Grand minima and equatorial symmetry breaking in axisymmetric dynamo models

Transcription

1 Astron. Astrophys. 33, (1998) ASTRONOMY AND ASTROPHYSICS Grand minima and equatorial symmetry breaking in axisymmetric dynamo models John M. Brooke 1,, Jaan Pelt 3,, Reza Tavakol, and Andrew Tworkowski 1 Manchester Computing, The University, Manchester M13 9PL, UK Mathematics Department, The University, Manchester M13 9PL, UK 3 Department of Physical Sciences, Astronomy Division, University of Oulu, P.O. Box 333, FIN-971 Oulu, Finland Tartu Observatory, EE, Toravere, Estonia Astronomy Unit, School of Mathematical Sciences, Queen Mary & Westfield College, Mile End Road, London E1 NS, UK Received 8 April 1997 / Accepted 1 November 1997 Abstract. We consider the manner in which time-periodic solutions of an axisymmetric dynamo model can undergo breaking of equatorial symmetry, i.e. loss of pure dipolar or quadrupolar symmetry. By considering the symmetry group underlying the solutions, we show that the fluctuations responsible for the symmetry-breaking can be constrained such that they are in resonance with the former solution. They can then be amplified until they are comparable in magnitude to the former solution. If the bifurcation is supercritical, the amplitude of the fluctuation is stabilised and a stable mixed parity limit cycle is formed. If it is subcritical it gives rise to a recently identified form of intermittency, called icicle intermittency. This produces episodes in which the original solution and the fluctuation are almost exactly synchronised and the fluctuation grows exponentially in amplitude, interrupted by brief episodes where synchronicity is lost and the amplitude of the fluctuation declines rapidly by several orders of magnitude. During these latter episodes there is a significant dip in the amplitude of the total magnetic field. This model-independent analysis can produce quantitative predictions for the behaviour of this bifurcation and we provide evidence for this behaviour by analysing timeseries from four different mean-field dynamo models, where intermittency is observed without the need for stochastic, or chaotically driven, forcing terms in the dynamo equations. We compare these results with recent work on intermittency in dynamo models and consider their relevance to the intermittency present in solar and stellar cycles. Key words: MHD chaos Sun: magnetic fields stars: magnetic fields Send offprint requests to: John Brooke 1. Grand minima and intermittency 1.1. Modelling intermittent behaviour There has recently been a great deal of interest in dynamo models that show intermittent behaviour (e.g. Schmitt et al. 199). The motivation for this is often to explain certain features of the solar cycle, in particular the intervals of quiescence in sunspot activity (grand minima). This behaviour is often referred to as intermittent without giving a clear and unambiguous definition of its precise meaning. Leaving aside models involving stochasticity, there are in principle two classes of mechanisms that have been invoked to explain such behaviour, namely (i) those involving various notions of intermittency given in dynamical systems theory (DST) and (ii) those based on amplitude modulation, either periodic (Tobias 199) or aperiodic (Weiss 198, Tobias et al. 199, Tobias & Weiss 1997). Intermittency in the sense of DST involves the interruption of one type of behaviour (e.g. periodic oscillations) by bursts of a different type of behaviour (e.g. chaotic oscillations). As the bifurcation point is approached from above, the average interval between such bursts, τ, obeys the limit τ.aswemove past the bifurcation point from below, τ and the original behaviour can eventually no longer be observed (see Sect. 1. for further details). If such a model applies to the solar cycle, then the bursts can be interpreted as the intervals when this cycle is disrupted, i.e. the grand minima. In models of type (ii), a basic cycle undergoes amplitude modulation with intervals of very low amplitude which represent the grand minima. In the references cited above, the modulation is periodic or nearly periodic. If we define τ as above we would expect the variance about τ to be markedly less than for intermittency of class (i), particularly near the bifurcation point. Recent work in DST shows that intermittency is associated with symmetry breaking (see Sect. 1.), for example there could be a change in equatorial symmetry in the solar magnetic

2 3 J.M. Brooke et al.: Grand minima and equatorial symmetry breaking field during the grand minima (Knobloch & Landsberg 199, Sokoloff & Nesme-Ribes 199). Although the two classes of model are distinct, the limited time span of the solar observations makes it difficult to distinguish between them. Weiss (199) considers that the solar records (even extended by proxy data) are insufficient to answer fundamental questions such as whether the chaotic nature of the solar cycle is stochastic or deterministic, although recently it has been claimed that most of the behaviour can be attributed to deterministic chaos (Rozelot 199). Also we cannot study the nature of the intermittent bifurcation as the control parameters are varied (as one would in a fluid dynamics experiment), although we can hope to get some of this information by analysing the magnetic cycle in a range of solar type stars. Another important question is: to what extent is the occurrence of grand minima in the solar magnetic activity a special case? It has been argued that around 3% of solar type stars are undergoing grand minima type behaviour (Balunias & Jastrow 199). If this is the case, then for any proposed intermittency mechanism we must show that it is operative in a range in parameter space and also consider whether the intermittency is in some sense intrinsic to the nonlinear mechanisms effective in this range. In the recent studies modelling the grand minima, it is often stated that the particular model reproduces the solar behaviour qualitatively. This statement is problematic, in that both classes of model described above can make this claim, yet they are fundamentally different both mathematically and physically. Moreover