aa r X i v : . [ a s t r o - ph ] A p r On Predicting the Solar Cycle using Mean-Field Models
Paul J. Bushby [email protected]
Steven M. Tobias [email protected] ABSTRACT
We discuss the difficulties of predicting the solar cycle using mean-field mod-els. Here we argue that these difficulties arise owing to the significant modulationof the solar activity cycle, and that this modulation arises owing to either stochas-tic or deterministic processes. We analyse the implications for predictability inboth of these situations by considering two separate solar dynamo models. Thefirst model represents a stochastically-perturbed flux transport dynamo. Hereeven very weak stochastic perturbations can give rise to significant modulationin the activity cycle. This modulation leads to a loss of predictability. In the sec-ond model, we neglect stochastic effects and assume that generation of magneticfield in the Sun can be described by a fully deterministic nonlinear mean-fieldmodel — this is a best case scenario for prediction. We designate the outputfrom this deterministic model (with parameters chosen to produce chaoticallymodulated cycles) as a target timeseries that subsequent deterministic mean-field models are required to predict. Long-term prediction is impossible even if amodel that is correct in all details is utilised in the prediction. Furthermore, weshow that even short-term prediction is impossible if there is a small discrepancyin the input parameters from the fiducial model. This is the case even if the pre-dicting model has been tuned to reproduce the output of previous cycles. Giventhe inherent uncertainties in determining the transport coefficients and nonlinearresponses for mean-field models, we argue that this makes predicting the solarcycle using the output from such models impossible.
Subject headings: (magnetohydrodynamics:) MHD – Sun: activity – Sun: mag-netic fields DAMTP, Centre for Mathematical Sciences, University of Cambridge, Wilberforce Road, CambridgeCB3 0WA, U.K. Department of Applied Mathematics, University of Leeds, Leeds LS2 9JT, U.K.
1. Introduction
Magnetic activity in the Sun is known to play a central role in driving both long-termand short-term dynamics (Tobias 2002; Weiss 2002). The magnetic field is responsible forspectacular events such as sunspots, solar flares, and coronal mass ejections, and for heatingthe solar corona to high temperatures. Large-scale magnetic activity is known to be dom-inated by the eleven year activity cycle. This cycle has been systematically observed sincethe early seventeenth century and its properties are well documented (see e.g. Ossendrijver2003). Of particular current interest is the impact of magnetic activity on solar irradiancethat might have significant implications for the terrestrial climate (see Solanki et al mean-field dynamo models (Steenbeck, Krause & R¨adler 1966; Krause & R¨adler 1980), which describethe evolution of the mean magnetic field, parameterising the effects of the small-scale fieldsand flows in terms of tensor transport coefficients. These transport coefficients include α ij (which leads to a regenerative term in the mean-field equations — the so-called α -effect) andthe turbulent diffusivity ( β ijk ). We stress here that there is no mechanism within the theoryfor determining the form of these coefficients, except for flows at low magnetic Reynoldsnumber or with short correlation time, and in solar models these are usually chosen ina plausible but ad-hoc manner (often, for simplicity, adopting isotropic representations inwhich α ij = αδ ij and β ijk = βǫ ijk ). Much attention has been focused upon determiningthese transport coefficients in both the linear and nonlinear dynamo regimes from numericalsimulations (Cattaneo & Hughes 1996; Brandenburg & Subramanian 2005) but there is stillno consensus over the nature of these, even to within an order of magnitude (see Courvoisier,Hughes & Tobias 2006). Mean-field models have, however, proved successful in providingillustrations of the type of behaviour that might be expected to occur in the Sun (and otherstars). It is often argued that, although these models have no predictive power , understandingthe underlying mathematical form of the equations can lead to the identification of robustpatterns of behaviour.Many different models have been proposed for the solar dynamo. In the distributed 3 –dynamo model, the α -effect operates throughout the convection zone and interacts with thelatitudinal shear (or the sub-surface shear layer, see Brandenburg 2005) to generate magneticfield. Alternatively, the dynamo could be operating near the tachocline, where an α -effectmight be driven either by a tachocline-based instability or by turbulent convection. This,in conjunction with the strong shear, could drive an “interface” dynamo (Parker 1993). Fi-nally, there are flux transport models, in which the (so-called) Babcock-Leighton mechanismproduces an α -effect (or source term) at the surface. This