Failed ejection and oscillations of a current-carrying filament balanced by gravity

Problem
Solar filaments (prominences) are dense, cool plasma structures suspended in the solar corona, often associated with solar eruptions and coronal mass ejections (CMEs). While many models describe filament stability and eruption, a significant limitation in existing numerical simulations is the assumption of zero plasma beta, which neglects the effects of gravity. Realistic coronal conditions involve finite plasma pressure and gravitational stratification, which are crucial for understanding filament stability, mass loading, and thermal structuring. Furthermore, observational data on filament eruptions frequently show "failed" events where the ejection process is halted, and the filament falls back, a phenomenon requiring models that account for gravitational forces and realistic current configurations to fully capture the underlying physics.

Methodology
The authors investigate the post-destabilization evolution of a filament using a gravity-balanced model proposed by Solov'ev (2010). The study employs a dual approach:

1. Analytical Investigation: The authors derive analytical expressions for electric current density, plasma density, and their spatial distributions based on the Solov'ev model. In this model, a dense filament carrying an electric current is supported against gravity by the repulsive Lorentz force between the filament current and its sub-photospheric image current.
2. Numerical Simulation: A two-dimensional magnetohydrodynamic (MHD) model is constructed using the Lare2d numerical code. The model solves the resistive, compressible MHD equations in a Cartesian coordinate system, explicitly including gravity and resistivity. The simulation domain ($x \in [-10, 10]$ Mm, $y \in [0, 40]$ Mm) is resolved by a $2000 \times 4000$ grid. The initial equilibrium state is characterized by a magnetic field $B_0 = 10^{-3}$ T, mass density $\rho_0 \sim 1.3 \times 10^{-9}$ kg m$^{-3}$, and temperature $T \sim 13000$ K. A modified temperature profile is used to simulate the transition region between the photosphere and corona.

The system is destabilized through two distinct scenarios: (a) increasing the electric current in the filament (by increasing the magnetic field strength $B_0$) and (b) reducing the filament's mass density. The evolution of the system is tracked to observe the ejection, halt, and fallback dynamics.

Key Contributions

• Derivation of Analytical Expressions: The paper provides analytical formulas for the electric current density, plasma density, and their spatial distributions as functions of model parameters, including the total electric current and filament weight.
• Gravity-Balanced MHD Model: The study implements a 2D MHD model that explicitly balances the repulsive magnetic force against gravity, addressing the limitations of zero-beta models that neglect gravitational effects.
• Characterization of Failed Ejections: The authors systematically simulate and characterize the "failed ejection" phenomenon, where a destabilized filament is ejected, halted at a specific altitude, and subsequently falls back, entering an oscillatory cycle.
• Quantitative Dependence: The maximum ejection altitudes are determined as functions of both the magnitude of the destabilizing currents and the degree of filament plasma dilution.

Results

• Oscillatory Behavior: In both destabilization regimes (current increase and density reduction), the filament is ejected upward, halted at a certain altitude, and falls back. This cycle repeats with a period of approximately 600 seconds.
• Ejection Velocities: The maximum ejection velocities differ based on the destabilization mechanism: approximately 80 km s$^{-1}$ for the current-increase scenario and 40 km s$^{-1}$ for the density-reduction scenario.
• Current Sheet Formation: Beneath the ejected filament, a current sheet forms where magnetic reconnection occurs. This reconnection disrupts the magnetic field lines connecting the upper and lower currents.
• Parameter Dependence:
  • Increasing the current (magnetic field strength) leads to higher ejection altitudes.
  • Reducing the filament density also leads to higher ejection altitudes, with lower density ratios ($\rho/\rho_0$) resulting in greater heights.
  • The density-reduction scenario results in the filament reaching its maximum height earlier and remaining there longer (200 s) compared to the current-increase scenario (100 s), though the oscillation period remains similar.
• Asymmetry: In the current-increase case, the density structure beneath the filament eventually develops asymmetry, leading to a displacement of the filament from the vertical axis. In the density-reduction case, the filament oscillates along the vertical axis to a stable state.
• Validation: Comparative simulations with an extended vertical domain confirmed that the original numerical domain size does not influence the magnetic field configuration or the maximum height reached.

Significance
The paper demonstrates that models incorporating gravitational forces and finite plasma beta are essential for understanding the pre-eruption equilibrium and the specific conditions under which filaments lose stability. By reproducing the observed "failed eruption" phenomenon and the subsequent oscillatory behavior, the study provides a theoretical framework for interpreting observational data regarding filament formation and destabilization. The results highlight that mass loading and thermal structuring are critical factors influencing the threshold for instability and the nonlinear dynamics of the eruption process, offering a more realistic alternative to ideal vacuum models.