This dynamo logic has recently been pushed further, by using dynamo models to actually advance in time measurements of the solar surface magnetic field in order to produce a cycle forecast. This approach is justified if the surface magnetic field is indeed a significant source of the poloidal field to be sheared into a toroidal component in the upcoming cycle, so that using this approach to forecasting already amounts to a strong assumption on the mode of solar dynamo action. In the stochastically-forced flux-transport dynamo solution of Figure 25, a strong correlation materializes between the peak polar field at cycle minimum, and amplitude of the subsequent cycle (see panel C). This occurs because in this model the surface polar field is advected down by the meridional flow to the dynamo source region at the base of the convection, and ends up feeding back into the dynamo loop. In other types of dynamo models where this feedback of the surface field does not occur, no such correlation materializes. For more on these matters see Charbonneau and Barlet (2010).
It is particularly instructive to compare and contrast the forecast schemes (and cycle 24 predictions) of Dikpati et al. 2006 (see also Dikpati and Gilman, 2006) and Choudhuri et al. 2007 (see also Jiang et al., 2007). Both groups use a dynamo model of the Babcock–Leighton variety (Section 4.8), in conjunction with input of solar magnetic field observations in a manner often (and incorrectly) described as “data assimilation”. The model parameters are adjusted to reproduce the known amplitudes of previous sunspot cycles, and the model is then integrated forward in time beyond this calibration interval to provide a forecast.
Table 1 details the various modelling components associated with each forecasting scheme. Both are remarkably similar, differing at the level of what one would usually consider modelling details, and do about as well at reproducing amplitude of past cycles over their respective calibration intervals. Yet, they end up producing cycle 24 amplitude forecasts that stand at opposite ends of the very wide range of cycle 24 forecasts produced by other techniques. A cycle 24 with SSN = 80 would place it amongst the weakest of the past century (cycles 14 and 16), while SSN = 180 would rank it on par with the two highest cycle amplitude on record (cycles 4 and 19; see Figure 22).
|Authors / Ref.||Dikpati et al. (2006)||Choudhuri et al. (2007)|
|Dynamo model||kinematic axisymmetric||kinematic axisymmetric|
|Magnetic diffusivity||Eq. (17),||Eq. (17),|
|plus high- surface layer|
|Differential rotation||solar-like parameterization||solar-like parameterization|
|Eqs. (15) – (16),||Eqs. (15) – (16),|
|all other parameters same||all other parameters same|
|Meridional circulation||single cell per quadrant||single cell per quadrant|
|closing at||closing at|
|Poloidal source term||data-driven surface forcing||subsurface -effect|
|plus weak tachocline -effect||plus buoyancy algorithm|
|Nonlinearity||algebraic -quenching||algebraic -quenching|
|only in tachocline -effect||in subsurface -effect|
|Solar data||time series of total sunspot area||DM Index|
|used to (continuously) drive||used to reset amplitude of|
|parametric surface forcing of||at “solar minimum”|
|Calibration interval||Cycles 16 – 23||Cycles 21 – 23|
|Cycle 24 forecast||SSN = 155 – 180||SSN = 80|
Much criticism has been leveled at these dynamo model-based cycle forecasting schemes, and sometimes unfairly so. To dismiss the whole idea on the grounds that the solar dynamo is a chaotic system is likely too extreme a stance, especially since (1) even chaotic systems can be amenable to prediction over a finite temporal window, and (2) input of data (even if not via true data assimilation) can in principle lead to some correction of the system’s trajectory in phase space. More relevant (in my opinion) has been the explicit demonstration that (1) very small changes in some unobservable and poorly constrained input parameters to the dynamo model used for the forecast can introduce significant errors already for next-cycle amplitude forecasts (see Bushby and Tobias, 2007, also Yeates et al., 2008); (2) the exact manner in which surface data drives the model can have a huge impact on the forecasting skill (Cameron and Schüssler, 2007). Consequently, the discrepant forecasts of Table 1 indicate mostly that current dynamo model-based predictive schemes still lack robustness. True data assimilation has been carried out using highly simplified dynamo models (Kitiashvili and Kosovichev, 2008), and clearly this must be carried over to more realistic dynamo models.
Finally, one must also keep in mind that other plausible explanations exist for the relatively good precursor potential of the solar surface magnetic field. In particular, Cameron and Schüssler (2008) have argued that the well-known spatiotemporal overlap of cycles in the butterfly diagram (see Figure 3), taken in conjunction with the empirical anticorrelation between cycle amplitude and rise time embodied in the Waldmeier Rule (Figure 22D; also Hathaway, 2010, Section 4.6), could in itself explain the precursor performance of the polar field strength at solar activity minimum. Given the unusually extended minimum phase between cycles 23 and 24, it will be very interesting to revisit all these model results once cycle 24 reaches its peak amplitude.
Living Rev. Solar Phys. 7, (2010), 3
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