Phase Space Analysis: The Equilibrium of the Solar Magnetic Cycle

Transcription

1 Solar Phys (2008) 250: DOI /s Phase Space Analysis: The Equilibrium of the Solar Magnetic Cycle D. Passos I. Lopes Received: 26 October 2007 / Accepted: 30 May 2008 / Published online: 21 July 2008 Springer Science+Business Media B.V Abstract We present the results of a statistical study of the solar cycle based on the analysis of the superficial toroidal magnetic field component phase space. The magnetic field component used to create the embedded phase space was constructed from monthly sunspot number observations since The phase space was split into 32 sections (or time instants) and the average values of the orbits on this phase space were calculated (giving the most probable cycle). In this phase space it is shown that the magnetic field on the Sun s surface evolves through a set of orbits that go around a mean orbit (i.e., the most probable magnetic cycle that we interpret as the equilibrium solution). It follows that the most probable cycle is well represented by a van der Pol oscillator limit curve (equilibrium solution), as can be derived from mean-field dynamo theory. This analysis also retrieves the empirical Gnevyshev Ohl s rule between the first and second parts of the solar magnetic cycle. The sunspot number evolution corresponding to the most probable cycle (in phase space) is presented. Keywords Solar cycle: models Sunspots: magnetic fields Sunspots: statistics 1. Introduction The solar cycle is one of the most studied time series in astrophysics; but it is still a mine of information. Understanding the solar cycle is important because through it we can infer properties of the solar structure and phenomena within. Moreover, the influence of the Sun s D. Passos I. Lopes ( ) CENTRA, Departmento de Física, Instituto Superior Técnico, Av. Rovisco Pais, Lisboa, Portugal lopes@fisica.ist.utl.pt D. Passos dariopassos@fisica.ist.utl.pt I. Lopes Departamento de Física, Universidade de Évora, Colégio António Luis Verney, Évora, Portugal

2 404 D. Passos, I. Lopes radiative and magnetic variability on the Earth s climate system is very poorly understood (Reid, 2000; Marsh and Svensmark, 2003) and as the solar cycle is the main solar variability driver, it is important to understand it and its connection to Earth s climate. The most used proxy to represent the solar cycle is the sunspot number, SSN, because the data records cover a large time period. It is currently believed that the mechanism responsible for this cycle is a magnetic dynamo operating in the convection zone (Parker, 1955; Solanki, 2003). The low dynamics produced by the dynamo action presents features, such as magnetic field reversal, that are also observed in recent laboratory dynamo experiments (Berhanu et al., 2007; Dubrulle et al., 2007). All solar physics areas have explored the solar cycle, ranging from helioseismology (e.g., Chaplin et al., 2007; Houdek et al., 2001; Libbrecht and Woodard, 1990) to irradiance changes, period variations, and neutrino flux (e.g., Usoskin and Mursula, 2003; Solanki and Krivova, 2005; Walther, 1999; Rogers and Richards, 2006; Schatten, 2003). In this work, we propose a method to compute the most probable solar magnetic cycle using the phase space of the toroidal magnetic field on the Sun s surface. It is shown that the solar magnetic cycle that is originated by the action of a complex interaction of different physical mechanisms (e.g., magnetic fields, differential rotation, and turbulent convection) evolves in the direction of an equilibrium state. In dynamical system analysis this equilibrium state is known as an attractor. The variability of the solar magnetic fields implies that the equilibrium in never reached, but rather the system oscillates randomly around the equilibrium solution. Furthermore, it is shown that the most probable cycle is well described as a limit cycle of a van der Pol oscillator. Our results are in agreement with a previous dimensional analysis of a mean-field dynamo theory that shows that the evolution of the toroidal magnetic field over time can be described as a van der Pol oscillator (e.g., Mininni, Gomez, and Mindlin, 2001; Pontieri et al., 2003). In Section 2 we discuss some basic properties of the solar magnetic cycle observed over the past three centuries and in Section 3 we compute the most probable solar cycle for this period based on the phase space. We discuss some of the properties of this phase space and introduce the basic approximation of the derivation of the van der Pol equation that determines the evolution of the toroidal magnetic field at the surface. Finally, in Section 4 we discuss the results obtained in this work. 2. Simple Solar Cycle Analysis In this work we used the monthly average international SSN since 1750 to the present (see Figure 1). We split the data into individual cycles and we did not consider any overlapping. The identification of the minima and maxima was the same one suggested in the database from where the data came (NOAA). For each cycle we made a Gaussian fit and retrieved the amplitude, standard deviation, and residual error. In Figure 2 we present the evolution of these statistical parameters over the studied period. This figure shows that there is a correlation between the maximum amplitude curve and residual error and an anticorrelation between cycle length and residual error. This translates to the already known relation between the cycle length and intensity: Usually shorter cycles tend to be high in amplitude and longer cycles tend to be lower in amplitude. Moreover, what we would like to refer to here is that the residual fit error can be interpreted as a measure of the asymmetry between the cycle s rise and fall. The bigger the residual error, the bigger is the asymmetry between these two phases. This implies that a symmetrical function such as a circular function (e.g., sine) or a

