Dikpati and Gilman (2001) work with what are effectively the mean field dynamo equations including meridional circulation. They design their “tachocline effect” in the form of a latitudinal parameterization of the longitudinallyaveraged 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 shallowwater theory. In analogy with meanfield theory, the resulting effect is assumed to be proportional to kinetic helicity but of opposite sign (see Equation (21)), and so is here predominantly positive at midlatitudes in the Northern solar hemisphere. In their dynamo model, Dikpati and Gilman (2001) use a solarlike differential rotation, depthdependent magnetic diffusivity and meridional circulation pattern much similar to those shown on Figures 5, 6, and 11 herein, and the usual ad hoc quenching formula (cf. Equation (24)) is introduced as the sole amplitudelimiting nonlinearity.

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 timelatitude diagrams of the toroidal field at the coreenvelope interface, and surface radial field. This is a solarlike solution with a midlatitude surface meridional (poleward) flow speed of , envelope diffusivity , and a coretoenvelope magnetic diffusivity contrast .

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, solarlike 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 selfexcited. Like conventional 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 dynamogenerated magnetic field at least up to equipartition strength, a potentially serious difficulty also shared by the BabcockLeighton models discussed in Section 4.8 below.
The applicability of shallowwater 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 timedependent latitudinal distribution, and (ii) be intricately dependent on , in a manner that is unlikely to be reproduced by a simple amplitudelimiting quenching formula such as Equation (24). Linear calculations carried out to date in the framework of shallowwater MHD indicate that the purely hydrodynamical 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 nonlinear 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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