Dr Roger B. Scott

picture of Roger Scott

Roger B. Scott
Postdoctoral Fellow
Department of Mathematics
University of Dundee
Nethergate, DD1 4HN
Dundee, Scotland, UK
Fulton G19
rbscott at dundee.ac.uk
01382 388315

Global Magnetic Topology of the Solar Corona

This page serves as a working summary of ongoing research into surveying the topological structure of global solar magnetic fields. Work on this and related topics is being actively assembed for publication. Individual manuscripts are listed below, along with updated information regarding publication status.

Related Manuscripts

  1. Magnetic Structures at the Boundary of the Closed Corona: Interpreting S-web Arcs
    • A theory paper about S-web arcs and the underlying coronal topology.
    • Submitted: 11 May, 2018 -- arXiv preprint available here.
  2. HQVseg: A computational tool for volume segmentation of coronal models
    • Methodology paper for partitioning global flux domains and identifying quasi-separatrix structures relavent to interchange reconnection.
    • In preparation: preprint available here.
  3. On the Magnetic Squashing Factor and the Lie Transport of Tangents

Summary and preliminary findings

In the current phase of this study we attempt to identify persistent, generic coronal features that are likely to contribute to material exchange between the open and closed corona, through a process referred to as interchange reconnection. Subsequent related work will deal with the details of the reconnection process for particular configurations -- for now it is sufficient to identify which structures are likely to play a meaningful role. To this end, we require a metric for characterizing global magnetic field and identifying regions that are (1) susceptible to the formatio of strong electric currents, and (2) associated with the boundary(ies) between open and closed magnetic flux. To this end we have identified the magnetic squashing factor as a suitable metric, and we use this to characterize the field throughout the volume. We then use image segmentation techniques to partition the domain into subvolumes, whose interfaces coincide with quasi-separatrix layers (QSLs). Once partitioned, the domains can be sorted and their interfaces can be characterized in a logical manner.

Using these techniques we have identified a preliminary categorization criteria for QSLs, which allows for immediate identification of typical volumetric composition based on the (usually much simpler) structuring at the outer boundary of the domain. We identify we we call high-Q segments, and we divide these into four categories, based on how they intersect with the heliospheric current sheet (HCS). Fully detatched segments are free-standing arcs of high-Q, with no connection to the HCS, and these correspond to separatrix curtains from closed field structures embedded entirely within the open flux domain. Partially detatched segments are high-Q arcs that have a single intersection with the HCS (or one of its child segments) but terminate in a region surrounded by low-Q. These correspond to separatrix curtains that intersect the global helmet streamer, and are bounded by a single null in the open corona, with the other bounding null in the closed corona (or below the photosphere).

In addition to fully and partially detatched segments, we identify a third family of structures, whose main characteristic is that they are bounded at both ends by an intersection with another high-Q arc. We refer to these as fully attached, and within this family, there are 3 subclasses, which refer to the number of intersections that each makes with the HCS. These are referred two as n2, n1, and n0, referrering to the number of vertices shared with the HCS.

Our preliminary analysis has focused on identifying arcs at the outer boundary of the domain, categorizing them based on the aformentioned system, and then following these arcs into the volume to see what conclusions can be formed for the types of structures associated with each. In the case of fully and partially detatched segments, the association with expected underlying structures is easy to verify. For fully attached segments we find the following correspondence. n2 (twice connected) arcs are usually associated with narrow (possibly singular) corridors of open flux at the photospheric level, that opens slowly and continuously with height above the photosphere and eventually expands to become the heliospheric current sheet. n1 and n0 arcs invariably occur along a QSL structure with an underyling closed field separatrix dome that completely detatches adjacent open-flux domains. Within this configuration, the n1 arcs are then typically related to the separatrix curtain of the dome, while the n0 arcs are related to singular corridors along the already detatched open-field region.

