Sunspot simulations with MURaM -- I. Parameter study using potential field initial conditions

This study utilizes the MURaM radiative MHD code to demonstrate that sunspot simulations initialized with potential magnetic fields and intensified bottom strengths (specifically 160 kG) best reproduce observed magnetic and dynamic properties, though achieving fully developed penumbrae and realistic field strengths likely requires higher numerical resolution and careful management of total magnetic flux.

The Gist

Imagine the Sun as a giant, churning pot of boiling soup. Sometimes, huge magnetic "knots" get tangled up in this soup and rise to the surface. When they do, they create dark, cool spots called sunspots. These spots aren't just empty holes; they have a dark center (the umbra) and a fuzzy, striped outer ring (the penumbra).

For years, scientists have tried to simulate these sunspots on supercomputers to understand how they work. But there was a problem: their computer models kept making sunspots that looked a bit "off." The magnetic fields were too flat, and the flow of gas didn't match what telescopes actually see.

In this new paper, a team of researchers decided to try a different recipe. Instead of forcing the magnetic fields to behave a certain way at the top of their simulation, they started with a "potential" field—a natural, relaxed magnetic shape—and let the physics do the rest. They ran dozens of simulations, tweaking the ingredients to see what made the best sunspot.

Here is what they found, explained with some everyday analogies:

1. The "Strength" of the Knot Matters

Think of the magnetic field as a rubber band.

• Weak Rubber Bands (20–40 kG): When they started with a weak magnetic "knot," the simulation barely made a sunspot at all. It just looked like a few stretched-out bubbles. There was no fuzzy outer ring (penumbra) and no outward flow of gas.
• Strong Rubber Bands (160 kG): When they cranked up the magnetic strength to 160 kG, the simulation finally created a sunspot that looked real. It had a dark center and a long, thin, striped outer ring that stretched out for miles.

2. The "Flow" Problem: Two-Way Traffic vs. One-Way Streets

In a real sunspot, gas usually flows outward from the center, like water draining from a bathtub but in reverse (this is called the Evershed flow).

• The Old Problem: Previous simulations often forced this flow to happen artificially.
• The New Discovery: In these new simulations, the flow was more complex.
  • Low Resolution (Blurry Camera): When the computer simulation was "blurry" (low resolution), the gas didn't just flow out. It flowed out in the middle and outer parts of the ring, but flowed in near the center. It was like a two-way street where cars were driving in opposite directions. This actually matches what we see in the early stages of real sunspot formation!
  • High Resolution (4K Camera): When they zoomed in with a super-sharp resolution, the simulation finally produced the classic, smooth outward flow (Evershed flow) in the outer parts, just like a real, mature sunspot.

3. The "Box" Size and Extra Ingredients

The researchers also tried changing the size of their virtual "box" and adding a uniform magnetic field to cancel out some of the flux (like adding salt to balance a soup).

• The Result: These changes didn't really change how the sunspot behaved inside the spot. The sunspot was like a self-contained storm; it didn't care much about the weather outside its immediate neighborhood. However, these changes did affect the magnetic field in the quiet areas around the spot.

4. The "Too Strong" Trap

They tried putting too much magnetic flux into the simulation (more than $10^{22}$ Maxwell).

• The Result: The center of the sunspot became unrealistically magnetic, like a magnet so strong it would rip a fridge door off. This told them that for a realistic simulation, they need to stick to a specific amount of magnetic "stuff" (around $10^{22}$ Maxwell).

The Big Takeaway

The main lesson from this paper is that resolution is key.

Imagine trying to paint a detailed portrait. If you use a thick brush (low resolution), you can't see the fine details, and the picture looks messy. If you use a fine brush (high resolution), the details emerge.

The researchers found that to get a sunspot that looks exactly like the ones we see in the sky—with the perfect striped ring and smooth outward flow—you need:

1. A very strong initial magnetic field (160 kG).
2. A specific amount of magnetic flux.
3. Crucially: A very high-resolution computer model.

Without the high resolution, the sunspot is stuck in an "adolescent" phase, showing two-way flows. With high resolution, it matures into the stable, flowing structure we observe. This helps scientists understand that the "messy" early stages of sunspot formation are actually a natural part of the process, driven by the emergence of new magnetic energy from deep inside the Sun.

Technical Summary

1. Problem Statement

Previous Magnetohydrodynamic (MHD) simulations of sunspots have struggled to reproduce observed magnetic field distributions accurately. A primary limitation in existing models (e.g., Rempel 2012) is the reliance on artificial top boundary conditions that force the magnetic field to be more horizontal at the photospheric level.

• Consequences of Artificial Boundaries: These conditions result in sunspots with an umbral-penumbral boundary inclination of $\approx 60^\circ$ (significantly more horizontal than the observed $\approx 45^\circ$) and a vertical magnetic field component ($B_z$) at the umbra boundary that is lower than observed values ($\approx 1.8$ kG).
• Goal: The authors aim to explore alternative initial conditions and parameter variations to better reproduce the magnetic and dynamic properties of observed sunspots, specifically focusing on the formation of the penumbra and the Evershed flow, without relying on artificial top boundary forcing.

2. Methodology

The study utilizes the MURaM (MPS/University of Chicago Radiative Magneto-hydrodynamics) code. The authors conducted a comprehensive parameter study using potential field initial conditions (proposed by Nordlund 2015) rather than self-similar or flux-emergence setups.

