Skip to main content

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


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.

This is a preview of subscription content, access via your institution.

Figure 1
Figure 2
Figure 3
Figure 4
Figure 5
Figure 6


Download references


The author thanks NSO for support to attend the 26th Sac Peak workshop, D.H. Mackay for useful discussions during this research, and the anonymous referee for helpful suggestions. Numerical simulations used the STFC and SRIF funded UKMHD cluster at the University of St Andrews. Magnetogram data from NSO/Kitt Peak were produced cooperatively by NSF/NSO, NASA/GSFC, and NOAA/SEL, and SOLIS data are produced cooperatively by NSF/NSO and NASA/LWS. The observed CME catalogue used is generated and maintained at the CDAW Data Center by NASA and The Catholic University of America in cooperation with the NRL. SOHO is a project of international cooperation between ESA and NASA.

Author information

Authors and Affiliations


Corresponding author

Correspondence to A. R. Yeates.

Additional information

Solar Origins of Space Weather and Space Climate

Guest Editors: I. González Hermández, R. Komm, and A. Pevtsov


Appendix A: Computational Grid

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

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

(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
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
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.

Rights and permissions

Reprints and Permissions

About this article

Cite this article

Yeates, A.R. Coronal Magnetic Field Evolution from 1996 to 2012: Continuous Non-potential Simulations. Sol Phys 289, 631–648 (2014).

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI:


  • Coronal mass ejections, theory
  • Magnetic fields, corona
  • Magnetic fields, models
  • Magnetic fields, photosphere
  • Solar cycle, models