surface α -effect is coupled to theradial shear in the tachocline (where another α -effect may be operating) via a meridional flow(Choudhuri, Sch¨ussler & Dikpati 1995; Dikpati & Charbonneau 1999). The relative meritsof these models are discussed elsewhere in the literature (see, e.g. Charbonneau 2005) —the only comment we make here is that this plethora of models arises because of the lack ofavailable constraints on the form of the transport coefficients in the mean-field formalism.We note further that it is not clear that any of the above scenarios capture the essentialdynamo processes correctly or that these processes can ever be captured by a mean-fieldmodel.It is also possible to construct predictions of solar activity without using dynamo theory,and there is a long literature describing these predictive methods (see e.g. Zhang 1996;Hathaway, Wilson & Reichmann 1999; Sello 2003; Zhao et al α -effect). As with all current mean fieldmodels these turbulent transport effects have been parameterised in a plausible but ad-hocmanner, and are unconstrained by observations and indeed theory. The simplest predictive 4 –scheme proposed by Dikpati et al (2006) therefore takes the form of a parameterised linearsystem forced by boundary observations. The implicit underlying philosophy here is that byreducing the correct physics for the generation of the solar activity cycle (i.e. a nonlinearself-excited dynamo) to such a scheme, predictions about future solar activity can be made.In this paper we shall investigate the predictability of various dynamo models. Wedemonstrate that even when all the nonlinear physics of the solar dynamo is removed, prob-lems remain for prediction owing to the increased importance of stochastic effects — evenvery weak stochastic perturbations can produce significant modulation in these linear-typemodels. We also discuss the best-case scenario for prediction where stochastic effects can beignored, and demonstrate that in these cases prediction is still difficult owing to uncertaintiesin the input parameters of these parameterised mean-field models.The paper is organised as follows. In the next section we describe (in a general way)the importance of modulation and the role of stochasticity and nonlinearity in solar dynamomodels. In section 3 we investigate a flux transport model and demonstrate how the presenceof even extremely weak noise can render predictions useless. In section 4 we consider the“best-case” scenario for prediction where noise does not play a role in the modulation — wedemonstrate that more accurate prediction schemes may arise by using basic timeseries anal-ysis techniques rather than from constructing mean-field models of the solar cycle. Finally,in section 5 we discuss the implications of our work for predictions of the solar cycle.
2. Problems for prediction and mechanisms for modulation
In this section, we discuss the problems that must be overcome by schemes designed toyield a prediction of future solar magnetic activity. Some of these problems arise owing tothe nature of solar magnetic activity whilst others arise from the lack of a detailed theorythat is capable of describing solar magnetic activity in such extreme conditions as those thatexist in the solar interior.It is clear that if the solar cycle were strictly periodic, with a constant amplitude, thenit would be straightforward to predict future behaviour. However, all measurements of solarmagnetic activity (both direct observations and evidence from proxy data) indicate thatthe variations in the magnetic activity do not follow a periodic pattern. Departures fromperiodicity may be driven either by perturbations or by modulation. For the case of a weaklyperturbed periodic system, the dynamics is essentially captured by the periodic signal, withthe small perturbations playing a secondary role. We distinguish this behaviour from amodulated signal in which there are significant departures from periodicity (often occurring 5 –on longer timescales), with large variations in the observed amplitude of the signal. All theevidence from direct observations indicates that the solar cycle is strongly modulated. Theamplitude of the solar cycle varies enormously over long timescales, an extreme example ofthis modulation was a period of severely reduced activity in the seventeenth century knownas the Maunder Minimum. Proxy data from records of terrestrial isotopes, such as Be and C (see e.g. Beer 2000, Weiss & Tobias 2000, Wagner et al are captured by the differential equations of dynamo theory, with no random elements.Stochastic processes are those that occur on an unresolved length or timescale, and so cannot be described by the differential equations without including a random element into themodel.It is well known that stochastic modulation can arise even if the deterministic physicsthat leads to the production of the basic cycle is essentially linear. This parameter regimeis generally considered to be a good one for prediction, since