3 Phase Space Analysis: The Equilibrium of the Solar Magnetic Cycle 405 Figure 1 Monthly sunspot numbers since Overlayed is the maximum of each cycle found after a Gaussian fit to each one. The dashed curve represents the evolution of the maxima (dots connected by splines). Gaussian does not take into account the difference between the rise and fall of the sunspot number during the solar magnetic cycle, making them inappropriate for describing the phenomenon. A way of finding more information about a function that better describes the solar cycle is by looking to its phase space. 3. Phase Space Analysis: The Most Probable Solar Cycle According to Pontieri et al. (2003), the sunspot number is proportional to the square root of the magnetic field generated by the solar dynamo. Thus, we can get a time series of a quantity that is proportional to the magnetic field, B(t) ±SSN 2, where the signs follow the approximate 11-year inversion of the field (Bracewell, 1953). Moreover, since sunspots are a direct consequence of the toroidal magnetic field that emerges at the surface, this B(t) can be more correctly interpreted as proportional to the toroidal magnetic field component. To create a phase space, we also need to compute db/dt =[B(t + ) B(t )]/2, where in our case = 6 months is the time step. We smooth the SSN signal (using a FFT filter) in order to clear small-time-scale sharp perturbations (smaller than 2 years) and leave only the average behavior of the cycle (see Figure 3). To compute the average magnetic cycle, we divided the phase space into 32 sections (i.e., 32 different temporal instants). For each section we calculated the average B(t) and db/dt. We also calculated the statistics for the step direction and length in B(t) and db/dt. The average direction vector for each section is superimposed in Figure 4 and gives us the most probable path in phase space.

4 406 D. Passos, I. Lopes Figure 2 (a) Evolution of the fits amplitudes. (b) Evolution of the fits standard deviation. (c) Error associated with each fit. We can see that the data revolve around a central trajectory (the most probable cycle) but with some degree of randomness that depends on the phase of the solar magnetic cycle. In this diagram we can clearly see the change in rates in the rising and falling intensities of the field, which are given by db/dt. For example, in Figure 3(b), in the upper left and lower right quadrants, we can see that db/dt makes a small kind of depression. This depression corresponds to the bump seen at the beginning of the waning part of the sunspot cycle and seems to be an intrinsic characteristic of the system. This result seems to indicate that the evolution of the solar cycle is very likely regulated by a set of magnetic hydrodynamic equations (e.g., dynamo theory) that forms a dynamical system that evolves after enough time to an attractor (Priest, 1984). In such a case, the attractor can be represented as a limit closed curve on the phase space, here represented by the most probable cycle. This kind of analysis also allows us to gain better insight into the behavior inside the solar cycle as a dynamical system and not from cycle to cycle independently.