in DST there are several forms of intermittency and considerable theoretical effort has gone into producing signatures (e.g. Bergé et al. 198) that can distinguish between them. Such signatures would be necessary 1 requirements for determining which models best describe the phenonemon of the grand mimima. In this paper we consider a recently-identified intermittency mechanism, referred to as icicle intermittency (see Sect., 3) and show that it occurs in a range of mean field axisymmetric dynamo models. We analyse the behaviour of this mechanism in detail for two main reasons. Firstly we wish to provide criteria by which such mechanisms can be identified from a timeseries of magnetic activity. We thus attempt to make predictions which can be confirmed or refuted by observational evidence. Secondly we present tools and methods of analysis which can apply to other intermittency mechanisms. On the basis of the analysis presented here, and in view of recent work (Brooke & Moss 199, Tworkowski et al. 1997, Tobias 199, Schlussler 199, Covas et al. 1997, Covas & Tavakol 1997) it seems as if intermittency is in some sense a common phenonemon in dynamo models and therefore the crucial theoretical problem becomes one of classification and prediction, rather than just demonstrating that intermittent type behaviour is possible. 1.. Mechanisms for intermittency The term intermittency derives originally from fluid dynamics, where it meant bursts of spatially and temporally incoherent behaviour in an otherwise laminar flow (Batchelor & Townsend 199). The term was given an apparently different meaning in dynamical system theory by Pomeau & Manneville (198), who described intermittency as a periodic signal interrupted by bursts of irregular behaviour. They also produced a scheme for classifying intermittency. By sampling the behaviour stroboscopically at the period of the signal (thus producing a Poincaré map) they reduced the analysis of the intermittency to the loss of stability of a fixed point. The classification was in terms of the way in which the largest eigenvalue of the mapping crossed the unit circle. A crossing at +1 was termed typei,at-1type III and at a pair of complex conjugate values type II intermittency. Scaling laws were deduced for each type, which could then be checked against experimental results. Another form of intermittency that has potential applications to astrophysical dynamo theory is the on-off mechanism (Platt et al. 1993a). In its simplest form this involves two coupled oscillators, one of which is near to a Hopf bifurcation (representing the birth of an oscillating magnetic field) and the other which produces a chaotic or stochastic signal. The oscillators are coupled and the chaotic oscillator moves the other below and above the Hopf bifurcation point (Platt et al 1993b). Here the fixed point represents the magnetically quiescent state. Unlike the Pomeau-Manneville mechanism, the destabilisation of the fixed point appears stochastic or chaotic, and the classification into types I, II or III is not relevant. Despite this, scaling laws can be derived, and different forms of on-off intermittency have recently been distinguished by this method (Lai 199). On-off intermittency can be shown to be one possible manifestation of a blowout bifurcation (Ott & Sommerer 199). Here an attractor in an invariant subspace loses stability to perturbations transverse to the subspace, while remaining stable to perturbations within it. Symmetries of the system give rise to invariant subspaces, e.g. quadrupolar and dipolar equatorial symmetries. Thus we have a link between intermittency and symmetry-breaking, via the on-off mechanism. Recently several studies of intermittent behaviour in meanfield dynamos have exhibited a form of behaviour similar to the on-off mechanism, but where both oscillators are periodic or quasi-periodic. This intermittency will be referred to as an icicle intermittency because it produces self-similar cascades of oscillations in equatorial symmetry as measured by the parity P (see Eq. ()), which resemble icicles formed by freezing water on the branches of a tree or stalactites in a cave (Fig. 1). This icicle intermittency can be shown to be genuinely chaotic (Brooke & Moss 199, Brooke 1997a, hereafter referred to as Papers 1 and respectively) Structure of this paper 1 but not always sufficient in practice We anticipate that this paper will interest two (hopefully overlapping) groups of readers. The first consists of those interested

3 J.M. Brooke et al.: Grand minima and equatorial symmetry breaking 31 parity parity time energy energy time Fig. 1. Plots of parity P (top) and energy (bottom) illustrating the icicle intermittency. Oscillations in P grow away from the pure quadrupolar symmetry P = 1. When they reach a sufficient value they are reinjected back towards pure quadrupolar symmetry. These epsiodes resemble icicles or stalactites. The energy signal is reduced during these episodes. in comparing intermittent timeseries derived from theory with observations. The second group are those interested in the dynamics underlying the behaviour of axisymmetric dynamos, of whatever model. In Sect. we give a qualitative description of the intermittency we are describing. Sect..1 describes the two dynamo models from which the results of Sect. are derived. In Sect. 3 we examine the theoretical connections between various phenomena observed in axisymmetric dynamos and we give a model independent theory which makes specific predictions about the timeseries of magnetic energy. Sect. gives a quantitative examination of these predictions using the techniques of phase dispersion analysis (PDA). Sect. summarises our conclusions, places them in the context of other studies of intermittency and suggests further areas for research.. The model.1. Dynamo models in shells and tori Our description of the icicle intermittency develops from studies of two different mean field dynamo (mfd) models (Tworkowski et al. 1997, Paper 1). They use the standard mfd dynamo equation (Krause & Rädler 198) of the form B t = (u B + αb) (η t B), (1) where u and B are the mean velocity and magnetic fields. As usual, η t is a