Simulation Setup

• Initial Conditions: A potential magnetic field was extrapolated from a Gaussian distribution at the bottom of the box ($B_{z,0}$). To mimic the observed vanishing field outside the spot, a uniform opposing vertical field ($B_{opp}$) was subtracted.
• Boundary Conditions:
  • Top: Open to inflows, closed to outflows, with magnetic field extrapolated as a potential field (no artificial inclination forcing).
  • Bottom: Generally open to mass flows, with magnetic fields extrapolated symmetrically. One specific run ('160_opp000b') tested a closed bottom boundary within the magnetic trunk.
• Parameters Varied:
  • Initial Magnetic Field Strength ($B_0$): 20, 40, 80, and 160 kG.
  • Magnetic Flux: Gaussian flux ($F_{Gauss}$) and total flux ($F_{tot}$) varied around $10^{22}$ Mx.
  • Opposing Field ($B_{opp}$): 0 to 300 G.
  • Box Width: 49.152 Mm (standard) vs. 98.304 Mm (wide).
  • Resolution: Low resolution (96 km horizontal / 32 km vertical) vs. High resolution (32 km / 16 km).
  • Radiative Transfer: Grey (low res) vs. Non-grey with 4 opacity bins (high res).

Data Processing

• Simulations were analyzed over a reference time of $\approx 4$ hours (or specific later times for high-res runs).
• Azimuthal averages of intensity, magnetic field vectors ($B_z, B_r, |B|$), inclination, and velocities ($v_r, v_z$) were calculated.
• Umbral and spot boundaries were defined by intensity contours ($I = 0.5 I_{qs}$ and $I = 0.9 I_{qs}$).

3. Key Contributions

1. Systematic Parameter Study: This is the first extensive parameter study of sunspot formation using potential field initial conditions with varying $B_0$, resolution, and box sizes in the MURaM code.
2. Resolution Dependence: The study highlights the critical role of numerical resolution in reproducing penumbral fine structure and flow patterns.
3. Validation of Potential Initial Conditions: It demonstrates that potential field initial conditions, combined with specific flux and field strength parameters, can yield realistic sunspot morphologies without artificial top boundary forcing.
4. Flux Emergence Dynamics: The paper links the observed bi-directional flows in simulations to ongoing flux emergence, offering a physical mechanism for early penumbra formation.

4. Key Results

A. Morphology and Size

• Field Strength ($B_0$): Increasing $B_0$ from 20 kG to 160 kG systematically increases the size of the spot, umbra, and penumbra.
  • $B_0 \le 40$ kG: No penumbra forms; only elongated convective cells with inflows.
  • $B_0 = 80$ kG: Intermediate state with shorter penumbral filaments.
  • $B_0 = 160$ kG: Long, slender penumbral filaments form.
• Penumbra Fraction: Even the best runs ($B_0=160$ kG) produce penumbral fractions ($r_{pu}/r_s \approx 47-52\%$) smaller than observed stable sunspots ($\approx 60\%$). This suggests the simulations represent early stages of sunspot formation rather than mature, stable spots.
• Flux Content: Runs with $F_{Gauss} > 10^{22}$ Mx result in unrealistically strong vertical fields ($B_z > 5$ kG) in the umbra. The most realistic results come from $F_{Gauss} \approx 10^{22}$ Mx.

B. Magnetic Field Properties

• Vertical Field ($B_z$): The best runs ($160$ kG, $10^{22}$ Mx) produce a $B_z$ profile that decreases realistically with radius, though the maximum central field ($\approx 4.2-4.5$ kG) remains higher than typical observations.
• Inclination: The inclination angle increases with radius, reaching $\approx 80^\circ$ at the spot boundary for $B_0=160$ kG, which is closer to observations than previous artificial-boundary models.
• Opposing Field & Box Size: Varying $B_{opp}$ or increasing box width affects the background field outside the spot but has little influence on the internal spot dynamics or structure.

C. Surface Flows (Dynamics)

• Low Resolution ($96/32$ km):
  • $B_0 \le 40$ kG: Pure inflows and downflows (no Evershed flow).
  • $B_0 \ge 80$ kG: Bi-directional flows appear. The inner penumbra shows strong inflows and downflows, while the outer penumbra shows outflows. This mimics observations of early penumbra formation (counter-Evershed flows).
• High Resolution ($32/16$ km):
  • Simulations produce a mix of filaments with bi-directional flows and filaments exhibiting pure Evershed flows (radial outflows along the entire filament length).
  • Outflow speeds are higher in high-resolution runs, consistent with previous findings by Rempel (2012).
• Flux Emergence: Most successful runs show ongoing flux emergence ($\approx 8-13\%$ per hour). The study suggests that flux emergence at the spot's outskirts drives the bi-directional flows and is a precursor to stable penumbra formation.

D. Failed Configurations

• Closed Bottom Boundary: Suppressing mass flows at the bottom of the magnetic trunk prevents the formation of a realistic penumbra and outflows.
• Excessive Flux: Fluxes $> 10^{22}$ Mx lead to unphysically strong magnetic fields.

5. Significance and Conclusions

• Best Configuration: The most realistic sunspot simulations are achieved with an initial potential field, a total flux of $\approx 10^{22}$ Mx, a strong initial field strength of 160 kG, and an open bottom boundary.
• Physical Insight: The simulations suggest that the bi-directional flows observed in early penumbral stages are driven by flux emergence. The transition to stable, pure Evershed flows may require higher numerical resolution and/or the natural formation of a magnetic canopy (which was not fully achieved in these runs).
• Future Directions: The authors speculate that embedding the sunspot in a much larger simulation domain to allow for natural small-scale dynamo action and canopy formation could enable the development of fully mature Evershed flows even at lower resolutions.
• Overall Impact: This work validates the potential field initial condition approach and identifies numerical resolution as a critical bottleneck for reproducing fully developed sunspot penumbrae in MHD simulations.