any nonlinear effects are onlyplaying a secondary role. However, in this stochastically-perturbed case, the small randomfluctuations that lead to the modulation will have large short-term effects and render pre-diction extremely difficult, if not impossible. Conversely, if the modulation arises purely asa result of deterministic processes, then the underlying physics is nonlinear (or potentiallynon-autonomous) and this leads to difficulty in prediction owing to the possible presence ofdeterministic chaos and (more importantly) the difficulty of constructing accurate nonlinearmodels with large numbers of degrees of freedom.In the next two sections we demonstrate the problems for prediction for dynamo modelsin both of the classes described above. In the next section we describe a flux transport modelof the same type as the one used in the prediction scheme of Dikpati et al (2006) and wedemonstrate that even very small random fluctuations can produce significant modulation,leading to extreme difficulties for prediction. We then, in section 4, go on to describe a modelwhere the modulation arises owing to the presence of deterministic chaos and show that inthis case, prediction using model fitting is a poor way to proceed, but some prediction ispossible if it is possible to reconstruct the attractor for activity. 6 –
3. Prediction using a stochastically-perturbed flux transport dynamo model3.1. The dynamo model
We assume initially that the modulated solar magnetic activity can be described bya stochastically-perturbed mean-field dynamo model. In this model, nonlinear effects areplaying a secondary role, and all the modulation is being driven by the stochastic effects.The aim of this section is to assess whether or not models of this type can be used to makemeaningful predictions of the solar magnetic activity. In these models, the evolution of thelarge-scale magnetic field is described by the standard mean-field equation (see, for example,Moffatt 1978), ∂ B ∂t = ∇ × ( α B + U × B − β ∇ × B ) . (1)Here, B represents the large-scale magnetic field and U corresponds to the mean velocityfield, β is the (turbulent) magnetic diffusivity, and the α B term corresponds to the mean-field α -effect. Using the well-known αω approximation, we solve this equation numerically inan axisymmetric spherical shell (0 . R ⊙ ≤ r ≤ R ⊙ and 0 ≤ θ ≤ π ). In solving Equation (1)we need to ensure that B remains solenoidal (i.e. ∇ · B = 0). To achieve this, we decomposethe magnetic field into its poloidal and toroidal components, B = B ( r, θ, t ) e φ + ∇ × ( A ( r, θ, t ) e φ ) , (2)where B ( r, θ, t ) denotes the toroidal (azimuthal) field component and the scalar potential A ( r, θ, t ) relates to the poloidal component of the magnetic field. So, rather than solvingEquation (1) directly, the problem has been reduced to solving two coupled partial differentialequations for the scalar quantities A ( r, θ, t ) and B ( r, θ, t ). We adopt idealised boundaryconditions, in which A = B = 0 at θ = 0 and θ = π and r = 0 . R ⊙ and A and B aresmoothly matched to a potential field at r = R ⊙ .This particular dynamo model is closely related to the flux transport model describedby Dikpati & Charbonneau (1999). The large-scale velocity field, U , is given by U = u r ( r, θ ) e r + u θ ( r, θ ) e θ + Ω( r, θ ) r sin θ e φ , (3)where Ω( r, θ ) is a prescribed analytic fit to the helioseismologically-determined solar rotationprofile (see, for example, Bushby 2006) and u r and u θ correspond to a prescribed meridionalcirculation. We assume that the meridional circulation pattern in each hemisphere consists 7 –of a single cell, with a polewards flow at the surface and an (unobservable) equatorwardsflow at the base of the convection zone — the flow is confined to the region R b ≤ r ≤ R ⊙ .The functional form that we adopt for this flow is similar in form to the one described byDikpati & Charbonneau (1999), u r ( r, θ ) = U o (cid:18) R ⊙ r (cid:19) (cid:20) −
23 + 12 c ξ . − c ξ . (cid:21) ξ sin θ (cid:0) θ − sin θ (cid:1) , (4) u θ ( r, θ ) = U o (cid:18) R ⊙ r (cid:19) (cid:2) − c ξ . − c ξ . (cid:3) cos θ sin θ, (5)where ξ ( r ) = [( R ⊙ /r ) − c = 4[ ξ ( R b )] − . , c = 3[ ξ ( R b )] − . , and U o is some characteristicflow speed. This flow pattern can be stochastically perturbed by setting R b = 0 . R ⊙ + ǫ ( t ),where ǫ ( t ) is a time-dependent, randomly fluctuating variable in the range − . R ⊙ ≤ ǫ ≤ . R ⊙ . The aim here is to assess whether or not such weak stochastic variations in theflow pattern could give rise to significant modulation in the activity cycle, and if so whatare the consequences for prediction.In order to complete the specification of the model, we need to choose plausible func-tional forms for the α -effect and the turbulent magnetic diffusivity. It should be emphasisedagain that these mean-field coefficients are poorly constrained by theory and observations,although plausible assumptions can be made. Defining β o to be a