turbulent diffusivity, which we take to be a constant. We assume the nonlinearity to be of α-quenching type, crudely representing the dynamical feedback of the Lorentz force on the small scale motions. In order to examine the structural stability of our results as the exact form of this nonlinearity is varied (particularly as its functional form is not well established), in addition to the commonly adopted α = α cos θ 1+B, () we also considered two other functional forms, given by Eqs. () and () of Tavakol et al. (199), with the behaviour α B 3 for large B. α = α cos θ B ( ) 3+B 1+B + B 3 tan 1 ( B ), (3) B due to Kitchatinov (1987) and α = α cos θ 1 3B ( 1 B 3(1 + B ) (1 B ) tan 1 (B ) B ), () from Rüdiger and Kitchatinov (1993). In each case α is a constant. The field is split into toroidal and poloidal components by writing B(η, θ, t) = (A ˆφ)+B ˆφ. () where ˆφ is the azimuthal unit vector. Fields with the symmetry B φ (r, θ, t) = B φ (r, θ, t) are called dipolar while those with B φ (r, θ, t) =B φ (r, θ, t) are called quadrupolar. Dipolar and quadrupolar are often referred to as antisymmetric (A) and symmetric (S) respectively, from the symmetry of the toroidal

4 3 J.M. Brooke et al.: Grand minima and equatorial symmetry breaking component. The parity can be measured by the global parity parameter P = E(S) E (A) E (S) + E (A) () (see Brandenburg et al. 1989a). Here E (A) is the energy of the antisymmetric (dipolar) part of the field and E (S) is the energy of the symmetric (quadrupolar) part... Spherical shell models The timeseries 1, and 3 of Fig. a d were derived by integrating (1) in spherical shell geometry. We assume a uniform radial rotational shear, Ω, and the two dynamo numbers that control the system are C ω = Ω R3 /η t, C α = α R/η t, where R is the radius of the outer boundary. Our unit of time is a global diffusion time, R /η t. We took C ω = 1 for the calculations described here, and the models are always substantially supercritical. The thickness of the shell is represented by the parameter r, the ratio of the radii of the inner and outer shell boundaries. The outer boundary condition was taken to be that the field fitted smoothly onto a vacuum exterior solution, and the inner was taken to be a linear superposition of perfectly conducting and penetrative magnetic boundary conditions measured by a parameter F [, 1]. The perfectly conducting and penetrative boundary conditions can be recovered by setting F to and 1 respectively. See Tavakol et al. (199) for further details. Series 1 is derived using () with r =., where C α =. and F =.9. Series is derived using (3) with r =., C α =1.and F =.7. Series 3 was derived using () with r =., C α =., and F =.71. These parameter values were chosen to give typical examples of different forms of intermittent behaviour. For an account of behaviour as the parameters are varied systematically see Tworkowski et al Our aim here is to provide a deeper analysis of the intermittent behaviour of these typical timeseries..3. Torus models Time series of Fig. a d was derived from solving (1) in a torus surrounded by vacuum. This was used to model a dynamo in a thick accretion disk. Details of the toroidal coordinate system and non-dimensionalisation of Eq. (1) are given in Brooke & Moss 199. The units of time are d /η t where d is the major radius of the torus. The torus was thick, i.e. its minor radius was of the same order as its major radius. The form of the alpha-quenching nonlinearity was () and we used a Keplerian rotation law (since the torus does not extend to the rotation axis the Keplerian law can apply to the whole torus). The two conventional dynamo parameters are C α = α d/η t and C ω = Ω d /η t. The boundary condition at the surface of the torus is that the field fits smoothly onto a vacuum field. We apply a regularity condition at the centre of the coordinate system (situated within the volume of the torus). In Paper 1 we give a thorough analysis of the results, here we only use series which shows two very long icicles. Series is obtained by setting C α =.8 and C ω = 1... Intermittent behaviour in the models Three out of four of these series show the icicle mechanism clearly, as well as the presence of other forms of intermittent behaviour. The intermittency manifests itself in changes in both the symmetry and energy (see Fig. 1). It is seen most clearly if the field is resolved into symmetric (S) and antisymmetric (A) components according to the symmetry of the toroidal field about the equator. These can also be referred to as quadrupolar (S) and dipolar (A), respectively. There are also strong theoretical reasons for adopting this procedure (Knobloch & Landsberg 199). In Fig. a d we plot the timeseries of log E, where E is the magnetic energy of each component (throughout this paper all logarithms are to base e). Time is measured in nondimensionalised units related to the diffusion time of the magnetic field (see Sect..1). These plots reveal the structure of the intermittency in the periods of low amplitude particularly clearly. If E is plotted instead, a period of low amplitude oscillations appears merely as a flat signal, hiding the fact that oscillatory behaviour is still occurring. It can be seen in all four plots that one of the components has a much stronger signal than the other and that quite small changes in the stronger signal coincide with dramatic changes in the weaker signal. This is shown most clearly in series and which exhibit episodes where the weaker component grows by over 1 orders of magnitude with a remarkably constant exponential growth rate. These features are the cascades of oscillations termed icicles. In Paper 1 it is shown that these represent oscillations which maintain the same periodic structure as their amplitude grows exponentially, a structure known as a cascade (Mandelbrot 1977). In series this behaviour is interrupted by bursts where the stronger signal (S in this case) loses its periodicity. These occur at the start of the timeseries, where they may represent a transient, then between times 93 and 1 and again between 187 and 193. In Fig. 1 a series is shown where the icicle behaviour is uninterrupted over the whole of