4.4 Truncated models

The “illustrative” nature of solar dynamo models is nowhere more clearly on display than in truncated or reduced models where some or all of the detailed spatial structure of the system is completely disregarded, and only temporal variations are explicitly considered. This is sometimes rationalised as a truncation or spatial integration of the equations of a more realistic inhomogenous system; in other cases, no such rationalisation is provided, representing the solar dynamo by an infinite, homogeneous or periodic turbulent medium where the amplitude of the periodic large-scale magnetic field varies with time only.

In the present subsection we deal with models that do keep one spatial variable (typically, the latitude), so growing wave solutions are still possible – these models, then, are still dynamos even though their spatial structure is not in a good correspondence with that of the solar dynamo.

This approach in fact goes back to the classic migratory dynamo model of Parker (1955) who radially truncated (i.e., integrated) his equations to simplify the problem. Parker seems to have been the first to employ a heuristic relaxation term of the form − Br ∕τd in the poloidal field equation to represent the effect of radial diffusion; here, τd = d2∕ηT is the diffusive timescale across the thickness d of the convective zone. His model was recently generalized by Moss et al. (2008) and Usoskin et al. (2009b) to the case when the α-effect includes an additive stochastic noise, and nonlinear saturation of the dynamo is achieved by α-quenching. These authors do not make an attempt to predict solar activity with their model but they can reasonably well reproduce some features of the very long term solar activity record, as seen from cosmogenic isotope studies.

Another radially truncated model, this time formulated in a Cartesian system, is that of Kitiashvili and Kosovichev (2009). In this model stochastic effects are not considered and, in addition to using an α-quenching recipe, further nonlinearity is introduced by coupling in the Kleeorin–Ruzmaikin equation (Zel’dovich et al., 1983) governing the evolution of magnetic helicity, which in the hydromagnetic case contributes to α. Converting the toroidal field strength to relative sunspot number using the Bracewell transform, Equation (3View Equation), the solutions reproduce the asymmetric profile of the sunspot number cycle. For sufficiently high dynamo numbers the solutions become chaotic, cycle amplitudes show an irregular variation. Cycle amplitudes and minimum–maximum time delays are found to be related in a way reminiscent of the Waldmeier relation.

Building on these results, Kitiashvili and Kosovichev (2008Jump To The Next Citation Point) attempt to predict solar cycles using a data assimilation method. The approach used is the so-called Ensemble Kalman Filter method. Applying the model for a “postdiction” of the last 8 solar cycles yielded astonishingly good results, considering the truncated and arbitrary nature of the model and the fundamental obstacles in the way of reliable prediction discussed above. While the presently available brief preliminary publication leaves several details of the method unclear, the question may arise whether the actual physics of the model considered has any significant role in this prediction, or we are dealing with something like the phase space reconstruction approach discussed in Section 3.3 above where basically any model with an attractor that looks reasonably similar to that of the actual solar dynamo would do. Either way, the method is remarkable, and the prediction for cycle 24 of a maximal smoothed annual sunspot number of 80, to be reached in 2013, will be worth comparing to the actual value.

In order to understand the origin of the predictive skill of the Boulder model, Cameron and Schüssler (2007Jump To The Next Citation Point) studied a radially truncated version of the model, wherein only the equation for the radial field component is solved as a function of time and latitude. The equation includes a source term similar to the one used in the Boulder model. As the toroidal flux does not figure in this simple model, the authors use the transequatorial flux Φ as a proxy, arguing that this may be more closely linked to the amplitude of the toroidal field in the upcoming cycle than the polar field. They find that Φ indeed correlates quite well (correlation coefficients r ∼ 0.8 –0.9, depending on model details) with the amplitude of the next cycle, as long as the form of the latitude dependence of the source term is prescribed and only its amplitude is modulated with the observed sunspot number series (“idealized model”). But surprisingly, the predictive skill of the model is completely lost if the prescribed form of the source function is dropped and the actually observed latitude distribution of sunspots is used instead (“realistic model”). Cameron and Schüssler (2007Jump To The Next Citation Point) interpret this by pointing out that Φ is mainly determined by the amount of very low latitude flux emergence, which in turn occurs mainly in the last few years of the cycle in the idealized model, while it has a wider temporal distribution in the realistic model. The conclusion is that the root of the apparently good predictive skill of the truncated model (and, by inference, of the Boulder model it is purported to represent) is actually just the good empirical correlation between late-phase activity and the amplitude of the next cycle, discussed in Section 2.1 above. This correlation is implicitly “imported” into the idealized flux transport model by assuming that the late-phase activity is concentrated at low latitudes, and therefore gives rise to cross-equatorial flux which then serves as a seed for the toroidal field in the next cycle. So if Cameron and Schüssler (2007Jump To The Next Citation Point) are correct, the predictive skill of the Boulder model is due to an empirical precursor and is thus ultimately explained by the good old Waldmeier effect (cf. Section 1.3.3)

The fact that the truncated model of Cameron and Schüssler (2007Jump To The Next Citation Point) is not identical to the Boulder model obviously leaves room for doubt regarding this conclusion. In particular, the effective diffusivity represented by the sink term in the truncated model is ∼ km2 s–1, significantly higher than in the Boulder model; consequently, the truncated model will have a more limited memory, cf. Yeates et al. (2008). The argument that the cross-equatorial flux is a valid proxy of the amplitude of the next cycle may be correct in such a short-memory model with no radial structure, but it is dubious whether it remains valid for flux transport models in general. In an attempt to appreciate the importance of the cross-equatorial flux in their model, Dikpati et al. (2008a) find that while this flux does indeed correlate fairly well (r = 0.76) with the next cycle amplitude, the toroidal flux is a much better predictor (r = 0.96). At first sight this seems to make it unlikely that the former can explain the latter; however, part of the difference in the predictive skill may be due to the fact that Φ shows much more short-term variability than the toroidal flux.

In any case, the obvious way to address the concerns raised by Cameron and Schüssler (2007Jump To The Next Citation Point) and further by Schüssler (2007) in relation to the Boulder model would be to run that model with a modified source function incorporating the realistic latitudinal distribution of sunspots in each cycle. The results of such a test are not yet available at the time of writing this review.

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