Potential-field source-surface model

Potential-Field Source-Surface Model

Field	Solar physics, Space weather
Origin	1969
Key people	Kenneth Schatten, John Wilcox, Norman Ness; Martin Altschuler, Gordon Newkirk
Purpose	Modeling the solar coronal magnetic field and predicting solar wind properties

The potential-field source-surface model (PFSS) is a widely used coronal magnetic field model that assumes the magnetic field in the solar corona is current-free up to a spherical outer boundary, the "source surface," where field lines are forced to be radial to mimic the action of the solar wind. ^{[1] [2]} In practice, the source-surface radius is often taken to be about 2.5 R_☉ solar radii, a value that emerged from eclipse constraints and comparisons with interplanetary measurements and that later studies found to work well for many rotations. ^{[2] [3] [4]} Solar physicists use PFSS to map open magnetic field regions that correlate with coronal holes, to estimate the heliospheric current sheet shape, and to drive space-weather models of the solar wind. ^{[5] [6]}

History

Two groups introduced the approach in 1969. Kenneth Schatten, John Wilcox, and Norman Ness linked photospheric fields to a notional outer "source surface" and showed that the sector structure of the interplanetary magnetic field reflects the polarity pattern at that surface. ^[1] Martin Altschuler and Gordon Newkirk developed spherical-harmonic methods to solve Laplace's equation, and proposed setting the scalar potential to a constant on a spherical surface around 2.5 solar radii to capture the corona's transition to a solar-wind-dominated regime. ^[2] J. Todd Hoeksema compared PFSS predictions with interplanetary magnetic field observations and found an optimal source-surface radius near 2.5 ± 0.25 solar radii for many intervals in solar cycle 21 (1976–1986). ^{[3] [4]}

During the 1990s, Y.-M. Wang and N. R. Sheeley Jr. connected PFSS open-flux geometry to solar wind speed, which underpinned the Wang–Sheeley–Arge forecasting framework that pairs a PFSS coronal solution with empirical wind-speed relations. ^{[5] [6]}

Model

A synoptic magnetogram from the Helioseismic and Magnetic Imager aboard the Solar Dynamics Observatory, showing the radial magnetic field at the solar photosphere used as input for PFSS models,

PFSS assumes a quasi-static, current-free magnetic field from the photosphere (${\displaystyle r=R_{\odot }}$) to a spherical "source surface" at (${\displaystyle r=R_{\rm {ss}}}$).

The magnetic field is written as the gradient of a scalar potential, (${\displaystyle \mathbf {B} =-\nabla \Phi }$), with (${\displaystyle \nabla ^{2}\Phi =0}$) in the modeling shell.

Boundary conditions are

• at ${\displaystyle r=R_{\odot }}$, the radial field ${\displaystyle B_{r}(R_{\odot },\theta ,\phi )}$ matches a synoptic magnetogram map of the photosphere
• at ${\displaystyle r=R_{\rm {ss}}}$, the field is purely radial, so ${\displaystyle B_{\theta }=B_{\phi }=0}$, which is equivalent to ${\displaystyle \Phi =}$ constant on the source surface. ^{[2] [5]}

With these conditions, the solution can be expanded in spherical harmonics,

${\displaystyle \Phi (r,\theta ,\phi )=\sum _{\ell =1}^{\ell _{\max }}\sum _{m=-\ell }^{\ell }\left[a_{\ell m},r^{\ell }+b_{\ell m},r^{-(\ell +1)}\right]Y_{\ell m}(\theta ,\phi ),}$

and the coefficients ${\displaystyle a_{\ell m}}$ and ${\displaystyle b_{\ell m}}$ are determined by matching the observed photospheric ${\displaystyle B_{r}}$ and enforcing vanishing tangential components at ${\displaystyle r=R_{\rm {ss}}}$. ^{[2] [7]}

Two families of numerical solvers are commonly used. Many implementations employ a spherical-harmonic expansion with a truncation (${\displaystyle \ell _{\max }}$) determined by the input map resolution. However, these methods can produce ringing artifacts near sharp magnetic structures and exhibit sensitivity at high latitudes. ^{[5] [7]} Finite-difference solvers, which operate on remeshed latitude grids or deformed spherical grids, help mitigate these artifacts and allow for non-spherical outer boundaries. ^{[7] [8]}

PFSS solutions are driven by synoptic magnetograms assembled from line-of-sight magnetic field measurements. Common data sources include the Wilcox Solar Observatory, NSO GONG, SOHO/MDI, and SDO/HMI magnetogram maps. ^{[5] [4]} Operational and research software packages include the SolarSoft PFSS package maintained by Marc DeRosa and collaborators, ^[9] NASA CCMC's PFSS 1.0 service, ^[10] and the open-source Python library pfsspy. ^[11]

