Addressing the open flux problem with a non-spherical solar coronal magnetic field model

To resolve the long-standing open-flux problem where traditional models underestimate solar magnetic flux, this paper introduces a Non-Spherical Potential Field (NSPF) model featuring a Non-Spherical Source Surface that successfully reproduces complex coronal topologies and aligns extrapolated open flux with in-situ measurements.

The Gist

The Big Problem: The Sun's Invisible "Leak"

Imagine the Sun as a giant, glowing balloon. Inside, it's hot and chaotic. Outside, it's surrounded by a magnetic "force field" that controls how solar wind (a stream of charged particles) escapes into space.

Scientists have a big problem: The "Leak" Mismatch.

• The Theory: When we use our best computer models to calculate how much magnetic "stuff" (flux) is leaking out of the Sun, the number is too small.
• The Reality: When we send spacecraft (like the Parker Solar Probe) out into space to measure it, they find way more magnetic flux than the models predict.

It's like trying to calculate how much water is leaking from a swimming pool by looking at the drain, but the actual amount of water missing is double what your math says. This is called the "Open Flux Problem."

The Old Solution: The "Perfect Sphere" Mistake

For decades, scientists have used a model called PFSS (Potential Field Source Surface).

• How it works: Imagine the Sun is a ball, and we draw a perfect, invisible sphere around it (like a giant glass bubble).
• The Rule: Inside the bubble, the magnetic field is complex and twisted. But the moment a magnetic line hits this glass bubble, the model forces it to go straight out, like a laser beam.
• The Flaw: The Sun isn't a perfect sphere, and its magnetic field isn't uniform. By forcing everything to be a perfect sphere, the model misses the "nooks and crannies" where the magnetic field actually opens up. To fix the "leak" problem, scientists used to just shrink the glass bubble smaller and smaller, but that made the physics look fake (like squishing a balloon until it pops).

The New Solution: The "Smart, Shapeshifting Bubble"

The authors of this paper (led by Ziqi Wu and Jiansen He) invented a new model called NSPF (Non-Spherical Potential Field).

Instead of a rigid, perfect glass bubble, they created a smart, shapeshifting bubble (the Non-Spherical Source Surface).

Here is how they built it, using a simple analogy:

1. The "Magnetic Umbrella" Analogy

Imagine the Sun's magnetic field lines are like the ribs of a giant umbrella.

• The Old Model: It assumed the umbrella was perfectly round. If the wind (solar wind) was strong, the model just said, "Okay, the whole umbrella is open," but it didn't account for the fact that some ribs are weaker than others.
• The New Model: The authors realized that where the umbrella is weak (the "cusp" or the tip of a streamer), the wind can blow the fabric up and open a hole.
• The Innovation: They let the "bubble" surface dip down exactly where the magnetic field is weak and the plasma is escaping. It's like a trampoline that sags under your weight. The surface isn't a sphere; it's a concave bowl sitting right under the escaping solar wind.

2. The Three-Layer Cake

The new model is built like a three-layer cake to handle different parts of the Sun's atmosphere:

• Layer 1 (The Bottom): The photosphere (the Sun's surface). This is where the magnetic field starts.
• Layer 2 (The Middle): The "Smart Bubble" (The Non-Spherical Source Surface). This is where the magic happens. The model calculates exactly where the magnetic field is weak enough to let the solar wind escape. It creates a "valley" in the magnetic field right under the escaping streams.
• Layer 3 (The Top): The Interplanetary Space. Once the wind escapes the bubble, it follows the "Parker Spiral" (a spiral shape caused by the Sun's rotation, like water spraying from a spinning garden hose).

Why This Matters: Solving the Mystery

By using this "shapeshifting bubble" instead of a rigid sphere, the model finally solved the puzzle:

1. It found the missing leak: Because the bubble dips down into the weak spots, it allows more magnetic field lines to escape naturally. This matches the high numbers measured by the spacecraft.
2. It looks more real: When they compared the model to actual pictures of the Sun (taken by telescopes), the magnetic loops looked exactly like the real sunspots and loops, not the fake, tiny loops of the old models.
3. It predicts the wind better: When they traced the path of the solar wind back to the Sun, the new model pointed to specific, compact areas on the Sun's surface. The old models were guessing wildly, sometimes pointing to the North Pole for a wind stream that clearly came from the South.

The Takeaway

Think of the old model as trying to fit a square peg in a round hole, or trying to wrap a gift with a rigid, spherical box. It didn't fit the Sun's messy, complex reality.

The new NSPF model is like wrapping that gift with flexible, stretchy fabric. It molds itself to the shape of the Sun's magnetic field, dipping down where the wind escapes and rising where it doesn't.

In short: The Sun's magnetic field isn't a perfect sphere. It's a bumpy, wiggly landscape. By letting the "boundary" of the model wiggle and dip along with the real physics, the scientists finally figured out where all that missing magnetic energy was hiding. This helps us better predict space weather, which protects our satellites and power grids on Earth.

Technical Summary

1. Problem Statement

The paper addresses the long-standing "open flux problem" in solar physics.

