4.7 Models based on flux tube instabilities

4.7.1 From instability to $a$ -effect

To date, stability studies of toroidal flux ropes stored in the overshoot layer have been carried out in the framework of the thin-flux tube approximation (Spruit, 1981 ). It is possible to construct “stability diagrams” taking the form of growth rate contours in a parameter space comprised of flux tube strength, latitudinal location, depth in the overshoot layer, etc. One such diagram, taken from Ferriz-Mas et al. (1994 ), is reproduced in Figure 16 . The key is now to identify regions in such stability diagrams where weak instability arises (growth rates $>~~ 1 yr$ ). In the case shown in Figure 16 , these regions are restricted to flux tube strengths in the approximate range $60 -150 kG$ . The correlation between the flow and field perturbations is such as to yield a mean azimuthal electromotive force equivalent to a positive $a$ -effect in the N-hemisphere (Ferriz-Mas et al., 1994 ; Brandenburg and Schmitt, 1998 ).

Figure 16:

Stability diagram for toroidal magnetic flux tubes located in the overshoot layer immediately beneath the core-envelope interface. The plot shows contours of growth rates in the latitude-field strength plane. The gray scale encodes the azimuthal wavenumber of the mode with largest growth rate, and regions left in white are stable. Dynamo action is associated with the regions with growth rates $~ 1 yr$ , here labeled I and II (diagram kindly provided by A. Ferriz-Mas).

4.7.2 Representative solutions

Dynamo models relying on the non-axisymmetric buoyant instability of toroidal magnetic fields were first proposed by Schmitt (1987 ), and further developed by Ferriz-Mas et al. (1994 ); Schmitt et al. (1996 ), and Ossendrijver (2000a ) for the case of toroidal flux tubes. These dynamo models are all mean-field-like, in that the mean azimuthal electromotive force arising from instability of the flux tubes is parametrized as an $a$ -effect, and the dynamo equations solved are then the same as those of the conventional $a_O_$ mean-field model (see Section 4.2.2), including various forms of algebraic $a$ -quenching as the sole amplitude-limiting nonlinearity. As with mean-field models, the dynamo period presumably depends sensitively on the assumed value of (turbulent) magnetic diffusivity, and equatorward propagation of the dynamo wave requires a negative $a$ -effect at low latitudes.

4.7.3 Critical assessment

Although it has not yet been comprehensively studied, this dynamo mechanism has a number of very attractive properties. It operates without difficulty in the strong field regime (in fact it requires strong fields to operate). It also naturally yields dynamo action concentrated at low latitudes, so that a solar-like butterfly diagram can be readily produced from a negative $a$ -effect even with a solar-like differential rotation profile, at least judging from the solutions presented in Schmitt et al. (1996 ) and Ossendrijver (2000a ,b ).

Difficulties include the need of a relatively finely tuned magnetic diffusivity to achieve a solar-like dynamo period, and a relatively finely-tuned level of subadiabaticity in the overshoot layer for the instability to kick on and off at the appropriate toroidal field strengths (compare Figures 1 and 2 in Ferriz-Mas et al., 1994 ). The non-linear saturation of the instability is probably less of an issue here than with the $a$ -effect based on purely hydrodynamical shear instability (see Section 4.5 above), since, as the instability grows, the flux ropes leave the site of dynamo action by entering the convection zone and buoyantly rising to the surface.

The effects of meridional circulation in this class of dynamo models has yet to be investigated; this should be particularly interesting, since both analytic calculations and numerical simulations suggest a positive $a$ -effect in the Northern hemisphere, which should then produce poleward propagation of the dynamo wave at low latitude. Meridional circulation could then perhaps produce equatorward propagation of the dynamo magnetic field even with a positive $a$ -effect, as it does in true mean-field models (cf. Section 4.4). At any rate, further studies of dynamo models relying on this poloidal field regeneration mechanism should be vigorously pursued.