9 Conclusion: Making Sense of the Observations

We are entering an exciting new age in our exploration of the solar interior. The continuous monitoring of global solar oscillations with high-resolution helioseismic instruments is still a relatively recent endeavor, covering roughly half of a (22-yr) solar activity cycle. Further monitoring will improve our understanding of rotational and structural variations and may reveal new patterns. Local helioseismology (Section 2.2) is even younger, and as the field continues to mature it promises ever greater insights into convective flows, magnetic activity, and global circulations below the photosphere. These helioseismic investigations together with increasingly powerful computing resources are fostering progressively more sophisticated and realistic numerical and theoretical models of solar interior dynamics.

Where does this interplay between models and observations now stand? Meaningful comparisons between the convective patterns found in global, 3D simulations and those thought to exist in the upper solar convection zone are now becoming feasible. Solar sub-surface weather (SSW) maps obtained from local helioseismology and large-scale structures inferred from the correlation tracking of surface features both reveal evolving patterns comparable to those seen in global simulations (Section 3.5). Further investigations are required to strengthen this connection and to understand where it currently appears to break down, most notably in flows associated with active regions.

Still, the main point of contact between global convection simulations and solar observations remains the internal rotation profile. There is little doubt that convection drives differential rotation in the solar envelope. Even a cursory look at the angular velocity profile inferred from helioseismology (Figure 1 ) clearly reveals a profound difference between the dynamics in the convection zone and in the stably-stratified radiative zone below. This, together with sound speed inversions, provides a dramatic validation of solar structure theory as a whole, although there are still discrepancies which must be understood, particularly in light of new elemental abundance determinations. The question is: How does convection redistribute angular momentum in such a systematic way?

Global simulations suggest that the solar differential rotation is maintained both through the Reynolds stress and through inhomogeneous convective heat transport, the latter of which can establish a thermal wind (Sections 4.3 and 6.3). Whereas the Reynolds stress dominates in the upper convection zone, the differential rotation in the lower convection zone is nearly in thermal wind balance. A realistic model must therefore take into account both momentum and heat transport by turbulent convection under the influence of rotation, stratification, and magnetism.

The global redistribution of angular momentum by the Reynolds stress is dominated by extended downflow lanes which are oriented north-south and which are confined primarily to low latitudes (Section 6.3). These exist amid a more intricate, evolving downflow network which becomes more isotropic at high latitudes and which fragments into an ensemble of disconnected and intermittent plumes at deeper layers (Figures 9 and 12 ). There is a possibility that these convective patterns (and their associated transport properties) may change as the resolution is further increased and the parameters achieve more solar-like conditions (Section 7.1). However, observations of granulation in the solar photosphere demonstrate that such patterns can persist in solar parameter regimes. Furthermore, the close correspondence between observations and simulations of granulation suggest that the essential dynamics of solar convection can indeed be captured using large-eddy simulation approach (e.g., Stein and Nordlund, 1998 , 2000 ; Keller et al., 2004 ; Vögler et al., 2005 ; Rincon et al., 2005 ). The rotation profiles in global simulations are in good agreement with helioseismic inversions in their gross features, if not in their finer details. A conspicuous shortcoming of current simulations is the absence of a self-consistently maintained tachocline. This can likely be attributed to insufficient spatial resolution and temporal duration to accurately capture the wide range of processes which may be occurring near the base of the convection zone (Section 8).

Another difficulty in many (but not all) simulations is a tendency to spin up the poles, producing a high-latitude prograde polar vortex which is not found in helioseismic inversions (Section 6.3). This is mainly due to axisymmetric circulations which tend to conserve their angular momentum; simulations which do not produce a polar vortex exhibit meridional circulation patterns which are confined to low latitudes. These results suggest that the meridional circulation in the solar envelope may not extend all the way to the poles.

The meridional circulation is driven by small differences between relatively large forces which are nearly in balance. This leads to large spatial and temporal variations in numerical simulations and possibly also in the Sun (Section 6.4). Near the surface, simulations typically exhibit poleward circulations at low latitudes in rough agreement with photospheric measurements and helioseismic inversions, although the latitudinal extent of these circulations is generally less in the simulations. Near the base of the convection zone, penetrative convection simulations yield equatorward circulation as is assumed in flux-transport dynamo models (Section 6.4). This equatorward circulation arises from the rotational alignment of downflow plumes, which also produces poleward angular momentum transport in the overshoot region (Section 6.3).

