Simulations of thermal convection in rotating spherical shells have produced many examples of sustained dynamo action (Gilman and Miller, 1981; Gilman, 1983; Glatzmaier, 1984, 1985a,b; Kageyama and Sato, 1997; Christensen et al., 1999; Roberts and Glatzmaier, 2000; Zhang and Schubert, 2000; Ishihara and Kida, 2002; Busse, 2002; Glatzmaier, 2002; Brun et al., 2004). Most of these are concerned with relatively laminar flows or parameter regimes which are more characteristic of the Earth’s core than the solar interior. Simulations of turbulent convection and dynamo action in more solar-like parameter regimes have recently been investigated by Brun et al. (2004). Results are illustrated in Figures 20 and 21.
In light of the complex topologies evident in Figures 20 and 21 it is no surprise that the axisymmetric field components are relatively small. Fluctuating fields account for of the total magnetic energy in Case M3. Furthermore, there is no clear separation of spatial or temporal scales and nonlinear correlations between fluctuating field components are not small in any sense, calling into question many of the assumptions often used in mean-field dynamo theory (see Section 4.5).
Magnetic fields on the Sun are also complex but they exhibit striking regularities, most notably those associated with the 22-year activity cycle (see Section 3.8). Furthermore, the axisymmetric component of the poloidal field on the Sun is predominantly dipolar, at least during solar minimum. This degree of order amid complexity has not yet been achieved with global simulations. Case M3, for example, does not exhibit cyclic behavior and the mean poloidal field involves dipolar, quadrupolar, and higher-order components (see Brun et al., 2004). Cyclic dipolar dynamos have however been achieved in other parameter regimes. A key element appears to be a strong differential rotation. When the kinetic energy of the differential rotation exceeds that of the convection, cyclic dynamos are more likely. This is a conclusion reached over two decades ago by Gilman (1983) and has generally been borne out in the later work cited at the beginning of this section. Furthermore, many cyclic dynamos operate in the strong field regime in which the magnetic energy exceeds the convection kinetic energy by an order of magnitude or more (e.g., Christensen et al., 1999; Zhang and Schubert, 2000; Ishihara and Kida, 2002). This is appropriate for planetary interiors but not for the Sun, where the rotational influence is much weaker. In Case M3, the kinetic energy in the differential rotation and in the convection are roughly equal while the magnetic energy is about an order of magnitude less.
The importance of a strong differential rotation in achieving cyclic behavior is consistent with mean-field theory where it is known that cycles are more readily achieved with - dynamos than dynamos (Ossendrijver, 2003). If meridional circulation is neglected, the cycle period is determined by the magnitude of , which in turn is proportional to the kinetic helicity to a first approximation (Section 4.5). To the author’s knowledge, all numerical simulations of thermal convection in rotating spherical shells which have achieved sustained, cyclic, dipolar dynamos (e.g., Gilman, 1983; Glatzmaier, 1985a,b; Kageyama and Sato, 1997; Zhang and Schubert, 2000; Ishihara and Kida, 2002; Busse, 2002) are dominated by so-called banana cells (see Section 6.1). This is either because of low resolution and correspondingly low Reynolds numbers or because of the strong rotational influence characteristic of planetary applications, or both. The Coriolis force acting on these relatively laminar flows induces kinetic helicity which in turn produces efficient poloidal field regeneration via the -effect12. An unrealistically large effective value of was identified as a possible reason why early solar dynamo simulations produced cycle periods of only , significantly less than the 22-year period of the solar activity cycle (Gilman, 1983; Glatzmaier, 1985a). In more turbulent parameter regimes, nonlinear correlations are likely to be reduced, implying a smaller . Thus, if cyclic, dipolar dynamos can be achieved in such parameter regimes, there is reason to believe that their periods may be more comparable to the Sun. However, this remains to be seen. Another difficulty exhibited by the early solar dynamo simulations of Gilman (1983) and Glatzmaier (1985b) is that the toroidal field tended to migrate poleward over the course of a cycle rather than equatorward as in the Sun (Section 3.8). This was attributed to the sign of the kinetic helicity, which determines the propagation direction of dynamo waves in mean-field theory (e.g., Ossendrijver, 2003; Charbonneau, 2005). However, it is well known that the sign of the kinetic helicity in rotating convection simulations reverses near the base of the convection zone (Sections 6.2 and 6.3). For this and other reasons (mainly having to do with the storage and amplification of toroidal flux), it has been argued that the lower convection zone and, in particular, the tachocline likely play a key role in the solar dynamo (e.g., Weiss, 1994; Ossendrijver, 2003).
Global convection simulations have not yet achieved a rotational transition region comparable to the solar tachocline. A realistic modeling effort would require very high spatial resolution (see Section 5.1) and may involve long-term processes which would be difficult to capture in a 3D simulation (see Section 8.5). However, a tachocline can be incorporated in a global model in an approximate way, for example by imposing solid body rotation in the interior via boundary conditions or body forces. This is the frontier for global, 3D, solar dynamo simulations.
The presence of a tachocline in a global simulation may promote more regular, cyclic behavior by providing a reservoir for field storage and a mechanism for field amplification, possibly up to super-equipartition values as is though to occur in the Sun (e.g., Fisher et al., 2000; Fan, 2004). Coupling to the radiative interior may also act to regularize the dynamo by providing thermal, mechanical, and electromagnetic inertia. For example, one of the important contributions of global convection simulations to geodynamo theory over the past decade has been the realization that an electrically conducting core adds stability to the dipolar dynamo, preventing overly sporadic and frequent reversals (e.g., Glatzmaier, 2002). Furthermore, the regularity of the solar cycle suggests that essentially linear processes such as dynamo waves may prevail over more chaotic turbulent processes, meaning the relatively quiescent tachocline may set the rhythm of the solar dynamo.
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