• Context: The Potential Field Source Surface (PFSS) model is the standard method for extrapolating the 3D coronal magnetic field from photospheric magnetograms. It assumes a current-free corona and a spherical "source surface" (typically at $2.5 R_\odot$) where field lines become radial.
• The Discrepancy: PFSS models consistently underestimate the total open magnetic flux compared to in-situ measurements (e.g., from the Parker Solar Probe, PSP).
• Limitations of Current Solutions:
  • Simply lowering the spherical source surface radius (e.g., to $1.5 R_\odot$) increases open flux but creates unphysical geometries, forcing coronal loops to be unrealistically small and introducing artificial polarity reversals.
  • Previous attempts to use non-spherical source surfaces (e.g., isogauss surfaces or ellipsoids) have been too computationally complex or lacked a fully practical formulation for real solar magnetograms.
  • Existing models struggle to simultaneously reproduce complex coronal loop structures, correct open flux budgets, and accurate solar wind source mappings.

2. Methodology: The Non-Spherical Potential Field (NSPF) Model

The authors propose a novel Non-Spherical Potential Field (NSPF) model that replaces the rigid spherical source surface with a dynamic, non-spherical one.

• Core Concept: The model introduces a Non-Spherical Source Surface (NSSS) defined as an isosurface of the total magnetic field strength ($|B|$).
  • Physical Justification: The NSSS naturally forms concave structures beneath helmet streamers (where plasma pressure exceeds magnetic tension) and allows field lines to open more realistically near open-closed boundaries.
• Model Architecture: The model consists of three concentric layers:
  1. Potential Field Layer: Extends from the photosphere to the NSSS.
  2. Current Sheet Layer: Extends from the NSSS to an exit sphere ($10 R_\odot$). This layer mimics the Schatten Current Sheet (SCS) approach, enforcing a uniform polarity to create a global Heliospheric Current Sheet (HCS) before restoring original polarities.
  3. Interplanetary Layer: Extends beyond $10 R_\odot$, where field lines follow the Parker spiral configuration.
• Numerical Implementation:
  • Solver: Uses the Finite Element Method (FEM) via the DOLFINx (FEniCS) library to solve the Laplace equation ($\nabla^2 u = 0$) for the magnetic scalar potential. This allows for handling irregular, non-spherical boundaries.
  • Workflow:
    1. Compute an initial potential field using a standard spherical source surface.
    2. Extract the NSSS as an isosurface of $|B|$ from this initial solution.
    3. Re-calculate the potential field within the domain bounded by the photosphere and the extracted NSSS.
    4. Apply the current sheet layer and Parker spiral extension.
  • Optimization: Instead of a fully iterative self-consistent loop (which was found to cause the surface to shrink), the authors select the optimal NSSS configuration by varying the initial spherical radius and matching the resulting open flux to observational constraints.

3. Key Contributions

• Novel Geometry: First practical implementation of a purely non-spherical potential field model for the solar corona using FEM, moving beyond the spherical constraint of PFSS.
• Physical Realism: The NSSS naturally creates concave shapes under streamer bases, allowing for a more realistic distribution of open field regions without artificially compressing all coronal loops.
• Integrated Workflow: A comprehensive pipeline that links photospheric magnetograms directly to Interplanetary Magnetic Field (IMF) predictions and solar wind source tracing.
• Open Flux Resolution: Successfully resolves the open flux problem by generating sufficient open flux through geometric flexibility rather than arbitrary parameter tuning.

4. Results

The model was tested using Carrington Rotation 2282 (March 2024, solar maximum) and compared against standard PFSS and PFSS+PFCS models using multi-instrument data (SDO/AIA, SOHO/LASCO, and PSP).

• Coronal Topology:
  • The NSPF model (with an initial radius of $2.2 R_\odot$) produced magnetic topologies that matched EUV and white-light coronagraph observations significantly better than standard PFSS models.
  • It correctly reproduced complex structures, such as separated helmet streamers and the face-on Heliospheric Current Sheet (HCS), which spherical models failed to capture accurately.
• Open Magnetic Flux:
  • The NSPF model generated a total open flux ($13.01 \times 10^{22}$ G cm$^2$) comparable to the best-performing PFSS+PFCS model ($13.65 \times 10^{22}$ G cm$^2$ with $R_{SS}=1.5 R_\odot$).
  • Crucially, unlike the PFSS+PFCS model which forces all loops below a uniform height, the NSPF model allowed loop heights to vary physically ($0.05 - 1.08 R_\odot$).
• IMF and Solar Wind Mapping:
  • IMF Prediction: The NSPF model showed superior agreement with PSP in-situ measurements, particularly in predicting the timing and location of polarity reversals (HCS crossings). It achieved a Relative Normalized Root Mean Squared Error (RMSE) of 0.10, compared to 0.09 for the best PFSS+PFCS model but with better physical consistency.
  • Source Regions: The NSPF model mapped solar wind footpoints to compact, localized regions near polarity separatrices. In contrast, the PFSS+PFCS model produced large-scale, unphysical drifts of footpoints between hemispheres.

5. Significance

• Solving the Open Flux Problem: The study demonstrates that the open flux deficit is largely a geometric issue. By allowing the source surface to deform based on magnetic field strength (mimicking plasma escape conditions), the model naturally generates the required open flux without violating physical constraints on loop sizes.
• Improved Space Weather Modeling: The NSPF model provides a more accurate initial condition for global MHD simulations and improves the reliability of solar wind source tracing, which is critical for space weather forecasting.
• Future Applications: The framework is adaptable for future constraints using high-resolution observations (e.g., from the Proba-3 mission) and machine learning, offering a robust foundation for understanding solar-heliospheric magnetic coupling.

In conclusion, the NSPF model represents a significant advancement over traditional PFSS models by integrating non-spherical geometry into potential field extrapolation, successfully reconciling coronal observations with in-situ solar wind measurements.