Magnetic Field Model

As a preliminary study, we have chosen to focus on a potential field source surface model for the global magnetic field. The model is constructed by using GONG magnetogram data in conjuction with the DuMFriC code developed by A. R. Yeates. The code solves the time-dependent magnetofrictional approximation to the mhd equations by integrating the vector potential while assimilating magnetogram data to account for ongoing flux emergence and evolution of the photospheric boundary condition. For our purposes we are only using the initial condition in order to calculate the PFSS field extrapolation, however in future studies we intend to apply the method to the time-dependent solution and so we have developed the pipeline with this in mind. The PFSS is calculated using a grid resolution of [60, 180, 360] for coordinates [r, cos-theta, phi], spanning a domain of r in [1, 2.5], costheta in [-1,1], and phi in [0, 2pi]. We use a gaussian filter of width 0.002 (normalized to the size of the domain). As the resulting PFSS defines the vector potential at cell faces, we must interpolate to get the magnetic field at grid center for the qslSquasher code. Once finished the input magnetic field has a grid resolution of [61, 180, 360]. We have used this technique to get PFSS models for gong magnetorgram data corresponding to the following list of dates. Mostly taken on 01 Jan of each year these represent an unbiased sample. The exception is the 29 July data, which does not follow the pattern as it was the first sample dataset that was included with the DumFRiC source code when provided by A. R. Yeates as an example setup.

  • 01 Jan, 2009
  • 01 Jan, 2010
  • 01 Jan, 2011
  • 01 Jan, 2012
  • 01 Jan, 2013
  • 01 Jan, 2014
  • 29 Jul, 2014
  • 01 Jan, 2015
  • 01 Jan, 2016
  • 01 Jan, 2017

Q estimation

3d rendering of Q isosurface Q visualization r=2.5
The 3D squashing factor can be visualized using isosurfaces in ParaView. Slices of Q are easier to visualize. This slice is of the magnitude of Q (no open/closed information encoded) at the top boundary

The above summarised work relies on a method for quickly and efficiently estimating the magnetic squashing degree throughout a 3D mangetic domain. In a previous paper we considered the method of Tassev & Savcheva and found the technique to be theoretically sound and computationally expedient. We have proceeded, therefore, using the QSL Squasher code, developed by Svetlin Tassev. The code has been adapted slightly to better suit our purposes, but its fundamental method is the following. A list of coordinate axes and values for the magnetic field are provided as a regular grid. Additionally, the extent and resolution of the desired output are provided in a configuraiton file. The code is written in C++ with OpenCL, VexCL, and Boost packages, which provide GPU computing functionality, and must be compiled separately for each configuration. Adaptive refinement is performed by linearizing the output along a hilbert curve and then checking for the local change in the diagonal distance between field line end-points (referred to as the field-line edge length, FLEDGE). Under refinement, the code inserts an additional point between any two that fail the criteria, so in principle the number of points after each refinement level goes as 2^n (n being the number of refinements). The output of the code is a list of coordinates and associated Q values. For our purposes, this has been adapted so that the reported value of Q is negative if a given field line terminates at the outer boundary (open field) and positive otherwise. We visualise the result using the metric slog10q, which is simply log10(abs(Q))*sign(Q).

An animated 3D figure showing slices through the volume with field lines threading the domain is given here.

Volume Segmentation

Mask of QSLs at top boundary Mask of QSLs at bottom boundary.
The QSL mask identifies regions of high-Q which will be used to partition the rest of the domain.