the timeseries of 8 units. Series 1 shows a more regular pattern and here the trend in amplitude of the growing component (in this case S) obeys a polynomial rather than an exponential law. In Sect. we show that this trend can be removed to reveal pure periodic behaviour. Also in series 1 the icicles grow typically by at most orders of magnitude, although near time 7 there is a dramatic plunge and recovery. Series 3 shows behaviour much more akin to the usual picture of on-off intermittency. Periods of coherent periodic behaviour in the symmetric component coincide with crashes in the amplitude of the antisymmetic. Otherwise both components oscillate irregularly at comparable orders of magnitude. The general picture is of two nonlinearly coupled oscillators. Mostly these are singly or multiply-periodic and are coherent with each other (this will be quantified in Sect. ). In the icicle behaviour one component has a generally greater amplitude than the other and this amplitude remains constant throughout

5 J.M. Brooke et al.: Grand minima and equatorial symmetry breaking 33 Log of energy antisymmetric energy Andrew Log of antisymmetric energy Log of energy symmetric energy Andrew Log of symmetric energy (a) time in diffusion units (b) time in diffusion units Andrew3 Log of antisymmetric energy Calpha.8 Log of antisymmetric energy Andrew3 Log of symmetric energy Calpha.8 Log of symmetric energy (c) time in diffusion units (d) time in diffusion units 1. Fig. a d. Plots of log E vs time for the timeseries described in Sect. : a, b, c, d show series 1,, 3 and respectively. The upper panel of each plot shows the antisymmetric, and the lower the symmetric, component of the field. The horizontal axis shows time and the vertical axis, magnetic energy. See text for discussion most of the icicle episode, while the weaker signal increases over several orders of magnitude. When the weaker component becomes comparable in magnitude to the stronger there is a perceptible disturbance in the stronger signal, while the amplitude of the weaker decays very rapidly by several orders of magnitude and the cycle restarts. In series 3, on the other hand, the signals are mostly of comparable magnitude and irregular but there are intervals where the symmetric signal becomes periodic and the antisymmetric drops by 1 orders of magnitude. These forms of intermittent behaviour persisted when the signs of both C α and C ω were reversed. We have not observed such behaviour (so far) when C α and C ω have the same sign.

6 3 J.M. Brooke et al.: Grand minima and equatorial symmetry breaking 3. Theory of the icicle intermittency 3.1. Symmetry-breaking, period-doubling and intermittency An important feature of the timeseries introduced in the previous section is the presence of (i) symmetry breaking, (ii) intermittency and (iii) some form of subharmonic bifurcation (we present evidence of this latter in Sect. ). To understand this we note that a survey of published results on nonlinear axisymmetric dynamos reveals certain characteristic patterns of behaviour. One is the Ruelle-Takens route to chaos via the destabilisation of a torus (e.g. Tobias et al. 199, Torkelsson & Brandenburg 199b, Brooke & Moss 199, Paper 1, Feudel et al. 1993). These models range from low order systems of ordinary differential equations (ODEs) to numerical solutions of time dependent partial differential equations (PDEs) with spatial derivatives in dimensions. We can summarise this transition in two ways, firstly by its cyclic behaviour and secondly by the structure of the attractor in phase space. periodic doubly-periodic chaotic limit cycle -torus strange attractor The motion on the -torus (doubly-periodic) has two frequencies that can be related by either a rational or an irrational number. The first gives periodic behaviour with modulated oscillations (see Sect. 1.1), the second gives quasi-periodic behaviour which can appear chaotic but can easily be distinguished by its Fourier spectrum which has distict peaks and no continuous component (see Glazier & Libchaber (1988) for a review). The spectrum of Lyapunov exponents also distinguishes quasiperiodic and chaotic behaviour. This sequence can sometimes also show the phenomenon known as torus-doubling where the torus doubles up in phase space in the manner of a loop being folded in a figure of eight. The torus in phase space behaves somewhat like a flux-tube in a Zeldovitch-Vainshtein dynamo. Since the orbits now go round each part of the doubled torus, a new period twice as long as the main period of the former torus appears, and so this is a subharmonic, or period-doubling, bifurcation. This phenonemon was studied in dynamical systems journals in the early 198s (Arneodo et al 1983, Kaneko 1983) and such behaviour has recently been seen in a 7th order system of ODEs, designed to model the solar dynamo (Feudel et al 1993), and in a numerical solution of a partial differential equation (Paper 1). Another important recurring pattern is that of breaking of an equatorial symmetry of the dynamo solutions. Since the commencement of the study of the stability of mean-field dynamo models (e.g. Brandenburg et al. 1989ab) it has been shown that as the dynamo bifurcates from the trivial solution B =, the stable solutions typically have pure symmetry whether they are steady or oscillating (pitchfork or Hopf bifurcations respectively). As the dynamo parameters increase (in this case C α and/or C ω ) these solutions can become unstable. There is then the possibility of oscillations periodic in both parity and energy (Brandenburg et al 1989ab), or chaotic oscillations in both (Torkelsson & Brandenburg 199abc, Paper 1), and the stability can switch to a solution of opposite parity via an intermediate mixed mode solution (Brandenburg et al 1989ab). Such behaviour has also been studied in low order models and PDE s with one spatial dimension. Jennings & Weiss (1991) (referred to hereafter as JW) showed that these symmetries had to be broken in definite sequences and put the symmetry-breaking on a firm footing by revealing the underlying structure of an 8th order Abelian group which determines the bifurcation structure of all axisymmetric