Preamble

The task that I have been assigned is to set the scene for the discussions that follow: to present my view of the principal issues that had confronted us before the meeting when trying to understand the dynamics of the solar tachocline. Most of what I write here is enlarged upon, and in some cases superseded by, the chapters that follow, in which references to most of the original publications can also be found. Nevertheless, I trust that it can serve as a useful elementary introduction to the subject, setting it into its wider astronomical context.

The tachocline is interesting to astrophysicists for a variety of reasons, the most important being (i) that it couples the radiative interior of the Sun, where nearly 90% of the angular momentum resides, to the convection zone, which is being spun down by the solar wind, (ii) that it controls conditions at the lower boundary of the convection zone, and is therefore an integral component of the overall rotational dynamics of the convection, and (iii), perhaps most relevant to the interests of the greater proportion of the participants of the workshop, it is now generally recognized as being the seat of the solar dynamo. It plays some role in shaping the evolution of the Sun, and it must be taken into account when interpreting the helioseismological diagnostics of the solar structure.

The combination of differential rotation and toroidal fields believed to exist in the solar tachocline should be unstable to global MHD modes, typically dominated by longitudinal wavenumber m = 1 modes for toroidal fields of peak value 30 kG and higher, and a broader range of low m values for weaker fields. For toroidal field bands, the high field instability takes the form of a ‘tipping’ of the band away from coincidence with circles of latitude. For a wide range of toroidal fields and differential rotations, and in both the overshoot and radiative parts of the tachocline, the unstable modes grow in a time short compared to a solar cycle, and are therefore of interest for the solar dynamo problem, as well as for creation of longitude-dependent magnetic patterns seen at the solar surface. The latitudinal momentum transport by Reynolds and Maxwell stresses associated with unstable modes provides a way to mix angular momentum in latitude, and help limit the thickness of the tachocline.

Introduction

The study of global MHD instabilities of differential rotation and toroidal fields that might be present in the solar tachocline began with Gilman & Fox (1997). Their original motivation was to see whether the magnetic field could destabilize the differential rotation of the tachocline, estimated to be stable to hydrodynamical disturbances by itself.

Two distinct classes of magnetic confinement models exist for the solar tachocline. The ‘slow tachocline’ models are associated with a large-scale primordial field embedded in the radiative zone. The ‘fast tachocline’ models are associated with an overlying dynamo field. I describe the results obtained in each case, their pros and cons, and compare them with existing solar observations. I conclude by discussing new lines of investigation that should be pursued, as well as some means by which these models could be unified or reconciled.

Introduction

Magnetic fields in the tachocline

Two distinct possible origins for solar magnetic fields in the tachocline region can be identified. The Ohmic decay timescale of a large-scale dipolar field embedded in the radiative interior is much larger than the estimated age of the Sun (Cowling 1945; Garaud 1999), so that a fraction of the magnetic flux initially frozen within the accreting protostellar gas is likely to persist today. In parallel, according to the standard dynamo field theory, small-scale magnetic fields are thought to be constantly generated by fluid motions within the solar interior. Optimal conditions for the generation of large-scale fields require the combination of large-scale azimuthal shear and small-scale helical motion, which are both naturally found in the region of the tachocline (Parker 1993; Ossendrijver 2003; Tobias 2005).

The discovery by Spruit of a new small-scale turbulent dynamo has significantly changed the tachocline model proposed by Gough & McIntyre (1998). The small-scale dynamo is shear driven, is characteristic of stably stratified flows, and is mediated by the kink or ‘tipping’ instability elucidated for such flows by R. J. Tayler. The dynamo works best in high latitudes and supports turbulent Maxwell stresses large enough to dominate the angular momentum transport, taking over from the pure mean meridional circulation (MMC) proposed by Gough & McIntyre (1998). What survives from the Gough & McIntyre proposal is the laminar thermomagnetic boundary layer at the tachopause, essential for the confinement of the interior field Bi by high-latitude downwelling. That downwelling is, however, itself confined within a double boundary layer at the tachopause. The thermomagnetic boundary layer sits just underneath a modified Ekman layer, in which the turbulent Maxwell stress of the small-scale dynamo diverges.

