1 Introduction

Magnetic fields on the Sun exist in a large variety of phenomena and interact in various ways with plasma and radiation. In the convection zone large and small scale magnetic fields are generated. These magnetic fields are partially transported into the outer layers of the Sun, i.e., into the chromosphere and the corona. The most prominent example of a magnetic phenomenon is a sunspot as seen in the photosphere. A typical sunspot has a lifetime of a few weeks and has a size of about 30 granules. The magnetic field strength spans from 1000 to 3000 Gauss in the deep photosphere, summing up to a magnetic flux of some 1022 Mx, typically. For an extensive review of the sunspot structure, we refer the reader to an instructive overview by Solanki (2003Jump To The Next Citation Point).

The magnetic field of a sunspot extends into the interior as well as into the outer layers of the Sun. The most detailed information of sunspots is obtained in the photosphere. The topology of the magnetic field above and beneath the photosphere is poorly understood. In particular our knowledge of the magnetic field extension into the interior presents a theoretical challenge. Direct measurements of the sub-photospheric structure are impossible, but at least for the larger scales, indirect methods are being explored in the framework of local helioseismology (cf. Gizon and Birch, 2005Jump To The Next Citation Point).

Sunspots are central to our understanding of solar magnetism in several aspects. Sunspots are the most prominent manifestation of the large scale cyclic solar magnetic field. Understanding their subsurface structure as well as the processes of formation, dynamic evolution, and decay is crucial for connecting them to the dynamo and flux emergence processes in the solar convection zone (Fan, 2009Jump To The Next Citation Point; Charbonneau, 2010). On smaller scales sunspots provide an ideal environment for studying magnetoconvection for a variety of different field configurations. While quiet Sun and plage regions have been modelled very successfully for almost 3 decades using 3D radiative MHD simulations (see the review by Nordlund et al., 2009Jump To The Next Citation Point), such models were only applied to sunspots in the past five years. The combination of detailed models with the wealth of high resolution observations has substantially advanced our understanding of sunspot structure over the past decade.

In this paper we aim to review our current understanding of sunspots. We approach the problem from different perspectives. In Section 2 we start out by characterizing the global structure of a sunspot. We describe model ideas for the sunspot structure, and describe how sunspots can be treated in a static (non-dynamic) configuration. In Section 3 we discuss the dynamic fine structure of umbra and penumbra. We summarize the key observational facts models have to explain, give an overview about several idealized models in use and summarize the recent progress in radiative MHD modeling of sunspots. In Section 4 the paradigm of sunspot formation by rising and emerging magnetic flux tubes is addressed. We also summarize observations and models of the moat region surrounding sunspots. In Section 5 we give a brief summary of our current understanding of subsurface structure and flow fields surrounding sunspots as derived from helioseismic inversions. We summarize our knowledge on sunspot modeling and conclude on our present understanding in Section 6.

We also point to previous reviews on this subject by Solanki (2003Jump To The Next Citation Point), Thomas and Weiss (2004, 2008), and Scharmer (2009Jump To The Next Citation Point). Models of flux emergence are discussed in more detail by Fan (2009Jump To The Next Citation Point), results from helioseismic inversions by Gizon and Birch (2005Jump To The Next Citation Point), Kosovichev (2006Jump To The Next Citation Point), Moradi et al. (2010Jump To The Next Citation Point), and Gizon et al. (2010aJump To The Next Citation Point).

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