models. This was an important advance because it made the bifurcation structure independent of the model. The particular model can affect the stability of the branches however. Using the procedure of numerical integration forward in time from an initial condition (adopted by all the mean-field studies quoted here) only stable solutions can be detected. We can detect unstable branches which have pure symmetry by integrating over one hemisphere only and imposing suitable boundary conditions at the equator, but we cannot follow unstable mixed parity solutions in this way. It should not be thought that because a solution is unstable it can be ignored, since the aforementioned grouptheory analysis shows that unstable mixed parity bifurcations have an essential role in changing symmetry from quadrupolar to dipolar and vice versa. An extensive study of these symmetrybreaking phenomena, in which unstable branches are followed, is given in Jennings (1991). Building on the analysis of JW, we show that there is a form of intermittency (the icicle mechanism), which unites the three important dynamical behaviours described above and this will be found generally in axisymmetric dynamo models provided that the stability criteria are appropriate. In this sense it is model-independent but may not be seen in every model. It predicts a sequence of behaviour as the parameters near the bifurcation point of the intermittency and also certain characteristic features in the intermittent timeseries. This is the link with observation. 3.. The symmetry breaking bifurcation We follow the notation of JW and examine how solutions with pure quadrupolar or dipolar symmetry (symmetries q and d respectively) and periodic time behaviour (symmetry t τ i ) break the symmetry about the equator (parity symmetry). Here τ is the period and θ is the angle measured from the equator in some axisymmetric geometry. Our notation here differs from that of JW in that we measure θ from the equator rather than the pole, and we explicitly include the period in the notation for the time symmetry. t τ i :(θ, t) (θ, t + τ/), (a, b) ( a, b) d : (θ, t) ( θ, t), (a, b) (a, b) q : (θ, t) ( θ, t), (a, b) ( a, b) Thus d and q are instantaneous symmetries valid at any epoch t. Multiplication of these symmetries gives two time-space symmetries t τ q = dt τ i and tτ d = qtτ i (the group is Abelian), t τ q :(θ, t) ( θ, t + τ/), (a, b) ( a, b) t τ d :(θ, t) ( θ, t + τ/), (a, b) (a, b)

7 J.M. Brooke et al.: Grand minima and equatorial symmetry breaking 3 The symmetries of pure dipolar and pure quadrupolar periodic solutions (denoted d o and q o respectively) are d o q o d, t τ i,tτ q q, t τ i,tτ d From the point of view of the group action, solutions with the above properties can be regarded as a point which is fixed with respect to the three symmetries given above. The point is, however, not fixed under the action of other members of the group, e.g. a point with q o symmetry is not fixed under d or t τ q. We propose to examine the breaking of equatorial symmetry by considering the behaviour of a small perturbation of the opposite symmetry, e.g. we give a q o solution a perturbation with dipolar parity. If the solution is stable this perturbation must decay, however in JW it is pointed out the this solution can break symmetry to a solution with either t τ i or tτ d alone. This solution cannot have both symmetries, or by the product structure of the group, it must then have q symmetry and there is no symmetrybreaking. We now consider how these symmetries constrain the possible dipolar perturbations. For t τ i we consider a quadrupolar field with a dipolar perturbation (the latter is indicated by the δ prefix), ( t ) ( t+ τ/ ) ( t + τ ) +b + δb b δb +b + δb +b δb b + δb +b δb ( +a + δa a + δa )( )( ) a δa +a + δa. +a δa a + δa Here we represent the symmetry changes by an array notation where the upper row represents the values of a or b respectively (see Sect..1) at θ (i.e. northern hemisphere) and the bottom row the corresponding values at θ. The notation is intended to indicate that the absolute values are the same at each half period (due to the periodicity) with the corresponding symmetries shown by the signs. We notice that the t τ i symmetry (applied to the total field) constrains the perturbation to have period τ/(n 1) where n =1,, 3...with τ being the period of the unperturbed field. Also the overall field does not now have the t τ d symmetry which it had before it was perturbed. Thus the field has lost both q and t τ d symmetry while retaining tτ i symmetry. If we constrain the field to have t τ d symmetry, again applied to a quadrupolar field with dipolar perturbation, we obtain ( t ) ( t+ τ/ ) ( t + τ ) +b + δb b + δb +b + δb +b δb b δb +b δb ( +a + δa a + δa ) ( ) ( ) a + δa +a + δa +a + δa a + δa Thus when we sample at periods of τ/, the dipolar perturbation appears to be constant. This means that its period must be τ/n where n =1,, 3... or else a perturbation that is constant in time. This result also holds for t τ q symmetry applied to (7) (8) a dipolar field with quadrupolar perturbation. In both cases the pure spatial symmetry q or d and the pure time symmetry t τ i are lost and we are left with a spatio-temporal symmetry. Thus the symmetries of the solution select the period of the perturbations. If we subject the solution to noise, the symmetries can select from the sea of fluctuations those with the correct periods and we have the possibility that these fluctuations will be amplified by resonance with the former (pure parity) solution. Both t τ i and t τ q have a fixed point which is obtained by sampling at a period of τ. This motivates the notation we have adopted and will be important in the analysis of intermittency (see Sect. 1.). By allowing spatial symmetry breaking while keeping the symmetries t τ i or tτ q, we are selecting the extra degrees of freedom that are permitted to the system. There are many other possible perturbations that can break the spatial symmetry. For example the above analysis leaves no place for subharmonics of the unperturbed solution, in particular perturbations with period τ which are essential if we are to link symmetry-breaking