The -quenching expression (24) used in the preceding section amounts to saying that dynamo action saturates once the mean, dynamo-generated field reaches an energy density comparable to that of the driving turbulent fluid motions, i.e., , where is the turbulent velocity amplitude. This appears eminently sensible, since from that point on a toroidal fieldline would have sufficient tension to resist deformation by cyclonic turbulence, and so could no longer feed the -effect. At the base of the solar convective envelope, one finds , for , according to standard mixing length theory of convection. However, various calculations and numerical simulations have indicated that long before the mean field reaches this strength, the helical turbulence reaches equipartition with the small-scale, turbulent component of the magnetic field (e.g., Cattaneo and Hughes, 1996, and references therein). Such calculations also indicate that the ratio between the small-scale and mean magnetic components should itself scale as , where is a magnetic Reynolds number based on the microscopic magnetic diffusivity. This then leads to the alternate quenching expressionstrong -quenching or catastrophic quenching. Since in the solar convection zone, this leads to quenching of the -effect for very low amplitudes for the mean magnetic field, of order . Even though significant field amplification is likely in the formation of a toroidal flux rope from the dynamo-generated magnetic field, we are now a very long way from the demanded by simulations of buoyantly rising flux ropes (see Fan, 2004).
A way out of this difficulty was proposed by Parker (1993), in the form of interface dynamos. The idea is beautifully simple: If the toroidal field quenches the -effect, amplify and store the toroidal field away from where the -effect is operating! Parker showed that in a situation where a radial shear and -effect are segregated on either side of a discontinuity in magnetic diffusivity (taken to coincide with the core-envelope interface, see Panel A of Figure 9), the dynamo equations support solutions in the form of travelling surface waves localized on the discontinuity in diffusivity. The key aspect of Parker’s solution is that for supercritical dynamo waves, the ratio of peak toroidal field strength on either side of the discontinuity surface is found to scale with the diffusivity ratio aset al. (2004), see also Panels B and C of Figure 9), and, while differing in important details, nonetheless confirmed Parker’s overall picture.
The next obvious step is to construct an interface dynamo in spherical geometry, using a solar-like differential rotation profile. This was undertaken by Charbonneau and MacGregor (1997). Unfortunately, the numerical technique used to handle the discontinuous variation in at the core-envelope interface turned out to be physically erroneous for the vector potential describing the poloidal field8 (see Markiel and Thomas, 1999, for a discussion of this point), which led to spurious dynamo action in some parameter regimes. The matching problem is best avoided by using a continuous but rapidly varying diffusivity profile at the core-envelope interface, with the -effect concentrated at the base of the envelope, and the radial shear immediately below, but without significant overlap between these two source regions (see Panel B of Figure 10). Such numerical models can be constructed as a variation on the models considered earlier, and stand somewhere between Parker’s original picture (see Panel A of Figure 9) and the models with spatially localized -effect and shear (see Panels B and C of Figure 9). In conjunction with a solar-like differential rotation profile, making a working interface dynamo model is markedly trickier than if only a radial shear is operating, as in the Cartesian models discussed earlier (see Charbonneau and MacGregor, 1997; Markiel and Thomas, 1999). Panel A of Figure 10 shows a butterfly diagram for a numerical interface solution with , , and a core-to-envelope diffusivity contrast . A magnetic energy time series for this solution is plotted in Panel D of Figure 8, together with a solution with a smaller diffusivity contrast (see Panel C of Figure 8). The poleward propagating equatorial branch is precisely what one would expect from the combination of positive radial shear and positive -effect according to the Parker-Yoshimura sign rule9. Here the -effect is (artificially) concentrated towards the equator, by imposing a latitudinal dependency for , and zero otherwise.
Zhang et al. (2003a) have presented results for a fully three-dimensional interface dynamo model where, however, dynamo solutions remain largely axisymmetric when a strong shear is present in the tachocline. They use an -effect spanning the whole convective envelope radially, but concentrated latitudinally near the equator, a core-to-envelope magnetic diffusivity contrast , and the usual algebraic -quenching formula. Unfortunately, their differential rotation profile is non-solar. However, they do find that the dynamo solutions they obtain are robust with respect to small changes in the model parameters. The next obvious step here is to repeat the calculations with a solar-like differential rotation profile.
So far the great success of interface dynamos remains their ability to evade -quenching even in its “strong” formulation, and so produce equipartition or perhaps even super-equipartition mean toroidal magnetic fields immediately beneath the core-envelope interface. They represent the only variety of dynamo models formally based on mean-field electrodynamics that can achieve this without additional physical effects introduced into the model. All of the uncertainties regarding the calculations of the -effect and magnetic diffusivity carry over from to interface models, with diffusivity quenching becoming a particularly sensitive issue in the latter class of models (see, e.g., Tobias, 1996a).
Interface dynamos suffer acutely from something that is sometimes termed “structural fragility”. Many gross aspects of the model’s dynamo behavior often end up depending sensitively on what one would normally hope to be minor details of the model’s formulation. For example, the interface solutions of Figure 10 are found to behave very differently if either
Compare also the behavior of the solutions discussed here to those discussed in Markiel and Thomas (1999). Once again the culprit is the latitudinal shear. Each of these minor variations on the same basic model has the effect that a parallel mid-latitude dynamo mode, powered by the latitudinal shear within the tachocline and envelope, interferes with and/or overpowers the interface mode. This interpretation is not inconsistent with the robustness claimed by Zhang et al. (2003a), since these authors have chosen to omit the latitudinal shear throughout the convective envelope in their model. Because of this structural sensitivity, interface dynamo solutions also end up being annoyingly sensitive to choice of time-step size, spatial resolution, and other purely numerical details. From a modelling point of view, interface dynamos lack robustness.
© Max Planck Society and the author(s)