4.9 Numerical simulations of solar dynamo action

Ultimately, the solar dynamo problem should be tackled as a (numerical) solution of the complete set of MHD partial differential equations in a rotating, stratified spherical domain undergoing thermally-driven turbulent convection in its outer 30% in radius. The first full-fledged attempts to do so go back some some thirty years, to the simulations of Gilman and Miller (1981); Gilman (1983Jump To The Next Citation Point); Glatzmaier (1985a,b). These epoch-making simulations did produce cyclic dynamo action and latitudinal migratory patterns suggestive of the dynamo waves of mean-field theory. However, the associated differential rotation profile turned out non-solar, as did the magnetic field’s spatio-temporal evolution. In retrospect this is perhaps not surprising, as limitations in computing resources forced these simulations to be carried out in a parameter regime far removed from solar interior conditions. Later simulations taking advantages of massively parallel computing architectures did managed to produce tolerably solar-like mean internal differential rotation (see, e.g., Miesch and Toomre, 2009, and references therein), as well as copious small-scale magnetic field, but failed to generate a spatially well-organized large-scale magnetic component (see Brun et al., 2004). Towards this end the inclusion of a stably stratified fluid layer below the convecting layers is now believed to be advantageous (although not strictly necessary, see Brown et al., 2010) as it allows the development of a tachocline-like shear layer where magnetic field produced within the convection zone can accumulate in response to turbulent pumping from above, and be further amplified by the rotational shear (see Browning et al., 2006, also Tobias et al., 2001, 2008, and references therein, for related behavior in local cartesian simulations).

Some of these simulations are now beginning to yield regular polarity reversals of the large-scale magnetic components. Figures 20Watch/download Movie and 21View Image present some sample results taken from Ghizaru et al. (2010Jump To The Next Citation Point), see also Brown et al. (2009) and Käpylä et al. (2010). Figure 20Watch/download Movie is an animation in Mollweide latitude-longitude projection of the toroidal magnetic component 0.02R ⊙ below the nominal interface between the convecting layers and underlying stable layers in one of these simulations. This toroidal component reaches some 2.5 kG here, and shows a very clear global antisymmetry about the equator, despite strong spatiotemporal fluctuations produced by convective undershoot. The cyclic variation of this large-scale field is quite apparent on the animation, with polarity reversals approximately synchronous across hemispheres.

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Figure 20: flv-Movie (9247 KB) Latitude-Longitude Mollweide projection of the toroidal magnetic component at depth r∕R = 0.695 in the 3D MHD simulation of Ghizaru et al. (2010Jump To The Next Citation Point). This large-scale axisymmetric component shows a well-defined overall antisymmetry about the equatorial plane, and undergoes polarity reversals approximately every 30 yr. The animation spans a little over three half-cycles, including three polarity reversals. Time is given in solar days, with 1 s.d. = 30 d.

Figure 21View ImageA shows, for the same simulation as in Figure 20Watch/download Movie, a time-latitude diagram of the zonally-averaged toroidal component, now constructed at a depth corresponding to the core-envelope interface in the model. This is again assumed to be the simulation’s equivalent to the sunspot butterfly diagram. This simulation was run for 255 yr, in the course of which eight polarity reversals have taken place, with a mean (half-)period of about 30 yr. Note the tendency for equatorward migration of the toroidal flux structures, and the good long-term synchrony between the Northern and Southern hemispheres, persisting despite significant fluctuations in the amplitude and duration of cycles in each hemiphere. Figure 21View ImageB shows the corresponding time-evolution of the zonaly-averaged radial surface magnetic component, again in a time-latitude diagram. The surface field is characterized by a well-defined dipole moment aligned with the rotational axis, with transport of surface fields taking place from lower latitudes and (presumably) contributing to the reversal of the dipole moment. Compare these time-latitude diagrams to the sunspot butterfly diagram of Figure 3View Image and synoptic magnetogram of Figure 4View Image, and reflect upon the similarities and differences.

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Figure 21: (A) Time-latitude diagram of the zonally-averaged toroidal magnetic component the core-envelope interface (r∕R = 0.718) and (B) corresponding time-latitude diagram of the surface radial field, in the 3D MHD simulations presented in Ghizaru et al. (2010Jump To The Next Citation Point). Note the regular polarity reversals, the weak but clear tendency towards equatorial migration of the deep toroidal magnetic component, and the good coupling between the two hemispheres despite marked fluctuations in successive cycles. The color scale codes the magnetic field strength, in Tesla.

Although much remains to be investigated regarding the mode of dynamo action in these simulations, some encouraging links to mean-field theory (Section 3.2.1) do emerge. The fact that a positive toroidal component breeds here a positive dipole moment is what one would expect from a turbulent α-effect (more precisely, the αϕϕ tensor component) positive in the Northern hemisphere. A posteriori calculation of the mean electromotive force ′ ′ ℰ = ⟨u × B ⟩ does reveal a clear hemispheric pattern, with ℰϕ having the same sign in both hemisphere, but changing sign from one cycle to the next, again consistent with the idea that the turbulent α-effect is the primary source of the large-scale poloidal component. Likewise, having a well-defined axisymmetric dipolar component being sheared by an axisymmetric differential rotation is consistent with the buildup of a large-scale toroidal component antisymmetric about the equatorial plane.

On the other hand, calculation of the r and 𝜃-components of the mean electromotive force indicates that the latter contributes to the production of the toroidal field at a level comparable to shearing of the poloidal component by differential rotation, suggestive of what, in mean-field electrodynamics parlance, is known as an α2Ω dynamo. Calculation of the α-tensor components also reveals that the latter do not undergo significant variations between maximal and minimal phases of the cycle, suggesting that α-quenching is not the primary amplitude-limiting mechanism in this specific simulation run. Although it would premature to claim that these simulations vindicate the predictions of mean-field theory, to the level at which they have been analyzed thus far, they do not appear to present outstanding departures from the mean-field Weltanschau.

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