Schüssler (1981) and Yoshimura (1981) modeled the torsional oscillation as a result of the Lorentz force due to dynamo waves; according to the latter paper, the phenomenon would be important only close to the surface, and would have only equatorward, not poleward, moving bands. LaBonte and Howard (1982) objected to the Yoshimura model on the grounds that it would predict a strong correlation between the strength of the surface magnetic field and that of the velocity signal, which did not seem to be the case in the observations.
Küker et al. (1996) used a different mechanism to generate the torsional oscillation signal in their model, considering it as the response of the Reynolds stress on the time-dependent dynamo magnetic field rather than a direct effect of the large-scale Lorentz force. This model gave a very weak poleward branch for the torsional oscillation signal.
Once the flows had been shown observationally to penetrate well below the surface, Durney (2000) suggested that, “the pattern of torsional oscillations appear to have the potential of critically discriminating between different dynamo models as, e.g., the Babcock–Leighton and interface models.”
Covas et al. (2000) used a model in which the observed rotation profile was imposed and the rotation variations arises from the action of the Lorentz force of the dynamo-generated magnetic field on the angular velocity. They were able to simulate approximately solar-like patterns of zonal flow bands and magnetic activity. In subsequent papers they focused on the the possibility of so-called “spatio-temporal fragmentation” allowing cycles of different periods in different regions, and in calculations with no density stratification in the convection zone they found this to be feasible (Covas et al., 2001a). The effect was not too sensitive to uncertainties in the rotation law (Covas et al., 2001b, 2002), and somewhat sensitive to the boundary conditions at the outer surface (Tavakol et al., 2002). Adding density stratification (Covas et al., 2004) did not substantially change the results, though the amplitude of the oscillations in the deeper layers of the convection zone did decrease as the density gradient increased. However, they did find that introducing quite a small amount of -quenching (magnetic feedback on turbulent convection) would suppress the torsional oscillation effect.
Spruit (2003) modeled the torsional oscillation pattern as a “geostrophic flow” driven by temperature variations near the surface associated with magnetic activity, and therefore having its greatest amplitude at the surface and falling to 1/3 of its surface value at . This model also accounts for the observed inflows into the activity belts. There are some problems in reconciling this model with the observations; it is difficult to see how the observed depth-dependent phase pattern could arise from a surface-originated cause, and the existence of the flows even at epochs where there are no active regions is also hard to explain, though Spruit suggested that the flows might be produced by unobserved small-scale and short-lived magnetic regions.
Rempel (2007) used a mean-field flux-transport dynamo model, with a model-derived differential rotation profile and meridional flow, to investigate the effects of various driving mechanisms for the torsional oscillation. The author concluded that the poleward-propagating branch of the pattern could be explained by a periodic forcing at mid-latitudes without any underlying migration of buried polar field. On the other hand, in this type of model the observed equatorward-propagating branch could not be reproduced without adding a thermal forcing after the manner of the Spruit (2003) model. Howe et al. (2006b) compared such a model with the observations, and found it not to be completely consistent with the observed interior behavior of the flows at lower latitudes.
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