characteristic value ofthe turbulent magnetic diffusivity within the solar convection zone, we adopt a similarspherically-symmetric profile to that adopted by Dikpati & Charbonneau (1999), β ( r ) = 12 ( β o − β c ) (cid:20) (cid:18) r − . R ⊙ . R ⊙ (cid:19)(cid:21) + β c , (6)where erf corresponds to the error function and β c (here taken to be 1% of β o ) represents themagnetic diffusivity below the turbulent convection zone. Following Dikpati & Charbonneau(1999), rather than prescribing a simple functional form for α we neglect the α -effect termin the toroidal ( B ) field equation and replace the corresponding αB term in the poloidal ( A )equation by a non-local, nonlinear source of poloidal flux, S ( r, θ, t ) = S o (cid:20) (cid:18) r − . R ⊙ . R ⊙ (cid:19)(cid:21) (cid:20) − erf (cid:18) r − R ⊙ . R ⊙ (cid:19)(cid:21) (7) " (cid:18) B (0 . R ⊙ , θ, t ) B o (cid:19) − sin θ cos θB (0 . R ⊙ , θ, t ) . S o is a characteristic value of this poloidal source and B o represents the (somewhatarbitrarily chosen) field strength at which this non-local source becomes suppressed by themagnetic field. This source term parameterises the contribution to the poloidal magneticflux due to the decay of active regions — the non-locality reflects the fact that active regionsare believed to form as the result of buoyant magnetic flux rising from the base of theconvection zone to the solar photosphere. See Dikpati & Charbonneau (1999) for a moredetailed discussion of this source term, though again it must be stressed that the functionalform and the nonlinear dependence are chosen in a plausible yet ad-hoc manner. In order to carry out numerical simulations, we first non-dimensionalise this flux trans-port model. By using scalings similar to those described by Dikpati & Charbonneau (1999),it can be shown that the model solutions are fully determined by two non-dimensional pa-rameters (once other parameters such as B o have been selected). Denoting the equatorialangular velocity at the solar surface by Ω eq , these non-dimensional parameters are the Dy-namo number, D = S o Ω eq R ⊙ /β o , and the magnetic Reynolds number corresponding to themeridional flow, Re = U o R ⊙ /β o . Here, we set D = 7 × and Re = 5600. In the absenceof stochastic noise, this set of parameters produces a strong circulation-dominated dynamoin which the magnetic energy is a periodic function of time. Although the dynamo numberis not weakly supercritical, nonlinear effects are not strong enough here to produce a modu-lated activity cycle — the primary role of the nonlinearity is to prevent the unstable dynamomode from growing exponentially. We term such a model a “linear-type” model.When weak stochastic effects are included in the model, the resulting activity cycle isindeed weakly modulated. This is illustrated in Figure 1, which shows the time-dependenceof this solution. The time-series clearly illustrates that, although the amplitude of the “cycleminimum” only appears to be weakly time-dependent, there are significant variations in thepeak amplitude of the magnetic energy time-series. These variations are qualitatively similarto those observed by Charbonneau & Dikpati (2000), who considered large amplitude randomfluctuations in the flow pattern within the solar convection zone — the peak amplitude ofthese fluctuations was comparable with the peak amplitude of the flow. In this particularmodel, we have shown that even very weak stochastic variations in the centre of mass of theflow pattern can still produce significantly modulated behaviour. These stochastic effectsare expected to become increasingly significant for dynamo numbers approaching critical.So, these models are obviously highly sensitive to the addition of stochastic noise.In the absence of stochastic noise, the attractor (in phase space) for this solution is 9 –two-dimensional, and the future behaviour of the solution at any instant in time is entirelydetermined by the current position of the system on the attractor. The same is not true whenthis system is perturbed by stochastic effects, and it clearly becomes much more difficultto predict the future behaviour of the system. Since the attractor of this stochasticallyperturbed solution cannot be unambiguously defined, another possible way of assessing the“predictability” of this solution is to look for a correlation between successive cycle maxima.Defining T n to be the magnitude of the n th cycle maximum, Figure 2 shows T n +1 as afunction of T n . It is clear from this scatter plot that there is no obvious correlation betweenthe amplitudes of successive cycle maxima in this case. Since the modulation is being drivenentirely by random stochastic forcing, this result is not surprising. This lack of correlationsuggests that the behaviour of previous cycles cannot be used to infer the magnitude ofthe following one. This implies that even weak stochastic effects may seriously reduce thepossibilities for solar cycle prediction in this linear-type regime.
