# 109. How Many Active Regions Are Necessary to Predict the Solar Dipole Moment?

Contributed by Tim Whitbread. Posted on September 10, 2018

Tim Whitbread1, Anthony Yeates1, & Andrés Muñoz-Jaramillo2,3,4
1. Department of Mathematical Sciences, Durham University, Durham, DH1 3LE, UK
2. Southwest Research Institute, 1050 Walnut Street, #300, Boulder, CO 80302, USA
3. National Solar Observatory, 3665 Discovery Drive, Boulder, CO 80303, USA
4. High Altitude Observatory, National Center for Atmospheric Research, 3080 Center Green, Boulder, CO 80301, USA

A major goal of solar physics research is to be able to predict the amplitude and timing of future solar cycles. Most successful is the precursor method, which uses the state of the Sun’s polar magnetic field or geomagnetic activity at solar minimum to predict the following cycle. Here we consider the Sun’s axial dipole moment (equivalent to the polar field at cycle minimum) to assess the effect of individual active regions on the polar field, and hence the amplitude of the subsequent cycle.

Nagy et al.[1] recently found that a single large, abnormally oriented (i.e. anti-Hale/anti-Joy) region can have a drastic effect on future cycles, in the most extreme cases inducing a grand minimum period. Jiang et al.[2] suggested that a single large region emerging at low latitudes or across the equator could have caused the unusually weak axial dipole moment at the end of Cycle 23, and consequently the weak Cycle 24. We test these claims by simulating the evolution of real active regions from Kitt Peak/SOLIS synoptic magnetograms from Cycles 21, 22 and 23 using a surface flux transport model[3].

To quantify the contribution of each region to the overall evolution of the dipole moment, we use the relative axial dipole moment $D_{\rm rel}^{\left(i\right)}$, which is defined as:
$D^{\left(i\right)}_{\rm rel}\left(t\right) = \frac{D^{\left(i\right)}\left(t\right)}{D_{\rm tot}\left(t_{\rm end}\right) - D_{\rm tot}\left(t_{\rm start}\right)} ,$
for region $i$, where $D_{\rm tot}\left(t\right)$ is the dipole moment of the full simulation with all regions included, and $D^{\left(i\right)}\left(t\right)$ is the dipole moment contribution of a single region simulated individually. The times $t_{\rm start}$ and $t_{\rm end}$ are the start and end of each cycle respectively, so that $D_{\rm rel}^{\left(i\right)}\left(t_{\rm end}\right)$ reflects the proportional contribution of region $i$ to the end-of-cycle axial dipole moment. A positive $D_{\rm rel}\left(t_{\rm end}\right)$ corresponds to a strengthening of the axial dipole moment at the end of the cycle, whilst a negative $D_{\rm rel}\left(t_{\rm end}\right)$ corresponds to a weakening.

Figure 1| Evolution of the axial dipole moment for Cycles 21 to 23. Each profile is obtained by: (a) including the biggest n contributors to the axial dipole moment, or (b) removing the biggest n contributors to the axial dipole moment. Color intensity is indicative of the number of regions used in each simulation. The gray curve shows the observed axial dipole moment. Vertical dashed lines indicate start/end points of cycles as used in this study.

Initially we assess the effect of the largest contributors from each cycle. Figure 1a shows the axial dipole moment when the top n contributors are included in the simulation. This shows how the axial dipole moment is the cumulative product of many individual regions. Notice that in Cycle 23, even when the top 100 contributors are included, the polar field is still unable to reverse. If we instead remove the top 10 strongest regions from the simulation (Figure 1b), we discover that the amplitude of the final axial dipole moment is overestimated in Cycles 21 and 23, and underestimated in Cycle 22. This demonstrates the impact of the strongest regions from the three cycles. In particular, in Cycle 23, the top 10 contributors made a net negative (weakening) contribution. So the polar field at the end of Cycle 23 could have been stronger had the strongest few regions emerged with different properties or not emerged at all. Ultimately, this supports the conclusions of Refs [1] and [2]: that a small number of `rogue’ regions can have a large effect on the polar field and next cycle, although the cumulative contribution of the smaller contributors cannot be ignored.

Figure 2| Final $D_{\rm rel}$ for each region from Cycle 21 against absolute emergence latitude (left panel), magnetic flux (middle panel) and initial $D_{\rm rel}$ (right panel). Markers are sized by absolute final $D_{\rm rel}$, and coloured by flux (left panel) and absolute latitude (middle and right panel).

Next we consider how the final contribution of a region depends on its properties at emergence. Figure 2 shows scatterplots for Cycle 21 (the results are consistent across all three cycles). We find that the largest contributors emerge below ±20°, and they do not necessarily need large fluxes. Furthermore, we see in the right-hand panel that the amplification in dipole moment contribution depends strongly on emergence latitude. This is clear in Figure 3, where we have calculated the slope of the lines of points in the right-hand panel from each 5° latitudinal bin. In our flux transport model there is a strong Gaussian latitudinal dependence, also found by Jiang et al.4.

Figure 3| Ratio between final and initial $D_{\rm rel}$ for 5° latitudinal bins for Cycles 21 (pink), 22 (yellow) and 23 (green). Error bars show standard error. Markers are plotted at the midpoint of each 5° bin. The blue curve is a Gaussian fit to the data.

For full details of this study, see Ref. [5]: Whitbread et al. (2018).

### References

[1] Nagy, M., Lemerle, A., Labonville, F., Petrovay, K. & Charbonneau, P. 2017, Solar Phys., 292, 167
[2] Jiang, J., Cameron, R. H. & Schüssler, M. 2015, ApJL, 808, L28
[3] Yeates, A. R., Baker, D. & van Driel-Gesztelyi, L. 2015, Solar Phys., 290, 3189
[4] Jiang, J., Cameron, R. H. & Schüssler, M. 2014, ApJ, 791, 5
[5] Whitbread, T., Yeates, A. R. & Muñoz-Jaramillo, A., 2018, ApJ, 863, 116