Parametric survey of longitudinal prominence oscillation simulations
^{1} School of Astronomy and Space Science, Nanjing University, 210093 Nanjing, PR China
email: zhangqm@pmo.ac.cn
^{2} Key Laboratory for Dark Matter and Space Science, Purple Mountain Observatory, CAS, 210008 Nanjing, PR China
^{3} Key Lab of Modern Astronomy and Astrophysics, Ministry of Education, PR China
^{4} Centre for mathematical Plasma Astrophysics, Department of Mathematics, KU Leuven, Celestijnenlaan 200B, 3001 Heverlee, Belgium
Received: 8 November 2012
Accepted: 12 April 2013
Context. Longitudinal filament oscillations recently attracted increasing attention, while the restoring force and the damping mechanisms are still elusive.
Aims. We intend to investigate the underlying physics for coherent longitudinal oscillations of the entire filament body, including their triggering mechanism, dominant restoring force, and damping mechanisms.
Methods. With the MPIAMRVAC code, we carried out radiative hydrodynamic numerical simulations of the longitudinal prominence oscillations. We modeled two types of perturbations of the prominence, impulsive heating at one leg of the loop and an impulsive momentum deposition, which cause the prominence to oscillate. We studied the resulting oscillations for a large parameter scan, including the chromospheric heating duration, initial velocity of the prominence, and field line geometry.
Results. We found that both microflaresized impulsive heating at one leg of the loop and a suddenly imposed velocity perturbation can propel the prominence to oscillate along the magnetic dip. Our extensive parameter survey resulted in a scaling law that shows that the period of the oscillation, which weakly depends on the length and height of the prominence and on the amplitude of the perturbations, scales with √R/g_{⊙}, where R represents the curvature radius of the dip, and g_{⊙} is the gravitational acceleration of the Sun. This is consistent with the linear theory of a pendulum, which implies that the fieldaligned 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 significant for short prominences. The oscillation damps with time in the presence of nonadiabatic processes. Radiative cooling is the dominant factor leading to damping. A scaling law for the damping timescale is derived, i.e., τ~ 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. We also found that mass drainage significantly reduces the damping timescale when the perturbation is too strong.
Key words: Sun: filaments, prominences / Sun: oscillations / methods: numerical
© ESO, 2013
1. Introduction
Solar prominences, or filaments that appear on the solar disk, are cold and dense plasmas suspended in the corona (TandbergHanssen 1995; Labrosse et al. 2010; Mackay et al. 2010). They are formed above the magnetic polarity inversion lines. The denser material is believed to be supported by the magnetic tension force of the dipshaped magnetic field lines (Kippenhahn & Schlüter 1957; Kuperus & Raadu 1974; Guo et al. 2010; Zhang et al. 2012; Xu et al. 2012; Su & van Ballegooijen 2012). These fascinating phenomena attracted many simulation efforts from different aspects, such as their formation, oscillations, and eruptions. For the formation, the chromospheric evaporation plus coronal condensation model has been studied widely with onedimensional (1D) simulations (e.g., Müller et al. 2004; Karpen et al. 2005; 2006; Karpen & Antiochos 2008; Antolin et al. 2010; Xia et al. 2011; Luna et al. 2012b), where no backreaction on the field topology is accounted for. This was for the first time extended to 2.5D by Xia et al. (2012), who simulated the in situ formation of a filament in a sheared magnetic arcade and showed that the condensation selfconsistently forms magnetic dips while ensuring forcebalance states. This finding strengthens the analysis performed for prominence formation and evolutions, as adopted by many authors to date. Once a prominence is formed, it might be triggered to deviate from its equilibrium position and start to oscillate.
Observations demonstrate that prominences are hardly static. In addition to smallamplitude oscillations (Okamoto et al. 2007; Ning et al. 2009), largeamplitude and longperiod prominence oscillations have been observed (e.g., Eto et al. 2002; Isobe & Tripathi 2006; Gilbert et al. 2008; Chen et al. 2008; Tripathi et al. 2009; Hershaw et al. 2011; Bocchialini et al. 2011). The observations of the prominence oscillations led to the comprehensive research topic of prominence seismology (Blokland & Keppens 2011a,b; Arregui & Ballester 2011; Arregui et al. 2012; Luna & Karpen 2012; Luna et al. 2012a), and the longterm oscillations were considered as one of the precursors for coronal mass ejections (CMEs; Chen et al. 2008; Chen 2011). Of particular interest in this paper are the longitudinal oscillations along the axis of prominences/filaments, which were first presented in the simulation results of Antiochos et al. (2000) discovered from Hα observations by Jing et al. (2003). The phenomenon was investigated in more detail in Jing et al. (2006) and Vršnak et al. (2007). Such largeamplitude oscillations are triggered by smallscale solar eruptions near the footpoints of the main filaments, such as minifilament eruptions, subflares, and flares. The initial velocities of the oscillations are 30–100 km s^{1}. The oscillation period ranges from 40 min to 160 min and the damping times are ~2–5 times the oscillation period (Jing et al. 2006).
Unlike the transverse oscillations, whose restoring force is known to be the magnetic tension force, the dominant restoring force for the longitudinal oscillations still awaits to be clarified. Jing et al. (2003) proposed several candidates for the restoring force, i.e., gravity, pressure imbalance, and magnetic tension force. Vršnak et al. (2007) suggested that the restoring force is the magnetic pressure gradient along the filament axis. With radiative hydrodynamic simulations, Luna & Karpen (2012) and Zhang et al. (2012) suggested that the gravity component along the magnetic field is the main restoring force. Li & Zhang (2012), on the other hand, suggested that both gravity and magnetic tension force contribute to the restoring force. As for the damping mechanism, it really depends on the oscillation mode. For the vertical oscillations, Hyder (1966) proposed that the magnetic viscosity contributes to the decay. For the horizontal transverse oscillations, Kleczek & Kuperus (1969) proposed that the induced compressional wave in the surrounding corona acts to seemingly dissipate the oscillatory power. More damping mechanisms have been proposed, such as thermal conduction, radiation, ionneutral collisions, resonant absorption, and wave leakage (see Tripathi et al. 2009 and Arregui et al. 2012, for reviews). For the longitudinal oscillations, Zhang et al. (2012) found that nonadiabatic terms such as the radiation and the heat conduction contribute to the damping, but they might not be sufficient to explain the observed shorter timescale. In their simulations the chromospheric heating was switched off, so that the prominence mass was nearly fixed. Conversely, Luna & Karpen (2012) studied the prominence oscillations while keeping the chromospheric heating and the resulting chromospheric evaporation. As a result, the prominence was growing in length and mass during oscillations. The authors found that there are two damping timescales, a short one for the initial stage and a longer one later. The analytical solution indicates that the mass accumulation can explain the fast damping of the initial state. For the later slower damping, they suggested nonadiabatic effects such as radiation and heat conduction. A quantitative survey is necessary to clarify how different geometrical and physical parameters of the prominence affect the damping timescale.
Within the framework of gravity serving as the restoring force for the longitudinal oscillations of the filament, in this paper we perform a parameter survey with the aim to clarify how the geometry of the magnetic field affects the oscillation period and how the combined effects of radiation and heat conduction contribute to the damping of the oscillations. We describe the numerical method in Sect. 2. After showing the effects of the perturbation type in Sect. 3, we display the results of our parameter survey in Sect. 4. Discussions and summary are presented in Sects. 5 and 6.
2. Numerical method
Highresolution observations indicate that a filament/prominence consists of many thin threads that are believed to be aligned to the individual magnetic tubes (Lin et al. 2005). Since the magnetic field inside the filament is quite strong (Schmieder & Aulanier 2012), the plasma beta is very low (β ~ 0.01–0.1) (Antiochos et al. 2000; DeVore & Antiochos 2000; Aulanier et al. 2006), and the thermal conduction is strongly prevented across the field lines, the dynamics inside different magnetic tubes can be considered to be independent. Therefore, the formation and evolution of a filament thread can be treated as a 1D hydrodynamic problem. Following Xia et al. (2011), the 1D radiative hydrodynamic equations, shown as follows, are numerically solved by the stateoftheart MPI adaptive mesh refinement versatile advection code (MPIAMRVAC; Keppens et al. 2003; 2012), where ρ is the mass density, T is the temperature, s is the distance along the loop, v is the velocity of plasma, p is the gas pressure, ε = ρv^{2}/2 + p/(γ − 1) is the total energy density, n_{H} is the number density of hydrogen, n_{e} is the number density of electrons, and g_{∥}(s) is the component of gravity at a distance s along the magnetic loop, which is determined by the geometry of the magnetic loop. Furthermore, γ = 5/3 is the ratio of the specific heats, Λ(T) is the radiative loss coefficient for the optically thin emission, H(s) is the volumetric heating rate, and κ = 10^{6}T^{5/2} erg cm^{1} s^{1} K^{1} is the Spitzer heat conductivity. As in the previous works mentioned in Sect. 1, we assumed a fully ionized plasma and adopted a onefluid model. Considering the helium abundance (n_{He}/n_{H} = 0.1), we took ρ = 1.4m_{p}n_{H} and p = 2.3n_{H}k_{B}T, where m_{p} is the proton mass and k_{B} is the Boltzmann constant. The above equations are different from those in Luna & Karpen (2012) in that a uniform cross section was assumed here for the flux tube for simplicity, whereas expanding flux tubes based on given, immobile 3D magnetic fields were adopted in Luna & Karpen (2012). The radiative hydrodynamic Eqs. (1)–(3) were numerically solved with the MPIAMRVAC code, where the heat conduction term is solved with an implicit scheme separately from other terms. To include the radiative loss, we interpolated with the secondorder polynomial to compile a highresolution table based on the radiativeloss calculations using updated element abundances and better atomic models over a wide temperature range (Colgan et al. 2008). The corresponding values in this table are systematically higher by almost two times than the previous radiative loss function adopted by Luna & Karpen (2012).
It is often believed that a prominence is hosted at the dip of a magnetic loop, supported by the magnetic tension force. Therefore, we adopted a loop geometry with a magnetic dip that is symmetric about the midpoint, as shown in Fig. 1. The loop consists of two vertical legs with a length of s_{1}, two quartercircular shoulders with a radius r (the length of each arc, s_{2} − s_{1}, is πr/2), and a quasisinusoidalshaped dip with a halflength of w. The height of the dip is expressed as y = D − Dcos(πx/2w) if the local coordinates (x, y) are centered at the midpoint of the dip. The dip has a depth of D below the apex of the loop. This geometry determines the fieldaligned component of the gravity, whose distribution along the left half of the magnetic loop is expressed as follows: (4)where the gravity at the solar surface g_{⊙} = 2.7 × 10^{2} m s^{2}, the total length of the loop L, the length of each vertical segment s_{1} = 5 Mm, and s_{2} = s_{1} + πr/2 Mm. The total length of the dip is 2w = L − 2s_{2}. The fieldaligned component of the gravity in the right half is symmetric to the left half. The parameter h = s_{1} + r − D gives the height of the central dip above the lower boundary.
Fig. 1 Magnetic loop used for the 1D radiative hydrodynamic simulations of the prominence oscillations. The horizontal and the vertical sizes are not to scale. 

