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

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6. Results: optimal driving frequencies

6.1. Sausage modes: [FORMULA]

The complex optimal driving frequencies are deduced as the nodes of the function G (26). Comparing equations (25) and (26) yields only a difference in the kind of Hankel functions. For finding the eigenvalues we use the Hankel functions of the first kind (outgoing), and to find optimal driving frequencies we use Hankel functions of the second kind (incoming).

The Hankel functions of the 1st and 2nd kind have a connecting equation which tells us exactly where to look for optimal driving frequencies in the case of no resonance. Since:


(where the asterisk denotes the complex conjugate) and since the complex conjugated solutions for a frequency equal the solutions for the complex conjugated frequency, it follows that the optimal driving frequencies are the complex conjugates of the eigenfrequencies in the case where [FORMULA] and as long as they are lying above the slow continuum (there is no Alfvén resonance when [FORMULA]).

These optimal driving frequencies with a positive imaginary part give, as noted by Keppens (1996) only an apparent 100% absorption. At these frequencies, one drives with an amplitude that grows in time, so that the lag of the amplitude growth of the outgoing wave with respect to the incoming perturbance appears as a total absorption.

The only sausage eigenmode for which it is interesting to look for its corresponding optimal driving frequency, when [FORMULA], is the d-mode (mode corresponding with the slow surface mode in the uniform case). Indeed this mode is lying in the slow continuum and the simple way to deduce the corresponding optimal driver is no longer valid. We do indeed find for the corresponding optimal driver of this mode a negative imaginary part, which implies no apparent 100% absorption, but a physically interesting optimal driving frequency, based only on the slow continuum.

6.2. Kink modes: [FORMULA]

In the search for the kink modes, we used the arguments put forward in Goossens & Hollweg (1993) and Keppens (1996). The general results there are that
1) total absorption at a real frequency only takes place when the equilibrium is fine-tuned, since total absorption requires the simultaneous equality of resistances (real parts of impedances) and reactances (imaginary parts)
2) under most conditions, the difference [FORMULA] is small, which is why leaky modes, optimal driving frequencies, and the real driving frequencies corresponding to maxima in the absorption coefficients are all closely connected.

This means that, when looking for optimal driving frequencies in the [FORMULA] -case, we can use the eigenfrequencies as input-frequencies and expect to find the optimal driving frequencies in the neighbourhood of these eigenfrequencies. The complete spectrum of optimal driving frequencies for [FORMULA] is shown in Fig. 11.

[FIGURE] Fig. 11. Optimal driving spectrum for [FORMULA] for a non-uniform photospheric flux tube with the same equilibrium parameters as before. Compare with Fig. 9 to notice that the real parts of the different modes are quasi identical.

The trig modes can be treated in the same way as most of the sausage modes by taking the complex conjugate of the eigenfrequencies. Hence the trig optimal driving frequencies are again only apparent optimal drivers and are physically uninteresting. Even the first trig mode that does enter the Alfvén continuum has a positive imaginary part and is therefore an apparent optimal driver.

All the other modes are influenced by the slow or/and the Alfvén continuum and for all of the other modes, we do find a corresponding optimal driving frequency, with a real part very close to the corresponding eigenfrequency. Comparison of the upper parts of Fig. 9 and 11 hardly shows any difference in the real parts of these modes. In general we could say that the damping is larger for the optimal driving frequencies, especially for small [FORMULA].

When we consider a driven problem, the purely real driving frequency can never equal one of these optimal driving frequencies, since they all have a non-zero imaginary part. This means that when we drive with a frequency, we can never expect total absorption of the modes. We can only look for driving frequencies at which maximal (not 100 %) absorption occurs.

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© European Southern Observatory (ESO) 1998

Online publication: February 4, 1998