The effects of compositional stratification in the helium settling layer under the tachopause are considered. It is concluded that Gough & McIntyre's (1998) ‘polar pits’ to burn lithium are dynamically impossible and that the tachopause is not only sharp but globally horizontal. That is, the tachopause, as marked by the top of the helium settling layer, follows a single heliopotential to within a very tiny fraction of a megametre from equator to pole.

Afterthoughts

Even the most casual of readers of this book will have noticed that the subject of the solar tachocline is highly controversial, in the best traditions of our science: we are all well aware that the tachocline constitutes an important physical structure in the solar interior, but we are not at all in agreement about any of the details. While this makes for a good deal of excitement – much in evidence both at the workshop and in this book – I did early on recognize that a straightforward summary of the workshop was therefore an impossibility; and my strong belief is that it is very premature for me to act as a ‘referee’ judging the merits of the various points of view expressed by my co-authors of this volume. This does not mean of course that I will not venture an opinion when appropriate – but it does mean that, in many cases, ex cathedra declarations of what is correct, and what is incorrect, are entirely premature.

For these reasons, I thought it would be more appropriate for me to step back from the fray, and to discuss some of the larger issues related to the tachocline, most especially those that I believe will play a key role in further developments of this subject; and to explain, whenever appropriate, why exactly it is that a definitive result remains to be obtained.

The tachocline is believed to play a crucial role in the dynamo that maintains magnetic activity in the Sun. We first review the observational properties of the 11-year activity cycle and the 22-year magnetic cycle, as well as of the recurrent grand minima, with a characteristic 200-year timescale, that are revealed by proxy records. Then we discuss dynamo mechanisms, including differential rotation (the ω-effect), the net effect of gyrotropic motions (the α-effect) and flux transport by both large-scale motions (e.g. meridional flows) and small-scale processes (e.g. turbulent transport). Next we consider the location of the solar dynamo, comparing models with dynamo action distributed throughout the convection zone, located near the surface or (most likely) concentrated near the interface between the convective and radiative zones. Local pockets of strong field can then escape from the vicinity of the tachocline and emerge through the photosphere as active regions. The nonlinear back-reaction of the magnetic field affects transport coefficients (both α and the turbulent diffusivity β) and also drives the zonal flows that are observed. Furthermore, it provides a mechanism for the modulation associated with grand minima. We conclude with our picture of the relationship between convection, differential rotation and the dynamo in the tachocline.

Observations

The Sun exhibits cyclic magnetic activity, as do other slowly rotating stars with deep convective envelopes. This activity is manifested in the sunspot cycle, which has an average period of 11 years, as shown in Figure 13.1.

Helioseismic inversions suggest that the tachocline straddles the base of the convection zone, incorporating the overshoot region and extending into the stably stratified radiative interior. Thus, the upper tachocline is dominated by penetrative convection while the lower tachocline is a stably stratified shear flow under the influence of rotation and magnetism. We review the nature of the turbulence that is likely to exist in these two disparate regions, focusing on the interaction between turbulence and differential rotation. It is argued that turbulent angular momentum transport is likely to be poleward throughout the tachocline, tending to suppress the latitudinal differential rotation maintained by turbulent stresses in the overlying convective envelope. Meanwhile, vertical angular momentum transport in the lower tachocline may be anti-diffusive, tending to amplify the vertical shear. The turbulent alignment of convective plumes may also drive an equatorward meridional circulation in the upper tachocline where it overlaps with the overshoot region.

Introduction

The solar tachocline lies near the base of the solar convection zone. This is a well-known result of course, but it is essential to establish precisely what near means in this context. Helioseismic structure inversions reveal a stiff transition between the nearly adiabatic stratification of the convection zone and the strongly subadiabatic stratification of the radiative interior, mediated by only a narrow region of convective overshoot. As others have argued in this volume, tachocline dynamics is very sensitive to where the rotational shear occurs relative to this structural transition.

Solar activity takes place in narrow bands of latitude that move like solitary waves from mid-latitudes toward the solar equator. This behaviour points to the existence of a thin layer in the Sun that may serve as a waveguide. With its grand minima, the cycle is intermittent in a way that does not occur in the simplest chaos models. To be useful as a primitive model of the cycle, a differential equation should be of high enough order to display such strong intermittency. These and other features of solar fluid dynamics led to the adumbration of an intermediate shear layer between the convection zone and the radiative core. This layer, like the weather layers in planetary atmospheres, produces coherent structures – sunspots and perhaps vortices. Similar layers may play a role in stellar activity in cool stars other than the Sun and perhaps even in hot stars if their atmospheres are turbulent.

