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

TL;DR

The paper addresses the mismatch between observed sunspot magnetic distributions and previous MHD simulations by adopting potential-field initial conditions and exploring a parameter space with varying bottom-field strength $B_0$, magnetic flux, opposing flux $B_ extrm{opp}$, box size, and resolution using the MURaM code. The main approach combines a potential-field extrapolated initial state with targeted parameter variations to assess impacts on spot size, penumbral development, flow patterns, and magnetic profiles. Key findings show that larger $B_0$ increases spot size and total penumbral extent, but penumbrae remain underdeveloped compared to observations, and pure Evershed flows are not always present; high-resolution runs can produce filaments with Evershed-like flows, and unrealistically strong fields arise at very high fluxes. The study concludes that potential-field initialization with $B_0\approx160$ kG and $F_ extrm{Gauss}=10^{22}$ Mx best captures early sunspot formation, while fully realistic penumbrae may require higher resolution or embedding the spot in a larger canopy to enable natural field inclination and stable penumbral flows.

Abstract

Context. Existing sunspot simulations fail to reproduce the observed magnetic field distribution due to an artificially increased $B_{hor}$ at the upper boundary. Aims. We explore alternative ways to better reproduce the magnetic and dynamic properties of observed sunspots. Methods. We use the radiative MHD code MURaM. As initial conditions, we placed a potential magnetic field into small-scale dynamo simulations and used potential-field extrapolation at the top. Results. We find that: (1) Simulations with increasing initial magnetic field strengths (20, 40, 80, and 160 kG) show larger spots, umbrae, and penumbrae. (2) The penumbral-to-spot sizes are smaller than those measured in observed sunspots. (3) In none of the runs are pure Evershed (radially outward) flows. Instead, bi-directional flows with inflows in the inner penumbra and outflows in the outer penumbra were measured, similar to early observations of penumbra formation for runs with $\ge80$ kG at 96/32 km resolution, whereas runs with 40 kG or less showed pure inflows. (4) Simulations with 160 kG at 32/16 km resolution contain filaments with bi-directional and Evershed flows. (5) Simulations with fluxes $>10^{22}$ Mx show unrealistically strong fields in the umbra. (6) The best runs with 160 kG and $10^{22}$ Mx give realistic radial profiles of $B_z$ and $B_r$, although stronger fields than observed. (7) Increasing the width of the box and reducing the overall flux by subtracting a uniform opposing vertical field have little influence on internal spot dynamics and fields, but change the mean vertical field outside the spot. Conclusions. Simulations of small ($10^{22}$) sunspots with an initial potential field and intensified bottom magnetic field strength best reproduce observations of the initial stages of sunspot formation. Numerical resolution may be critical for achieving fully developed penumbrae.

Figures (8)

• Figure 1: Quadrants of bolometric intensity maps for four selected simulation runs with different initial magnetic field strengths. From top-left clock-wise: $B_0=20$ kG, 40 kG, 80 kG and 160 kG. The magnetic flux is the same for all four runs: $F=10^{22}$ Mx.
• Figure 2: Azimuthally averaged intensity $I/I_\textrm{qs}$ (solid lines) as a function of radius in Mm and fraction of the spot radius. The colours indicate the simulations listed in the inset legend. For the older runs (with names ending with 'o', red), the bolometric intensity is shown, and for the other runs (blue), the continuum intensity is shown. Vertical dashed lines show the position of the umbral (left panel with $R<10$ Mm, right panel $R<1$) and spot boundaries (left panel $R>10$ Mm, right panel $R=1$ black). The horizontal dashed lines in the right panel show the average umbral intensity. Values corresponding to the location of the vertical dashed lines are given in Table \ref{['tab:res']}, columns $r_\textrm{u}$, $r_\textrm{s}$, $r_\textrm{u/s}$.
• Figure 3: Vertical magnetic field component, $B_\mathrm{z}$, of selected simulations. $B_\mathrm{z}$ from an observation (green) is also displayed for comparison.
• Figure 4: Azimuthal averages of the vertical magnetic field component, $B_\mathrm{z}$ (top left), the radial magnetic field component, $B_\mathrm{r}$ (top right), the inclination (bottom left), and the intensity, $I_\textrm{out}$ (bottom right), of our more realistic simulations (blue, red, black and grey), and of an observation (green), and an old reference simulation (gold). Horizontal dashed lines refer to $B_\mathrm{z}=1867\,$G, the observed critical value for penumbral-type convection to operate (top left), and to an inclination of $90\degr$, meaning a horizontal field (bottom left).
• Figure 5: Surface velocity map of sunspot run '160_opp000'. Upper left: Vertical flows, $v_\mathrm{z}$, downflows in red, upflows in blue. Lower right: Radial velocities, $v_\mathrm{r}$, inflows red (towards the spot centre), outflows blue. The black contours are the same as shown in Fig. \ref{['fig:I_out_map']}.