Several extensions to the basic PFSS model have been developed to relax its assumptions and better match specific observational data. The Schatten current-sheet source-surface (CSSS) and related models incorporate sheet and volume currents above the potential field domain to better represent the heliospheric current sheet and reduce discrepancies in open magnetic flux calculations. ^[12] The source-surface geometry has also been generalized beyond a simple sphere, including oblate or prolate shapes that can improve agreement with white-light coronal streamers and in situ magnetic sector structure for some solar rotations. ^[13] Studies have also suggested that the source-surface height varies with the solar cycle, creating a "breathing source surface" that is positioned at lower heights near solar minima and higher heights near solar maxima. ^{[14] [15]}

Validation and performance

Ulysses spacecraft observations of solar wind speed as a function of heliographic latitude during solar minimum. Slow wind (≈300 km/s) is confined to equatorial regions, while fast wind (≈800 km/s) originates from polar regions. Red and blue colors indicate outward and inward polarities of the heliospheric magnetic field, respectively.

PFSS maps the distribution of open versus closed magnetic field lines in the solar corona. Open-field footpoints at the photosphere correspond broadly to coronal holes observed in EUV and He I 10830 Å images, while the tallest closed loops trace the helmet streamer belt and pseudostreamers. ^[5] These connections enable forecasting of the heliospheric current sheet and estimation of solar wind sources. Operational systems such as the Wang–Sheeley–Arge model compute coronal solutions with PFSS, then assign wind speeds to open field lines using expansion-factor and boundary-distance predictors. ^[6] In a comprehensive review, Thomas Wiegelmann and colleagues noted that PFSS has been "successful in modeling the global" structure of the coronal field for many purposes. ^[5]

Comparisons with global magnetohydrodynamic models show that PFSS often reproduces the large-scale magnetic topology and sector structure, especially when the coronal field is near potential. However, it underestimates some cusp heights and open flux, and yields shorter field lines on average compared to more sophisticated models. ^{[17] [5]} Eclipse white-light and UV Ly-α synoptic maps have been used to tune or validate the source-surface height and the heliospheric neutral line location inferred from PFSS, typically favoring values near or slightly above 2.5 solar radii during solar minima. ^[18]

A persistent challenge known as the "open flux problem" remains unresolved. Studies find that PFSS and other coronal models tend to underestimate total unsigned open flux compared to in-situ measurements, particularly near solar maximum. Setting the source surface very low can improve flux estimates but then over-expands open-field footpoints at the photosphere. ^{[19] [20]}

PFSS remains attractive because it is "simple to develop and implement," as Pete Riley and coauthors noted in a comparison study, and it resolves global structure with low computational cost. ^[17] However, the same authors cautioned that PFSS assumptions are "seldom, if ever, met" in reality, since coronal currents, magnetic reconnection, and non-spherical source-surface shapes actually occur. ^[17] Reviews emphasize that while PFSS has been "successful in modeling the global" magnetic field on large scales, more sophisticated force-free or MHD models are required to address currents, energy, and dynamics in and around active regions. ^[5]

PFSS serves as the coronal backbone of the Wang–Sheeley–Arge model. Early speed predictions relied on the flux-tube expansion factor derived from the PFSS solution, but later work showed that the distance from an open-field footpoint to the nearest coronal hole boundary correlates more strongly with observed wind speed for many time intervals. ^{[6] [21]}

See also

• Wang–Sheeley–Arge model
• Parker spiral
• Heliospheric current sheet
• Coronal hole
• Force-free magnetic field
• Magnetohydrodynamics

