---
author:
- 'M. Sch[ü]{}ssler'
bibliography:
- 'pap.bib'
title: 'Are solar cycles predictable?'
---

Introduction
============

Taken at face value, the question posed in the title has to be answered in the affirmative: one cannot deny that there is on the market a whole lot of predictions of future solar activity levels. A quick (and unsystematic) search in the Smithonian/NASA Astrophysical Data System (ADS) reveals that there are 50% more hits for the combination of “prediction” and “solar cycle” in title or abstract than for “solar dynamo”. In fact, there are alone 281 such hits for publications since 2004 (status: July 16, 2007). Interestingly, the intersection of both sets, i.e., papers dealing both with solar-cycle prediction [*and*]{} with the solar dynamo comprises less than 5% of the papers on prediction. This is not very surprising because, until recently, solar dynamo models have not been considered to have reached a state of maturity to be used for predictive purposes.

Most prediction methods in the literature can be categorized into one of two classes [cf. @Wilson:1994]:

1.  [*Extrapolation methods,*]{} based on statistics or pattern recognition: most relevant information about the system is assumed to be contained in the available data (e.g., the sunspot number record), so that the future can be extrapolated from the past. The simplest example is harmonic analysis [e.g., @Echer:etal:2004], but also concepts of nonlinear dynamics are used [e.g., @Sello:2001].

2.  [*Precursor methods,*]{} assuming that certain physical quantities measured during the descending or minimum phase of an activity cycle contains information about the strength of the next cycle [e.g., @Lantos:Richard:1998; @Hathaway:etal:1999; @Schatten:2003].

The overall success of the various methods in predicting the future is rather limited (e.g., see Figure 14.2 in Wilson 1994 and Figure 6 of Lantos & Richard 1998). However, if the historical record of solar activity is considered, some precursors show remarkable levels of correlation with the strength of the following cycle. For instance, @Ohl:1966 took the minimum level of geomagnetic variations (as measured, for instance, by the [*aa*]{} index) as a precursor for the strength of the next cycle. This method does not have any adjustable parameters and yet provides a correlation coefficient of $r=0.91$ for solar cycles 12-22 [@Hathaway:etal:1999]. The method of @Thompson:1993, which is also based upon geomagnetic variations, even yielded $r=0.97$ for the same cycles, but utterly failed in actually predicting cycle 23: the predicted value for the sunspot number of $R=160$ turned out to be more than 30% too high! This result reminds us that a high correlation coefficient for post-dicting the past does not necessarily imply a high skill of the method for predicting the future.

Solar cycle predition and dynamo models
=======================================

The paper by @Schatten:etal:1978 is (to my knowledge) the first paper in which the words “dynamo theory” and “sunspot number prediction” appear together in the title. The authors argue that, in the framework of the Babcock-Leighton dynamo model, the polar magnetic field of the Sun around solar minimum should be a predictor for the strength of the next cycle: since for such models the polar field is thought to reflect the global dipolar poloidal field from which the toroidal field of the next cycle is being generated by differential rotation, the strength (or magnetic flux) of this toroidal field should by higher for a stronger polar field. Since measurements of the polar fields are rather uncertain and consistent data series are available only since a few decades, a stringent test of the suggestion of @Schatten:etal:1978 could not be carried out so far. Various proxies for the polar field have also been considered, but with inconclusive results [@Layden:etal:1991]. Nevertheless, the method has been used to predict a rather weak solar cycle 24 [@Svalgaard:etal:2005; @Schatten:2005].

