Numerical models of sunspot formation and fine structure

Sunspots are central to our understanding of solar (and stellar) magnetism in many respects. On the large scale, they link the magnetic field observable in the photosphere to the dynamo processes operating in the solar interior. Properly interpreting the constraints that sunspots impose on the dynamo process requires a detailed understanding of the processes involved in their formation, dynamical evolution and decay. On the small scale, they give an insight into how convective energy transport interacts with the magnetic field over a wide range of field strengths and inclination angles, leading to sunspot fine structure observed in the form of umbral dots and penumbral filaments. Over the past decade, substantial progress has been made on both observational and theoretical sides. Advanced ground-and space-based observations have resolved, for the first time, the details of umbral dots and penumbral filaments and discovered similarities in their substructures. Numerical models have advanced to the degree that simulations of entire sunspots with sufficient resolution to resolve sunspot fine structure are feasible. A combination of improved helioseismic inversion techniques with seismic forward modelling provides new views on the subsurface structure of sunspots. In this review, we summarize recent progress, with particular focus on numerical modelling.