5.7 Solar cycle predictions based on dynamo models

The idea that measurements of the solar surface magnetic field in the descending phase of a cycle can be used to forecast the amplitude (and/or timing) of the next cycle goes back many decades, but it is Schatten et al. (1978) who explicitly justified this procedure on the basis of dynamo models, which led to a wide variety of dynamo-inspired precursor schemes (see Hathaway et al., 1999, for a review).

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 αΩ dynamo solution of Figure 25View Image, 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. 2006Jump To The Next Citation Point (see also Dikpati and Gilman, 2006) and Choudhuri et al. 2007Jump To The Next Citation Point (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 22View Image).

Table 1: Two dynamo-based solar cycle forecasting schemes
Authors / Ref. Dikpati et al. (2006) Choudhuri et al. (2007)
Dynamo model kinematic axisymmetric kinematic axisymmetric
Babcock–Leighton Babcock–Leighton
Core-CZ interface r∕R = 0.7 r∕R = 0.7
Magnetic diffusivity Eq. (17View Equation), Δ η = 300 Eq. (17View Equation), Δ η = 104
plus high-η surface layer
Differential rotation solar-like parameterization solar-like parameterization
Eqs. (15View Equation) – (16View Equation), w ∕R = 0.05 Eqs. (15View Equation) – (16View Equation), w ∕R = 0.015
all other parameters same all other parameters same
Meridional circulation single cell per quadrant single cell per quadrant
closing at r∕R = 0.71 closing at r∕R = 0.635
Poloidal source term data-driven surface forcing subsurface α-effect
plus weak tachocline α-effect plus buoyancy algorithm
Nonlinearity algebraic α-quenching algebraic α-quenching
only in tachocline α-effect in subsurface α-effect
Solar data time series of total sunspot area DM Index
used to (continuously) drive used to reset amplitude of A
parametric surface forcing of A 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 3View Image), taken in conjunction with the empirical anticorrelation between cycle amplitude and rise time embodied in the Waldmeier Rule (Figure 22View ImageD; 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.

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