Although they refer to dynamo theory (in fact, to a vague notion of the Babcock-Leighton model), the proponents of the polar field precursor have never actually used a mathematical dynamo model to support their suggestion, neither in the original paper [@Schatten:etal:1978] nor in any of the follow-up papers. In a very crude way, such an attempt has been made only very recently by @Choudhuri:etal:2007. These authors use a Babcock-Leighton-type flux-transport dynamo model and arbitrarily rescale the poloidal field at 4 cycle minima according to measured polar field values (from the Mount Wilson and Wilcox solar observatories). It is not surprising (in fact, almost trivial) that the toroidal fields of the respective following cycles reflect the value of the scaling factor. In fact, any linear or mildly nonlinear model would lead to the same result, so that @Choudhuri:etal:2007 effectively do not go beyond the original suggestion of @Schatten:etal:1978. This is also demonstrated by @Brandenburg:Kaepylae:2007 who obtain practically the same result with a heavily truncated toy model. Therefore, such a crude approach to ‘data assimilation’ does not provide more information than simple correlation studies and, in particular, does not furnish constraints for dynamo models.

The approach of @Dikpati:etal:2006 and @Dikpati:Gilman:2006, hereafter referred to as the DDG model, is the first serious attempt to use a mean-field dynamo model to predict solar cycle strength. These authors use an axisymmetric (longitude-averaged) flux-transport dynamo model in a spherical shell with a solar-like meridional flow (poleward at the surface) and a low turbulent diffusivity in the convection zone. The differential rotation is chosen according to the helioseismic measurements; it generates toroidal magnetic flux near the bottom of the convection zone, the amount of which which is taken as the predictor quantity. In such models, the dynamo loop is usually closed by assuming an $\alpha$-effect relating the toroidal field to a near-surface source term for the poloidal field. In the DDG model, this kind of closure is replaced by a source term that directly reflects the observed emergence of tilted bipolar magnetic regions: the source with Gaussian latitude profile drifts between 35 deg and 5 deg latitude during a sunspot cycle and its strength is scaled with the historical record of observed sunspot areas since 1876 (RGO data plus extensions since 1976). Through this data assimilation procedure, the source term reflects the actual variations of the flux emergence at the surface and incorporates them into the evolution of the model. The DDG model provides amazingly high correlation coefficients between the amount of low-latitude toroidal magnetic flux in the deep convection zone calculated (‘predicted’) by the model and the strength (maximum sunspot number) of the corresponding cycle; values up to $r=0.99$ are obtained.

The success of the DDG model is surprising given the various assumptions and parametrizations entering the model, for instance: (1) arbitrary prescription of the (unknown) meridional flow pattern in the deep convection zone, (2) a strong radial drop of the turbulent magnetic diffusivity between the surface layers and the deeper parts of the convection zone, (3) schematic prescription of the profile, width and latitude drift of the poloidal field source. This has led @Bushby:Tobias:2007 to argue that the correlations obtained by DDG are either fortuitous or the result of parameter tuning, claiming that it is “impossible to predict the solar cycle using the output of such models”. They give two examples to support this claim: a) a flux-transport model with stochastic fluctuations of the meridional flow, and 2) an interface dynamo with a strong nonlinearity (back-reaction on the differential rotation). While the arguments of @Bushby:Tobias:2007 certainly apply to their kind of “ab-initio” dynamo models with a closed dynamo loop, it is not so clear how much weight they carry concerning the data assimilation approach of DDG: in fact, using the observed flux emergence takes account of at least part of the random fluctuations and nonlinearities certainly inherent in working of the solar dynamo, namely those associated with the connection between the toroidal field deep in the convection zone and the surface field. Precisely these variations in the source strength eventually determine the modulation of the cycle amplitudes in the DDG model, but in a non-trivial way (as exemplified by the correctly reproduced drop of activity from cycle 19 to 20, in spite of a strong source amplitude provided by the flux emergence in cycle 19). Other fluctuations, such as variations of the meridional flow, could also be incorporated into the model once sufficiently detailed and extended measurements become available.

Even if the claims of @Bushby:Tobias:2007 would apply to the DDG model, the question remains why this model provides such high correlations. Could it really be parameter tuning? This can be tested in a straightforward manner: take the DDG model and use a source with random amplitudes for 12 cycles; then tune the model parameters such that the predictor (toroidal field) reproduces the actual maxima of the last 9 cycles (with a correlation coefficient exceeding 0.95, say). I very much doubt that the DDG model has such a degree of flexibility and I would dare to predict that this will turn out to be an impossible task indeed!

