4 Theoretical Aspects of Small Amplitude Oscillations:
Periods and Spatial Distribution

The usual interpretation of small amplitude oscillations is that some external agent excites MHD waves in the form of periodic disturbances of the cold plasma. MHD waves can be propagating or standing. In the first case, there is a periodic disturbance of the particles of the prominence plasma that may propagate in the medium. In the second case, the wave is confined to a region with fixed boundaries, thus producing the positive interference of propagating waves. Theoretical models usually consider small amplitude perturbations superimposed on an equilibrium configuration. Then the properties of propagating/standing MHD waves are analyzed. In the case of standing waves, we usually refer to the MHD eigenmodes of the system or to the modes for short.

Following our previous discussion of observations (Section 3.4), oscillations may affect individual threads, groups of threads or even larger areas of a prominence. The wave information (period, wavelength, phase speed, damping time) obtained from the analysis of this kind of events has been presented in Sections 3.5 and 3.6. Given that the main purpose of studying prominence oscillations is to gain a more profound understanding of their nature via seismological studies, it is necessary to study these oscillations theoretically. The information one expects to derive from these works consists of the main wave properties (period, wavelength, phase speed, damping time, spatial distribution, …). They can then be compared with the observationally determined values. The theory also allows us to determine the temporal variation of the perturbed magnetic field strength and its orientation, the perturbed density, temperature, etc., which means that these variables constitute another source of comparison with observations that will hopefully be exploited in the near future.

Theoretical works are here divided into five groups that reflect widely different choices of prominence equilibrium models: (a) simple, “toy” prominence models (Section 4.1); (b) models in which the prominence is represented as a plasma slab of finite width surrounded by the solar corona (Section 4.2); (c) line current prominence models (Section 4.3); (d) models of infinitely long prominence threads (Section 4.4); and (e) models concerned with the oscillations of prominence threads of finite length (Section 4.5).

 4.1 Oscillations of very simple prominence models
 4.2 Oscillations of prominence slabs
 4.3 Oscillations of line current models
 4.4 Fine structure oscillations (infinitely long thread limit)
  4.4.1 Individual thread oscillations
  4.4.2 Collective thread oscillations
 4.5 Fine structure oscillations (finite length threads)

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