with period-doubling (see Sect. 3.1). For this we need the fixed point at periods τ to become unstable to a period-doubling bifurcation, i.e. the fixed point splits into points which are fixed points at period τ. We therefore consider a perturbation with period τ where the perturbation changes periodically from dipolar to quadrupolar symmetry every quarter period, i.e. τ/, ( t ) ( t+ τ/) ( t + τ ) ( t+3τ/) +δb + +δb δb + δb +δb + δb δb + +δb ( ) ( ) ( )( ) +δa + +δa δa + δa δa + +δa +δa + δa q d q d with the matrix for epoch t +τ (not shown) being identical to that for epoch t. Thus we split the perturbation into quadrupolar and dipolar components, and indicate the dipolar component with a prime ( ). The δ prefix to each field value indicates that we are considering this as a perturbation but we do not show here the unperturbed field. This is because this perturbation can be considered to apply to either a quadrupolar or a dipolar field. We have explicitly shown the values where each component passes through and underneath each array we give the instantaneous symmetry of the field at that phase of the oscillation. In between these phases the perturbation will have mixed parity. The perturbation can be described as a dipolar and a quadrupolar oscillation of period τ separated by a constant phase-lag of τ/. Owing to this symmetry between the quadrupolar and dipolar components, this perturbation could be involved in the destabilisation of a q o or a d o solution. It cannot be accounted for within the group structure of JW, because as well as the symmetry t τ i it has a symmetry orbit, t τ d t τ q (9) t τ d... (1) To see this consider (9) comparing the field at t with that at t+τ, and the field at t + τ/ with that at t +3τ/. Our reasons for introducing this perturbation are twofold. Firstly it allows us to describe the behaviour of the icicle in-

8 3 J.M. Brooke et al.: Grand minima and equatorial symmetry breaking termittency described in Sect.. The solution here alternately returns to and moves away from the original invariant subspace, thus the oscillations in parity alternate between values very close to +1 and values which increasingly move towards -1, thus producing the phenomenon of the icicles described in Sect. 1.. Secondly it allows us to extend the theory of JW to situations where subharmonic, and in particular period-doubling, bifurcations are possible. Other subharmonic bifurcations are also possible, e.g. d o with period τ, but the present choice extends the theory of JW in a simple and clearly motivated way and the prediction of orbiting symmetries has been confirmed (Brooke 1997b). In examining the stability of a fixed point in this way, we are essentially making a Floquet stability analysis, similar to that used to analyse a Hopf bifurcation, but with the addition of considerations of parity symmetry, i.e. of spatial structure. This leads directly to the analysis of intermittency by the Pomeau- Manneville mechanism. In the Hopf bifurcation a fixed point loses stability to a limit cycle. If the original fixed point is obtained from the Poincare map of a periodic orbit (by sampling stroboscopically at the given period), there arises the possibility of resonances between the original limit cycle and the newly formed one. We have seen above that symmetry constraints can lead to the periods satisfying resonance criteria. The Hopf bifurcation can occur supercritically or subcritically. In the former case a stable limit cycle is created, in the latter unstable oscillations appear which result in intermittency. If the resonance is 1:1, then the intermittency will be of type I. This resonance is produced where the symmetry t τ i is present (see above). Type III intermittency is produced where the resonance between the original and bifurcated periods is in the ratio 1: (e.g. by the perturbation given by (9)). If the resonances are other than this, or if the periods are incommensurate, then we have intermittency of type II (see e.g. Manneville (199) Ch.). The essential feature of the intermittent episodes we have termed icicles is that the solutions can be resolved into quadrupolar and dipolar oscillations which are synchronized and the amplitude of one component increases monotonically (in the simplest manifestation exponentially). The symmetry considerations of this section provide a theoretical explanation of this synchronicity. The bifurcation contains therefore all the potential richness exhibited by the Hopf bifurcation plus the additional features of spatial symmetry breaking. In Paper it is shown that this bifurcation can be simply described as on-off intermittency modified by resonances. This linking of the Pomeau-Manneville and on-off mechanisms is an important conclusion and provides a basis for classifying and quantifying symmetry-breaking intermittency on the basis of known results. If the Hopf bifurcation is supercritical a mixed parity solution will be observed, where the quadrupolar and dipolar oscillations are synchronized but with constant amplitude. We would expect the period of the oscillations in parity to be related to the period of the pure parity solution by a simple expression with integer coefficients. It could be that a particular model can switch from mixed parity periodic oscillations to the icicle intermittency as the structure of the model varies. In the case of type III intermittency (unstable period-doubling bifurcation) we have a mechanism that unites the three important dynamical effects mentioned at the start of this section, namely period-doubling, intermittency and breaking of equatorial symmetry.. Testing the predictions in the dynamo models.1. Phase Dispersion Analysis (PDA) In order to detect the resonances and period changes characteristic of the bifurcation described above we must be able to detect the fundamental period of the icicle episodes in the timeseries of Sect.. Our analysis of periodicity in timeseries is based on phase dispersion analysis (hereafter referred to as PDA). PDA is based on the observation that periodic components in a timeseries show up as phase-process diagrams with reduced dispersion (see Fig. ). By this we mean that we select a period and