So, after all, the DDG model cannot be brushed away off-handedly. We need to understand where its predictive skill comes from, since this might tell us something important about the solar dynamo: for instance, does the evolution of the surface flux during a cycle play a crucial role in the dynamo process (and affect the strength of subsequent cycles) or is it just a superficial epiphenomenon of the hidden dynamo?

A simple flux transport model
=============================

Given its parametrization of poorly know properties (such as internal meridional flow and turbulent diffusivity), is it conceivable that the details of the interior in fact do not matter for the correlations obtained by DDG? If that would be the case, then a pure surface transport model driven by the same source (emerging flux) as used in the DDG model should already contain and reveal the relevant information. In a recent paper [@Cameron:Schuessler:2007], we have therefore considered a very simple (almost trivial) axisymmetric surface flux transport model for the radial magnetic field component as a function of latitude and time. Flux is fed into the system by a source term analoguous to the DDG source and we follow its subsequent evolution under the influence of a poleward meridional flow and turbulent diffusion. We have considered various quantities as predictors. In the spirit of the Babcock-Leighton model, the amount of magnetic flux diffusing over (or reconnecting at) the equator is the most relevant quantity: only this part of the emerged flux represents the global dipole field that acts as the poloidal field source for the toroidal field of the next cycle. If we match as closely as possible the procedures and parameter choices in the DDG model, we indeed find that the cross-equator flux during cycle $n$ is correlated with the maximum sunspot number of cycle $n+1$ with $r=0.9$. The result turns out to be fairly robust with respect to parameter variations; values up to $r=0.95$ can be reached by ‘tuning’. Incidentally, taking the polar field strength as a predictor results in much lower correlation coefficients.

How do the high correlation doefficient with the cross-equator flux come about? For instance, why is the comparatively weak cycle 20 correctly ‘predicted’ although a large amount of surface flux has emerged during the preceding cycle 19? Fig. \[fig:blowup\] explains how this works. The cross-equator magnetic flux peaks a few years before sunspot minimum during the time when the source flux is supplied at low latitudes, so that 1) the distance to the equator is smaller and 2) the meridional flow (antisymmetric to the equator) is less forceful. Both effects lead to a a strongly increasing rate of magnetic flux diffusing over the equator when the source approaches the equator. As a consequence, the amplitude of the predictor is determined by the amount of flux emergence (sunspot area) in the declining phase of the cycle. The relatively high flux emergence during the declining phase of cycle 18 leads to a large value of the predictor for cycle 19. Significantly less flux erupted during this phase in cycle 19, so that the predictor for cycle 20 is lower, although the total flux emergence during cycle 19 is much higher than that of cycle 18. This result already indicates that the predictor could be rather sensitive to the definition of the source latitudes in the model.

Having understood that the level of activity a few years before minimum mainly determines the amplitude of the predictor in our simple surface-transport model, we may ask ourselves whether we need the model at all in order to make a prediction. In fact, taking the level of recorded sunspot number three years before minimum and correlating it with the strength of the next maximum for all cycles since 1750 leads to a value of $r=0.89$ for the correlation coefficient.

On the other hand, can we possibly improve the flux-transport model? We have detailed information about the areas and latitudes of individual sunspot groups for the whole period since 1874 [RGO, SOON and Russian data, see @Balmaceda:etal:2005], so that we can replace the schematic source term of DDG by a procedure that separately takes into account each sunspot group in the data. The surprising result is a dramatic drop of the correlation between the cross-equator flux and the strength of the following cycle: with $r=0.33$, the predictive skill is almost gone. In fact, the predictor now correlates better with the strength of the ongoing cycle than with the next cycle.

The origin of the predictive skill
==================================

The results sketched in the previous section leave us with puzzling questions. Why is there predictive skill in the flux-transport model with the schematic source and why does it completely vanish for more realistic input data? Why does the 3-year precursor based upon sunspot numbers work reasonably well? Has any of this anything to do with the Babcock-Leighton dynamo scheme?

