4.1 The solar dynamo: a brief summary of current models

Extensive summaries of the current standing of solar dynamo theory are given in the reviews by Petrovay (2000), Ossendrijver (2003), Charbonneau (2010), and Solanki et al. (2006). As explained in detail in those reviews, all current models are based on the mean-field theory approach wherein a coupled system of partial differential equations governs the evolution of the toroidal and poloidal components of the large-scale magnetic field. The large-scale field is assumed to be axially symmetric in practically all current models. In some nonlinear models the averaged equation of motion, governing large-scale flows is also coupled into the system.

In the simplest case of homogeneous and isotropic turbulence, where the scale l of turbulence is small compared to the scale L of the mean variables (scale separation hypothesis), the dynamo equations have the form

∂B ----= ∇ × (U × B + αB ) − ∇ × (ηT × ∇B ). (13 ) ∂t
Here B and U are the large-scale mean magnetic field and flow speed, respectively; ηT is the magnetic diffusivity (dominated by the turbulent contribution for the highly conductive solar plasma), while α is a parameter related to the non-mirror symmetric character of the magnetized plasma flow.

In the case of axial symmetry the mean flow U may be split into a meridional circulation Uc and a differential rotation characterized by the angular velocity profile Ω0(r,πœƒ):

U = Uc + rsinπœƒ Ω0 eΟ•,

where r, πœƒ, Ο• are spherical coordinates and eΟ• is the azimuthal unit vector. Now introducing the shear

Ω = r sin πœƒ∇ Ω , Ω = − sgn dΩ0-⋅ |Ω |, 0 dr

assuming α β‰ͺ ΩL and ignoring spatial derivatives of α and ηT, Equation (13View Equation) simplifies to the pair

∂A- = αB − (U ⋅ ∇ )A − (∇ ⋅ U )A + η ∇2A , (14 ) ∂t c c T
∂B ∂A 2 ∂t--= Ω ∂x-− (Uc ⋅ ∇ )B − (∇ ⋅ Uc)B + ηT ∇ B, (15 )
where B and A are the toroidal (azimuthal) components of the magnetic field and of the vector potential, respectively, and ∂A∂x- is to be evaluated in the direction 90° clockwards of βƒ—Ω (along the isorotation surface) in the meridional plane. These are the classic αΩ dynamo equations, including a meridional flow.

In the more mainstream solar dynamo models the strong toroidal field is now generally thought to reside near the bottom of the solar convective zone. Indeed, it is known that a variety of flux transport mechanisms such as pumping (Petrovay, 1994) remove magnetic flux from the solar convective zone on a timescale short compared to the solar cycle. Following earlier simpler numerical experiments, recent MHD numerical simulations have indeed demonstrated this pumping of large scale magnetic flux from the convective zone into the tachocline below, where it forms strong coherent toroidal fields (Browning et al., 2006). As this layer is also where rotational shear is maximal, it is plausible that the strong toroidal fields are not just stored but also generated here, by the winding up of poloidal field. The two main groups of dynamo models, interface dynamos and flux transport dynamos, differ mainly in their assumptions about the site and mechanism of the α-effect responsible for the generation of a new poloidal field from the toroidal field.

In interface dynamos α is assumed to be concentrated near the bottom of the convective zone, in a region adjacent to the tachocline, so that the dynamo operates as a wave propagating along the interface between these two layers. While these models may be roughly consistent and convincing from the physical point of view, they have only had limited success in reproducing the observed characteristics of the solar cycle, such as the butterfly diagram.

Flux transport dynamos, in contrast, rely on the Babcock–Leighton mechanism for α, arising due to the action of the Coriolis force on emerging flux loops, and they assume that the corresponding α-effect is concentrated near the surface. They keep this surface layer incommunicado with the tachocline by introducing some arbitrary unphysical assumptions (such as very low diffusivities in the bulk of the convective zone). The poloidal fields generated by this surface α-effect are then advected to the poles and there down to the tachocline by the meridional circulation – which, accordingly, has key importance in these models. The equatorward deep return flow of the meridional circulation is assumed to have a significant overlap with the tachocline (another controversial point), and it keeps transporting the toroidal field generated by the rotational shear towards the equator. By the time it reaches lower latitudes, it is amplified sufficiently for the flux emergence process to start, resulting in the formation of active regions and, as a result of the Babcock–Leighton mechanism, in the reconstruction of a poloidal field near the surface with a polarity opposed to that in the previous 11-year cycle. While flux transport models may be questionable from the point of view of their physical consistency, they can be readily fine-tuned to reproduce the observed butterfly diagram quite well.

It should be noted that while the terms “interface dynamo” and “flux transport dynamo” are now very widely used to describe the two main approaches, the more generic terms “advection-dominated” and “diffusion-dominated” would be preferable in several respects. This classification allows for a continuous spectrum of models depending on the numerical ratio of advective and diffusive timescales (for communication between surface and tachocline). In addition, even at the two extremes, classic interface dynamos and circulation-driven dynamos are just particular examples of advection or diffusion dominated systems with different geometrical structures.

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