  It is found that both microflare-sized impulsive heating at one leg of the
loop and a suddenly imposed velocity perturbation can propel the prominence to
oscillate along the magnetic dip. An extensive parameter survey results in a
scaling law, showing that the period of the oscillation, which weakly depends
on the length and height of the prominence, and the amplitude of the
perturbations, scales with $\sqrt{R/g_\odot}$, where $R$ represents the
curvature radius of the dip, and $g_\odot$ is the gravitational acceleration of
the Sun. This is consistent with the linear theory of a pendulum, which implies
that the field-aligned component of gravity is the main restoring force for the
prominence longitudinal oscillations, as confirmed by the force analysis.
However, the gas pressure gradient becomes non-negligible for short
prominences. The oscillation damps with time in the presence of non-adiabatic
processes. Compared to heat conduction, the radiative cooling is the dominant
factor leading to the damping. A scaling law for the damping timescale is
derived, i.e., $\tau\sim l^{1.63} D^{0.66}w^{-1.21}v_{0}^{-0.30}$, showing
strong dependence on the prominence length $l$, the geometry of the magnetic
dip (characterized by the depth $D$ and the width $w$), and the velocity
perturbation amplitude $v_0$. The larger the amplitude, the faster the
oscillation damps. It is also found that mass drainage significantly reduces
the damping timescale when the perturbation is too strong.