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Our simulations start from a thermal and forcebalanced equilibrium state where the background heating is balanced by radiative loss and thermal conduction, and the plasma in the loop is quiescent. The simulations are divided into three steps. (1) Prominence formation: A prominence forms and grows near the center of the magnetic dip as chromospheric material is evaporated into the corona and condensates due to thermal instability after chromospheric heating is introduced near the footpoints of the loop. (2) Prominence relaxation: the prominence relaxes to a thermal and forcebalanced equilibrium state as the localized heating is halted and the chromospheric evaporation ceases. (3) Prominence oscillation subjected to perturbations: The prominence starts to oscillate with a damping amplitude after perturbations are introduced. In step 1, which lasts for a time interval of Δt_{1}, the heating term H(s) in Eq. (3) is composed of two terms, the steady background heating H_{0}(s) and the localized chromospheric heating H_{1}(s), which are expressed as follows: where the quiescent heating term H_{0} is adopted to maintain the hot corona with the amplitude E_{0} = 3 × 10^{4} erg cm^{3} s^{1} and the scaleheight H_{m} = L/2, and the localized heating term H_{1} is adopted to generate chromospheric evaporation into the corona with the amplitude E_{1} = 10^{2} erg cm^{3} s^{1}, the transition region height s_{tr} = 6 Mm, and the scale height λ = 10 Mm. The heating is taken to be symmetric so that it forms a static prominence near the magnetic dip center, so that we can easily control the manner how the prominence is triggered to oscillate. Our methodology is different from that in Luna & Karpen (2012), who used asymmetric heating that spontaneously leads to the oscillation once the prominence is formed. In step 2, H_{1} is switched off. Owing to the absence of the chromospheric evaporation, the gas pressure inside the magnetic loop drops, so the compressed prominence expands until a new equilibrium is reached, which takes less than approximately 2.4 h. In step 3, a perturbation is introduced to the prominence to trigger its oscillation. Note that H_{0} remains throughout the simulations.
From the observational point of view, there might be two types of perturbations. The first one is an impulsive momentum injected into the magnetic loop as the magnetic reconnection near the footpoints rearranges the magnetic loop rapidly. The second is impulsive heating due to subflares (e.g., Jing et al. 2003; Vršnak et al. 2007; Li & Zhang 2012) or microflares (Fang et al. 2006) near the footpoints of the magnetic loop where a large amount of magnetic energy is impulsively released through magnetic reconnection. The gas pressure is greatly increased, which could propel the prominence to oscillate along the dipshaped field lines. In our 1D simulations, we separated the two effects to see their difference. In one case, a velocity perturbation with the following distribution was imposed to the prominence, (7)where s_{pl} and s_{pr} are the coordinates of the left and right boundaries of the prominence, δ = 10 Mm is the buffer zone that allows that the perturbation velocity varies smoothly in space, and v_{0} is the perturbation amplitude. In the other case, impulsive heating (H_{2}), as described as follows, was introduced near the righthand footpoint of the magnetic loop, (8)where the heating spatial scale s_{scale} = 2.5 Mm, the peak location s_{peak} = 245 Mm, the heating timescale t_{scale} = 5 min, and the peak time t_{peak} = 15 min. The heating increases to the peak for 15 min and then decreases to 0.
As for the boundary conditions, all variables at the two footpoints of the magnetic loop are fixed, which is justified because the density in the low atmosphere is more than four orders of magnitude higher than that in the corona. The same approach has been adopted by Ofman & Wang (2002) and Xia et al. (2011), assuming that the coronal dynamics has little effect on the low atmosphere. The approach was verified by Hood (1986) with the parameters being far from the marginal stability. The violation of the rigid wall conditions in certain cases was discussed by van der Linden et al. (1994).
3. Effects of the perturbation type
To check how the two types of perturbation described in Sect. 2 influence the characteristics of the prominence oscillations, we performed simulations of oscillations that are excited by the two types of perturbation while keeping Δt_{1} = 7.2 h, r = 20 Mm, D = 10 Mm, and L = 260 Mm.
In case A, the prominence oscillation is triggered by a velocity perturbation over the whole prominence body. With v_{0} = − 40 km s^{1} (the minus means that the velocity is toward the left), the temporal evolution of the plasma temperature distribution along the magnetic loop is displayed in the left panel of Fig. 2. In response to the perturbation, the prominence, signified by the low temperature, starts to oscillate around the equilibrium position. The oscillation amplitude decays with time. Fitting the trajectory of the mass center of the oscillating prominence with a decayed sine function (9)we find the initial amplitude A_{0} = 34.9 Mm, the oscillation period P = 84.3 min, and the damping timescale τ = 272 min. Assuming that the prominence thread has a crosssection area of ~ 3.14 × 10^{14} cm^{2} (Lin et al. 2005), the initial kinetic energy of the oscillating prominence thread is estimated to be ~7.2 × 10^{23} erg. The single decayed sine function used for fitting the Hα observations (Jing et al. 2003; Vršnak et al. 2007; Zhang et al. 2012) fits the simulated observations very well. In contrast, a combination of Bessel function and an exponential decay function is necessary to fit the initial overtone in the simulations of Luna & Karpen (2012), which results from the continual mass accumulation.
In case B, the prominence oscillation was triggered by the impulsive heating that was deposited near the right leg of the magnetic loop to mimic a microflare near the prominence. To do this, an impulsive heating term H_{2}(s) in Eq. (8) was added to the heating term H in Eq. (3), where s_{peak} = 245 Mm, meaning the heating is concentrated at a height of 15 Mm above the right footpoint of the magnetic loop.
The right panel of Fig. 2 depicts the temporal evolution of the temperature distribution along the magnetic loop with E_{2} = 0.24 erg cm^{3} s^{1}. With the typical crosssection area of a prominence thread being ~3.14 × 10^{14} cm^{2}, the corresponding total energy deposited into the single magnetic loop E_{heating} is 1.8 × 10^{25} erg. This value is reasonable since observations indicate that the total energy of a microflare is 10^{26}–10^{27} erg or even more (e.g., Shimizu et al. 2002; Hannah et al. 2008; Fang et al. 2010), and several percent of the released energy dissipates into one prominence thread. From another point of view, in the framework of the magnetic reconnection model for microflares, the magnetic energy release rate is estimated to be B^{2}v_{in}/(4πL). With the magnetic field B ~ 20 G, the reconnection inflow speed v_{in} being about 0.1 times the Alfvén speed, which is about 1000 km s^{1} (Jiang et al. 2012), and the spatial size L = 10″, the energy release rate is estimated to be ~0.88 erg cm^{3} s^{1}, which is on the order adopted here. Fitting the trajectory of the oscillating prominence with the damped sine function as shown in Eq. (9) yields A_{0} = 35.8 Mm, P = 84.3 min, and τ = 268 min. The corresponding initial velocity is also –40 km s^{1}. This indicates that a typical microflare near the leg of the magnetic loop hosting a prominence thread can excite the prominence longitudinal oscillations with an initial velocity of tens of km s^{1}. The corresponding kinetic energy is only ~7.2 × 10^{23}/1.8 × 10^{25}, i.e., ~4% of the deposited thermal energy. The remaining ~96% of the energy deposit contributes to the heating of the chromosphere.
Fig. 2 Comparison of the evolutions of the loop temperature between the two types of perturbations. The left panel corresponds to the case with velocity perturbations with v_{0} = −40 km s^{1} and the right panel to the case with localized heating perturbations with E_{2} = 0.24 erg cm^{3} s^{1}. 

