4.5 Models based on shear instabilities

We now turn to a recently proposed class of dynamo models relying on the latitudinal shear instability of the angular velocity profiles in the upper radiative portion of the solar tachocline (Dikpati and Gilman, 2001

). Although the study of dynamo action in this context has barely begun, results published so far (see Dikpati and Gilman, 2001

; Dikpati et al., 2004

) make this class of models worthy of further consideration.

4.5.1 From instability to $a$ -effect

Dikpati and Gilman (2001 ) work with what are effectively the mean field $a_O_$ dynamo equations including meridional circulation. They design their “tachocline $a$ -effect” in the form of a latitudinal parameterization of the longitudinally-averaged kinetic helicity associated with the planforms they obtain from a linear hydrodynamical stability analysis of the latitudinal differential rotation in the part of the tachocline coinciding with the overshoot region (see Dikpati and Gilman, 2001 ). Figure 14 shows some typical latitudinal profiles of kinetic helicity for various model parameter settings and azimuthal wavenumbers, all computed in the framework of shallow-water theory. In analogy with mean-field theory, the resulting $a$ -effect is assumed to be proportional to kinetic helicity but of opposite sign (see Equation (21 )), and so is here predominantly positive at mid-latitudes in the Northern solar hemisphere. In their dynamo model, Dikpati and Gilman (2001 ) use a solar-like differential rotation, depth-dependent magnetic diffusivity and meridional circulation pattern much similar to those shown on Figures 5 , 6 , and 11 herein, and the usual ad hoc $a$ -quenching formula (cf. Equation (24 )) is introduced as the sole amplitude-limiting nonlinearity.

Figure 14:

A sample of longitudinally-averaged kinetic helicity profiles associated with the linearly unstable horizontal planforms of azimuthal order $m$ (as indicated) in the shallow-water model of Dikpati and Gilman (2001

). The parameters $s2$ and $s4$ control the form of the latitudinal differential rotation, and are equivalent to the parameters $a2$ and $a4$ in Equation (18

) herein. The parameter $G$ is a measure of stratification in the shallow-water model, with larger values of $G$ corresponding to stronger stratification (and thus a stronger restoring buoyancy force); $G -~ 0.1$ is equivalent to a subadiabaticity of $- 4 ~ 10$ (diagram kindly provided by M. Dikpati).

4.5.2 Representative solutions

Many representative solutions for this class of dynamo models can be examined in Dikpati and Gilman (2001 ) and Dikpati et al. (2004 ), where their properties are discussed at some length. Figure 15 shows time-latitude diagrams of the toroidal field at the core-envelope interface, and surface radial field. This is a solar-like solution with a mid-latitude surface meridional (poleward) flow speed of $17 m s-1$ , envelope diffusivity $jT = 5 × 1011 cm2 s-1$ , and a core-to-envelope magnetic diffusivity contrast $Dj = 10 -3$ .

Figure 15:

Time-latitude “butterfly” diagrams of the toroidal field at the core-envelope interface (left), and surface radial field (right) for a representative dynamo solution computed using the model of Dikpati and Gilman (2001

). Note how the deep toroidal field peaks at very low latitudes, in good agreement with the sunspot butterfly diagram. For this solution the equatorial deep toroidal field and polar surface radial field lag each other by $~ p$ , but other parameter settings can bring this lag closer to the observed $p/2$ (diagrams kindly provided by M. Dikpati).

Note the equatorward migration of the deep toroidal field, set here by the meridional flow in the deep envelope, and the poleward migration and intensification of the surface poloidal field, again a direct consequence of advection by meridional circulation, as in the mean-field dynamo models discussed in Section 4.4 in the advection-dominated, high $Rm$ regime. The three-lobe structure of each spatio-temporal cycle in the butterfly diagram reflects the presence of three peaks in the latitudinal profile of kinetic helicity (see Figure 14

4.5.3 Critical assessment

While these models are only a recent addition to the current “zoo” of solar dynamo models, they have been found to compare favorably to a number of observed solar cycle features. In many cases they yield equatorward propagating dominant activity belts, solar-like cycle periods, and correct phasing between the surface polar field and the tachocline toroidal field. These features can be traced primarily to the advective action of the meridional flow. They also yield the correct solution parity, and are self-excited. Like conventional $a_O_$ models relying on meridional circulation to set the propagation direction of dynamo waves (see Section 4.4.2), the meridional flow must remain unaffected by the dynamo-generated magnetic field at least up to equipartition strength, a potentially serious difficulty also shared by the Babcock-Leighton models discussed in Section 4.8 below.

The applicability of shallow-water theory to the solar tachocline notwithstanding, the primary weakness of these models, in their present form, is their reliance on a linear stability analysis that altogether ignores the destabilizing effect of magnetic fields. Gilman and Fox (1997 ) have demonstrated that the presence of even a weak toroidal field in the tachocline can very efficiently destabilize a latitudinal shear profile that is otherwise hydrodynamically stable (see also Zhang et al., 2003b ). Relying on a purely hydrodynamical stability analysis is then hard to reconcile with a dynamo process producing strong toroidal field bands of alternating polarities migrating towards the equator in the course of the cycle, especially since latitudinally concentrated toroidal fields have been found to be unstable over a very wide range of toroidal field strengths (see Dikpati and Gilman, 1999 ). In the MHD version of the shear instability studied by P. Gilman and collaborators, the structure of the instability planforms is highly dependent on the assumed underlying toroidal field profile, so that the kinetic helicity can be expected to (i) have a time-dependent latitudinal distribution, and (ii) be intricately dependent on $<B >$ , in a manner that is unlikely to be reproduced by a simple amplitude-limiting quenching formula such as Equation (24 ). Linear calculations carried out to date in the framework of shallow-water MHD indicate that the purely hydrodynamical $a$ -effect considered here is indeed strongly affected by the presence of an unstable toroidal field (see Dikpati et al., 2003 ). However, progress has been made in studying non-linear development of both the hydrodynamical and MHD versions of the shear instability (see, e.g., Cally, 2001 ; Cally et al., 2003 ), so that the needed improvements on the dynamo front are hopefully forthcoming.