3. Stability analysis
3.1. Global stability analysis
The stability analysis is done in terms of the Lagrangian displacement expressed in the Frenet basis vectors as
with the unperturbed arclength , the equilibrium location and the Lagrangian displacement components , and in the tangential direction , the normal direction and the binormal direction of the unperturbed flux tube, respectively.
Linearisation of the time dependent thin flux tube equations with respect to and separation of the time dependence as
for the tangential and normal components. The decoupled binormal component of the perturbation equation yields stable Alfvén waves running along the flux tube. is a differential operator of second order whose coefficients , ,... are functions of the unperturbed arclength . For details see Schmitt (1995, 1998). For later application it is useful to transform the independent variable of Eq. (25) from the arclength to the horizontal distance . Except for the redefinition Eq. (25) remains unaltered. The coefficients are given in detail in the Appendix. Using the equilibrium model shown in Fig. 3 we obtain coefficients with steep gradients in the prominence-corona transition region and largely constant values in the prominence and coronal part of the flux tube Fig. 4.
It can be shown that the operator is self-adjoint thus yielding only real eigenvalues . The equilibrium is stable, if all eigenvalues are positive, and unstable, if at least one eigenvalue exists with .
As a consequence of the self-adjointness of it is possible to define the change of the potential energy due to perturbations as
which can be used to test the numerically obtained eigenfunctions and eigenvalues by a simple integration.
3.2. Boundary conditions and eigenfunctions
For solving the eigenvalue problem, boundary conditions have to be specified. The symmetry of the equilibrium model causes a symmetry of the coefficients with respect to the center of the prominence. The coefficient is antisymmetric, the other coefficients are symmetric. The structure of Eq. (25) allows for two different symmetries of the eigenfunctions with respect to :
As a consequence, it is sufficient to consider only one half of the flux tube and to use the boundary conditions
at the center of the prominence. We assume that the foot point of the arch is kept fixed in the photosphere of the sun, which implies the boundary condition for the normal component of the Lagrangian displacement. The tangential component does not need to vanish because a plasma flow along the flux tube induced by the perturbation shall not be excluded. However, taking into account the increase of density in the photosphere, which causes a decrease of any flow speed, the fixed boundary condition leads to similar results. We checked this numerically by modeling the transition from the corona to the photosphere and solving the eigenvalue problem with the less restrictive condition . No significant difference was found.
The first six eigenfunctions together with their eigenvalues are shown in Fig. 5. Note the steep decrease of the normal displacement and the tangential displacement near the origin for the antisymmetric and symmetric fundamental mode, respectively, displayed on an appropriate scale in Fig. 6. It is conspicuous that in all models only the eigenvalue of the antisymmetric fundamental mode with , is very small compared to the other eigenvalues and may become negative, which implies instability. Thus only the sign of this eigenvalue is important for the stability of the configuration. In the following section we discuss the dependence on the different prominence parameters of this eigenvalue.
The variation of the smallest eigenvalue as a function of the different model parameters
is investigated in order to study their influence on the stability of the configuration. For this purpose, different prominence models have been considered, where only one parameter was varied and the other parameters were kept fixed.
The variation of the smallest eigenvalue as a function of height, width and effective density contrast for prominences with a central temperature of and a corona temperature of is shown in Fig. 7. From Fig. 7 we deduce the tendency that an increase of the height and the effective density contrast causes a decrease of the critical width at which the prominence becomes unstable. The loss of stability with increasing height corresponds to the observational fact that prominences with normal polarity are usually observed at lower heights.
Fig. 8 is the counterpart of Fig. 7 for a prominence embedded in a corona of . A comparison with Fig. 7 shows the stabilizing effect of increasing the coronal temperature. This effect is correlated to the destabilization by increasing the height because the coronal part of the prominence solution is determined by the external pressure scale height. Thus an increase of temperature (and pressure scale height) is equivalent to a decrease of the prominence height. A similar result is obtained by increasing the central temperature of the prominence.
In Fig. 9 the variation of the eigenvalues with width and effective density contrast is shown for a prominence with a central temperature of , embedded in a corona. Compared to Fig. 8 the critical widths are increased up to about in the model with the effective density contrast of 10.
