Potential-field source-surface model

Two groups introduced the approach in 1969. Kenneth Schatten, John Wilcox, and Norman F. 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]

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]