Simulation of a partially erupting flux rope
cme_1 cme_2 cme_3

Background

Coronal mass ejections or CMEs are spectacular eruptions which occur very frequently in the corona of our Sun. These eruptions are driven either by emergence of magnetic flux or due to cancellation of magnetic flux due to shearing. They carry away, into the heliosphere, solar plasma with masses typically around 1.6 X 1012 kg with an average velocity of 400-500 km/s. Their speeds can however reach as high as 3.5Mm/s! During the solar maximum most CMEs originate above active regions where as at other times they originate from erupting polar crown filaments and inside helmet streamers. Emerging solar active regions with strong photospheric twist or equivalently helicity are known to repeatedly flare and produce homologous coronal mass ejections. X class flares usually originate from strongly rotating sunspots (Schrijver, 2009) and may indicate emergence of a highly twisted flux rope. Homologous CMEs frequently give rise to "cannibalism" or CME-CME interactions which are known to be one of the most energetic and geo-effective space weather phenomena. Cannibal CMEs are usually fast ones (speeds > solar wind speed~400 km/s) since they are released into regions where the magnetic field has been thrown open by a preceeding CME and therefore has less magnetic tension to overcome.

The solar corona is magnetically dominated and has a very low gas pressure compared to the magnetic pressure i.e., a very low plasma beta (0.07-0.2) in contrast to the solar interior with a high plasma beta (>14). This means that Alfven velocities are extremely high in the Corona (~4 Mm/s) and magnetic field changes in the photosphere propagate to the corona almost instantaneously. Most parts of the corona are magnetically confined. However hot regions with low magnetic fields (i.e., current sheets) can have a plasma beta >1 leading to leakage of plasma across the current sheet or cusps. A low plasma beta also means that the magnetic fields is the corona should be force free because there is no gas pressure available to balance the Lorentz force.

Our understanding of the energetic phenomena of CMEs have increased by leaps and bounds due to data from space missions like SOHO, SDO, STEREO etc. Modeling efforts have also gained momentum due to availability of more computational power, a necessary requirement for the corona due to very small Alfven travel times. It is generally agreed upon that CMEs occur due a loss of equilibrium of the magnetic system initially in quasi static equilibrium and driven from below by the photosphere. We use the phrase "loss of equilibrium" as opposed to "instability" since the latter typically involve growth rates which can be analytically calculated. As for the CMEs, the analytical models can only predict the condition under which the loss of equilibrium can take place rather than describing the state in the aftermath of the loss of equilibrium. To understand the loss of equilibrium scenario let us discuss a few conceptual models:

  1. Mass loading model: Imagine a coiled spring (magnetic field lines) compressed by a heavy mass (prominence). If the heavy mass is shifted then the spring uncoils rapidly (Low 1996, 1999)
  2. Tether release model:The coiled spring again, restrained by tethers. If we proceed to cut the tethers, then the total force on the spring is distributed amongst remaining tethers. We continue the process till the strain on the tethers become large enough to break them and thus the spring uncoils (Forbes & Isenberg 1991)
  3. Tether straining model:Instead of cutting some of the tethers, we increase the force on them (e.g., by jacking up the spring against the tethers) till the tethers break and releases the spring (Antiochos , 1998).
twisted torus
Fig1. - The twisted torus before emerging through the bottom boundary at R=1 Rs. The foot points of the potential magnetic field at the bottom boundary are shown in color.

CME modelers believe that at the center of any CME lies a flux rope. The flux rope can either form in the corona or emerge from below the surface in form of an active region. NASA's SDO recently caught the act of formation of a flux rope. The dipped field lines of the flux rope are potential regions for supporting high density prominence material. The flux rope often gets destabilized due to "torus instabilty" which is an ideal MHD expansion instability of an toroidal current and the overlying poloidal magnetic field system. Torus instability (Bateman, 1978; Kliem & Toeroek, 2006; Isenberg & Forbes, 2007) is an expansion instability of a flux rope that occurs when the external strapping field confining the flux rope decreases with height above the surface at a sufficiently steep rate. If the flux rope is highly twisted then the flux rope axis also undergoes a writhe referred to as "helical kink instability". The helical kink instability is expected to develop for a line-tied coronal flux rope if the total winds of the field line twist about the axis exceed a critical value between the line-tied ends (Hood & Priest, 1981; Toeroek & Kliem, 2003). Even though torus and kink instabilities are ideal MHD phenomena, the presence of reconnection modifies the state from which the loss of equilibrium takes place. For example, the tether-cutting reconnections occurring below the flux rope add twisted flux to the rising flux rope thereby intensifying the loop current.


Twisted flux rope emergence model for initiation of homologous CMEs

twisted torus
Fig2. - The sigmoid-cusp-sigmoid transition in the morphology of the heated field lines. The red fieldlines belong to the first erupting rope where as the yellow fieldlines belong to the second erupting rope. Time is indicated in hours.

