4.5 Models based on shear instabilities

We now turn to a recently proposed class of flux transport 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, 2001Jump To The Next Citation Point; Dikpati et al., 2004Jump To The Next Citation Point). These authors work with what are effectively the mean field α Ω dynamo equations including meridional circulation. They design their “tachocline α-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. The analysis is carried out in the framework of shallow-water theory (see Dikpati and Gilman, 2001Jump To The Next Citation Point). In analogy with mean-field theory, the resulting α-effect is assumed to be proportional to kinetic helicity but of opposite sign (see Equation (19View Equation)), and ends up predominantly positive at mid-latitudes in the Northern solar hemisphere. In their dynamo model, Dikpati and Gilman (2001Jump To The Next Citation Point) use a solar-like differential rotation, depth-dependent magnetic diffusivity and meridional circulation pattern much similar to those shown in Figures 5View Image, 6View Image, and 10View Image herein. The usual ad hoc α-quenching formula (cf. Equation (23View Equation)) is introduced as the sole amplitude-limiting nonlinearity.

4.5.1 Representative solutions

Many representative solutions for this class of dynamo models can be examined in Dikpati and Gilman (2001Jump To The Next Citation Point) and Dikpati et al. (2004Jump To The Next Citation Point), where their properties are discussed at some length. Figure 14View Image 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 ηT = 5 × 1011 cm2 s−1, and a core-to-envelope magnetic diffusivity contrast Δ η = 10− 3. 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, when operating 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 for this model.

View Image

Figure 14: Time-latitude “butterfly” diagrams of the toroidal field at the core-envelope interface (top), and surface radial field (bottom) for a representative dynamo solution computed using the model of Dikpati and Gilman (2001Jump To The Next Citation Point). 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 ∼ π, but other parameter settings can bring this lag closer to the observed π ∕2 (diagrams kindly provided by M. Dikpati).

4.5.2 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. The model can be adjusted to 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 α Ω 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 to be discussed in Section 4.8 below.

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). Achieving dynamo saturation through a simple amplitude-limiting quenching formula such as Equation (23View Equation) is then also hard to justify. 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.

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