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

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 used 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) 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, consistent with to early observations of penumbra formation for runs with $\ge80$ kG at 96/32 km resolution, whereas runs with $\le40$ 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 beyond 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.