With the benefit of hindsight, the answer to these questions seems amazingly simple, almost trivial. Let us first remind ourselves that there is a third possibility for the origin of predictive skill in linear models like the DDG approach or ours: besides 1) intrinsic validity of the model and 2) sheer luck or parameter tuning, there could be 3) correlations in the input data themselves. We shall see below that such correlations indeed exist and that they probably are responsible for the correlations obtained with most precursor methods and also with our simple flux-transport model.

![Schematic illustration of the amplitude-dependent shift of the minimum between overlapping, asymmetric sunspot cycles and its influence on a precursor quantity (sunspot activity 3 years before minimum). A stronger follower cycle (solid curve) with a shorter rise time leads to an earlier minimum (M1) and a higher predictor (P1) than a weaker subsequent cycle (dashed curve, minimum M2, predictor P2) with a longer rise time. Both alternatives for the follower cycle start at $t=11\,$yr [from @Cameron:Schuessler:2007]. []{data-label="fig:idea"}](idea){width="0.9\linewidth"}

It turns out that a combination of two well-known properties of the solar cycle explains (or, at least, contributes a significant part to) the predictive skill of precursor-type models:

1.  [*overlapping of cycles:*]{} active regions belonging to the new cycle start to appear in mid latitudes while there is ongoing flux emergence near the equator connected with the old cycle.

2.  [*Waldmeier’s rule:*]{} stronger cycles tend to rise faster towards sunspot maximum [@Waldmeier:1935].

The important point is that both properties make the level and the timing of the formal solar minimum (epoch of minimum sunspot number) depend on the strength of the following cycle. This is exactly the correlation in the sunspot number (or area) data that eventually leads to the predictive skill of precursors. Fig. \[fig:idea\] illustrates schematically how this comes about. Given are time profiles of overlapping sunspot cycles according to an empirical functional form that reproduces both the rise and decay parts of a cycle, including Waldmeier’s rule [@Hathaway:etal:1994; @Li:1999]. The figure shows the effect of the strength of the following cycle (dotted curves) on the summed activity levels around minimum activity between the cycles. The faster rise of a stronger follower cycle leads to an earlier and higher sunspot minimum in the summed activity curve (solid line) than in the case of a weaker follower cycle (dashed line). In the case shown, the time shift of the minimum epoch is about one year. Since a sunspot cycle is defined as the time between adjacent minima, the activity in the declining phase of the first cycle, (i.e., in a fixed time interval relative to the respective solar minimum epoch) is considerably larger when the follower cycle is stronger than when it it weaker. When a precursor is taken relative to sunspot minimum (e.g., our choice of the sunspot number 3 years before minimum), it is obvious that its level indeed will reflect the strength of the following cycle – without requiring any kind of direct physical connection between the precursor and the following cycle amplitude.

It is clear that the correlation in the input data (sunspot area record) explained above also underlies the predictive skill of our flux-transport model with a schematic source. In this case, we have assumed [following @Dikpati:etal:2006] a fixed latitude progression of the source centroid from 35 deg to 5 deg between two sunspot minima. Consequently, in the case of a strong follower cycle, higher activity levels in the descent phase (a few years before minimum) due to the correspondingly earlier minimum epoch are mapped to lower emergence latitudes and, therefore, lead to a higher amount of magnetic flux diffusing over the equator. If we directly take the emerging active regions with their actual emergence latitudes from the available record, then the near-equator flux emergence comes only from the preceding cycle, and thus correlates with its strength. As a consequence, our precursor does not show predictive skill any more and, in fact, only reflects the strength of the ‘old’ cycle. We are not in a position to claim that the same explanation also holds for the correlations found with the DDG model, but this can be tested by replacing their schematic source term by the existing sunspot group data with actual emergence latitudes.

