Influence of photospheric magnetic conditions on the catastrophic behaviors of flux ropes in active regions
Abstract
Since only the magnetic conditions at the photosphere can be routinely observed in current observations, it is of great significance to find out the influences of photospheric magnetic conditions on solar eruptive activities. Previous studies about catastrophe indicated that the magnetic system consisting of a flux rope in a partially open bipolar field is subject to catastrophe, but not if the bipolar field is completely closed under the same specified photospheric conditions. In order to investigate the influence of the photospheric magnetic conditions on the catastrophic behavior of this system, we expand upon the 2.5 dimensional ideal magnetohydrodynamic (MHD) model in Cartesian coordinates to simulate the evolution of the equilibrium states of the system under different photospheric flux distributions. Our simulation results reveal that a catastrophe occurs only when the photospheric flux is not concentrated too much toward the polarity inversion line and the source regions of the bipolar field are not too weak; otherwise no catastrophe occurs. As a result, under certain photospheric conditions, a catastrophe could take place in a completely closed configuration whereas it ceases to exist in a partially open configuration. This indicates that whether the background field is completely closed or partially open is not the only necessary condition for the existence of catastrophe, and that the photospheric conditions also play a crucial role in the catastrophic behavior of the flux rope system.
1 Introduction
Largescale solar explosive phenomena, such as prominence/filament eruptions, flares and coronal mass ejections (CMEs), are widely considered to be different manifestations of the same physical process (e.g. Low, 1996; Archontis & Török, 2008; Chen, 2011; Zhang et al., 2014), which is believed to be closely related to solar magnetic flux ropes (e.g. Low, 2001; Török et al., 2011). Many theoretical analyses have been made to investigate the eruptive mechanisms of magnetic flux ropes so as to shed light on the physical processes of solar eruptive activities (Forbes & Priest, 1995; Chen & Shibata, 2000; Kliem & Török, 2006; Su et al., 2011; Longcope & Forbes, 2014). Van Tend & Kuperus (1978) concluded that a filament system loses equilibrium if the current in the filament exceeds a critical value. This process is called “catastrophe”, which occurs via a catastrophic loss of equilibrium. Catastrophe has been suggested to be responsible for flux rope eruptions by many authors (Priest & Forbes, 1990; Forbes & Isenberg, 1991; Isenberg et al., 1993; Lin, 2004; Zhang & Wang, 2007; Kliem et al., 2014). During catastrophe, magnetic free energy is always released by both magnetic reconnection and the work done by Lorentz force (Chen et al., 2007; Zhang et al., 2016). It was also demonstrated in previous studies that catastrophe has close relationship with instabilities (e.g. Démoulin & Aulanier, 2010; Kliem et al., 2014).
In previous studies, a 2.5 dimensional ideal MHD model in Cartesian coordinates was used to investigate the evolution of the equilibrium states associated with a flux rope embedded in bipolar magnetic fields. It was found that no catastrophe occurs for the flux rope of finite cross section in a completely closed bipolar configuration (Hu & Liu, 2000), consistent with the conclusion in analytical analyses (Forbes & Isenberg, 1991; Forbes & Priest, 1995). If the background bipolar field is partially open, however, the magnetic system is catastrophic (Hu, 2001). The equilibrium solutions are then bifurcated: the flux rope may either stick to the photosphere (lower branch solution) or be suspended in the corona (upper branch solution). If the control parameter exceeds a critical value, the flux rope jumps upward from the lower branch to the upper branch, which is called “upward catastrophe” (Zhang et al., 2016). Here control parameters characterize physical properties of the magnetic system; any parameter can be selected as the control parameter provided that different values of this parameter will result in different equilibrium states (Kliem et al., 2014; Zhang et al., 2016). Whether a system is catastrophic depends on how its equilibrium states evolve with the control parameter. Recently, Zhang et al. (2016) found that there also exists a “downward catastrophe”, i.e., a sudden jump from the upper branch to the lower branch, during which magnetic energy is also released, implying that the downward catastrophe might be a possible mechanism for energetic but noneruptive activities, such as confined flares (e.g. Liu et al., 2014; Yang et al., 2014), but observational evidence is being sought.
Since catastrophe could account for many different solar activities, it is important to investigate what influences the existence and properties of the catastrophe. Previous studies have demonstrated that whether the background bipolar field is completely closed or partially open greatly influences the catastrophic behavior of the flux rope system. A question arises as to whether this is the only affecting factor . Due to the limit of current observing technologies, coronal magnetic configurations, corresponding to the background fields around the flux rope, can not be directly measured. What can be observed is the photospheric magnetic conditions. To reveal the influence of the photospheric magnetic conditions on the catastrophe of a flux rope system could not only help to better understand the decisive factors for catastrophe, but also shed light on the flare/CME productivity of active regions (e.g. Romano & Zuccarello, 2007; Schrijver, 2007; Wang & Zhang, 2008; Chen & Wang, 2012; Liu et al., 2016). By numerical simulations in spherical coordinates, Sun et al. (2007) found that if the global photospheric flux is concentrated too close to the magnetic neutral line, the system losses its catastrophic behavior. Many solar eruptive activities originate from active regions (Su et al., 2007; Chen et al., 2011; Sun et al., 2012; Titov et al., 2012), the spatial scale of which is small as compared with the solar radius, hence Cartesian coordinates suit the simulations of flux ropes in active regions. In this paper, we use the same 2.5 dimensional ideal MHD model in Cartesian coordinates as in previous studies (Section 2) to simulate the evolution of the equilibrium states under different photospheric conditions with the background field either partially open (Section 3) or completely closed (Section 4). Finally, a discussion about the implications of the simulation results is given in Section 5.
