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4.5 The solar dynamo

As discussed in Sections 1 and 4.2, the magnetic fields which drive solar variability are thought to be generated by fluid motions in the convection zone and tachocline. Kinetic energy is converted to magnetic energy via hydromagnetic dynamo processes and this flux subsequently emerges from the surface, playing a central role in the dynamics of the solar atmosphere and heliosphere.

There are many recent reviews on the solar dynamo so there is no need for a detailed discussion here. For a comprehensive overview of solar dynamo theory as a whole an excellent place to start is the recent article by (Ossendrijver, 2003Jump To The Next Citation Point). Mean-field models of the solar activity cycle are reviewed in these volumes by Charbonneau (2005Jump To The Next Citation Point). Tobias (2004) focuses on the role of the solar tachocline in particular. Further details and perspectives on solar and stellar dynamos are provided by Weiss (1994Jump To The Next Citation Point), Mestel (1999Jump To The Next Citation Point), Schrijver and Zwaan (2000), and Rüdiger and Hollerbach (2004Jump To The Next Citation Point). Dynamo theory from a more general astrophysical perspective has been reviewed comprehensively by Moffatt (1978Jump To The Next Citation Point), Parker (1979Jump To The Next Citation Point), Childress and Gilbert (1995), and most recently by Brandenburg and Subramanian (2004Jump To The Next Citation Point).

Some insight into the nature of solar dynamo processes may be obtained from the evolution equation for the mean field, which is just the longitudinal average of Equation (42View Equation):

@ ---<B> = f^c <BM >. \~/ _O_ + \~/ × (<vM> × <B > + E - j \~/ × <B >), (20) @t
where E is the turbulent emf, arising from the non-axisymmetric field components:
' ' E = <v× B >. (21)
The first term on the right-hand side of Equation (20View Equation) is the familiar _O_-effect; differential rotation converts poloidal field <B > M to toroidal field <B > f and amplifies it, extracting energy from the rotational shear. The second term represents advection of magnetic flux by the meridional circulation. Although the meridional circulation may redistribute and amplify magnetic flux, it cannot produce an exchange of energy between the mean toroidal and poloidal field components.

The term involving E represents field generation by turbulent convection or other processes, such as shear instabilities (see Section 8.2). Note that our derivation of Equation (20View Equation) involves no additional approximations beyond the standard anelastic (or compressible) MHD equations. However, this equation is the starting point for mean-field dynamo theory in which additional approximations are made in order to make the system more tractable. In many mean-field models the rotation profile _O_ and the meridional circulation <vM > are specified and the Lorentz force is neglected, making the approach kinematic. Some type of parameterization is then introduced for the turbulent emf E and Equation (20View Equation) is solved for <B >.

The simplest and most common parameterization may be derived by exploiting the linearity of the induction equation in B (neglecting Lorentz force feedback on v) and by assuming scale separation between the mean and fluctuating fields. The problem can be further simplified by assuming that the fluctuations are pseudo-isotropic, meaning that their statistics are invariant under rotation of the coordinate system but not necessarily invariant under reflection. In this case the turbulent emf may be represented in terms of the mean field as:

E = a&lt;B &gt;- jt \~/ × &lt;B &gt;, (22)
which is valid to lowest order in the ratio of fluctuating scales to mean scales (Moffatt, 1978Jump To The Next Citation Point). The term involving a on the right-hand-side of Equation (22View Equation) represents the amplification of mean fields by fluctuating motions, which is widely known as the a-effect. The final term in Equation (22View Equation) represents turbulent diffusion with an effective diffusivity given by jt. If the assumptions of homogeneity and pseudo-isotropy are relaxed, a and jt become pseudo-tensors and can represent more general transport processes such as magnetic pumping (Ossendrijver, 2003Jump To The Next Citation Point). In general, a and jt vary with latitude and radius and may depend on other parameters of the problem such as the rotation rate _O_ 0 and the strength of the mean field 2 <B >. For example, in many mean-field models, a and jt are quenched (reduced in amplitude) as _O_0 or 2 <B > become large (e.g., Rüdiger and Hollerbach, 2004Jump To The Next Citation PointCharbonneau, 2005Jump To The Next Citation Point).