4. Predictions using a deterministic dynamo model4.1. The dynamo model
In the previous section, we demonstrated that even very weak stochastic perturbations tothe meridional flow pattern can lead to a loss of predictability in a linear-type flux transportdynamo model. In that model, the modulation of the activity cycle was driven entirelyby stochastic effects. As discussed in Section 2, the only other possible scenario is thatthe observed modulation is driven by nonlinear effects. This scenario, where the observedmodulation is deterministic in origin, is the “best-case” scenario for prediction, as in this casethe entirely unpredictable stochastic elements may be ignored. We stress again that, giventhat solar magnetic activity is significantly modulated, either deterministic or stochasticmodulation must be considered in any realistic model (predictive or otherwise) of the solarcycle. So, in this section, we completely neglect stochastic effects and assume that theobserved (chaotic) modulation in the solar magnetic activity can be described by a fullydeterministic model in which any activity modulation (e.g. solar-like “Grand minima”) isdriven entirely by nonlinear effects. The model that we use was described in detail in tworecent papers (Bushby 2005, 2006), so we only present a brief description here. The exactdetails of the model are unimportant for our main conclusions.Like the flux transport dynamo model from the previous section, this model describes anaxisymmetric, mean-field, αω -dynamo in a spherical shell. Unlike the previous model, thismodel represents an “interface-like” dynamo that is operating primarily in the region aroundthe base of the solar convection zone. It is worth mentioning again that (as discussed in the 10 –introduction) there is still no general consensus regarding which of these dynamo scenariosis more likely to be an accurate representation of the solar dynamo. For this interface-likedynamo model, we neglect meridional motions, since they are poorly determined near thebase of the solar convection zone. Like several earlier models (e.g. Tobias 1997; Moss &Brooke 2000; Covas et al V ( r, θ, t ), the large-scalevelocity field is given by U = [Ω( r, θ ) r sin θ + V ( r, θ, t )] e φ , (8)where (as in the previous model) Ω( r, θ ) represents an analytic fit to the solar differentialrotation. Whilst the evolution of the large-scale magnetic field is again governed by Equa-tion (1), an additional evolution equation is required for the velocity perturbation, V . Thisequation is given by ∂V∂t = 1 µ o ρ [( ∇ × B ) × B ] · e φ + 1 r ∂∂r (cid:20) νr ∂∂r (cid:18) Vr (cid:19)(cid:21) (9)+ 1 r sin θ ∂∂θ (cid:20) ν sin θ ∂∂θ (cid:18) V sin θ (cid:19)(cid:21) , where ρ represents the fluid density (here taken to be constant), µ o is the permeability offree space and ν represents the (turbulent) fluid viscosity.In order to complete the model, the spatial dependence of the transport coefficients ( α , β and ν ) must also be specified. Again, we emphasise that there are no direct observationalconstraints relating to these coefficients — as noted in the introduction, there is no consensusas to their form and there is still a debate as to their order of magnitude (and even their sign).Having said that, it is possible to make some plausible assumptions for an “interface-like”dynamo model (see Bushby 2006 for more details). The precise choices of these parametersare unimportant for our main conclusions.Having set up this model, it is possible to choose a set of parameters so that thesolutions do reproduce some salient features of the solar dynamo (Bushby 2005, 2006). Westress here that, although the parameters have been chosen in a plausible manner, thisdynamo model should not be regarded as an accurate representation of the solar interior 11 –and is subject to many uncertainties. Furthermore we stress again that this is the casewith all mean-field solar dynamo models. However we use this model as a useful tool toanalyse the possibility of producing predictive models of the solar cycle. We proceed bychoosing fiducial parameters and profiles for the turbulent transport coefficients that leadto “solar-type” magnetic activity, with chaotically modulated cycles and recurrent “GrandMinima”. We then integrate this model forward in time to produce a timeseries and designatethis timeseries as the “target” run, which any subsequent model should be able to predict.This target run is shown in Figure 3, which shows a timeseries of the activity togetherwith a reconstruction of the dynamo attractor in phase space. Although this solution ischaotically