5 Data-driven and Data-inspired Models of Emerging Flux

With the ever growing body of work on flux emergence, the field has matured to a point where lessons drawn from observations, theory and numerical experiments are beginning to be applied to modeling of actual emerging flux regions on the Sun. Some of the numerical model efforts in this line of research use observational data to drive their simulations, while others take inspiration from the observations to construct idealized scenarios.

5.1 Flux emergence models with one-way coupling

The first data-driven models of flux emergence did not use observational data. Abbett and Fisher (2003*) carried out anelastic simulations of the rise of twisted Ω-loops in the convection zone. Due to limitations of the analestic model, the computational domain of these simulations has a top boundary in the subphotospheric layers. The MHD variables from horizontal planes of the self-contained analestic simulations were then used to set the ghost cell values of a 3D compressible MHD code, which has a computational domain that captures the photosphere and corona (see Figure 61*). This way the compressible MHD simulations of flux emergence are driven by evolution in the anelastic simulations. As noted in their paper, an assumption made is that the MHD quantities do not vary significantly in the near-surface layers of the convection zone (i.e., the region skipped between the anelastic and compressible models). Fully compressible simulations of flux emergence in granular convection have since shown otherwise (e.g., Cheung et al., 2007a; Abbett, 2007; Martínez-Sykora et al., 2008*, 2009; Tortosa-Andreu and Moreno-Insertis, 2009; Fang et al., 2010, 2012). Nevertheless, the study by Abbett and Fisher (2003*) showed that data-driven simulations using different codes to capture different subdomains and physical regimes is possible.

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Figure 61: 3D rendering illustrating the one-way coupled model of flux emergence by Abbett and Fisher (2003). MHD quantities from anelastic simulations of the rise of Ω-loops in the convection zone are used to drive the bottom boundary of a compressible MHD code. Image reproduced with permission, copyright by AAS.

A more recent example of a model of flux emergence with one-way coupling is the work by Chen et al. (2014*). As bottom boundary data for their model of the formation of AR coronal loops, they used MHD quantities sampled at the base of the photospheric layer in a fully-compressible, radiative MHD simulation of AR formation (Rempel and Cheung, 2014*, see Sections 3.4.6 and 3.5). The numerical model that provided the bottom boundary data captures the top 16 Mm of the convection zone but only the first few hundred km of the solar atmosphere. In contrast, the computational domain of the Chen et al. (2014*) model has a bottom boundary at the photosphere and extends to a height of 73 Mm. In this data-driven simulation, they were able to model the formation of million degree coronal loops above the model AR (see Figure 62*).

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Figure 62: This model of the formation of AR corona by Chen et al. (2014*) is driven at the bottom boundary (z = 0) by a radiative MHD simulation of AR-scale flux emergence (Rempel and Cheung, 2014). The greyscale in panels (a) and (b) shows synthetic magnetograms at the photospheric layer. The yellow shading in all four panels shows EUV images of AR coronal loops synthesized for the 193 Å channel of SDO/AIA. Image reproduced with permission from Chen et al. (2014), copyright by ESO.

5.2 Formation of current layers in emerging flux regions

Motivated by the discovery of serpentine field lines, bald patches and Ellerman bombs in emerging flux regions (Pariat et al., 2004, see also Section 3.4.3 in this paper), Pariat et al. (2009) developed an MHD model for driving current build-up in an emerging flux region. To do so, they constructed a potential field extrapolation from a SoHO/MDI magnetogram of an emerging flux region. Using this potential field as an initial condition for an MHD calculation, the introduced Maxwell stresses in the field by applying a expansion flow transverse to the bottom boundary. They did so while imposed a line-tied bottom boundary condition for the magnetic field. While this boundary condition does not truly mimic an emerging flux scenario, it served as a instructive experiment to test where currents would form due to photospheric driving. They reported that normalized current densities (j∕B) were found to be especially intense at bald patches, where U-loops are expected to be pinched off following magnetic reconnection and where Ellerman bombs are expected to occur. They note, however, that currents also develop at different locations along separatrix surfaces and conclude that reconnection is not confined to the vicinity of bald patches.

Figure 63: mpeg-Movie (8155 KB) 3D rendering from a simulation of an eruption following emergence of a twisted flux tube into a pre-existing sunspot. This is an idealized scenario to mimic the evolution of NOAA AR 10930, which produced an X3.4 flare with an Earth-directed CME. Observations of AR 10930 showed the formation of a delta spot following emergence of a parasitic bipolar region. Reproduced with permission from Fan (2011*), copyright by AAS.