2 Basic equations and the initial and boundary conditions
A Cartesian coordinate system is used and a magnetic flux function is introduced to denote the magnetic field as follows:
(1) 
Neglecting the radiation and heat conduction in the energy equation, the 2.5D MHD equations can be written in the nondimensional form:
(2)  
(3)  
(4)  
(5)  
(6) 
where denote the density, velocity, temperature and magnetic flux function, respectively; and correspond to the zcomponent of the magnetic field and the velocity, which are parallel to the axis of the flux rope; is the normalized gravity, is the characteristic ratio of the gas pressure to the magnetic pressure, where and is the vacuum magnetic permeability and gas constant, respectively; , , , and are the characteristic values of density, temperature, length and magnetic flux function, respectively. The initial corona is isothermal and static with
(7) 
In this paper, the background field is taken to be bipolar, either partially open or completely closed (see Sections 3 and 4 for details). It is assumed to be symmetrical relative to the axis. The lower boundary corresponds to the photosphere; at the lower boundary is always fixed at the value of the background field except during the emergence of the flux rope. There is a positive and a negative surface magnetic charge located at the photosphere within and , respectively. The photospheric magnetic flux distribution is characterized by the distance between the inner edges of the two charges () and the width of the charges (). With different values of and , different background configurations can be calculated by complex variable methods accordingly (see Sections 3 and 4).
The magnetic properties of the flux rope are characterized by the axial magnetic flux passing through the cross section of the flux rope, , and the annular magnetic flux of the rope of per unit length along direction, , which is simply the difference in between the axis and the outer boundary of the flux rope. Here we select as the control parameter, i.e. we analyze the evolution of the equilibrium solutions of the system versus with a fixed . The varying represents an evolutionary scenario, e.g., flux emergence (Archontis & Török, 2008) or fluxfeeding from chromospheric fibrils (Zhang et al., 2014). It should be noted that, if not changed manually, and of the rope should be maintained to be conserved, which is achieved by the numerical techniques proposed by Hu et al. (2003).
With the initial conditions, equations (2) to (6) are solved by the multistep implicit scheme (Hu, 1989) to allow the system to evolve to equilibrium states. In order to investigate the influence of the photospheric magnetic conditions on the catastrophic behavior of the flux rope system, we calculate the evolution of the flux rope in different background configurations in the following procedures. Starting from a background configuration with given and , we let a magnetic flux rope emerge from the central area of the base. Following Hu & Liu (2000) and Hu (2001), the emergence of the flux rope is assumed to begin at in the central area of the base and end at s, after which the flux rope are fully detached from the base. The emerging speed is uniform, and then the emerged part of the flux rope is bounded by at time , where
(8) 
and Mm is the radius of the rope. The relevant parameters at the base of the emerged part of the flux rope () are specified as:
(9)  
(10)  
(11)  
(12)  
(13) 
where is the flux function of the background field, and is a constant controlling the initial magnetic properties of the emerged rope. The values of range from 2.0 to 4.0 for different cases. The outer boundary of the emerged rope is determined by After the emergence of the rope, we obtain an equilibrium state with the flux rope sticking to the lower boundary. Starting from such a state, new equilibrium solutions with different but the same are calculated, and thus we obtain the evolution of the flux rope in equilibrium states as a function of in the given background configuration, as described by the geometric parameters of the flux rope, including the height of the rope axis, , and the length of the current sheet below the rope, . Similar procedures are repeated for background configurations with different and to obtain the evolutionary profiles of the flux rope under different photospheric flux distributions. The influence of the photospheric conditions could then be revealed by comparing the evolutions of the flux rope under different background configurations (see Section 3 and Section 4). Note that since we adjust in our simulation to calculate different equilibrium solutions, the value of is insignificant, which only influences the initial magnetic properties.
If the flux rope breaks away from the photosphere, a vertical current sheet will form beneath it. In our numerical scheme, any reconnection will reduce the value of at the reconnection site. Therefore, by keeping invariant along the newly formed current sheet, reconnections, including both numerical and physical magnetic reconnections, are completely prevented across the current sheet.
3 Simulation results in partially open bipolar field
First we analyze the influence of photospheric flux distributions on the magnetic system associated with partially open bipolar background fields. Following Hu (2001), the background magnetic field can be cast in the complex variable form
(14) 
where , the position of the neutral point of the partially open bipolar field is (, ), and
(15) 
The magnetic flux function is then calculated by
(16) 
and the flux function at the photosphere can be derived as
(17) 
where is the total magnetic flux emanating upward from the positive charge per unit length along the zaxis. Note that is independent of the distance .
The magnetic configurations of the background fields are shown in
d
he two magnetic surface charges are denoted by the thick lines in the figures.
)22(c) and 22(g)22(i) show the background field configurations for , , , , , Mm, respectively, with the same Mm, whereas
)23(d) for , , , Mm with the same Mm. The corresponding photospheric distributions of the normal component of the magnetic field, , are plotted in
)23(h), respectively. The ratio of the magnetic flux of the open component to the total flux of the background field is determined by
(18) 
where
(19) 
is the flux function at the neutral point , corresponding to the flux of the open component. For the background fields with different and , is always selected to be 0.8, and the resultant varies slightly among different cases. The computational domain is taken to be Mm, Mm, with symmetrical condition used for the left side (). As mentioned above, at the lower boundary is always fixed to be except during the emergence of the flux rope. In the simulation, potential field conditions are used at the top ( Mm) and right ( Mm) boundaries, except for the location of the current sheet ( Mm, Mm), at which incrementequivalue extrapolation is used.
By the simulating procedures introduced in Section 2, the evolutions of the equilibrium states of the system consisting of a flux rope in the background configurations with different are calculated, as plotted in