Apart from explaining the predictive skill of a many precursor quantities measured during the descent phase or around solar minimum, the overlapping of cycles and Waldmeier’s rule also naturally accounts for a number of well-known correlations in the sunspot record, for instance: 1) strong cycles tend to be preceded by short cycles [e.g., @Solanki:etal:2002b], 2) minimum levels preceding strong cycles tend to be higher [@Hathaway:etal:2002], and 3) more asymmetric cycles tend to be followed by weaker cycles [e.g., @Lantos:2006].

Conclusions
===========

I think that the correlations introduced into the sunspot number and sunspot area records by the combination of cycles overlap and Waldmeier’s rule go a long way towards explaining the predictive skill of many precursor approaches as well as the correlations provided by the flux-transport models with a schematic source. Consequently, there is more to these models than just numerology or parameter-tuning. However, the key point is not so much to predict but to [*understand*]{} the solar cycle. So what have we learned in this respect? Not very much, I am afraid: the correlation introduced by the Waldmeier effect of overlapping cycles does not require any kind of physical relation between the surface fields of the previous cycle(s) and the strength of the following cycle; in particular, it cannot be taken as evidence in favour of a Babcock-Leighton type dynamo model. In fact, it can be shown that precursor methods successfully predict cycle sequences with randomly varying strength [@Cameron:Schuessler:2007]. On the other hand, these results do not exclude a physical connection between precursor and following cycle strength. For instance, the precursors could also be affected by flux emergence in high latitudes, e.g., in the form of ephemeral regions preceding the appearance of the first sunspots of the new cycle [e.g., @Harvey:1993; @Harvey:1994a], so that the new cycle would already directly affect the surface flux during the descending phase of the old cycle. These all remain valid possibilities, it is only that the predictive skill of precursor methods per se does not help us to decide which of these is in fact realised by the Sun.

In all such considerations we should not forget that all the relationships that may be used for prediction are ‘noisy’ and thus valid only in a statistical sense. The existence of grand minima like the Maunder minimum reminds us that the Sun has much more variability in store than simple statistical analysis of sunspot data would be able to predict. And even if a prediction method has a good correlation record for the past, it may completely fail for the next cycle. The split opinion of the NOAA/NASA Solar Cycle 24 Prediction panel[^1] about whether the coming cycle would be high or low provides a good illustration about the ‘state of the art’ – and may actually reflect intrinsic limitations as illustrated by the examples given by @Bushby:Tobias:2007.

So, where do we stand now concerning the question in the title? We have seen that, owing to the cycle overlap and the Waldmeier effect, predictor methods can obtain relevant information about the new cycle at the epoch around solar minimum. However, the underlying statistical relationships contain a significant amount of scatter, so that actual predictions are uncertain, as their mixed performance in the past clearly shows. The skill of such predictor schemes does not seem to provide constraints or relevant information about the working of the solar dynamo, apart from the trivial fact that a valid dynamo model ultimately will have to reproduce and explain the underlying correlations. Ever since reading the paper of @Legrand:Simon:1981, I had hoped that there would be more to learn.

In order to end on a more positive note, let me say that, of course, the last word on these matters is not spoken. It may turn out, after all, that the data assimilation model of @Dikpati:etal:2006 and @Dikpati:Gilman:2006 will pass with flying flags the crucial test of using the actual flux emergence events in its source - and that these results will be independently confirmed by others and without excessive parameter tuning. Then sceptics like @Bushby:Tobias:2007 or @Cameron:Schuessler:2007 would have a hard time to search for sources of the predictive skill other than the operation of a Babcock-Leighton-type dynamo. It would also be worthwhile to look for signatures of the new cycle during the post-maximum phase (or even before) of the ongoing cycle, e.g., by monitoring flux emergence and the evolution of large-scale magnetic patterns in mid/high latitudes and compare with surface flux-transport simulations. Observational data from SOLIS, Hinode, SDO and eventually Solar Orbiter will be particularly suitable for this purpose. A positive detection of such signatures could possibly extend the lead time for solar cycle prediction using precursors.

[^1]: see http://www.sec.noaa.gov/SolarCycle/SC24
