3.2 Toroidal to poloidal

In view of Cowling’s theorem, we have no choice but to look for some fundamentally non-axisymmetric process to provide an additional source term in Equation (11View Equation). It turns out that under solar interior conditions, there exist various mechanisms that can act as a source of poloidal field. In what follows we introduce and briefly describe the four classes of such mechanisms that appear most promising, but defer discussion of their implementation in dynamo models to Section 4, where illustrative solutions are also presented.

3.2.1 Turbulence and mean-field electrodynamics

The outer ∼ 30% of the Sun are in a state of thermally-driven turbulent convection. This turbulence is anisotropic because of the stratification imposed by gravity, and lacks reflectional symmetry due to the influence of the Coriolis force. Since we are primarily interested in the evolution of the large-scale magnetic field (and perhaps also the large-scale flow) on time scales longer than the turbulent time scale, mean-field electrodynamics offers a tractable alternative to full-blown 3D turbulent MHD. The idea is to express the net flow and field as the sum of mean components, ⟨u ⟩ and ⟨B ⟩, and small-scale fluctuating components ′ u, ′ B. This is not a linearization procedure, in that we are not assuming that ′ |u |∕|⟨u⟩| ≪ 1 or ′ |B |∕|⟨B ⟩| ≪ 1. In the context of the axisymmetric models to be described below, the averaging (“⟨⟩”) is most naturally interpreted as a longitudinal average, with the fluctuating flow and field components vanishing when so averaged, i.e., ⟨u′⟩ = 0 and ⟨B ′⟩ = 0. The mean field ⟨B ⟩ is then interpreted as the large-scale, axisymmetric magnetic field usually associated with the solar cycle. Upon this separation and averaging procedure, the MHD induction equation for the mean component becomes

∂⟨B ⟩ ′ ′ -∂t--= ∇ × (⟨u ⟩ × ⟨B ⟩ + ⟨u × B ⟩ − η ∇ × ⟨B ⟩), (13 )
which is identical to the original MHD induction Equation (1View Equation) except for the term ′ ′ ⟨u × B ⟩, which corresponds to a mean electromotive force ℰ induced by the fluctuating flow and field components. It appears here because, in general, the cross product u′ × B ′ usually will not vanish upon averaging, even though u′ and B ′ do so individually. Evidently, this procedure is meaningful if a separation of spatial and/or temporal scales exists between the (time-dependent) turbulent motions and associated small-scale magnetic fields on the one hand, and the (quasi-steady) large-scale axisymmetric flow and field on the other.

The reader versed in fluid dynamics will have recognized in the mean electromotive force the equivalent of Reynolds stresses appearing in mean-field versions of the Navier–Stokes equations, and will have anticipated that the next (crucial!) step is to express ℰ in terms of the mean field ⟨B ⟩ in order to achieve closure. This is usually carried out by expressing ℰ as a truncated series expansion in ⟨B ⟩ and its derivatives. Retaining the first two terms yields

ℰ = α : ⟨B ⟩ + β : ∇ × ⟨B ⟩. (14 )
where the colon indicates a tensorial inner product. The quantities α and β are in general pseudo-tensors, and specification of their components requires a turbulence model from which averages of velocity cross-correlations can be computed, which is no trivial task. We defer discussion of specific model formulations for these quantities to Section 4.2, but note the following:

The production of a mean electromotive force proportional to the mean field is called the α-effect, and it can as a source of both A and B, and thus offers a viable T → P mechanism. Its existence was first demonstrated in the context of turbulent MHD, but it also arises in other contexts, as discussed immediately below. Although this is arguably a bit of a physical abuse, the term “α-effect” is used in what follows to denote any mechanism producing a mean poloidal field from a mean toroidal field, as is almost universally (and perhaps unfortunately) done in the contemporary solar dynamo literature.

Other forms of turbulent mean electromotive forces are possible when the large-scale magnetic field develops variations on scales comparable to that of large-scale flows, notably angular velocity shears (see Rädler et al., 2003; Pipin and Seehafer, 2009, and references therein). This can lead to the appearance of an additional contribution on the RHS of Equation (14View Equation), of the general form δ × (∇ × ⟨B ⟩). Such a mean-field-aligned emf cannot contribute to the sustenance of ⟨B ⟩, but operating concurently with other inductive mechanisms, can in principle contribute to dynamo action.