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4. Parameter survey
The results in Sect. 3 reveal that the oscillation period does not strongly depend on the two types of perturbations, i.e., impulsive momentum and localized heating at one footpoint used in our investigation. We concentrated on the oscillation characteristics that follow the short transient/excitation phase previously obtained from simple decaying sinusoidal fitting. A small difference in the decay timescale exists between the two perturbation types. With the same initial velocity, the decay timescale is 4 min shorter in the case of impulsive heating than that in the case of impulsive momentum. However, the relative variation, 1.4%, is very small. Therefore, we can conclude that the oscillation is basically intrinsic and the characteristics of the oscillation depend on the prominence itself and the geometry of the magnetic loop in our case without mass accumulation, and the oscillations are excited by either impulsive momentum or localized heating. The prominence feature is only characterized by the thread length (l), and the geometry of the magnetic loop is characterized by r, D, and w as depicted in Fig. 1. Among the three geometrical parameters, h = s_{1} + r − D determines the height of the prominence, D and w determine the curvature of the magnetic dip. If other parameters are fixed, the length of the prominence is determined by the duration of the chromospheric evaporation in step 1, i.e., Δt_{1}, as described in Sect. 2. Moreover, the decay timescale might vary with the perturbation amplitude, therefore another parameter is the initial perturbation velocity v_{0}. In this section, we perform a parameter survey to investigate how each of the five parameters (Δt_{1}, r, D, w, and v_{0}) changes the oscillation period and the decay timescale. For each parameter, several cases with different values are simulated with other parameters fixed. In our simulations, we set r = 10 Mm, D = 5 Mm, w = 110 Mm, and v_{0} = −20 km s^{1} when varying Δt_{1}. We set Δt_{1} = 7.16 h, D = 5 Mm, w = 90 Mm, and v_{0} = −20 km s^{1} when varying r. We set Δt_{1} = 7.16 h, D = 5 Mm, r = 10 Mm, and v_{0} = −20 km s^{1} when varying w. We set Δt_{1} = 7.16 h, r = 20 Mm, w = 93.6 Mm, and v_{0} = − 20 km s^{1} when varying D. We set Δt_{1} = 7.16 h, r = 20 Mm, w = 93.6 Mm, and D = 10 Mm when varying v_{0}. Since the oscillation characteristics are found to be nearly insensitive to the perturbation type, we used the velocity perturbation to excite the oscillations in the survey.
4.1. Length and mass of the prominence
After finishing the first two steps of the simulations as described in Sect. 2, we obtained a quasistatic prominence. The dependence of the prominence length l on Δt_{1}, h, D, and w is shown in the four panels of the upper row of Fig. 3. It can be seen that l, which fits into the scaling law , increases with the duration of the heating time Δt_{1}. This is expected because more chromospheric plasma is evaporated into the corona when Δt_{1} increases. The length l decreases with h as l ~ h^{0.37}, which is probably because it takes a longer time for the more tenuous corona to condensate as the height of the magnetic dip increases, and therefore the effective heating time is shorter. The length l decreases with D as l ~ D^{0.21}, which can be understood as the prominence becoming more compressed as the magnetic dip becomes deeper. However, the length of the prominence does not vary considerably with w. Of course, w should not be too small, otherwise thermal instability would not occur. The lengths of these simulated prominence threads are consistent with the reported values, i.e., tens of Mm (Lin et al. 2005).
The dependence of the prominence mass M on Δt_{1}, h, D, and w is shown in the four panels of the lower row of Fig. 3. The dependence of M on Δt_{1}, h, and w is similar to l. Their difference is that l decreases with D whereas M does not change with D, which means that the plasma number density (10^{10}–10^{11} cm^{3}, and the corresponding density is 10^{14}–10^{13} g cm^{3}) is higher in the prominence with a deeper magnetic dip. A scaling law is obtained by fitting the data points, which is .
The above results are derived for a dipped magnetic loop filled via chromospheric evaporation with a limited lifetime, where the prominence thread can be sustained in the corona. For magnetic loops without a dip (e.g., MendozaBriceño et al. 2005) or with a shallow dip and asymmetric heating (e.g., Karpen et al. 2006), condensations repetitively form, stream along the magnetic field, and ultimately disappear after falling back to the nearest footpoint. Therefore, the mass and length of the prominence evolve dynamically, without reaching an equilibrium value.
Fig. 3 Scatter plots of the total length l (upper panels) and mass M (lower panels) of the prominences at the end of relaxation step as functions of Δt_{1}, h, D, and w. 

