6.5 Dynamo processes

The solar dynamo involves an intricate interplay of complex processes occurring over a wide range of spatial and temporal scales (see Section 4.5 and Section 5.1). Consequently, global convection simulations are a long way from making detailed comparisons with photospheric and coronal observations of magnetic activity. Still, they have provided important insight into several key elements of the global dynamo, particularly field generation in the convection zone (processes $0 -1$ in Figure 8

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 .

Figure 20:

The radial velocity $vr$ (a), the radial magnetic field, $Br$ (b), and the toroidal magnetic field, $Bf$ (c), are shown near the top of the computational domain ( $r = 0.95Ro .$ ) for Case M3 of Brun et al. (2004

). White and yellow tones denote outward flow (a), outward field (b), and eastward field (c) as indicated by the color tables.

As in Cartesian simulations of MHD convection (e.g., Brandenburg et al., 1996

; Cattaneo et al., 2003 ), radial field near the top of the computational domain is swept into downflow lanes by horizontally converging flows¹⁰ . The field distribution is intermittent and confined primarily to the downflow network (Figure 20

, panels a and b). Field within the network is of mixed polarity and is wrapped up by cyclonic vorticity, generating large gradients which promote magnetic reconnection. Magnetic helicity is generated locally but it is of mixed sign and no clear global patterns emerge¹¹ (cf. Section 3.8). A potential-field extrapolation of the radial field at the upper boundary exhibits a complex topology, with interconnected loops spanning a wide range of spatial scales (Figure 21

, panel a). This may be compared with photospheric extrapolations which are similarly complex (see Figure 4

Figure 21:

(a) Potential-field extrapolation of the radial magnetic field $Br$ at the outer boundary of Case M3. White lines represent closed loops while green and magenta lines indicate field which is outward and inward, respectively, at $2.5R o.$ , the boundary of the extrapolation domain. (b) Volume rendering of the toroidal field $Bf$ of Case M3 in a narrow latitude band centered at the the equator. The equatorial plane is tilted slightly with respect to the line of sight. Typical field amplitudes are 1000 and 3000 G in frames (a) and (b), respectively (from Brun et al., 2004

Toroidal fields are somewhat less intermittent and peak in the horizontally-diverging regions between downflow lanes (Figure 20

, panel c). These regions tend to be broadest at low latitudes where much of the toroidal field energy is concentrated. Differential rotation stretches fields into toroidal ribbons (Figure 21

, panel b) which generally reach higher amplitudes ( $~ 3000 G$ ) than poloidal fields ( $~ 1000 G$ ). The energy in the mean (axisymmetric) toroidal field exceeds that in the mean poloidal field by about a factor of three, indicating that an $_O_$ -effect is operating (cf. Section 4.5). However, the magnetic energy in the fluctuating (non-axisymmetric) poloidal and toroidal field components is comparable.

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 $98%$ 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 $a$ - $_O_$ dynamos than $a2$ dynamos (Ossendrijver, 2003 ). If meridional circulation is neglected, the cycle period is determined by the magnitude of $a$ , 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 $a$ -effect¹² . An unrealistically large effective value of $a$ was identified as a possible reason why early solar dynamo simulations produced cycle periods of only $1- 10 yr$ , 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 $a$ . 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.