Note that these figures contain the full information of including an external magnetic field as described above. The connection between (real) density contrast, external magnetic field strength and effective density contrast is given by Eq. (18) which shows that the effective density contrast decreases with increasing field strength. Accordingly, (see the results displayed in Figs. 7, 8 and 9) an external magnetic field has a stabilizing effect. These arguments hold only up to a critical field strength because for the central dip of the flux loop vanishes and inside the flux tube a density inversion will occur. Rayleigh-Taylor instability, however, is not allowed for by our one-dimensional stability formalism.
If an external magnetic field is included, a prominence model with a central temperature of , an external temperature of and a density contrast of about 100 can be stable up to a width of if we use .
The temperature profile we used is a very good approximation for the central part of the prominence. In the coronal part of the flux tube are other profiles than possible. In order to demonstrate that the temperature in the coronal part does not affect the stability strongly, we present the results obtained with temperature profiles, having coronal values and (Fig. 10). Comparison of the left graph of Fig. 7 with Fig. 10 shows that changing the temperature only affects the critical width of models with a high effective density contrast. Decreasing the temperature in the coronal part of the flux tube means decreasing the internal pressure scale height, leading to an increasing gas pressure (and internal plasma beta) inside the flux tube towards the foot points. This can only affect the stability if the internal plasma beta is sufficiently high enough which is only the case in models with high effective density contrast. But the models with low effective density contrasts are more reasonable if an external magnetic field is included. This means that our restriction to the prescribed temperature profile is a good approximation to more realistic temperature profiles as long as we only consider the dynamical stability.
As mentioned above, changing the external gas pressure (keeping the internal temperature profile, the external plasma beta, external temperature and the (real) density contrast fixed) leads to solutions with the same path but rescaled profiles of pressure and density inside and outside the flux tube. It can also be shown that all coefficients of Eq. (25) have the same dependence on the external pressure, which thus has no effect on the linear eigenvalue problem.
So far we presented the stability analysis for a simplified flux tube model in which we prescribed the temperature profile. As mentioned above the models of Degenhardt (1995) and Cramphorn (1996) have small widths (lower than ) and thus show no instabilities consistent with the results discussed here. The polytropic model of Degenhardt & Deinzer (1993) is found to be unstable.
3.4. Interpretation of the instability
In order to get an intuitive idea of the instability, it is useful to transform the eigenfunctions back into cartesian coordinates. For the eigensolution with the smallest eigenvalue the vertical and horizontal displacements of the flux tube and are shown in Fig. 11.
Obviously the central part of the prominence moves only horizontally (here to the left), whereas the coronal part moves down on the left side and up on the right side. The main point is the fact that the dense plasma of the prominence does not need to flow upward against gravity in order to flow out of the central dip. If this instability occurs, the plasma flows horizontally at without changing its potential energy.
3.5. Oscillations of prominences
Although we are primarily interested in stability considerations, the analysis also provides a variety of oscillatory modes. Oscillations of prominences have recently received much attention, both observationally and theoretically. Observational data of prominences show long-term oscillations with periods of about and short-term oscillations with periods of about (Tandberg-Hanssen 1995, Sütterlin et al. 1997). A classification of the modes has been achieved on the basis of idealized models by Joarder & Roberts (1992, 1993), Oliver et al. (1993), Oliver & Ballester (1995) and Joarder et al. (1997). In the stable case the smallest eigenvalue of our models corresponds to periods larger than . The next two oscillations are in the range of about and the higher order oscillations have periods below . Our modes can be compared to the modes classified by Oliver et al. They distinguish between kink and sausage modes, which correspond to our antisymmetric and symmetric modes. Comparing eigenfunctions and eigenvalues there is a close correspondence between their hybrid slow (mainly horizontal motion) and our antisymmetric fundamental mode. These modes can be observed in prominences at the limb because of their dominating horizontal motion. Binormal displacements of the flux tube correspond to Alfvén modes.
We note that the inertia of the external plasma accelerated by the moving flux tube influences the eigenfrequencies. The exact description of this effect is still under controversial debate (Moreno-Insertis et al. 1996). As long as this effect is parametrized as an enhanced inertia in the normal component of Eq. (25) (Spruit 1981) this leads only to longer oscillation periods, but does not change the sign of the eigenvalues . This can be proven in the following way: The left hand side of Eq. (25) is modified by the substitution whereas the right hand side remains unaltered. Thus Eq. (27) which determines the sign of remains unaltered, too. Spruit introduced assuming a potential flow around the tube.
© European Southern Observatory (ESO) 1999
Online publication: March 1, 1999