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4.2. Oscillation period and decay timescale
As the velocity perturbation is introduced to the quasistatic prominence, the prominence starts to oscillate. Fitting the trajectory of the oscillating prominence with the damped sine function shown in Eq. (9), we obtain the oscillation period (P) and the decay timescale (τ) for each case in the parameter survey.
The variations of P along with the parameters l, h, D, w, and v_{0} are shown in the upper row of Fig. 4. P increases slightly with l and v_{0}, and decreases slightly with h. However, it increases significantly with w and decreases with D. To fit the variations with a scaling law, we obtain . Therefore, the period of the prominence longitudinal oscillations relies dominantly on the geometry of the dip, especially its curvature. The range of P agrees with the reported values in previous studies (e.g., Jing et al. 2006).
The variations of τ along with the five parameters are shown in the lower row of Fig. 4. τ increases significantly with l and D, and decreases with w and v_{0}. In the cases of  v_{0}  = 70 and 80 km s^{1}, part of the prominence mass drains down to the chromosphere, which is why the triangles in the lowerright panel of Fig. 4 do not follow the trend of the data points denoted by the diamonds where  v_{0}  < 70 km s^{1}. The decay timescale does not vary significantly with h. To fit the variations with a scaling law, we obtain , where the cases with prominence drainage are not included in the fitting. The values of τ are also in the same order of magnitude as the observed ones.
Fig. 4 Scatter plots of the period P (upper panels) and damping time τ (lower panels) of the prominences in the oscillation step as functions of l, h, D, w, and v_{0}. The values of P and τ in the cases  v_{0}  = 70 and 80 km s^{1} that cause mass drainage at the footpoint of the coronal loop are marked with triangles in the right panels. 