It is particularly important to understand dynamics near the base of the convection zone from the perspective of dynamo theory. Meaningful comparisons between global convection simulations and observations of magnetic activity will only be possible if the simulations incorporate tachocline dynamics to some degree, either by resolving the relevant processes or by parameterizing them. Improved dynamo simulations are necessary to better understand fundamental elements of the solar activity cycle such as the butterfly diagram as well as more subtle aspects such as chirality patterns (Section 3.8). An accurate representation of magnetic activity may also be a prerequisite to reproducing flow patterns such as torsional oscillations which appear to be driven by the Lorentz force associated with the dynamo-generated field (Yoshimura, 1981 ; Schüssler, 1981 ; Kitchatinov et al., 1999 ; Durney, 2000b ; Covas et al., 2001 , 2004 ; Bushby and Mason, 2004 ). Capturing such processes in a 3D global convection simulation represents one of the most challenging and important frontiers of solar modeling.

The further exploration of tachocline dynamics is in itself a diverse and fascinating frontier which will be the focus of many theoretical, computational, and observational efforts in the coming years. The structure of the tachocline and its coupling to the convective envelope and radiative interior involves an intricate interplay between penetrative convection, instabilities, stably-stratified turbulence, and waves in the presence of rotational shear and magnetism.

The most compelling aspect of the tachocline, namely its thinness, can probably be attributed at least in part to magnetic fields. A fossil field permeating the radiative interior is currently the leading explanation for the nearly uniform rotation in this region inferred from helioseismology (Section 8.5). However, magnetic confinement models are still rather schematic and much more theoretical and numerical work is needed to verify and clarify the proposed mechanisms. Furthermore, relatively ’fast’ dynamics likely dominate in the upper tachocline where penetrative convection, instabilities, waves, and turbulence redistribute momentum and energy on timescales of months to years (fossil-field confinement models operate on timescales of $~ 106 yr$ ).

The depth and location of the tachocline clearly vary with latitude but the base of the convection zone and overshoot region apparently do not (Section 3.6). This implies that tachocline structure is not governed solely by penetrative convection and, furthermore, that instabilities and turbulence in the lower tachocline do not produce enough vertical mixing to substantially alter the background stratification. The prolate structure of the tachocline may be a result of latitudinal pressure gradients induced by the strong toroidal fields which are thought to exist at low and mid-latitudes over the course of the solar cycle (Dikpati and Gilman, 2001b ). Magnetic confinement models also exhibit larger tachocline depths at high latitudes due to the assumed dipolar structure of the poloidal field (the polar pit; see Section 8.5). However, it is unclear from these latter models why the tachocline may be prolate.

Temporal variations provide another means by which to investigate tachocline dynamics. In particular, helioseismic inversions have revealed a 1.3-year oscillation in the angular velocity, which appears to straddle the base of the convection zone at low latitudes (Section 3.3). This may arise from the interaction of gravity waves and shear (Section 8.4). Alternatively, it may be a manifestation of the MHD shear instabilities discussed in Section 8.2, which generally have an oscillatory component. A third possibility is that the tachocline oscillations arise from spatio-temporal fragmentation of the longer-period torsional oscillations (Covas et al., 2001 , 2004 ).

Distinguishing between these alternatives will require more detailed probing of tachocline structure. For example, the joint instability of a banded toroidal field and latitudinal differential rotation predicts the presence of a prograde jet which provides gyroscopic stabilization against the tipping of the band (Section 8.2). The search for such jets in helioseismic inversions is going on now and has produced a few possible candidates (Christensen-Dalsgaard et al., 2004 ).

The mere presence of a zonal jet in the tachocline does not necessarily indicate the gyroscopic stabilization of a toroidal band. Self-organization processes in rotating, stratified turbulence tend to produce banded zonal flows even in the absence of magnetic fields (Section 8.3). However, the sense of the jet can provide clues as to its origin. For example, gyroscopic stabilization requires a prograde jet whereas a breaking Rossby wave will produce a retrograde jet (e.g., McIntyre, 1998 ).

Probing the interior of a star is not easy, but we are making progress. Ambitious observing programs and modeling efforts promise more excitement in the near future.