The maculate Sun

Rotation and turbulence in stars are significant for an understanding of stellar evolution and for the fluid dynamics of accretion discs. We can watch these processes most closely in our own Solar System. Observations of the Sun, the giant planets and the earth reveal coherent structures whose study has been one of the most exciting adventures in the mathematical science of the twentieth century. (At a meeting in the Newton Institute, we ought to recall this.)

I recall here how latitude-dependent rotation imposed by the solar convection zone on the top of the radiation zone would burrow deep into the interior, owing to thermal diffusion, in any laminar and purely hydrodynamic model. Since helioseismology has shown that this differential rotation remains confined in a thin boundary layer, the tachocline, it means that the radiative spread is inhibited by another physical process; this process may be purely hydrodynamic (non-MHD), which is the scope of this chapter, or it may involve magnetic fields: those are considered by Garaud in Chapter 7 of this book. I will show that the confinement of the tachocline can be achieved through an anisotropic turbulent viscosity, whose cause and plausibility are discussed. Other hydrodynamic mechanisms are examined, such as internal gravity waves, which may also play a role in the tachocline. An alternative possibility is that the tachocline is fully embedded in the layer of penetrative convection, in which case no differential rotation would be applied on to the radiation zone.

Introduction

In 1990, I was invited with Ed Spiegel to give the principal lectures at the Woods Hole summer school. The theme of that year, ‘Stellar Fluid Dynamics’, was covered extensively by Ed, and I chose to focus on problems related to the rotation of stars. My last lecture, as it happened, was devoted to ‘flow between the Sun's convection and radiation zones and transport of chemicals’.

It is natural to associate the tachocline with the region of generation of a strong toroidal field by the winding-up of a weaker poloidal component. Here I discuss the break-up and subsequent escape of such a field via magnetic buoyancy instabilities. I consider the different modelling approaches that have been employed and discuss which have the most relevance in a solar context.

Introduction

For many years, a controversial issue of solar magnetism has been that of the location of the site (or sites) of the generation and storage of the Sun's predominantly toroidal magnetic field, which eventually escapes and rises to the surface, leading to active regions and, ultimately, to much of the exotic magnetic behaviour observed in the photosphere, chromosphere and corona. For two rather different reasons, the idea had been put forward that the bulk of the toroidal field must be stored either at the base of, or just beneath, the convection zone. From estimates of the rise times of magnetic flux tubes through the convection zone, Parker (1975) argued that the dynamo must operate only in the ‘very lowest levels of the convective zone’. Golub et al. (1981) (see also Spiegel & Weiss 1980) proposed a similarly deep-seated layer of toroidal field, but from arguments based instead on the expulsion of magnetic fields by convective motions. The discovery of the tachocline by helioseismology provides probably the most compelling evidence for pinning down the location of the solar toroidal field.

Over the past 25 years helioseismology has at last enabled us to probe the internal structure and dynamics of our local star, the Sun. Perhaps its greatest triumph has been to determine how the rotation varies in the solar interior. Although the bulk of the radiative zone, occupying the innermost 70% by radius, rotates more or less uniformly, the known variation with latitude of angular velocity at the surface persists down to the base of the outer convective envelope. Since it had previously been supposed that the Sun rotates sufficiently rapidly for the angular velocity to be constant on cylindrical surfaces in the convection zone it was a surprise to find that it is actually constant on conical surfaces. It came as an even greater surprise to discover that the transition between the differentially rotating exterior and the uniformly rotating interior is effected through an extremely thin layer – the tachocline – whose thickness is less than 4% of the solar radius.

This unexpectedly abrupt transition has forced us all to refine our ideas on the interactions between turbulent convection, rotation and magnetic fields, for it seems that these last play a key role in preventing the tachocline from spreading downwards into the radiative zone. To describe the internal structure of the tachocline requires an understanding of convective penetration, turbulent diffusion, mixing and angular momentum transport.