then plot the timeseries with t replaced by t modulo P (thus each t may now have multiple values of the timeseries). If the period chosen corresponds to a period in the timeseries, then the new curve will be well-defined. If not, the values will be scattered and no curve can be distinguished. This is what defines the phase dispersion in the PDA analysis. Instead of traditional Fourier spectra we compute spectra of phase-dispersions for different frequencies. It is important to note that prominent minima in the phase-dispersion spectra indicate actual periodicities in the data set. This differs from traditional Fourier spectra where maxima indicate harmonic components in the series (see Fig. 3). The particular statistic used to measure phase-dispersion is D (ν) = min a,a k,b k N ( f(ti ) a i=1 K a k cos(πνt i )+b k sin(πνt i ) ), (11) k=1 where f(t i ),i =1,...,N is the timeseries considered, ν is the particular frequency point in the spectrum and K is the number of harmonics in the model. Our approach is similar to that used by astronomers to seek periods from variable star light curves (see Martinez & Koen 199 and references therein). All the periodicity analysis was performed with help of a computer package Irregularly Spaced Data Analysis ISDA (Pelt 199). The methods used here apply potentially to any intermittent timeseries, with regular or irregular time steps (which is important in examining the observational records). In examining periodicity we often restrict the examination of the timeseries to episodes of icicle-like behaviour in which the A and S components are locked together. These are interrupted by bursts of chaotic behaviour which dominate the spectrum of the whole of the timeseries, especially in the weaker component. In Tables 1- which follow we indicate the start and end of the episodes which we analyse for periodicity and these can be compared with the plots in Fig..

9 .. Subcritical bifurcation of the torus dynamo J.M. Brooke et al.: Grand minima and equatorial symmetry breaking 37 From the study in Paper 1 it is clear that the bifurcation which produces the intermittency shown in Fig. 1 and timeseries is a subcritical bifurcation from a previously stable quadrupolar (S) solution. We have a two parameter control space (C α, C ω ), which we restrict to one parameter by fixing C ω = 1. As we vary C α from C α =. toc α =.8we go through the bifurcation point which from other evidence can be identified as C α.. See Paper for further details. By the arguments of the previous section, we expect to see a doubling of a basic period as we pass through C α =.. Since the behaviour of the quadrupolar attractor is doubly-periodic just before it loses stability, there will be more than one period which can provide the subharmonic bifurcation. It turns out that this is the longer period, which also gives the strongest signal. In Fig. 3 we show the principal components of the spectra of the S and A components in the coherent regions from t =3tot =, then from t =3tot =, derived from series of Sect. (see Fig. d). The spectra remain constant over both time intervals for all the features we mention here. There are also some interesting dynamical fluctuations that we do not describe here since their amplitude is very small and they do not appear to affect the dynamics of the bifurcation. The spectra are not copies of each other, but the parts where they are linked by the component of largest amplitude are. It is this period that will show the period-doubling bifurcation. In Table 1 we show this period for symmetric and antisymmetric components for values of the bifurcation parameter C α as it crosses the symmetry-breaking point. We determine the basic period by seeking the strongest minima in the phase dispersion spectrum. It can be quite clearly seen that the period doubles as we cross the bifurcation point C α =.. In Paper 1 it is shown that the Poincaré map also shows the doubling of the torus at this point. In Fig. 1 we see the icicles in the parity as the symmetry gets broken. Thus the predictions of Sect. 3 describe this bifurcation accurately. It can also be shown (see Paper ) that the characteristic length of the icicles scales as (C α C αb ) 1 where C αb is the bifurcation point. We say characteristic length, because the growth rate for each icicle is slightly different, as is the initial state from which the growth begins after the previous return..3. Supercritical bifurcation in the shell dynamo We examine the behaviour of the dynamo model responsible for series. It is clear that there is an unstable pure quadrupolar (S) solution. In Table we show the basic periods for S and A components for all the icicle episodes where the components are coherent. The periods are given to the level of accuracy guaranteed by the PDA techniques. The first point to which we draw attention is that the periods of the A and S components match each other to a high level of precision for each episode. The worst discrepancy is for the interval. to.91, where it is a deviation on the level of.1 of the basic period. Moreover this is in all cases smaller than the variation between episodes. This is con- Fig. 3. The phase dispersion spectrum for the exponentially growing regions of series (with the exponential trend removed). Top: from times 3 to, bottom: from times 3 to. The dominant frequency of the A and S components is the same. There are other frequencies which are different in the two spectra. It is interesting that the second icicle has a subharmonic signal whereas the first does not. This is typical of icicle-type behaviour, though the icicles appear similar in structure, there are subtle differences between each episode. firmed remarkably by comparing the episodes between times and where there is a change of period of the order of. of its magnitude, yet the A and S components maintain their closeness. The picture is of trajectories being reinjected towards the marginally unstable attractor, picking up a period from the oscillatory background in its neighbouurhood, then maintaining it throughout the coherent episode until the collapse of the coherence and the reinjection. This is remarkable behaviour, especially when it is considered that the A symmetry grows by up to 1 orders of magnitude in some of these epsiodes. This distinguishing feature of the icicle mechanism is, as far as we are aware, completely novel in the study of the intermittent destabilisation of an invariant subspace. The sudden change in period halfway through the timeseries is also noteworthy, and it will be of interest to see if it occurs more generally. We note here that these subtleties are lost when a spectrum of the whole timeseries is taken, all we see is an average of the basic periods. Not only that, but the incoherent bursts of the