We emerge a highly twisted flux rope in the shape of a torus into the lower corona where the ambient magnetic field is a potential arcade at a speed ~ 0.1% Alfven velocity as shown in Fig. 1. This is achieved by driving the bottom boundary by an electromotive force. Our simulations are carried out in a spherical domain extending from the solar surface to 5 Rs with a latitudinal extent of [-7.5, 7.5] deg and a longitudinal extent of [-10, 10] deg. The outer boundary is open where as the side walls are perfectly conducting. Even though artificial but this can be likened to presence of coronal holes in the flanks which provide a radial guide for the CMEs. We carry out full MHD simulations but in absence of the anisotropic Spitzer thermal conduction using the Magnetic flux Emergence (MFE) code. The extensive parameter search as been carried out using the faster but more dissipative open source PENCIL CODE.

The flux rope is so twisted that its axis kinks as soon as the subsurface anchors cross the lower boundary. The rotation of the axis of the torus enables it to find an opening in the overlying arcade and partially erupt into the corona. After this a second flux rope forms by reconnections between the legs of the original kinked tube and also due to continuous emergence of the subsurface torus. We have demonstrated reformation of the flux rope with a sigmoid morphology after the post flare loops have dissipated away. Some "super-active" regions like NOAA AR 8668 show repeated reformation of the soft X-ray sigmod after every eruption. The flux rope in our model erupts for the second time and then for third and the fourth. Each time the flux rope reforms itself (see Fig. 2). The last eruption is asymmetric with only a part of the flux rope erupting sideways. All the first three CMEs are classified as fast since their speeds are 650 km/s, 1400 km/s and 1800 km/s respectively. The greater acceleration of the following CME compared to its precursor is because the following flux rope is erupting into the field that has been opened up by the leading eruption and therefore has less downward magnetic tension to overcome. The second CME collides with the ejecta of the first CME before it reaches 3 Rs. However at the instant of collision between the first and the second CME the merged ejecta attains a speed that is greater than the first CME but slower than the second in order to conserve momentum. The merged CMEs exit the domain traveling at a speed of ~950 km/s (see Fig. 3).

Lagrangian Tracking
Fig3. - (a) Snapshots of the radial velocity in the central meridional plane the onset of the second CME and at a time when the second blob catches up with the first. (b) Height vs time of three Lagrangian points (ER1, ER2, ER3) each inside one of the three erupting ropes. (c) Velocity vs time for the same Lagrangian points tracked in (b). The red dashed lines in (b) and (c) indicate the exact time of the snapshots in (a).

Active regions (AR) appearing on the surface of the Sun are classified into α, β, γ and δ by the rules of the Mount Wilson Observatory, California on the basis of their topological complexity. Amongst these, the δ-sunspots are known to be super-active and produce the most X-ray flares. Delta-sunspots are formed when two sunspots of opposite polarity magnetic field appear very close to each other and reside in the same penumbra, the radial filamentary structure outside the umbral region of the strongest magnetic fields. Strong shear and horizontal magnetic fields often exist at the polarity-inversion line separating the two polarities. The subsurface processes which form the delta-sunspots are still debated. Early observational studies propose that delta-sunspots form from collision-merging of topologically separate dipoles, while numerical simulations by show that kink unstable magnetic flux-tube - helical field lines winding around a central axis - emerging from the subsurface can have a delta-sunspot like structure. The NOAA active region 11158 was the first delta sunspot which was extensively observed by the NASA/Solar Dynamics Observatory (SDO) space craft in February 2011. Below we show the evolution of the line-of-sight (LOS) magnetic field of the active region as recorded by the HMI instrument on board SDO. The delta sunspot can be seen forming due to the interaction of two opposite polarities (blue and yellow) at the center of the images.

NOAA AR11158
Fig4. - Evolution of the LOS magnetic field in the superactive delta sunspot NOAA 11158.

Recently, we performed a simulation of the Sun by mimicking the upper layers and the corona, but starting at a more primitive stage than any earlier treatment. We find that this initial state consisting of only a thin sub-photospheric magnetic sheet breaks into multiple flux-tubes which evolve into a colliding-merging system of spots of opposite polarity upon surface emergence. This numerical simulation, making no assumptions about the properties of sub-surface flux tubes, demonstrates the formation of a δ-sunspot from the collision of two or more young flux emerging regions developing in close vicinity. It is very similar to what is often seen in the solar photospheric magnetograms e.g., the widely studied active region with NOAA number 11158. In the right panel of Fig. 5, one can see how the initial flat and thin horizontal magnetic sheet has warped to form twisted magnetic tubes under the influence of rotating convection in the subsurface of the Sun. Earlier simulations assumed a uniform twist of the cylindrical flux tubes and often made regions of the initial tube selectively buoyant and did not include the effect of solar rotation on rising flux tubes. Inclusion of solar rotation in our simulation alleviated the need to include a twist parameter by hand. The solar rotation became responsible for imparting a non uniform twist along the length of the flux tubes. Since, the twist is non uniform only select regions of the twisted tube have higher magnetic flux density and are naturally buoyant compared to other regions.

Simulation domain Simulation domain
Fig5. - (Left panel) The initial state inside the box with a thin magnetic layer represented by the isosurface of Bρ-1/4. Few field lines in this layer are shown in green. Additionally, the ambient (arcade shaped) magnetic field lines are shown in cyan. The location of the photosphere is marked by convective granules represented by iso surfaces of vz, with red (yellow) representing upward (downward) vz. (Right panel) Same as (a) but at the time of the C1 flare. The field lines are colored according to plasma velocity orientation along them with red (blue) representing upward (downward) velocity.