Sunspot simulations with MURaM

Context. Existing sunspot simulations fail to reproduce the observed magnetic field distribution due to an artificially increased Bhor at the upper boundary. Aims. We explore alternative ways to better reproduce the magnetic and dynamic properties of observed sunspots. Methods. We used the MURaM radiative magnetohydrodynamic code. As the initial conditions, we placed a potential magnetic field into small-scale dynamo simulations and used potential field extrapolation as the top boundary conditions. Results. We find that (1) simulations with increasing initial magnetic field strengths (20 kG, 40 kG, 80 kG, and 160 kG) show increasing spot, umbral, and penumbral sizes; (2) penumbral-to-spot sizes are smaller than those measured in observed sunspots; (3) none of the runs show pure Evershed (radially outwards) flows and, instead, bi-directional flows with inflows in the inner penumbra and outflows in the outer penumbra were measured, consistent with observations of early stages of penumbra formation for runs with 80 kG or more and 96/32 km resolution, whereas runs with ≤40 kG showed pure inflows; (4) simulations with 160 kG and an increased resolution of 32/16 km contain filaments with bi-directional and Evershed flows; (5) simulations with fluxes > 1022 Mx show unrealistically strong fields in the umbra; and (6) best runs with 160 kG and 1022 Mx give realistic profiles of Bz and Br with radius, albeit with stronger fields than those typically observed. Finally, (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. Nonetheless, these choices affect the average vertical field beyond the spot. Conclusions. Simulations of small (1022 Mx) sunspots with an initial potential field and intensified magnetic field strength at the bottom of the box seem to best reproduce observational results of the initial stages of sunspot formation. Our findings also suggest that increased numerical resolution could be critical for achieving fully developed penumbrae.