modulated, it is certainly no more chaotic than the equivalent attractor for the Be data, which is a well-known proxy for solar magnetic activity (e.g. Beer 2000). Whilstthe nonlinear effects are significant enough to drive the modulation, they are actually verydifficult to detect. In this model, the cyclic component of the fluctuations in the differentialrotation (which are driven by the nonlinear Lorentz force) are small compared with themean differential rotation. This is consistent with observations of the (so called) torsionaloscillations in the solar convection zone. Finally, note once more that, since the modulationis driven entirely by nonlinear effects, this model is specified exactly.
The question is then posed as to whether any mean-field model can be constructed thatleads to meaningful predictions of the future behaviour of the target run. Clearly the bestchance for a mean-field model being capable of predicting the future behaviour of the targetrun is to use the exact model that led to the target run data. Hence we test this model first,as all subsequent models will be inferior to this. We proceed by setting the model parametersto be those that generated the long test run, and consider the behaviour of solutions thatare started from very similar points on the attractor. Some of the solutions are shown inFigure 4. This figure shows clearly that although the predictor solutions are able to trackthe target solution for a couple of activity cycles, the nature of the solutions means thatthe predictors and target solution diverge quickly after this time. This is not surprisingbehaviour. It is well-known that chaotic solutions have a sensitive dependence on initialconditions and that long-term prediction of such solutions is fraught with problems (see e.g.Tong 1995). What is clear is that simply using a model that is based upon mean-field theorywill not work in the long term even if the model is correct in every detail . One might be ableto predict one or two cycles ahead if one has solved the problem of constructing an exactrepresentation of the solar dynamo but as noted above this is not an easy task. 12 –We now turn to the related problem of short-term prediction. As discussed in theintroduction there are large uncertainties in the form and amplitude of the input parametersfor all mean-field models. What we investigate here is whether these uncertainties leadto significant difficulties in prediction even in the short term. Again we examine the bestcase scenario and consider a mean-field model for the predictor runs that has correctlyparameterised the form of all the input variables (differential rotation, α -effect, turbulentdiffusivity and nonlinear response). In addition these predictors have been given the correctinput values for all-but-one of the parameters. Hence the predictor models are exactly thesame as the target model with the exception of one input parameter that has been alteredby 5%. This would be a staggeringly good representation should it be possible to achievethis for solar activity. Furthermore we increase the chances of the predictor being able topredict the future behaviour of the target solution by matching the two timeseries over anumber of cycles. This is analogous to the procedure employed by Dikpati et al (2006) whocite support for their forecasting model by assuring that their model agrees with the solarcycle data for eight solar cycles — in reality this is not difficult to achieve with enoughmodel parameters at one’s disposal. Figure 5 shows the results of integrating the predictormodels for two different choices of incorrect parameter. Note that even though the predictorhas been designed to reproduce the target over a number of cycles and that the predictoris very closely related to the target, there is still a good chance that it can get the nextcycle incorrect, with significant errors in (particularly) the cycle amplitude. There are alsoclear variations in the cycle period, which obviously implies that the exact time betweensuccessive cycle maxima is also an unpredictable feature of the system.We stress again that any mean-field model of solar activity includes transport coefficientsthat are still uncertain possibly to an order of magnitude (and certainly not to 5% accuracy).Although the incredible success of global and local helioseismology is placing restrictions onthe form of the differential rotation and the meridional flows, it is unlikely in the foreseeablefuture that significant constraints will be put on the transport coefficients or their nonlinearresponse to the mean magnetic field.
