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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 (197719781983Jump To The Next Citation Point), Gilman and Miller (1981Jump To The Next Citation Point1986), and Glatzmaier (1984Jump To The Next Citation Point1985aJump To The Next Citation Point,bJump To The Next Citation Point1987Jump To The Next Citation Point). 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, 1970Jump To The Next Citation PointGilman, 1975Jump To The Next Citation Point). 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, 1995Tilgner and Busse, 1997Kageyama and Sato, 1997Jump To The Next Citation PointChristensen et al., 1999Jump To The Next Citation PointRoberts and Glatzmaier, 2000Jump To The Next Citation PointZhang and Schubert, 2000Jump To The Next Citation PointIshihara and Kida, 2002Jump To The Next Citation PointBusse, 2002Jump To The Next Citation PointGlatzmaier, 2002Jump To The Next Citation Point). 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. (1999Jump To The Next Citation Point) 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. (2004Jump To The Next Citation Point). Further details on the scientific results have been reported by Miesch et al. (2000Jump To The Next Citation Point); Elliott et al. (2000Jump To The Next Citation Point); Brun and Toomre (2002Jump To The Next Citation Point); DeRosa et al. (2002Jump To The Next Citation Point) and Brun et al. (2004Jump To The Next Citation Point).

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., 2000Jump To The Next Citation Point).

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Figure 9: The radial velocity near the top of the simulation domain is shown for Case M3 (Brun et al., 2004Jump To The Next Citation Point), Case F (Brun et al., 2005Jump To The Next Citation Point), and Case D2 (DeRosa et al., 2002Jump To The Next Citation Point). 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.

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