3.2.2 Hydrodynamical shear instabilities

The tachocline is the rotational shear layer uncovered by helioseismology immediately beneath the Sun’s convective envelope, providing smooth matching between the latitudinal differential rotation of the envelope, and the rigidly rotating radiative core (see, e.g., Spiegel and Zahn, 1992; Brown et al., 1989; Tomczyk et al., 1995; Charbonneau et al., 1999, and references therein). Stability analyses of the latitudinal shear within the tachocline carried out in the framework of shallow-water theory suggest that the latitudinal shear can become unstable when vertical fluid displacement is allowed (Dikpati and Gilman, 2001Jump To The Next Citation Point). These authors also find that vertical fluid displacements correlate with the horizontal vorticity pattern in a manner resulting in a net kinetic helicity that can, in principle, impart a systematic twist to an ambient mean toroidal field. This can thus serve as a source for the poloidal component, and, in conjunction with rotational shearing of the poloidal field, lead to cyclic dynamo action. This is a self-excited T → P mechanism, but it is not entirely clear at this juncture if (and how) it would operate in the strong-field regime (more on this in Section 4.5 below).

3.2.3 MHD instabilities

It has now been demonstrated, perhaps even beyond reasonable doubt, that the toroidal magnetic flux ropes that upon emergence in the photosphere give rise to sunspots can only be stored below the Sun’s convective envelope, more specifically in the thin, weakly subadiabatic overshoot layer conjectured to exist immediately beneath the core-envelope interface (see, e.g., Schüssler, 1996; Schüssler and Ferriz-Mas, 2003; Fan, 2009Jump To The Next Citation Point, and references therein). Only there are growth rates for the magnetic buoyancy instability sufficiently long to allow field amplification, while being sufficiently short for flux emergence to take place on time-scales commensurate with the solar cycle (Ferriz-Mas et al., 1994Jump To The Next Citation Point). These stability studies have also revealed the existence of regions of weak instability, in the sense that the growth rates are numbered in years. The developing instability is then strongly influenced by the Coriolis force, and develops in the form of growing helical waves travelling along the flux rope’s axis. This amounts to twisting a toroidal field in meridional planes, as with the Parker scenario, with the important difference that what is now being twisted is a flux rope rather than an individual fieldline. Nonetheless, an azimuthal electromotive force is produced. This represents a viable T → P mechanism, but one that can only act above a certain field strength threshold; in other words, dynamos relying on this mechanism are not self-excited, since they require strong fields to operate. On the other hand, they operate without difficulties in the strong field regime.

Another related class of poloidal field regeneration mechanism is associated with the buoyant breakup of the magnetized layer (Matthews et al., 1995). Once again it is the Coriolis force that ends up imparting a net twist to the rising, arching structures that are produced in the course of the instability’s development (see Thelen, 2000aJump To The Next Citation Point, and references therein). This results in a mean electromotive force that peaks where the magnetic field strength varies most rapidly with height. This could provide yet another form of tachocline α-effect, again subjected to a lower operating threshold. MHD versions of the hydrodynamical shear instability discussed in Section 3.2.2 have also been studied (see, e.g., Arlt et al., 2007bJump To The Next Citation Point; Cally et al., 2008; Dikpati et al., 2009, and references therein), but the fundamentally nonlinear nature of the flow-field interaction makes it difficult to construct physically credible poloidal source terms to be incoporated into dynamo models.

3.2.4 The Babcock–Leighton mechanism

The larger sunspot pairs (“bipolar magnetic regions”, hereafter BMR) often emerge with a systematic tilt with respect to the E-W direction, in that the leading sunspot (with respect to the direction of solar rotation) is located at a lower latitude than the trailing sunspot, the more so the higher the latitude of the emerging BMR. This pattern, known as “Joy’s law”, is caused by the action of the Coriolis force on the secondary azimuthal flow that develops within the buoyantly rising magnetic toroidal flux rope that, upon emergence, produces a BMR (see, e.g. Fan et al., 1993Jump To The Next Citation Point; D’Silva and Choudhuri, 1993; Caligari et al., 1995Jump To The Next Citation Point). In conjunction with the antisymmetry of the toroidal field giving rise to sunspots evidenced by Hale’s sunspot laws, this tilt is at the heart of the Babcock–Leighton mechanism for polar field reversal, as outlined in cartoon form in Figure 2View Image.

Physically, what happens is that the leading spot of the BMR is located closer to the equator, and therefore experiences greater diffusive cancellation across the equatorial plane with the opposite polarity leading spots of the other hemisphere, than the trailing spots do. Upon decay, the latter’s magnetic flux is preferentially transported to the polar region by supergranular diffusion and the surface meridional flow. The net effect is to take a formerly toroidal magnetic field and convert a fraction of its associated flux into a net dipole moment, i.e., it represents a T → P mechanism. With the polar cap flux amounting to less than 0.1% of the unsigned magnetic flux emerging in active regions throughout a cycle, the efficiency of this so-called Babcock–Leighton mechanism needs not be very high. Here again the resulting dynamos are not self-excited, as the required tilt of the emerging BMR only materializes in a range of toroidal field strength going from a few 104 G to about 2 × 105 G.


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