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5. Discussions
5.1. Restoring force
For an oscillating phenomenon, it is most important to determine the restoring force, which directly decides the oscillation period. In our 1D hydrodynamic simulations, the only forces exerted on the prominence are the gravity and the gas pressure gradient, both are restoring forces for the longitudinal oscillations. To compare their importance, we calculated the two forces in the case with Δt_{1} = 7.16 h, v_{0} = − 40 km s^{1}, r = 20 Mm, D = 10 Mm, and w = 93.6 Mm. The two forces were calculated when the prominence is the farthest from the equilibrium position. Although the plasma in prominences is hundreds of times denser than the ambient corona, it is not an ideal rigid body. For oscillations with higher modes, as studied by Luna et al. (2012a), the pressure gradient changes rapidly along the prominence thread. For the fundamentalmode oscillations in this paper, the prominence oscillates as a whole and the pressure gradient changes slightly along the thread. Therefore, for simplicity, we compared the overall magnitude of the two forces by a simple calculation instead of pointtopoint as in the simulations. The integral of the gravity force is quantified between the two ends of the prominence, i.e., , where a unit area is assumed for the cross section. The integral of pressure gradient force over the prominence is expressed as . The left and right boundaries of the prominence are defined to be where the density drops to 7 × 10^{14} g cm^{3}. Figure 5 displays the temporal evolution of the ratio F_{g}/F_{p}, from which it is seen that the gravitational force is generally about ten times stronger than the gas pressure gradient force.
Fig. 5 Temporal variation of F_{g}/F_{p} when the displacement of the prominence reaches maximum during each halfcycle for r = 20 Mm and D = 10 Mm. The velocity perturbation is –40 km s^{1}. 