In analogy with Equation (22View Equation), we will in this paper loosely refer to the a-effect in the general sense of field generation via the turbulent emf term in Equation (20View Equation). This does not necessarily imply that the parameterization in Equation (22View Equation) is an accurate one. In practice, solar dynamo processes may be much more subtle than this simple expression suggests (see Section 6.5). Still, the classical a-effect is a useful concept and remains an important ingredient of dynamo theory.

Unlike the _O_-effect, the a-effect can work both ways: it may convert toroidal field energy to poloidal field energy or vice versa. The field conversion and amplification process is often associated with vorticity and shear as in the classical scenario, first described by Parker (1955), in which field lines are lifted and twisted by helical eddies. In the special case of homogeneous, pseudo-isotropic turbulence, the a parameter is directly proportional to the mean kinetic helicity of the flow, Hk = <w.v > (Moffatt, 1978Jump To The Next Citation PointOssendrijver, 2003Jump To The Next Citation Point). Rotation induces vorticity and breaks the reflection symmetry of the fluid equations, so rotating flows are generally helical and tend to be efficient dynamos, although rotation is not required for sustained dynamo action (Cattaneo et al., 2003Jump To The Next Citation Point).

Although Equation (20View Equation) only strictly applies to the mean (longitudinally-averaged) field (or some other suitable spatial or ensemble average), similar processes also operate on fluctuating (non-axisymmetric) fields. All toroidal field structures are amplified to some extent by rotational shear and processes akin to the (generalized) a-effect generate magnetic energy on a wide range of spatial scales. Most solar dynamo models focus on the axisymmetric component of the field but observations indicate that the magnetic field structure in the solar photosphere and corona is quite complex, with a large non-axisymmetric component (see Section 3.8 and Figure 4View Image). Solar variability is dominated not by mean fields but by localized structures such as active regions, filaments, and coronal loops.

Our current paradigm for how the solar dynamo operates is illustrated in Figure 8View Image. The density stratification tends to make solar convection highly anisotropic, characterized by relatively weak, broad upflows amid a complex, evolving network of strong downflow lanes and plumes (0). Turbulent downflow plumes possess substantial vorticity and helicity which may amplify fields through the a-effect (1). These fields are then pumped downward by the anisotropic convection and accumulate in the overshoot region and tachocline (2). Intermittent plumes may dredge up some of this flux and return it to the convection zone where it may be further amplified and again pumped down. Differential rotation in the tachocline stretches and amplifies this disorganized field into strong, coherent toroidal flux tubes and sheets (3). As the field becomes stronger, it eventually becomes buoyantly unstable and rises toward the surface (4). The Coriolis force acting on these rising structures twists them in a systematic way which depends on latitude (5). Weaker structures may be shredded by turbulent convection in the envelope and the flux is then recycled (6). Stronger fields and configurations (e.g., twisted tubes) remain coherent throughout the convection zone and emerge from the surface as bipolar active regions (7). Large-scale poloidal fields may be generated by the a-effect (1) or by the turbulent diffusion of surface flux after the tubes have emerged (7). Due to the manner in which field is amplified by the _O_-effect (3) and to the tilts induced in surface active regions due to the Coriolis force (5), surface diffusion would tend to build large-scale poloidal fields opposite in sign to the prevailing field, eventually producing a global polarity reversal.

View Image

Figure 8: Schematic illustration of the solar dynamo. Numbers indicate particular processes as described in the text (courtesy N. Brummell).
This schematic picture of the solar dynamo is compelling but highly simplified. In actuality, each of the processes identified in Figure 8View Image is complex and researchers are only beginning to understand how they work in detail.


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