Coronal Magnetic Field Evolution from 1996 to 2012: Continuous Non-potential Simulations

Abstract

Coupled flux transport and magneto-frictional simulations are extended to simulate the continuous magnetic-field evolution in the global solar corona for over 15 years, from the start of Solar Cycle 23 in 1996. By simplifying the dynamics, our model follows the build-up and transport of electric currents and free magnetic energy in the corona, offering an insight into the magnetic structure and topology that extrapolation-based models cannot. To enable these extended simulations, we have implemented a more efficient numerical grid, and have carefully calibrated the surface flux-transport model to reproduce the observed large-scale photospheric radial magnetic field, using emerging active regions determined from observed line-of-sight magnetograms. This calibration is described in some detail. In agreement with previous authors, we find that the standard flux-transport model is insufficient to simultaneously reproduce the observed polar fields and butterfly diagram during Cycle 23, and that additional effects must be added. For the best-fit model, we use automated techniques to detect the latitude–time profile of flux ropes and their ejections over the full solar cycle. Overall, flux ropes are more prevalent outside of active latitudes but those at active latitudes are more frequently ejected. Future possibilities for space-weather prediction with this approach are briefly assessed.

Appendices

Appendix A: Computational Grid

Our computational grid is divided into latitudinal sub-blocks, each of which has uniform spacing in the stretched variables

$$ x=\phi/\Delta, \qquad y=-\log\bigl(\tan(\theta/2)\bigr)/\Delta, \qquad z=\log(r/ \mathrm{R}_{\odot})/\Delta $$

(16)

(van Ballegooijen, Priest, and Mackay 2000), where Δ is the equatorial grid spacing in longitude [ϕ]. The horizontal cell-sizes are dx=dy=1 for the equatorial sub-block, and double in each sub-block towards the poles (Figure 7). The vertical cell size is dz=1 for all sub-blocks. This introduction of sub-blocks with different spacing counters the problem of grid convergence toward the poles, since the horizontal cell area is Δ² r ²sin² θ dxdy. The sub-block boundaries in latitude are determined so that each grid cell is as large in horizontal area as possible, while never exceeding the equatorial cell area Δ² r ² (Figure 8). With a longitudinal resolution of 192 cells at the Equator and 12 at the poles, there are nine sub-blocks. (The number 12 is chosen due to the parallel architecture used.) The total number of grid cells in (x,y) is 18 936, as compared to 63 744 for a uniform, single-block grid with unit spacing.

Figure 7

Example of the variable grid with 48 cells at the Equator (compared to 192 in the actual simulations) and seven sub-blocks (compared to nine).

Figure 8

Horizontal cell area (relative to that at the Equator) as a function of latitude, where symbols (joined by solid lines) show the variable grid and dashed lines a uniform grid with dx=dy=1 everywhere.

A complication arising from the variable grid is that the different sub-blocks need to communicate ghost values of B _0r and B _0ϕ with one another at each timestep. This is analogous to the inter-level communications in Adaptive Mesh Refinement (AMR) codes, except that our grid is fixed in time. Restriction (from fine to coarse sub-blocks) is the simpler process, for which we use an area-weighted average over fine grid cells (Balsara 2001). Prolongation (from coarse to fine sub-blocks) is trickier since the coarse-block solution must be interpolated to get the fine-block ghost values at intermediate locations. We follow the Taylor expansion method of van der Holst and Keppens (2007), using monotonic van Leer slope estimates (Evans and Hawley 1988). These slopes are also computed throughout the grid and used for slope-limiting in the advection terms, in order to prevent spurious oscillations near sharp gradients. Our numerical tests indicate that numerical diffusion due to this “upwinding” is negligible compared to the physical diffusion D. Finally, after computing the update ∂ A ₀/∂t within each sub-block, we replace boundary values on coarser sub-blocks with those derived from finer sub-blocks. This is analogous to the “flux correction” of Berger and Colella (1989).

Appendix B: Global Boundary Conditions

The staggered grid requires ghost-cell values of two components of B ₀ outside each boundary. In longitude the domain is simply periodic. On the photosphere r=R_⊙, we fix ghost cell values of B _0θ , B _0ϕ by requiring that v _0r =0 on r=R_⊙ in the magneto-frictional model. On the outer boundary r=2.5R_⊙, we impose a radial outflow velocity (see Yeates et al., 2010a) that models the effect of the solar wind radially extending field lines, while still allowing horizontal fields to escape during flux-rope ejections. The resulting ghost-cell values of B _0θ , B _0ϕ do not play an important role and are simply set by zero-gradient conditions. On the latitudinal boundaries (located at approximately ± 89.33^∘ latitude) we need ghost cell values for B _0r and B _0ϕ . The latter are simply set to zero, while the ghost values of B _0r are chosen to satisfy Stokes’ Theorem given the integral of A _0ϕ around the latitudinal boundary.