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Since the gravity is the dominant restoring force, the overall motion of the prominence can also be described for simplicity as (10)where x = s − L/2 is the displacement of the prominence from the equilibrium position. It is not easy to solve this equation analytically. However, if the oscillation amplitude is much smaller than the halfwidth of the whole magnetic dip (w), we derive the approximation sin(πx/w) ≈ πx/w. Accordingly, the above equation is simplified to be (11)with the solution . The corresponding period is (12)This period can also be readily obtained if the prominence is taken in analogy to a pendulum whose period is (13)where R is the curvature radius of the dipped magnetic loop. With the shape of the loop being y = D − Dcos(πx/2w), the curvature radius at the loop center is approximated to be R = 2w^{2}/(Dπ^{2}). Substituting R into Eq. (13), we derive , the same as Eq. (12). Figure 6 compares the oscillation periods obtained from the hydrodynamic simulations (diamonds) and those estimated from Eq. (12) (solid line) when the two parameters, D and w, are changed. This shows that Eq. (12) is a very good approximation for estimating the period of the prominence longitudinal oscillation. Of course, it should be kept in mind that the derivation of Eq. (12) is based on the assumption that the dipped magnetic loop has a sinusoidal shape. More generally, the oscillation period is related to the local curvature radius R by the formula , as also demonstrated by Luna & Karpen (2012).
Recently, Luna et al. (2012a) extended the theoretical analysis of longitudinal prominence oscillations by including the effect of the pressure gradient force. They found that the ultimate fundamental frequency of the oscillations is found from , where ω_{g} and ω_{s} stand for the gravitydriven and pressuredriven frequencies, respectively. The ratio of the two frequencies , where R_{lim} denotes the critical value of the curvature radius (R) of the magnetic dip. If R ≪ R_{lim}, gravity dominates over pressure in the restoring force of longitudinal oscillations. The authors pointed out that the reported values of the curvature are low compared with R_{lim}, so that it is reasonable to ignore the effect of the pressure term in most cases. In our parameter survey, R_{lim} = 0.175(L − l)l ranges from 760 to 2100 Mm and the ratio R/R_{lim} ranges from 0.1 to 0.5. Hence, we confirm their theoretical results of gravity being the main restoring force for the fundamental mode in this parameter range.
For a prominence above the solar limb, all the parameters in Eq. (12) can be roughly measured. Combined with the results in this paper, the comparison between simulations and observations in Zhang et al. (2012) implies that Eq. (12) is a good approximation for estimating the oscillation period. For the prominence longitudinal oscillations on the solar disk, i.e., filament longitudinal oscillations, only the oscillation period can be unambiguously measured. Equation (12) then provides a diagnostic tool for inferring the geometry of the dipped magnetic loop. Especially when w can be roughly estimated from forcefree magnetic extrapolations, the depth of the dip, D, can be determined. At least, we can estimate the curvature radius of the dipped magnetic field, R, through Eq. (13). After determining R, Luna & Karpen (2012) proposed an approximate method for estimating the magnetic field in the prominence.
In addition to the dominant dependence on the geometric parameters, the oscillation period also weakly changes with length and height of the prominence, as well as with the initial velocity. This can be understood as follows: (1) dependence on the prominence length: because the prominence thread is shorter, the ratio of the gas pressure gradient to the gravity would increase as indicated by our simulations, therefore, the gas pressure gradient would contribute to the restoring force, resulting in a shorter oscillation period. (2) Dependence on the prominence height: as seen from Fig. 3, with other parameters the same, a high prominence has a shorter length. Therefore, for the same reason as in (1), the oscillation period would be shorter. (3) Dependence on the initial velocity: since sin(πx/w) is always smaller than πx/w in Eq. (10), the nonlinear term would naturally lead to a long period as the oscillation amplitude increases.
Fig. 6 Comparison of the periods of the prominence oscillations from simulations (diamonds) and theoretical analysis (solid line) as a function of the depth of the magnetic dip D (left panel) and the width of the dip w (right panel). Note that both axes are in logarithmic scale. 

