4.3 Explicit models

The current buzz in the field of model-based solar cycle prediction was started by the work of the solar dynamo group in Boulder (Dikpati et al., 2006Dikpati and Gilman, 2006Jump To The Next Citation Point). Their model is a flux transport dynamo, advection-dominated to the extreme. The strong suppression of diffusive effects is assured by the very low value (less than 20 km2 s–1) assumed for the turbulent magnetic diffusivity in the bulk of the convective zone. As a result, the poloidal fields generated near the surface by the Babcock–Leighton mechanism are only transported to the tachocline on the very long, decadal time scale of meridional circulation. The strong toroidal flux residing in the low-latitude tachocline, producing solar activity in a given cycle is thus the product of the shear amplification of poloidal fields formed near the surface about 2 – 3 solar cycles earlier, i.e., the model has a “memory” extending to several cycles. The mechanism responsible for cycle-to-cycle variation is assumed to be the stochastic nature of the flux emergence process. In order to represent this variability realistically, the model drops the surface α-term completely (a separate, smaller α term is retained in the tachocline); instead, the generation of poloidal field near the surface is represented by a source term, the amplitude of which is based on the sunspot record, while its detailed functional form remains fixed.

Dikpati and Gilman (2006Jump To The Next Citation Point) find that, starting off their calculation by fixing the source term amplitudes of sunspot cycles 12 to 15, they can predict the amplitudes of each subsequent cycle with a reasonable accuracy, provided that the relation between the relative sunspot numbers and the toroidal flux in the tachocline is linear, and that the observed amplitudes of all previous cycles are incorporated in the source term for the prediction of any given cycle. For the upcoming cycle 24 the model predicts peak smoothed annual relative sunspot numbers of 150 or more. Elaborating on their model, they proceeded to apply it separately to the northern and southern hemispheres, to find that the model can also be used to correctly forecast the hemispheric asymmetry of solar activity (Dikpati et al., 2007).

The extraordinary claims of this pioneering research have prompted a hot debate in the dynamo community. Besides the more general, fundamental doubt regarding the feasibility of model-based predictions (see Section 3.2 above), more technical concerns arose, to be discussed below.

Another flux transport dynamo code, the Surya code, originally developed by A.  Choudhuri and coworkers in Bangalore, has also been utilized for prediction purposes. The crucial difference between the two models is in the value of the turbulent diffusivity assumed in the convective zone: in the Bangalore model this value is 240 km2 s–1, 1 – 2 orders of magnitude higher than in the Boulder model, and within the physically plausible range (Chatterjee et al., 2004). As a result of the shorter diffusive timescale, the model has a shorter memory, not exceeding one solar cycle. As a consequence of this relatively rapid diffusive communication between surface and tachocline, the poloidal fields forming near the surface at low latitudes due to the Babcock–Leighton mechanism diffuse down to the tachocline in about the same time as they reach the poles due to the advection by the meridional circulation. In these models, then, polar magnetic fields are not a true physical precursor of the low-latitude toroidal flux, and their correlation is just due to their common source. In the version of the code adapted for cycle prediction (Choudhuri et al., 2007Jump To The Next Citation PointJiang et al., 2007Jump To The Next Citation Point), the “surface” poloidal field (i.e., the poloidal field throughout the outer half of the convection zone) is rescaled at each minimum by a factor reflecting the observed amplitude of the Sun’s dipole field. The model shows reasonable predictive skill for the last three cycles for which data are available, and can even tackle hemispheric asymmetry (Goel and Choudhuri, 2009). For cycle 24, the predicted amplitude is 30 – 35% lower than cycle 23.

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