Q-Maps -- Insight into the Topology of the Coronal Magnetic Field

The topology of the coronal magnetic field is essential for understanding solar energetic events and properties of the solar wind. Magnetic null points, separatrix and quasi-separatrix surfaces are likely preferred sites for magnetic reconnection. (Quasi-)separatrix surfaces serve as boundaries between topologically distinct flux systems having different properties. Knowledge of the magnetic topology of the corona is useful to the solar and space physics research community, as well as for space-weather forecasting.

It is possible to identify topological regions in the corona where the magnetic field configuration may be unstable to perturbation. Q, a geometrical parameter that describes the squashing factor of elemental flux tubes, is useful for understanding the coronal configurations that are relevant to space weather. Q-maps provide intuitive visualization of the topological magnetic features where reconnection may preferably occur. Such structures are thought to be important for launching the slow solar wind and have been used to support the S-Web theory for the slow wind.

Q-maps are computed from the results of a 3D coronal field model applied to photospheric magnetic field observations. The general method developed by Titov et al. (2008, 2011) is applied to current observations from the HMI (Helioseismic and Magnetic Imager) instrument on SDO (Solar Dynamics Observatory) and to archival observations from the MDI (Michelson Doppler Imager) instrument on SOHO (the Solar and Heliospheric Observatory).

What is a Q-Map?

Q characterizes the local divergence of nearby magnetic field lines. Q is large at separatrix surfaces and quasi-separatix layers. Closed-field regions of similar field-line connectivity are enclosed by high-Q lines. Open field regions with different photospheric sources are also separated by high-Q lines.

Squashing Factor For a given field line, an infinitesimal circle is mapped along field lines into an infinitesimal ellipse. The circle and ellipse are on the boundary surfaces of a finite volume defined by curvilinear coordinates (U1,U2) and (W1,W2) -- for example the photosphere and the source surface of a coronal field model.

The aspect ratio of the resulting ellipse defines the squashing factor of the elemental flux tube enclosing the given field line. Q=N^2/delta where N is the generalized norm and delta is the expansion / contraction factor (Titov et al, 2008).

When N is large and/or delta is very small, Q can become large. slog Q is especially convenient for analyzing the structure of magnetic configurations where
slog Q = sign(Br) * log [Q/2 + (Q^2/4 - 1)^0.5 ]

We have calculated slog Q at ten heights above the photosphere for each solar rotation from 1996 to the present.

Q-Map Products Based on Potential Field Source Surface Coronal Field Model

The figure shows the photosphere and corona for one complete solar rotation in August 2010, Carrington Rotation 2099. There are four panels. The values are computed from HMI observations of the photospheric field. Each panel is a map of the whole Sun at some height at or above the photosphere. These figures are frames of a movie.

The upper left panel is the smoothed radial magnetic field in the photosphere from which the coronal field is calculated. Positive field is white and negative black. The Sun was not too active in mid 2010. Q-Map Movie Frame Upper right: Foot points near the photosphere that open to the solar wind. Blue points indicate where the magnetic field polarity is positive (out of the Sun). Red points have negative polarity.

Lower left: slog Q map at 1.10 solar radii showing complex structures. Blue and red show positive and negative polarity. Dark colors are high values of Q. Lower right: slog Q at 2.499 Rs showing simplified structure higher in the corona. The neutral line separating blue and red is the base of the heliospheric current sheet.

HMI Synoptic Products

Synoptic maps of Q are available from the Solar Dynamics Observatory Joint Science Operations Center (JSOC). JSOC data series can be exported like any other SDO HMI or AIA data. The primary data series are described below.

Click on the button to go to the JSOC lookdata program for the indicated data series.

A movie of the synoptic Q-maps, photospheric field, and open-field lines footpoint can be found in HMI Synoptic Q-Maps

MDI Synoptic Products

The Michelson Doppler Imager instrument on SOHO began observing the Sun in 1996 and stopped taking synoptic magnetic observations in 2011. Data are available for most rotations from CR 1911 -- CR 2107. The data quality and original resolution are different for MDI than for HMI. The two instruments overlap from CR 2097 - CR 2107 in 2010 - 2011.

A movie of the MDI synoptic Q-maps, photospheric field, and open-field lines footpoint can be found in MDI Synoptic Q-Maps.

HMI Daily Products

Daily Q-maps are computed using a better measure of the global magnetic field for each particular day.

Standard synoptic maps are observed over a ~27.27-day time span and are assembled using only central meridian observations as they rotate beneath the Earth. Each longitude is observed at a different time. A synoptic frame, used for this product, is constructed for each day using 120 degrees of longitude from one magnetogram to replace the leftmost edge of the usual synoptic map. At least the left-most third of the Sun is observed simultaneously on that day. There are some issues associated with time-evolution and proper motion of magnetic features, but daily Q-maps can better reflect dynamic conditions as they evolve on the solar surface and in the corona.

Daily Q-map for 2018.08.25    The colorful figure shows the Q-map for August 25, 2018 at 1.001 Rs. The map is 360 degrees wide and extends from pole to pole The left 120 degrees of the input map has been constructed from a magnetogram taken at 12 UT on that day. The rest of the input frame is the standard synoptic map observed a few to many days earlier at central meridian. A map is computed each day by extending the map to the left by about 13 degrees. The color bar for slogQ is at the bottom, with blue (yellow) representing negative (positive) magnetic polarity.

Click on Figure to compare the Q-Maps for 2018 August 25 and 26.

Annual movies of daily Q-maps can be found in Yearly Synoptic Frames.

Q-Maps Based on Other Coronal Field Models

Alternative Coronal Models

Q-maps are computed using a data cube of coronal field values, which can be determined using any model. Contact the HMI Magnetic Team for information about additional models.

Active Region Q-Maps

We have computed high-resolution Q-Maps above a limited number of active regions using a non-linear force-free field (NLFFF) model applied to HMI vector magnetic field measurements. The figure shows results for a portion of AR 12673 on 6 September 2017 at 11:35 UT.

We may be able to fulfill requests for certain active regions of interest. Currently the processing runs in IDL and must be tailored for each application.

Q-Map Slices over AR 12673 Panel (a) shows the radial component of the photospheric field in a portion of AR 12673 observed 6 September 2017. The neutral line, in orange, is crossed by four green lines labeled (b) - (e). The lines are the bottom boundaries of vertical slices of log-Q maps shown in panels (b) - (e). In this case the field was determined using a Non-Linear Force-Free Field (NLFFF) model. See Zhao et al, 2014) for model details.

These data sets were prepared with support of NASA Grant NNX15AN49G to Stanford University
and was a collaboration between the Stanford Solar Observatories Group and Predictive Science, Inc.