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5.2. Damping mechanisms
When energy dissipation terms such as radiative cooling and heat conduction are removed from Eq. (3), as we did in a test simulation, we found that the prominence oscillation does not damp at all. When the two nonadiabatic terms are kept, the prominence oscillation always damps. To see the importance of the two terms, we calculated the time integrations of radiative loss (E_{R}) and thermal conduction (E_{C}) of the whole system after subtracting the corresponding values when the prominence is static at the center of the dip. Here E_{R} and E_{C} are the integrals of the radiative and the conductive terms in the energy equation Eq. (3), where the integrals are taken in the whole corona above the two footpoints. The evolutions of the ratio (E_{R}/E_{C}) for v_{0} = − 40, –50, and –60 km s^{1} are displayed in Fig. 7. The ratio is always higher than unity. Especially in the early stage of the oscillation when the amplitude is still large, E_{R} is even one order of magnitude larger than E_{C}. It is also revealed that as the initial velocity increases, E_{R} becomes increasingly important in most of the lifetime of the oscillation. Our results support the conclusions of Terradas et al. (2001; 2005) that radiative loss is responsible for the damping of the slow mode of prominence oscillations in the dipshaped magnetic configurations, which seems to be different from the case of slowmode waves propagating in the coronal loops where heat conduction contributes more to the damping (De Moortel et al. 2002a; 2002b).
Fig. 7 Temporal variations of E_{R}/E_{C} in the oscillation step for v_{0} = − 40, –50, and –60 km s^{1}. 

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The role of radiative cooling can be understood in a simple model as follows: since there are two segments of the corona in the magnetic loop, as the prominence oscillates, one part would be attenuated and the other be compressed. Assuming that the total length of the coronal part of the magnetic loop is unity, which includes part x, which is to the left of the prominence, and the other part 1 − x, which is to the right of the prominence. Hence, the densities of the corona on the two sides are proportional to 1/x and 1/(1 − x), respectively. The total optically thin radiative loss of the coronal part is proportional to x^{2} + (1 − x)^{2}, which is minimum when x = 0.5, i.e., when the prominence is situated at the equilibrium position. Whenever the prominence deviates from the loop center, the cooling becomes stronger, dissipating the kinetic energy of the oscillating prominence. The model is best illustrated by the relationship between the damping timescale (τ) and the initial amplitude of the oscillation, i.e., A_{0} in Eq. (9). As A_{0} increases, one of the two coronal parts is more severely compressed, so the radiative cooling x^{2} + (1 − x)^{2} deviates more strongly from the lowest value, i.e., it becomes higher. As a result, the oscillation decays more rapidly.
Based on the sinusoidal function, A_{0} ∝ v_{0}P. Substituting Eq. (12) into it, we obtain A_{0} ∝ v_{0}wD^{− 1/2}. With this, it is easy to understand the positive correlation between the decay timescale τ and D, and the negative correlation between τ and w as revealed by the lower row of Fig. 4. Along this line of thought, the dependence of the decay timescale on the prominence length can be explained as follows: Because the prominence thread is longer, the coronal part of the magnetic loop, which radiates the thermal energy, is shorter. More importantly, the longer thread, with the same initial velocity, has a higher kinetic energy. Therefore, it takes a longer time for the compressed coronal part to radiate it.
Fig. 8 Temporal evolution of the temperature along the magnetic loop when the initial velocity perturbation is as strong as v_{0} = − 80 km s^{1}. Note that the prominence passes the magnetic loop apex and drains down to the chromosphere at the left footpoint around t = 0.8 h. 

