7.4 Predicting future cycle amplitudes based on dynamo theory

Dynamo models for the Sun’s magnetic field and its evolution have led to predictions based on aspects of those models. Schatten et al. (1978) suggested using the strength of the Sun’s polar field as a predictor for the amplitude of the following cycle based on the Babcock (1961) dynamo model. In the Babcock model the polar field at minimum is representative of the poloidal field that is sheared out by differential rotation to produce the toroidal field that erupts as active regions during the following cycle. Diffusion of the erupting active region magnetic field and transport by the meridional flow (along with the Joy’s Law tilt of these active regions) then leads to the accumulation of opposite polarity fields at the poles and the ultimate reversal of the polar fields as shown in Figure 15View Image.

Good measurements of the Sun’s polar field are difficult to obtain. The field is weak and predominantly radially directed and thus nearly transverse to our line-of-sight. This makes the Zeeman signature weak and prone to the detrimental effects of scattered light. Nevertheless, systematic measurements of the polar fields have been made at the Wilcox Solar Observatory since 1976 and have been used by Schatten and his colleagues to predict cycles 21 – 24. These polar field measurements are shown in Figure 42View Image along with smoothed sunspot numbers. While the physical basis for these predictions is appealing, the fact that the necessary measurements are only available for the last three cycles is a distinct problem. It is unclear when the measurements should be taken. Predictions by this group for previous cycles have given different values at different times. The RMS differences between the published predictions and the observed cycle amplitudes suggest that these predictions are about as good as the geomagnetic precursor predictions. The polar fields are obviously much weaker during the current minimum. This has led to a prediction of Rmax(24) = 75 ± 8 by Svalgaard et al. (2005Jump To The Next Citation Point) – about half the size of the previous three cycles based on the polar fields being about half as strong. While in previous minima the strength of the polar fields (as represented by the average of the absolute field strength in the north and in the south) varied as minimum approached, this did not happen on the approach to cycle 24 minimum in late 2008. This suggests that the prediction made in 2005 still holds.

View Image

Figure 42: Polar magnetic fields as measured at the Wilcox Solar Observatory. The average of the north and south field strengths near the time of sunspot cycle minimum is expected to be an indicator for the amplitude of the next sunspot cycle.

Over the last decade dynamo models have started to include the effect of the Sun’s meridional circulation and found that it can play a significant role in the magnetic dynamo (cf. Dikpati and Charbonneau, 1999). In these models the speed of the meridional circulation sets the cycle period and influences both the strength of the polar fields and the amplitudes of following cycles. Two predictions have recently been made based on flux transport dynamos with assimilated data – with very different results.

Dikpati et al. (2006Jump To The Next Citation Point) predicted an amplitude for cycle 24 of 150 – 180 using a flux transport dynamo that included a rotation profile and a near surface meridional flow based on helioseismic observations. They modeled the axisymmetric poloidal and toroidal magnetic field using a meridional flow that returns to the equator at the base of the convection zone and used two source terms for the poloidal field – one at the surface due to the Joy’s Law tilt of the emerging active regions and one in the tachocline due to hydrodynamic and MHD instabilities. The diffusivity in the model is a function of depth with a surface diffusivity of 5 × 1012 cm2 s–1 falling to 5 × 1010 cm2 s–1 at r = 0.9R. They drive the model with a surface source of poloidal field that depends upon the sunspot areas observed since 1874. Measurements of the meridional flow speed prior to 1996 are highly uncertain (cf. Hathaway, 1996) so they maintained a constant flow speed prior to 1996 and forced each of those earlier cycles to have a constant period as a consequence. The surface poloidal source term drifted linearly from 30° to 5° over each cycle with an amplitude that depended on the observed sunspot areas. They based their prediction on the strength of the toroidal field produced in the tachocline. They found excellent agreement between this toroidal field strength and the amplitude of each of the last eight cycles (the four earlier cycles – during the initialization phase – were also well fit but not with the degree of agreement of the later cycles). The correlation they find between the predicted toroidal field and the cycle amplitudes is similar to that found with the geomagnetic precursors and polar field strength indicators. When they kept the meridional flow speed at the same constant level during cycle 23 they found Rmax(24) 180. When they allowed the meridional flow speed to drop by 40% as was seen from 1996 – 2002 they found Rmax(24) 150 and further predicted that cycle 24 would start late.

Choudhuri et al. (2007Jump To The Next Citation Point) predicted an amplitude for cycle 24 of 80 using a similar flux-transport dynamo but with the surface poloidal field at minimum as the assimilated data. They used a similar axisymmetric model for the poloidal and toroidal fields but with a meridional flow that extends below the base of the convection zone and a diffusivity that remains high throughout the convection zone. In their model the toroidal field in the tachocline produces flux eruptions when its strength exceeds a given limit. They compare the number of eruptions to the observed sunspot numbers and use this as the predictor for cycle 24. They assimilate data by instantaneously changing the poloidal field at minimum throughout most of the convection zone to make it match the dipole moment obtained from the Wilcox Solar Observatory observations (Figure 41View Image). They found an excellent fit to the last three cycles (the full extent of the data) and found Rmax(24) 80, in agreement with the polar field prediction of Svalgaard et al. (2005).

Criticism has been leveled against all of these dynamo-based predictions. Dikpati et al. (2006Jump To The Next Citation Point) criticized the use of polar field strengths to predict the sunspot cycle peak that follows by four years by questioning how those fields could be carried down to the low latitude tachocline in such a short time. Cameron and Schüssler (2007) produced a simplified 1D flux transport model and showed that with similar parameters to those used by Dikpati et al. (2006Jump To The Next Citation Point) the flux transport across the equator was an excellent predictor for the amplitude of the next cycle but the predictive skill was lost when more realistic parameterizations of the active region emergence were used. Yeates et al. (2008) compared an advection-dominated model like that of Dikpati et al. (2006) to a diffusion-dominated models like that of Choudhuri et al. (2007) and concluded that the diffusion-dominated model was better because it gave a better fit to the relationship between meridional flow speed and cycle amplitude. Dikpati et al. (2008a) returned with a study of the use of polar fields and cross equatorial flux as predictors of cycle amplitudes and concluded that their tachocline toroidal flux was the best indicator. Furthermore, they found that the polar fields followed the current cycle so that the weak polar fields at this minimum are due to the weakened meridional flow. The strongest criticism of these dynamo-based predictions was give by Tobias et al. (2006) and Bushby and Tobias (2007). They conclude that the solar dynamo is deterministically chaotic and thus inherently unpredictable.

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