Large-Scale Dynamics of the Convection Zone and Tachocline Mark S. Miesch High Altitude Observatory National Center for Atmospheric Research 3450 Mitchell Lane, Boulder, CO 80301 U.S.A. email: [email protected] http://www.hao.ucar.edu/~miesch Accepted on 15 February 2005 Published on 22 April 2005 http://www.livingreviews.org/lrsp-2005-1 Living Reviews in Solar Physics Published by the Max Planck Institute for Solar System Research Germany Abstract The past few decades have seen dramatic progress in our understanding of solar inte- rior dynamics, prompted by the relatively new science of helioseismology and increasingly sophisticated numerical models. As the ultimate driver of solar variability and space weather, global-scale convective motions are of particular interest from a practical as well as a the- oretical perspective. Turbulent convection under the influence of rotation and stratification redistributes momentum and energy, generating differential rotation, meridional circulation, and magnetic fields through hydromagnetic dynamo processes. In the solar tachocline near the base of the convection zone, strong angular velocity shear further amplifies fields which subsequently rise to the surface to form active regions. Penetrative convection, instabilities, stratified turbulence, and waves all add to the dynamical richness of the tachocline region and pose particular modeling challenges. In this article we review observational, theoretical, and computational investigations of global-scale dynamics in the solar interior. Particular empha- sis is placed on high-resolution global simulations of solar convection, highlighting what we have learned from them and how they may be improved. c Max Planck Society and the authors. 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Contents 1 A Turbulent Sun 5 2 Probing the Solar Interior 7 2.1 Global helioseismology .................................. 7 2.2 Local helioseismology ................................... 8 2.3 Surface and atmospheric observations ......................... 9 3 What Do We Observe? 11 3.1 Differential rotation of the solar envelope ....................... 11 3.2 The tachocline ...................................... 12 3.3 Torsional oscillations and other temporal variations in the solar rotation . 13 3.4 Meridional circulation .................................. 14 3.5 Giant cells, waves, and solar subsurface weather .................... 16 3.6 The base of the convection zone ............................. 18 3.7 Thermal asphericity and subsurface magnetic fields . 18 3.8 Solar magnetism ..................................... 19 4 Fundamental Concepts 22 4.1 Governing equations ................................... 22 4.2 Energetics ......................................... 22 4.3 Maintenance of differential rotation ........................... 24 4.3.1 Angular momentum redistribution ....................... 25 4.3.2 The Taylor–Proudman theorem and thermal wind balance . 27 4.4 Maintenance of meridional circulation ......................... 29 4.5 The solar dynamo .................................... 31 5 Modeling Solar Convection 34 5.1 The challenge ....................................... 34 5.2 3D numerical simulations ................................ 35 5.3 Reduced models ..................................... 36 5.4 Thin-shell approximations for the tachocline ...................... 39 6 What Do Global Simulations Tell Us about the Convection Zone? 41 6.1 Historical perspective .................................. 41 6.2 Convection structure ................................... 43 6.3 Differential rotation ................................... 47 6.4 Meridional circulation .................................. 53 6.5 Dynamo processes .................................... 58 6.6 Comparisons with mean-field theory .......................... 62 6.6.1 Mean-field hydrodynamics ............................ 62 6.6.2 Solar dynamo theory ............................... 63 7 How Can We Do Better? (With Global Simulations) 65 7.1 Resolution ......................................... 65 7.2 Subgrid-scale modeling .................................. 66 7.3 Boundary influences ................................... 69 8 Dynamics of the Tachocline and Overshoot Region 72 8.1 Convective penetration .................................. 72 8.2 Instabilities ........................................ 75 8.3 Rotating, stratified turbulence .............................. 80 8.4 Internal waves ....................................... 83 8.5 Tachocline confinement ................................. 87 9 Conclusion: Making Sense of the Observations 93 10 Acknowledgments 96 A Appendices 97 A.1 Notation .......................................... 97 A.2 The anelastic equations ................................. 98 A.3 Energy conservation in the anelastic approximation . 100 A.4 Angular momentum balance ............................... 102 A.5 Meridional circulation maintenance . 102 A.6 Potential vorticity and Rossby waves . 103 A.7 Gravity waves in a vertical shear flow . 106 References 139 Large-Scale Dynamics of the Convection Zone and Tachocline 5 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 char- acteristic 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 am- plifying 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