6.1 Historical perspective

The most conceptually straightforward approach to studying global-scale solar convection is to solve the nonlinear, 3D equations of motion in a rotating spherical shell of fluid heated from below and cooled from above. The first numerical models to do so were developed by Gilman (1977 , 1978 , 1983

), Gilman and Miller (1981

, 1986 ), and Glatzmaier (1984

, 1985a

, 1987

). The convection structure was dominated by traveling, columnar convection cells with a north-south alignment and a periodic longitudinal structure ( $m ~ 10$ ), similar to the preferred convection modes predicted by linear theory (Busse, 1970

; Gilman, 1975

). These became known as banana cells because of their elongated appearance, sheared into a crescent shape by the differential rotation they established. These pioneering studies yielded great insight into the nonlinear interaction between convection, rotation, and magnetic fields, but they had limited spatial resolution and were therefore restricted to relatively laminar flows, far from the highly turbulent parameter regimes thought to exist in the solar interior (see Section 5.1).

In the two decades since, many more simulations of convection in rotating spherical shells have appeared, but most have been concerned with physical conditions which are characteristic of the Earth’s outer core and other planetary interiors (e.g., Sun and Schubert, 1995 ; Tilgner and Busse, 1997 ; Kageyama and Sato, 1997 ; Christensen et al., 1999 ; Roberts and Glatzmaier, 2000 ; Zhang and Schubert, 2000 ; Ishihara and Kida, 2002 ; Busse, 2002 ; Glatzmaier, 2002 ). Relative to the Sun, the Earth is rapidly rotating (smaller Rossby number), weakly compressible (smaller density contrast) and highly magnetic (strong Lorentz force). Furthermore, the geometry of the convective shell is somewhat different and physical effects such as compositional gradients and radioactivity play an important role.

In order to revisit solar convection with the latest generation of scalable parallel supercomputers, Clune et al. (1999 ) developed a numerical model which is now known as the Anelastic Spherical Harmonic (ASH) code. The algorithm is similar to that described by Glatzmaier (1984) and solves the 3D anelastic equations described in Appendix A.2 using a pseudospectral method with spherical harmonic and Chebyshev basis functions. Recent ASH simulations have achieved much higher resolution and subsequently more turbulent parameter regimes than the pioneering studies by Gilman and Glatzmaier referred to above. In the remainder of this section, we will focus on results obtained with the ASH code. For a description of the numerical method see Clune et al. (1999 ) and Brun et al. (2004 ). Further details on the scientific results have been reported by Miesch et al. (2000 ); Elliott et al. (2000 ); Brun and Toomre (2002 ); DeRosa et al. (2002 ) and Brun et al. (2004 ).

The ASH code is dimensional and uses realistic values for the solar radius, luminosity, and mean rotation rate. The reference state is based on 1D solar structure models. Since global simulations cannot capture the complex dynamics occurring in the near-surface layers (see Section 7.3), the upper boundary of the computational domain is generally placed below the photosphere, at $0.96 -0.98R o .$ . For computational efficiency, the lower boundary is often placed at the base of the convection zone (see Section 7.3) but some simulations have included penetration into the radiative interior (Miesch et al., 2000 ).

Figure 9:

The radial velocity near the top of the simulation domain is shown for Case M3 (Brun et al., 2004

), Case F (Brun et al., 2005

), and Case D2 (DeRosa et al., 2002

). Bright and dark tones denote upflow and downflow as indicated by the color tables. Orthographic projections are shown with the north pole tilted $o 35$ toward the observer. The equator is indicated with a solid line. Magnified areas shown in the lower panels correspond to square $o 45$ patches which extend from latitudes of $10o N -55o N$ .