5. Predictions using a reconstruction of the attractor
Having established that there are difficulties in obtaining reliable predictions by fittingmean-field models (even if the modulation is deterministic in origin), it is of interest todetermine whether or not more reliable predictions could be obtained by utilising moregeneral timeseries analysis techniques. In order to reconstruct an attractor from a giventimeseries, it is necessary to define a corresponding phase space. There are various ways of 13 –doing this, but given (any) discrete timeseries, x ( t ), in which the data is sampled at intervalsof ∆ t , the vector X ( t ) = [ x ( t ) , x ( t − ∆ t ) ..., x ( t − ( d − t )] (10)defines a point in a d -dimensional “embedded” phase space (see, e.g., Farmer & Sidorowich1987; Casdagli 1989). Given a time T , the idea of a prediction algorithm is to find a mapping f such that f ( X ( T )) gives a good approximation to x ( T + ∆ t ). The predictive mappingtechnique that is used here uses a local approximation method (see, e.g., Casdagli 1989),which considers the behaviour of the nearest neighbours, in phase space, to X ( T ). By usinga least squares fit, the subsequent evolution of each of these neighbouring points in phasespace is used to construct a piecewise-linear approximation to the predictive map, f . Thisapproximate mapping can then be applied to X ( T ) to obtain an estimate for x ( T +∆ t ). Thisalgorithm can then be repeated to find estimates for x ( T + 2∆ t ) and subsequent points. Theoptimal value for d can be determined by minimising the error of this predictive algorithmover the known segment of the timeseries.The results of applying this predictor algorithm to the target solution are also shown inFigure 5, where the timeseries predictions are shown as crosses. The prediction is started fromthe cycle maximum before the mean-field predictor diverges from the target. Longer trainingtimeseries lead to a more densely-populated reconstructed attractor, which increases theprobability of making more accurate predictions. However, rather than using the entire targetrun, these predictions are based upon (approximately) 50 cycles — this will give a fairercomparison between these results and timeseries predictions that are based upon the realsunspot data. The application of the algorithm to earlier segments of the timeseries suggeststhat a value of d ≥
6. Conclusions
Solar magnetic activity arises as a result of a hydromagnetic dynamo — that much webelieve to be true. As yet, there is no consensus on the location of the dynamo, the dominant 14 –nonlinear or stochastic effects, or even the fundamental processes that are responsible forthe operation of such a dynamo. Although plausible mechanisms have been proposed, asyet none of these are entirely satisfactory. Against this background, there is a drive to beable to predict solar activity with greater accuracy, due to the importance of this activity indriving solar events.What we have demonstrated here is that no meaningful predictions can be made fromillustrative mean-field models, no matter how they are constructed. If the mean-field modelis constructed to be a driven linear oscillator then the small stochastic effects that lead to themodulation will have an extremely large effect on the basic cycle and make even short-termprediction extremely difficult. The second scenario, where the modulation arises as a result ofnonlinear processes rather than stochastic fluctuations, is clearly a better one for prediction— though here too, prediction is fraught with difficulties. Owing to the inherent nonlinearityof the dynamo system, long-term predictions are impossible (even if the form of the model iscompletely correctly determined). Furthermore, even short-term prediction from mean-fieldmodels is meaningless because of fundamental uncertainties in the form and amplitude ofthe transport coefficients and nonlinear response. Any deterministic nonlinear model thatproduces chaotically modulated activity cycles will be faced with the same difficulties.The equations that describe dynamo action in the solar interior are known to be nonlin-ear partial differential equations — the momentum equation is nonlinear in both the velocityand the magnetic field. One indication of the role played by nonlinear effects in the solardynamo is the presence of cyclic variations in the solar differential rotation (the “torsionaloscillations”). Furthermore estimates of the field strength at the base of the convection zoneconsistent with the observed formation of active regions yield fields of sufficient strength(10 − G) for the nonlinear Lorentz force to be extremely significant, whilst the flows arevigorously nonlinear and turbulent. It therefore seems extremely unlikely that the dynam-ics of the solar interior can be described by a forced linear system without throwing awaymuch (if not all) of the important physics. In this case it must be argued not only that thisdiscarded physics is irrelevant to the dynamo process but also that the parameterisation ofthe unresolved physics should not include a stochastic component, as this would have anextremely large effect on such a relinearised system.It is certainly tempting to try to use the observed magnetic flux at the solar surfaceas an input to a model for prediction (whether nonlinear or stochastic, mean-field or fullMHD). Certainly any fully consistent solar activity model constructed in the future shouldbe capable of reproducing the observed pattern of magnetic activity at the solar surface,although this will require a complete understanding not only of the generation process viadynamo action, but also the processes which lead to the formation and subsequent rise of 15 –concentrated magnetic structures from the solar interior to the surface. However it is notclear what role the flux at the solar surface plays in the basic dynamo process. Is it inherentto the process (as modelled by flux transport dynamos) or simply a by-product of the dynamoprocess that is occurring deep within the sun? Estimates suggest that between 5 and 10%of the solar flux generated in the deep interior makes it to the solar surface (e.g. Galloway& Weiss 1981). For the flux at the solar surface to be the key for dynamo action, it mustbe explained why the majority of the magnetic flux that resides in the solar interior playssuch a little part in the dynamics (to such an extent that it does not even appear as a smallstochastic perturbation to the large-scale flux transport dynamo).Finally it is important to stress that even if a model has been tuned so as to reproduceresults over a number of solar activity cycles, then there is a good chance of error in theprediction for the next cycle. Any advection-diffusion system in which one is free to specifynot only the sources and the sinks but also the transport processes can be tuned to reproduceany required features of activity. Moreover, the formulation of a prediction in terms of aparameterised mean-field model does not inherently put the prediction on a sounder scientificbasis than a prediction based on methods of timeseries analysis alone (some of which usevery sophisticated mathematical techniques). This, of course, is not to say that any givenprediction from such a model will be incorrect, just that the basis for making the predictionhas no strong scientific support.We would like to thank Nigel Weiss for useful discussions and for providing helpfulcomments and suggestions. PJB would like to acknowledge the support of PPARC. 16 –
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This preprint was prepared with the AAS L A TEX macros v5.2.
18 – -11 -10 -10 -10 -10 A ^ Fig. 1.— The time evolution of the stochastically-perturbed flux transport dynamo. Top:Timeseries for the mean of the squared toroidal field ( B ) at the base of the convection zone.Bottom: In this figure, B at the base of the convection zone is plotted against the mean ofthe squared values of the poloidal magnetic potential ( A ) at the surface of the domain. 19 – T _ ( n + ) Fig. 2.— The lack of correlation between successive maxima in the stochastically perturbedtimeseries (as shown in Figure 1). Defining T n to be magnitude of the n th maximum, thisplot shows the sequential behaviour of these maxima, plotting T n +1 as a function of T n . 20 – -5 -5 -5 -5 -5 -5 V^22•10 -8 -8 -8 -8 -8 A^268101214B^2
Fig. 3.— The time evolution of the target solution. Top: Timeseries for the mean of thesquared toroidal field ( B ) in the dynamo region. Bottom: An attractor for the targetsolution, in which B is plotted against the mean of the squared values of the poloidalmagnetic potential ( A ) and the velocity perturbation ( V ). 21 – Fig. 4.— Three timeseries showing the evolution of the mean squared toroidal field in thedynamo region. The solid line shows a segment of the target solution timeseries; the dashedand dotted lines show the time-evolution of solutions that are started from nearby points onthe same attractor. Although all solutions have the same model parameters, the timeseriesrapidly diverge after a couple of cycles. 22 –15.735 15.740 15.745 15.750 15.755 15.760Time678910 16.06 16.07 16.08 16.09Time789101112