Having calculated the magentic squashing degree we have a volumetric indication for the location of separatrix layers and quasi-separatrix layers, as well as a flag for open vs. closed flux domains. We then proceed to divide the volume into descrete domains, using QSLs as the surfaces bounding these domains. And because Q is conserved along field lines, such a construction naturally divides the domains such that each can be identified as a flux-tube. I.e., if the boundaries are isosurfaces of Q, they will always be tangent to B. Unfortunately, due to numerical errors in the calculation, this identification is not exact. Moreover, in the case of genuine separatrix surfaces, it is possible for Q to diverge in such a way that it is infinite in a region between adjacent pixels, but finite and relatively small at the sampled grid centers. In this case, however, the field line mapping is discontinous and so, it's properties on either side of the discontinuity are unrelated. Therefore, we expect the behavior across such an unresolved layer to be similar to that of a step function, and we detect such cases by inspecting the gradient of Q. The final identification of a QSL is then given by any region where the magnitude of the Sobolev vector (Q, grad lnQ) exceeds a threshold value (usually 10^2.5).

Region segmentation layer 0 Region segmentation layer 1 Region segmentation layer 2 Region segmentation layer 3 Region segmentation layer 4 Region segmentation layer 5
Descending down into the volume, the segmentation demonstrates the behavior of each flux domain.

From thresholding we develop a mask that is then smoothed using morphological operators (opening and closing). Then, at each boundary we can discretise the domains by simply labeling each simply connected region on that boundary . This technique is robust at the outer boundery where the field lines are predominantly radial (identically in the case of a potential field) but less robust within the volume and at the lower boundary. In particular, any numerical flaw in the realisation of a QSL will create a "leak" that will prevent the identification of individual domains. We therefore apply the labeling only at the boundaries and then "grow" the regions into the volume using a watershed algarithm. This is accomplished by using the boundary labels as a seed and then using Q as the scalar "elevation" for the 3D watershed (analogous to 2D but without the simple topographical visualization). The watershed is done in multiple steps, first with the QSL mask preserved and the open-closed flux distinction preserved. Then with the mask removed, and then with the OC mask relaxed (the regions can grow through opposite signed flux but cannot incorporate it). After region growing, every pixel in the volume identifies with an individual flux domain and retains information about the sign (open or closed) of the flux.

Logical Sorting

With the entire domain segmented we can identify individual QSL structures based on their proximity to each flux domain. To do this we estimate the typical thickness of the QSL mask by performing a distance transform using the QSL mask itself. Then, for each region, we perform an additional distance transform to get the smallest distance for each pixel in the domain to the nearest element of that region. From this we can then condluce that any element is "adjacent" to any other element if the distance transform between them is less than a threshold value comapared to the typical QSL distance. This then allows the identification of each flux domain with its surrounding QSL structure by considering the intersection of the QSL mask with the collection of points that self-identify as ``adjacent'' to the region. The QSL dividing any two domains is the intersection of two QSLs for a pair of domains. And if we think of QSLs as separatrix fans, then the separator at a fan-fan intersection can be found by considering the intersection of 3 and 4 flux domains. Working backward, one can find a QSL vertex and querry to see what domains it is close to, and the find the intersection of QSLs for all of those domains, which will recover the vertex and it's entire volume extent. Such techniques can be used to find separatrix layers that span the open closed boundary as in, eg, a a separatrix dome on the open closed boundary with a fan that forms a partially detached QSL segment.

Simple topologies / geometries

While the general topology of global fields can be quite complex, we can understand the typical structures as building upon a few simple representative configurations. Here we consider a subset of possible geometries.
Triple-null dome with fan-fan intersection of helmet streamer
3-null dome helmet intersection
A 3-null separatrix dome is coincident with the open-closed boundary and forms a fan-fan intersection with the helmet streamer. At high altitude, there is a partially detached arc that terminates on an open-field spine. This arc is part of the fan surface from a prone null, bounded by the spines of two upright nulls, whose fans form a separatrix dome that straddles the open-closed boundary..
Triple-null tunnel with fan-fan intersection of helmet streamer
3-null dome helmet intersection
A 3-null separatrix tunnel is coincident with the open-closed boundary and forms a fan-fan intersection with the helmet streamer. At high altitude, there is a partially detached arc that terminates on a bald patch separatrix surface. The arc is part of the fan surface of a prone null which, in the absence of a bald patch, must terminate in a closed field region, inside the helmet streamer.