10 38 J.M. Brooke et al.: Grand minima and equatorial symmetry breaking Andrew Log of Andrew Log of time in diffusion units Fig.. Plots of energy of the A (top) and S (bottom) components for timeseries (see Sect. for details). The icicle behaviour appears as a transient, and the attractor is periodic in both parity and energy. This is the result of a supercritical bifurcation, where the amplitude of the parity oscillations is a stable fixed point. Table 1. The variation of the basic period with the bifurcation parameter for the torus dynamo, C α. S is the symmetric (quadrupolar) component, A is the antisymmetric (dipolar). The periods are derived by applying PDA to icicle epsiodes from typical timeseries for each value of C α. Tmin is the time at the start of the icicle episode and Tmax is the time at which the coherence between the S and A components is lost. The period-doubling at the bifurcation point (C α =.) is clearly seen for both components. The timeseries for C α =.8 is shown in Fig. d. C α Symm Tmin Tmax Period Frequency. S A S A S A S A S A weaker component dominate the coherent epsiodes where its magnitude is typically several orders of magnitude smaller. The spectrum in the incoherent episodes is qualitatively different from that of the coherent episodes and dominates the spectrum of the whole timeseries. However we cannot tell from a series produced at a single point in parameter space whether this has been produced by a subcritical bifurcation as above, or whether we are seeing the supercritical bifurcation in reverse, i.e. at a parameter value before the stable mixed parity solution is established. Accordingly we vary the parameters slightly, to C α =1.81, C ω = 1, f =.7, r =.. These values should be compared with those given for timeseries in Sect.. The results are shown in Fig.. We now see that we are examining a supercritical bifurcation in reverse. The prediction of Sect. 3 is that the basic period of the stable solution will be related to that of the unstable pure parity solution by an integer multiple. We present the basic periods in Table 3. By comparing this with Table where the icicle intermittency is the asymptotic attractor we see that the mixed parity solution is associated with a doubling of the period. Thus the period-doubling bifurcation of Sect. 3 has been observed in both supercritical and subcritical forms. In order to illustrate our method of determining the basic period, we show the phase-portraits for the two different regions in Fig. (i.e. icicle and stable parity oscillation). Using the PDA methodology, we find that if we are observing the basic period we have a single pattern on the phase-process diagram over the range to 1. If we have a multiple of the basic period we see a repeated pattern instead. Also in Fig. we show a plot of the antisymmetric component against the symmetric component in the stable and unstable regions of the timeseries, after the exponentially growing trend is removed. We see an orbit expressing the coherence of the two opposite symmetries and in the stable region the orbit has doubled. The fact that the orbits are so well defined shows that the underlying periodicity remains constant throughout the period of exponential growth... Coherent behaviour in the other series If we examine series 1, we see a number of coherent episodes. The picture is of a just unstable dipolar mode. In Table we

11 J.M. Brooke et al.: Grand minima and equatorial symmetry breaking 39 Fig.. Top: periodic orbit in the icicle behaviour of series. The exponentially growing trend is removed and log EA is plotted against log ES. The fact that the orbit is so well defined shows that in the icicles we have an exponentially growing periodic oscillation. Bottom: the same for the stable parity oscillation in series. The orbit has now doubled showing the period-doubling feature of the bifurcation. Fig.. Top two plots: phase diagrams for the icicle behaviour in series at periods.17 and.73. The phase pattern is repeated between and 1 (corresponds to to π) in the top plot showing that.17 is the subharmonic of the basic period.73. The bottom two plots show the same for the stable mixed-parity oscillations. Here the phase diagram for.1 is not repeated and this is the basic period. show the coherence of both components, linked by the same basic period. The rising trend of the symmetric component is now approximated by a polynomial rather than an exponential law. We present phase dispersion spectra of the symmetric and antisymmetric signals in Fig. 7. The strongest signal in both spectra is the first harmonic of the basic frequency (.). The A component has a clearly visible signal at the basic frequency while the S component does not. From Table it can be seen that the transient at the start contains two basic periods, a possible interpretation would be that the shorter period comes from the solution which has just bifurcated. This is a stable solution periodic in parity. Further investigation shows that this is the supercritical bifurcation in reverse. An investigation of parameter space around this bifurcation indicates that there may be more than one bifurcation occurring in this region. It is possible that this additional complexity means that the icicles do not have the simple exponential trend of the previous example. This is currently under investigation. In series 3 (shown in Table ) we see only one brief interval where the symmetries are in coherence, around t = 8. The basic period of the antisymmetric signal is almost exactly twice that of the symmetric. There are three other time intervals where