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The first six cases (i.e.,  v_{0}  from 10 km s^{1} to 60 km s^{1}) in the lowerright panel of Fig. 4 show that the decay timescale decreases nearly linearly with the initial perturbation velocity. However, when v_{0} is higher than 70 km s^{1}, part of the prominence would pass the magnetic loop apex and drain down. The critical velocity for the prominence to reach the loop apex can be roughly estimated as km s^{1}. Therefore, the value of v_{criti} is 73 km s^{1} for D = 10 Mm. As revealed from our simulations, even when v_{0} = − 70 km s^{1}, mass drainage happens, although the amount of the drainage is much lower than that for v_{0} = − 80 km s^{1}. The temperature evolution along the loop for v_{0} = − 80 km s^{1} is presented in Fig. 8. Part of the prominence falls down to the left leg of loop, leading to the drainage of the prominence mass and kinetic energy as well, while the remaining part continues to oscillate along the dip. The oscillation period and the decay timescale in the cases with mass drainage are marked as triangles in Fig. 4. Their periods, ~90.6 min, are slightly shorter than the trend defined by other cases without mass drainage (diamonds), which is consistent with the weak positive correlation between P and the prominence length l. However, the damping timescales are greatly reduced compared to the trend defined by other cases without mass drainage, as seen from the lowerright panel of Fig. 4. This result, namely that mass drainage would greatly reduce the decay timescale, might explain the mismatch between the simulation and the observation of the decay of a prominence oscillation reported in Zhang et al. (2012).
6. Summary
In this paper, we carried out 1D hydrodynamic simulations of longitudinal prominence oscillations using the MPIAMRVAC code, extending earlier numerical simulations of prominence formation (Xia et al. 2011) and of prominence oscillations (Luna & Karpen 2012; Zhang et al. 2012). The simulations were divided into three steps: First, a prominence forms and grows near the center of the dipshaped coronal loop due to chromospheric heating and the subsequent thermal instability. Then, it relaxes to a quiescent state after the chromospheric heating is switched off. Subjected to two types of perturbations that mimic subflares, the prominence starts to oscillate along the dip. Within the framework of the evaporationcondensation model, we obtained scalinglaws for the prominence length (l) and mass (M), which are expressed as and , where Δt_{1} is the time duration of the chromospheric heating and evaporation, h is the prominence height, D is the depth of the magnetic dip. We found that l is insensitive to the half length of the magnetic dip (w) once w is large enough, about 60 Mm; M is insensitive to D and w. Both transient heating at one leg of the loop and an impulsive velocity perturbation applied to the prominence as a whole are capable of driving a coherent oscillation along the dip. The oscillation properties were found to be insensitive to the perturbation type in the regimes we studied. In the case of the transient heating, ~4% of the deposited energy is converted into the kinetic energy of the prominence. The longitudinal oscillations are sustained mainly by the tangential component of gravity, except when the prominence is short and the gas pressure gradient becomes important as well. Both simulations and linear analysis revealed that the period of oscillation (P) is 2, where R denotes the curvature radius of the dip, as also found by Luna & Karpen (2012). Other parameters, such as the length and height of the prominence, as well as the perturbation velocity, also affect P, though only slightly. The longitudinal oscillations damp in the presence of nonadiabatic effects, i.e., radiative loss and thermal conduction (Soler et al. 2009), among which radiative loss plays a leading role. With the parameter survey, we obtained a scalinglaw for the decay timescale τ, which is expressed as , where v_{0} is the initial velocity perturbation. We also found that prominence mass drainage, once it happens, significantly reduces the decay timescale, which may explain the mismatch between the simulations and the observations disclosed by Zhang et al. (2012).
These results are limited in application. According to this paper, the mass of a prominence thread is insensitive to the depth D and the width w of the magnetic dip. This is based on the prominence formation directly via chromospheric evaporation with a fixed lifetime Δt_{1}. According to Xia et al. (2011), the prominence would grow via siphon flow even when the localized heating is switched off, though the growth speed is much slower. Recently, Luna et al. (2012a) pointed out that the restoring force of the longitudinal oscillations depends on the depth of the magnetic dip. For shallow dips, gas pressure plays an important role, while gravity is the main factor for deep dips. Moreover, Li & Zhang (2012) suggested that magnetic tension may also contribute to the restoring force. As for the damping mechanisms, several other effects might need to be taken into account in the future simulations, such as the wave leakage and plasma viscosity (Ofman & Wang 2002). However, some will only be quantifiable in true multidimensional configurations, e.g., starting from the prominences formed in Xia et al. (2011).
Acknowledgments
The authors thank the anonymous referee for detailed and enlightening comments that improved the paper. Q. M. Zhang acknowledges C. Fang, M. D. Ding, W. Q. Gan, Y. P. Li, Z. J. Ning, S. M. Liu, D. J. Wu, H. Li, and L. Feng for discussions and suggestions throughout this work. R.K. acknowledges funding from the Interuniversity Attraction Poles Programme initiated by the Belgian Science Policy Office (IAP P7/08 CHARM). The research is supported by the Chinese foundations NSFC (11025314, 10878002, 10933003, and 11173062) and 2011CB811402.
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All Figures
Fig. 1 Magnetic loop used for the 1D radiative hydrodynamic simulations of the prominence oscillations. The horizontal and the vertical sizes are not to scale. 

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In the text 
Fig. 2 Comparison of the evolutions of the loop temperature between the two types of perturbations. The left panel corresponds to the case with velocity perturbations with v_{0} = −40 km s^{1} and the right panel to the case with localized heating perturbations with E_{2} = 0.24 erg cm^{3} s^{1}. 

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In the text 
Fig. 3 Scatter plots of the total length l (upper panels) and mass M (lower panels) of the prominences at the end of relaxation step as functions of Δt_{1}, h, D, and w. 

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In the text 
Fig. 4 Scatter plots of the period P (upper panels) and damping time τ (lower panels) of the prominences in the oscillation step as functions of l, h, D, w, and v_{0}. The values of P and τ in the cases  v_{0}  = 70 and 80 km s^{1} that cause mass drainage at the footpoint of the coronal loop are marked with triangles in the right panels. 

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In the text 
Fig. 5 Temporal variation of F_{g}/F_{p} when the displacement of the prominence reaches maximum during each halfcycle for r = 20 Mm and D = 10 Mm. The velocity perturbation is –40 km s^{1}. 

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In the text 
Fig. 6 Comparison of the periods of the prominence oscillations from simulations (diamonds) and theoretical analysis (solid line) as a function of the depth of the magnetic dip D (left panel) and the width of the dip w (right panel). Note that both axes are in logarithmic scale. 

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In the text 
Fig. 7 Temporal variations of E_{R}/E_{C} in the oscillation step for v_{0} = − 40, –50, and –60 km s^{1}. 

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In the text 
Fig. 8 Temporal evolution of the temperature along the magnetic loop when the initial velocity perturbation is as strong as v_{0} = − 80 km s^{1}. Note that the prominence passes the magnetic loop apex and drains down to the chromosphere at the left footpoint around t = 0.8 h. 

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In the text 