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 (11

). 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 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 of the influence of Coriolis forces on turbulent fluid motions. 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, $$ and $$ , and small-scale fluctuating components $' u$ , $' B$ . This is not a linearization procedure, in that we are not assuming that $' |u |/ ||« 1$ or $' |B |/||« 1$ . 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 $$ 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

$@ ' ' -@t-- = \~/ × (× + - j \~/ × ), (13)$

which is identical to the original MHD induction Equation (1

) except for the term $' ' $ , 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 $u'× B'$ will not necessarily 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 $$ in order to achieve closure. This is usually carried out by expressing $E$ as a truncated series expansion in $$ and its derivatives. Retaining the first two terms yields

$E = a : + b : \~/ × . (14)$

where the colon indicates an inner product. The quantities $a$ and $b$ 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:

Even if $$ is axisymmetric, the $a$ -term in Equation (14 ) will effectively introduce source terms in both the $A$ and $B$ equations, so that Cowling’s theorem can be circumvented.
Parker’s idea of helical twisting of toroidal fieldlines by the Coriolis force corresponds to a specific functional form for $a$ , and so finds formal quantitative expression in mean-field electrodynamics.

The production of a mean electromotive force proportional to the mean field is called the $a$ -effect. 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 “ $a$ -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.

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 that 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., 1999a , and references therein). While the latitudinal shear in the tachocline has been shown to be hydrodynamically stable with respect to Rayleigh-type 2D instabilities on a spherical shell (see, e.g., Charbonneau et al., 1999b , and references therein), further analysis allowing vertical displacement in the framework of shallow-water theory suggests that the latitudinal shear can become unstable when vertical fluid displacement is allowed (Dikpati and Gilman, 2001 ). 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). From the modelling point of view, it amounts to a mean-field-like $a$ -effect confined to the uppermost radiative portion of the solar tachocline, and peaking at mid-latitudes.

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, 2004 , 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., 1994 ). 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 thus 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, 2000a , 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 $a$ -effect, again subjected to a lower operating threshold.

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., 1993 ; D’Silva and Choudhuri, 1993 ; Caligari et al., 1995 ). This tilt is at the heart of the Babcock-Leighton mechanism for polar field reversal, as outlined in cartoon form in Figure 2 .

Mathematically, the Babcock-Leighton mechanism can be understood in the following manner; the surface distribution of radial magnetic field associated with a BMR ( $B0r(h,f)$ , say) can, as with any continuous function defined on a sphere, be decomposed into spherical harmonics:

Now at high $Rm$ , under the joint action of differential rotation and magnetic dissipation, all non-axisymmetric (i.e., $m /= 0$ ) terms will be destroyed on a timescale much faster than diffusive, a process known as axisymmetrization that is the spherical equivalent of the well-known process of magnetic flux expulsion by closed circulatory flows. Therefore, after some time the surface radial field will assume the form

where the $Pl(cos h)$ are the Legendre polynomials. Equation (16

) now describes an axisymmetric poloidal field. Since the BMR field was oriented in the toroidal direction prior to destabilization, rise, and emergence, the net effect is to produce a poloidal field out of a toroidal field, thus offering a viable $T --> P$ mechanism. 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 $4 10 G$ to about $5 2 × 10 G$ .