1 A Turbulent Sun

Measurements of plasma flows in the surface layers of the Sun by Doppler imaging, tracking of surface features, and helioseismic inversions reveal an intricate, rapidly evolving structure characteristic of a highly turbulent fluid (e.g., Toomre, 2002

). Small-scale ( $~ 1 -2 Mm$ ) granulation cells dominate the velocity and irradiance patterns, blanketing the solar surface with a network of relatively cool, dark downflow lanes surrounding brighter, broader upwellings of warmer fluid. The granulation patterns change continually and chaotically, driven by vigorous thermal convection under the influence of stratification, ionization, and radiative transfer effects as convective heat transfer gives way to radiation, moving energy outward through the extended solar atmosphere and into the interplanetary medium (e.g., Stein and Nordlund, 1998

, 2000

). Larger-scale convective patterns have also been detected including mesogranulation at $~ 5 Mm$ and supergranulation at $~ 30 Mm$ (Leighton et al., 1962

; November et al., 1981

; Muller et al., 1992

; Rast, 2003

; DeRosa and Toomre, 2004

). Local helioseismology reveals even more structure: swirling, converging, and diverging horizontal flows, meandering zonal jets, and global meridional circulations all of which evolve substantially over the course of months and years (Section 2.2).

Despite this seething complexity, the Sun exhibits some striking regularities. Among these is the latitudinal variation of the surface rotation rate, which is non-uniform; equatorial regions rotate with a period of about 27 days whereas polar regions rotate with a period of about 35 days. This differential rotation pattern is remarkably smooth and steady, monotonically decreasing from equator to pole and varying by not more than about 5% since the first systematic measurements were made by Carrington (1863 ) over a century ago. Another striking manifestation of order amid the chaos of solar convection is the solar activity cycle in which belts of magnetic activity regularly appear at mid latitudes, propagate toward the equator, and then vanish as new activity belts form at mid latitudes and repeat the process (Schrijver and Zwaan, 2000 ; Stix, 2002 ; Charbonneau, 2005 ). Other systematic patterns are also evident within the framework of this activity cycle, such as the orientation and chirality of individual active regions and the frequency and magnitude of eruptive events such as flares and coronal mass ejections (see Section 3.8).

Turbulent, electrically conducting flows such as solar granulation are generally capable of amplifying and maintaining magnetic fields through hydromagnetic dynamo action. This is the likely origin of much of the small-scale magnetic flux observed in the photosphere, sometimes referred to as the magnetic carpet or as the salt and pepper which dots high-resolution magnetograms of the solar disk (e.g., Schrijver and Zwaan, 2000 ). This small-scale flux concentrates in granular downflow lanes and evolves rapidly, continually replenishing itself in less than a day. However, it does not exhibit the emergence patterns and cyclic behavior characteristic of much larger active regions. Rather, the generation of small-scale magnetic flux locally by dynamo action within the solar surface layers and its advection by granulation and supergranulation is distinct from, but coupled to, the generation of larger-scale field which is manifested in the solar activity cycle (Simon et al., 2001 ). Thus, there is not one solar dynamo, but two: a local dynamo which continuously generates small-scale, relatively random magnetic fluctuations in the solar surface layers, and a global dynamo which maintains the larger-scale cyclic activity (Cattaneo, 1999 ).

The regularities in magnetic activity associated with the global dynamo likely have little to do with the granulation and supergranulation patterns observed in the photosphere. These motions are thought to be confined to the upper few percent of the solar interior ( $r > 0.97R o.$ ). Solar structure models and helioseismic inversions suggest that the solar envelope is convectively unstable over a much larger region, down to $r ~ 0.71Ro .$ (Christensen-Dalsgaard et al., 1991 ; Basu and Antia, 2001 ). Relative to granulation and supergranulation, the motions which occupy the bulk of the solar convection zone are thought to be larger-scale and slower, with turnover timescales comparable to the solar rotation period of about one month. These motions are thus more influenced by rotation which induces anisotropic momentum and heat transport, thus maintaining global-scale flows such as differential rotation and meridional circulations. Such flows are thought to play a key role in the global dynamo. Rotation also induces kinetic and magnetic helicity, another important ingredient in solar dynamo theory (Section 4.5).

Understanding the dynamics and dynamo processes occurring within the deep solar convection zone has far-reaching implications for understanding solar and stellar magnetism, evolution, structure, and variability. Furthermore, since much of solar variability is tied to cyclic magnetic activity, such insight is essential in order to gain a better understanding of how the Sun influences life on Earth through a variety of processes collectively known as space weather (Schrijver and Zwaan, 2000 ). However, large-scale convection motions in the Sun are notoriously difficult to observe directly because they are masked by the much more vigorous granulation in the near-surface layers (Section 3.5). We must instead rely on their indirect observational manifestations such as magnetic activity in the solar atmosphere and the internal rotation profile inferred from helioseismology (Section 3.1). The helioseismic investigations have proven particularly enlightening as they have revealed a narrow layer of strong radial shear in the solar angular velocity, where the differential rotation of the convective envelope undergoes a transition to nearly uniform rotation in the radiative interior. The discovery of this shear layer, now known as the solar tachocline, has had profound implications for solar dynamo theory.

In this review we will give a general overview of solar interior dynamics, focusing on large-scale motions in the convection zone and tachocline. Smaller-scale dynamics in the solar surface layers including granulation, supergranulation, and issues relating to the local dynamo are discussed elsewhere in this journal. Although we will often discuss the deep convection zone and tachocline in the context of the global dynamo, we make no attempt to cover all aspects of solar dynamo theory. More comprehensive discussions of solar dynamo modeling and of the evolution and emergence of magnetic flux in the convection zone can be found in these volumes in the reviews by Fan (2004 ) and Charbonneau (2005 ). Even with this restricted scope, the subject matter is vast and we must necessarily focus on some aspects more than others. Particular emphasis will be placed on 3D numerical simulations of turbulent convection. References are provided throughout should the reader wish to explore the subject matter further or to seek a different perspective.

This review is organized as a web-based reference in that it has a modular form and ample cross-referencing; the reader is encouraged to skip to the sections of most interest. Like most reviews, it is targeted mainly at non-specialists: students and interested researchers from other disciplines.

In Section 2 we describe the means by which we can potentially glean information about the solar interior; how do we know what we know? The most relevant observational results are then be reviewed in Section 3. There we discuss observational diagnostics of convection, mean flows, and dynamo processes in the solar envelope. We also define the tachocline and review what is known about it observationally. Some fundamental theoretical principles and modeling approaches are then discussed in Sections 4 and 5. Among these approaches, high-resolution numerical simulations of thermal convection in rotating spherical shells offer unique promise in elucidating the complex turbulent dynamics of the solar convection zone and we discuss their implications, current limitations, and future prospects in Sections 6 and 7. We then turn to the tachocline and the region of convective overshoot which forms the interface between the solar envelope and the radiative interior. Since most of the tachocline is thought to be stably-stratified, it exhibits qualitatively different dynamics relative to the convection zone, as we discuss in Section 8. We close with an attempt to tie it all together in Section 9 where we assess the current state of interplay between dynamical models and observations.