5.3 Active region eruptions following flux emergence

Inspired by the evolution of NOAA AR 10930, which erupted on December 13, 2006 with an X3.4 flare and an Earth-directed CME, Fan (2011) carried out idealized MHD simulations in a spherical subdomain. From observations of the AR 10930, it can be inferred that the formation of the delta spot associated with the eruption was formed following the emergence of a parasitic flux rope into a pre-existing spot. In order to restrict the time-step of the MHD simulation to reasonable values, the initial condition was constructed by smooth an MDI magnetogram so that the maximum field strengths decreased from 3 kG to 200 G (to decrease the Alfvén speed, which limits the simulation time-step). A twisted toroidal flux rope with an east-west orientation was emerged just south of the pre-existing sunspot in such a way that the following polarity of the new region reconnected with the pre-existing spot. This interaction lead to some erosion of the bootstrapping arcade field overlying the new flux rope, which became unstable erupted (see Figure 63Watch/download Movie). Qualitative comparison with observations (such as the locations of the double ribbon structure) suggests this is a plausible scenario for AR 10930. Since the configuration of the magnetic field was close to what is needed to instigate both the torus and the helical kink instabilities, the ultimate cause for the eruption in the simulation could not be attributed to solely one or the other.

Figure 64: mov-Movie (21784 KB) Left: Distribution of normalized current density (j/B, green) and magnetic field orientation (arrows) at a vertical cut along the dashed line shown in the right panel. Right: Magnetogram sequence (remapped from SDO/HMI data) used for the data-driven simulation of NOAA AR 11158 by Cheung and DeRosa (2012*). Image reproduced with permission, copyright by AAS.

To model the evolution of an eruptive active region over multiple days, Cheung and DeRosa (2012*) performed data-driven simulations using the magnetofriction (hereafter MF) method (Yang et al., 1986; Craig and Sneyd, 1986). This method assumes that plasma velocity v in the induction equation to be proportional to local value of the Lorentz force.14 This allows the magnetic field to evolve to a lower energy state while keeping field line connectivity (except at current layers where magnetic diffusion occurs). Cheung and DeRosa (2012*) used the MF code to model the evolution of NOAA AR 11158. During the course of its birth (into a relatively quiescent region on the Sun) and passage across the solar disk, AR 11158 launched multiple eruptions and large M- and X-class flares (see Schrijver et al., 2011; Sun et al., 2012).

For their data-driven model, Cheung and DeRosa (2012*) used an potential field extrapolation as an initial condition, and imposed Ex and Ey at the bottom boundary to drive the model forward in time. Since they only used longitudinal magnetograms (albeit remapped from SDO/HMI data), the horizontal electric field components are not well-constrained. So they performed numerical experiments under varying assumptions. In particular, they tested how the presence (or absence) of sustained twisting/shearing of the photospheric field affected the structure of the modeled AR. They found that in the case of applied twisting/shearing, multiple flux rope ejections emanated from the sheared arcade above the sharp polarity inversion line (see the animated version of Figure 64Watch/download Movie). The timing of these flux rope ejections do not coincide with the times of actual observed eruptions since the applied boundary condition was ad hoc. However, the model did give an enhanced horizontal field at the polarity inversion line following flux rope ejections. Such behavior was also reported in an observational study of AR 11158 using SDO/HMI data (Wang et al., 2012).

The time-evolutionary approach (whether using MF or MHD) should be clearly distinguished from other data-constrained modeling approaches like non-linear force-free field extrapolation (NLFFF; see Mikić and McClymont, 1994; McClymont et al., 1997*; DeRosa et al., 2009*, for extended discussions), which use individual vector magnetograms to extrapolate the instantaneous coronal field. In the work of Cheung and DeRosa (2012), the evolution of AR 11158 was modeled by driving the simulation with a temporal sequence of magnetograms to capture the AR from emergence to eruption (Gibb et al., 2014, used a similar approached for modeling AR 10977). This was motivated by previous studies using the MF approach to model the evolution of the global corona (Yeates et al., 2007, 2008; Yeates and Mackay, 2009a,b; Yeates et al., 2010). These studies demonstrate that the coronal magnetic configuration at a given time is dependent on prior photospheric evolution. While time-independent NLFFF models may be useful for studying the 3D coronal field in some scenarios (consult McClymont et al.*, 1997 and DeRosa et al.*, 2009 for critical assessments regarding NLFFF extrapolation), they do not provide information about how photospheric driving (as manifested in terms of the electric field) transports magnetic flux, energy, and helicity into the atmosphere. Since these central questions of flux emergence are not addressed, a detailed discussion of time-independent data-constrained models is outside the scope of this review.

To advance beyond data-inspired and (ad hoc) data-driven models of emerging and erupting ARs requires robust and reliable measurements of the photospheric vector velocity and magnetic fields. These quantities allow one to compute the − v × B electric field that is required for setting boundary conditions that faithfully describe transport processes associated with flux emergence. The measurement of such quantities is not only useful for MF models, but also for time-dependent MHD models that aim to capture the full dynamical consequences of emerging flux. Recent work on data-driven modeling is preceded by studies that attempt to constrain photospheric flows and their associated electric fields (Chae, 2001; Kusano et al., 2002; Démoulin and Berger, 2003; Longcope, 2004; Welsch et al., 2004, 2007; Schuck, 2005, 2006; Georgoulis and LaBonte, 2006; Ravindra et al., 2008). Advances in this direction of research depend on high-cadence vector magnetograms (e.g., from SDO/HMI or SOLIS) and observations of the atmospheric response to emerging flux. Reliable data sets, together with improvements in methods to constrain photospheric electric fields (Fisher et al., 2010, 2012), will likely stimulate substantial progress in data-driven modeling in the coming years.

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