of eigenmodes and the spectrum of optimal driving frequencies for 1D non-uniform flux-tubes. We used the equality of the scattered and the transmitted normal acoustic impedances to select eigenfrequencies and the equality of incoming and transmitted acoustic impedances to find the optimal driving frequencies. These impedance criteria prove extremely useful and are rather transparant to work with. This method allows for general complex frequencies and therefore for the possibility for acoustic leakage into the surroundings.

We recovered the results for uniform flux-tubes found by Cally (1986) for both sausage and kink modes. The overall picture is satisfyingly similar, although the equilibria are not identical. But some small differences were noticed. We found a few of the slightly leaky modes that seems to start the infinite, anti-Sturmian set of non-leaky body modes that Cally has found. And when [FORMULA] we completed the cubic surface perturbation (a) for small [FORMULA], while Cally found this for [FORMULA] only.

The recovery of these results for uniform tubes allowed us to extend to inhomogeneous tubes. We fixed the equilibrium-parameters and kept them the same throughout all the calculations. In general there seems to be a one-to-one correspondence with the modes for uniform tubes. Only the fast surface mode found for uniform tubes is not recovered for inhomogeneous tubes when [FORMULA]. When [FORMULA] all of the modes become leaky in a non-uniform tube due to the resonances.

Due to the non-uniformity and the associated absence of discontinuity, the clear distinction between surface and body-modes has faded, as seen for the fast surface mode that we found in the uniform case and followed in the transition to non-uniform tubes. The role of the boundary in the uniform tubes is reduced to merely a place where some of the derivatives of equilibrium variables are not continuous and is partly taken over by the Alfvén resonant point. A surface perturbation in a uniform tube is now shifted towards the resonance. The Alfvén resonance also marks the regions where oscillating or exponential behaviour is expected, as did the boundary of a uniform tube.

We defined the optimal driving frequencies as those frequencies for which 100% absorption occurs and looked for them starting from the eigenfrequencies. There are two kinds of optimal driving frequencies. The first category are the physically uninteresting, apparent optimal drivers, which have a positive imaginary part and thus an exponentially growing amplitude. Most of the sausage optimal drivers and some of the kink optimal driving frequencies (the so-called trig modes) fall into this category. They are found as the complex conjugates of the eigenfrequencies.

In addition to these, we also find a category of physically interesting optimal driving frequencies. We found only one genuine optimal driving frequency for the [FORMULA] -case. When [FORMULA], each eigenfrequency, lying in the Alfvén continuum has a corresponding genuine optimal driving frequency.

Since the transmitted normal acoustic impedance appears in both impedance criteria, eigenfrequencies and optimal driving frequencies are ultimately connected. By that the genuine optimal driving frequencies can be found by using the eigenfrequencies as a first approximation. Thus we could reevaluate the results obtained by Goossens & Hollweg (1993) and Keppens (1996), who realized this connection between eigenmodes, optimal driving frequencies and maxima in the absorption coefficients for real driving frequencies. For real driving frequencies total absorption can not be expected for the cases considered here, since all optimal driving frequencies found have a non-zero imaginary part.

Finally, we call for a similar investigation of the close connection between the slow continuum, the complex leaky modes, the optimal driving frequencies, etc., in twisted flux tube equilibria. The presence of the twist in the equilibrium may be important when studying their absorption and scattering properties. This will be addressed in a forthcoming paper.

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© European Southern Observatory (ESO) 1998

Online publication: February 4, 1998