### 4.3 Oscillations of line current models

A completely different approach, based on line current models of filaments, was taken by van den Oord
and Kuperus (1992), Schutgens (1997a,b), and van den Oord et al. (1998) in order to study filament
vertical oscillations. They used the model introduced by Kuperus and Raadu (1974), in which the
prominence is treated as an infinitely thin and long line, i.e., without internal structure. The interaction of
the filament current with the surrounding magnetic arcade and photosphere was taken into
account. Furthermore, both normal (NP) and inverse polarity (IP) configurations were considered.
When a perturbation displaces the whole line current representing the filament, that remains
parallel to the photosphere during its motion, the coronal magnetic field is also disturbed and the
photospheric surface current is modified. This restructuring affects the magnetic force acting on
the filament current. As a consequence, either this force enhances the initial perturbation and
the original equilibrium becomes unstable, or the opposite happens and the system is stable
against the initial disturbance. As a further complication, van den Oord and Kuperus (1992),
Schutgens (1997a,b), and van den Oord et al. (1998) took into account the finite travel time of
the perturbations between the line current and the photosphere and investigated the effect of
these time delays on the filament dynamics. For both NP and IP configurations, exponentially
growing or decaying solutions were found, which means that perturbations are amplified and the
equilibrium becomes unstable, or that oscillations are damped in time and the equilibrium is
stable.
Schutgens and Tóth (1999) considered an IP magnetic configuration in which the prominence is not
infinitely thin but is represented by a current-carrying cylinder. They solved numerically the
magnetohydrodynamic equations assuming that the temperature has a constant value (10^{6} K) everywhere.
The inner part of the filament is disturbed by a suitable perturbation that causes the prominence to move
like a rigid body in the corona, both vertically and horizontally, undergoing exponentially damped
oscillations. Horizontal and vertical motions can be studied separately since they are decoupled. It turns out
that the period and damping time of horizontal oscillations are much larger than those of vertical
oscillations. Some remarks about the damping mechanism at work in these models is presented in
Section 5.6.