References

1. Schatten, K. H.; Wilcox, J. M.; Ness, N. F. (1969), "A model of interplanetary and coronal magnetic fields", Solar Physics, 6 (3): 442–455, Bibcode:1969SoPh....6..442S, doi:10.1007/BF00146478
2. Altschuler, M. D.; Newkirk, G. (1969), "Magnetic fields and the structure of the solar corona. I: Methods of calculating coronal fields", Solar Physics, 9 (1): 131–149, Bibcode:1969SoPh....9..131A, doi:10.1007/BF00145734
3. Hoeksema, J. T. (1984), Structure and Evolution of the Large Scale Solar and Heliospheric Magnetic Fields (PhD thesis), Stanford University
4. Potential-Field Source-Surface Models, National Solar Observatory, 2018
5. Wiegelmann, T.; Petrie, G. J. D.; Riley, P. (2017), "Coronal Magnetic Field Models", Space Science Reviews, 210 (1–4): 249–274, Bibcode:2017SSRv..210..249W, doi:10.1007/s11214-015-0178-3
6. Arge, C. N.; Pizzo, V. J. (2000), "Real-time solar wind updates using the Wang–Sheeley model", Journal of Geophysical Research: Space Physics, 105 (A5): 10465–10480, doi:10.1029/1999JA000262
7. Tóth, G.; van der Holst, B.; Huang, Z. (2011), "Obtaining potential field solutions with spherical harmonics and finite differences", Astrophysical Journal, 732 (2): 102, arXiv: 1104.5672 , Bibcode:2011ApJ...732..102T, doi:10.1088/0004-637X/732/2/102
8. Kruse, M.; Heidrich-Meisner, V.; Wimmer-Schweingruber, R. F.; Hauptmann, M. (2020), "An elliptic expansion of the potential field source surface model", Astronomy & Astrophysics, 642: A171, arXiv: 2005.12843 , Bibcode:2020A&A...638A.109K, doi:10.1051/0004-6361/202037734
9. Schrijver, C. J.; DeRosa, M. L. (2003), "Photospheric and heliospheric magnetic fields", Solar Physics, 212 (1): 165–200, Bibcode:2003SoPh..212..165S, doi:10.1023/A:1022908504100
10. PFSS 1.0, NASA Community Coordinated Modeling Center
11. Stansby, D.; Yeates, A.; Badman, S. T. (2020), "pfsspy: A Python package for potential field source surface modelling", Journal of Open Source Software, 5 (54): 2732, Bibcode:2020JOSS....5.2732S, doi:10.21105/joss.02732
12. Zhao, X.; Hoeksema, J. T. (1995), "Prediction of interplanetary magnetic field strength" (PDF), Journal of Geophysical Research: Space Physics, 100 (A1): 19, 109–19, 116, Bibcode:1995JGR...100...19Z, doi:10.1029/94JA02266
13. Kruse, M.; Heidrich-Meisner, V.; Wimmer-Schweingruber, R. F. (2021), "Evaluation of a potential field source surface model with elliptical source surfaces via ballistic backmapping of in situ spacecraft data", Astronomy & Astrophysics, 645: A83, Bibcode:2021A&A...645A..83K, doi:10.1051/0004-6361/202039120
14. Lee, C. O.; Arge, C. N.; Odstrcil, D.; Pizzo, V.; Mays, M. L.; Curlin, R. J. (2011), "Coronal field opens at lower height during solar minima: Consequences for solar wind predictions" (PDF), Solar Physics, 269: 367–388, doi:10.1007/s11207-010-9699-9
15. Arden, W. M.; Norton, A. A.; Sun, X. (2014), "A "breathing" source surface for cycles 23 and 24", Journal of Geophysical Research: Space Physics, 119 (10): 1476–1485, Bibcode:2014JGRA..119.1476A, doi:10.1002/2013JA019464
16. McComas, D. J.; Elliott, H. A.; Schwadron, N. A.; Gosling, J. T.; Skoug, R. M.; Goldstein, B. E. (2003-05-15). "The three-dimensional solar wind around solar maximum". Geophysical Research Letters. 30 (10): 1517. Bibcode:2003GeoRL..30.1517M. doi: 10.1029/2003GL017136 . ISSN   1944-8007.
17. Riley, P.; Linker, J. A.; Mikić, Z.; Lionello, R.; Ledvina, S. A.; Luhmann, J. G. (2006), "A comparison between global solar magnetohydrodynamic and potential field source surface model results" (PDF), Astrophysical Journal, 653 (2): 1510–1516, Bibcode:2006ApJ...653.1510R, doi:10.1086/508565
18. Cora, A.; Mancuso, S.; Giordano, S. (2011), "Validation of the PFSS coronal magnetic field model through UVCS/SOHO synoptic observations", Memorie della Società Astronomica Italiana Supplementi, 19: 89–96
19. Linker, J. A.; Caplan, R. M.; Downs, C.; Riley, P.; Mikic, Z.; Lionello, R. (2017), "The open flux problem", Astrophysical Journal, 848 (1): 70, arXiv: 1708.02342 , doi: 10.3847/1538-4357/aa8e00
20. Badman, S. T.; Bale, S. D.; Rouillard, A. P.; McComas, D. J. (2021), "Measurement of the open magnetic flux in the inner heliosphere", Astronomy & Astrophysics, 650: A18, doi:10.1051/0004-6361/202039407
21. Riley, P. (2015), "On the role played by magnetic expansion factor in the location of the slow solar wind", Space Weather, 13: 154–169, doi:10.1002/2014SW001144