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Astron. Astrophys. 335, 329-340 (1998)

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4. Classification of eigenmodes

When a coronal loop is disturbed, the nature of its response is determined by the spectrum. For k and R expression (7) forms two separate eigenvalue problems for the fast and the Alfvén modes. The fast magnetosonic spectrum is governed by


whereas the Alfvén spectrum is governed by


In the latter equation x shows up as a parameter : for each value of x the equation forms a Sturm-Liouville eigenvalue problem. When we vary x, the corresponding discrete spectrum [FORMULA] is smeared out in a discrete set of continua.

Fig. 1 shows the eigenfrequencies of the first three fast eigenmodes together with the upper and lower bound of the Alfvén continuous spectrum as function of k. As seen from the expressions (9) and (10) which describe the solutions as a superposition of eigenmodes, the possible values of k are multiples of [FORMULA]. The corresponding eigenfrequencies of the fast waves are indicated in Fig. 1 as dots.

[FIGURE] Fig. 1. The eigenfrequencies of the first three fast eigenmodes (gray lines) together withe the upper and lower bound of the Alfvén continuum (black lines) as function of k.

In the model of a plasma slab which is inhomogeneous in the x-coordinate and where gravity is ignored, the x-component of the displacement vector [FORMULA] and the perturbation of total pressure P are described by the two following equations:


where D
and where [FORMULA] and [FORMULA] denote the slow and Alfvén frequencies. [FORMULA] and [FORMULA] are the cut-off frequencies representing the points where oscillatory behaviour changes to evanescent behaviour or vice versa. In an inhomogeneous medium [FORMULA] and [FORMULA] vary with x and define 4 frequency intervals. For a fixed value of x the following sequence of inequalities is valid:


For a pressureless plasma with a uniform magnetic field (p), the values of [FORMULA] and [FORMULA] are:



The first cut-off frequency [FORMULA] equals zero which leads to the absence of slow waves in our model. Furthermore, for the uncoupled case k, the second cut-off frequency [FORMULA] equals exactly the Alfvén frequency [FORMULA]. Therefore fast modes corresponding to an eigenfrequency above the Alfvén continuum are travelling waves in the exterior coronal environment. In an open system there is a continuous spectrum of these modes above the cut-off frequency [FORMULA] and so these 'leaky modes' radiate their energy away from the loop. In our closed box model of a coronal loop they are artificially kept in the neighbourhood of the loop.

The modes with frequency within the range of the continuous spectrum and consequently under the cut-off frequency in the exterior coronal environment, oscillate inside the coronal loop but are evanescent outside the loop. These solutions correspond to what we call the 'body modes' of the loop. For these modes the energy is not radiated away but is stored inside the loop.

For k the body modes couple to localized Alfvén waves and form essentially quasi-modes (Tirry & Goossens 1996). Due to the resonant coupling, small length scales are generated around the resonant point which enhance dissipation and hence the heating of the loop.

If especially body modes are excited, a lot of energy is stored and is possibly converted into heat through the coupling with the Alfvén modes (i.e. when k). On the other hand, when most of the dominant excited modes are leaky modes, the energy is radiated away in the coronal environment. Thus the nature of the dominant excited modes determines the efficiency of the resonant absorption by Alfvén waves.

However we have some remarks about the coupled case: If we steadily increase the value of k from zero, first of all the cut-off frequency [FORMULA] no longer equals the Alfvén frequency but is larger. As a consequence the continuum of leaky modes is shifted upwards.

Secondly the discrete set of body modes will resonantly couple to localized Alfvén continuum modes unless its oscillation frequency shifts out of the Alfvén continuum (see e.g. Tirry & Goossens 1996). Berghmans & Tirry (1997) calculated the continuous change of a quasi-mode frequency with respect to k and showed that for relatively small values of k, generally the frequency does not move out of the Alfvén continuum. More details about the possible consequences of the coupling will be studied in the related paper by De Groof et al. (1998).

Since the nature of the excited modes in the uncoupled case can only slightly differ from the one in the coupled case, we study in the next section the amount of kinetic energy in both body modes and leaky modes, excited by radially polarized footpoint motions with k. The ratio of the kinetic energy in the body modes to the kinetic energy in the leaky modes will reveal much information about the efficiency of the resonant absorption in the footpoint driven problem.

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© European Southern Observatory (ESO) 1998

Online publication: June 12, 1998