5 Phase Space Analysis: The Equilibrium of the Solar Magnetic Cycle 407 Figure 3 (a) Time evolution of B(t) with the original signal used in this analysis in gray and the smoothed signal (FFT filtered) in black. (b) Phase diagram of the smoothed data using = 6 months. Figure 4 Phase diagram divided into 32 sections represented by different colors. Superimposed are the average direction vectors of each section of the solar cycle. The direction vectors indicate the evolution during the cycle. It was noticed by Polygiannakis and Moussas (1996) that this diagram resembles the phase space diagram of a van der Pol oscillator and some authors have already been able to show that it is possible to get a van der Pol oscillator type for the toroidal magnetic field equation based only on a truncated version of dynamo theory (Mininni, Gomez, and Mindlin, 2001; Pontieri et al., 2003). These authors used mean-field theory and the expression for the induction equation (see, e.g., Priest,1984) in spherical coordinates B φ t A p t = rv p (B φ /r) + r [ (A p ê φ ) ] Ω + η ( 2 r 2) B φ, (1) = r 1 v p (ra p ) + αb φ + η ( 2 r 2) A p, (2) where v p represents the poloidal component of the average velocity field, Ω is the angular velocity, η is the magnetic diffusivity, and B φ and B p = (A p ê φ ) represent the toroidal and poloidal components of the magnetic field, respectively. The term Ω represents the differential rotation of the Sun. After some algebra, the authors showed that the toroidal

6 408 D. Passos, I. Lopes Figure 5 The black line represents the fit to all data points (in gray), the crosses represent the average cycle, and the dashed red line represents the fit to the average cycle. magnetic field, which is related to the eruptions of sunspots, can be written as with d 2 B φ dt 2 = ω 2 B φ μ db φ ( 3βB 2 dt φ 1 ) + λbφ 3, (3) ( ω 2 vp = 1 ) 2 Ω 0 α 0 S ɛ, (4) ), (5) ( vp μ = 2 1 α 0 S β = γ 0 μ, (6) ( vp λ = 1 ) γ 0 Ω 0τ α 0 S ɛ, (7) where α 0 is a typical plasma velocity, S is the dimensionless magnetic Reynolds number, Ω 0 is the frequency associated with differential rotation in dimensionless units, ɛ = 0.1 is a dimensionless constant associated with the length of the upper convective layers, γ 0 is a constant associated with magnetic buoyancy, and τ is the saturation time for the alpha effect in dimensionless units. Consult Pontieri et al. (2003) and reference therein for further details. Equation (3) can be described by using the dynamical system s standard form. In an analogous way to Mininni, Gomez, and Mindlin (2001), and using a least square fit method (with the Levenberg Marquardt algorithm), we were able to find that the parameters of Equation (3) that best fitted our average cycle are ω = , μ= , ξ = , λ= (8) For the average solar magnetic cycle we confirm that the limit curve obtained can be qualitatively described as the solution of a van der Pol type oscillator (see Figure 5).

7 Phase Space Analysis: The Equilibrium of the Solar Magnetic Cycle 409 Figure 6 The dashed line represents the smooth SSN experimental data used in this analysis and the thick black curve and dots represent the most probable cycle found by using this technique. The error bars represent 1σ of the distribution of B(t) in each sector. We can see that the average cycle fit is different from the van der Pol fit to all data points that were previously covered by Mininni, Gomez, and Mindlin (2001). If we do a fit to all data points we get ω = , μ= , (9) ξ = , λ= These values are very similar to those found in Mininni, Gomez, and Mindlin (2001) and differ only because we used a different sunspot data set. Although the parameters obtained here are qualitatively similar to the average cycle fit, they differ by almost 20%. This might be relevant in more complex and accurate models where physical quantities, such as those present in Equations (4)to(7), could be used, in principle, to impose boundaries to dynamo theories. With the B values of the most probable cycle we can do the inverse of the process described in the beginning of this section and plot the respective time evolution of the average SSN cycle (Figure 6). As would be expected we recover two average sunspot cycles. Interestingly, these two cycles follow the Gnevyshev Ohl s rule. In the context of the phase space this empirical rule can be a consequence of the system s constant search for equilibrium; that is, the trajectory tends to the attractor (average cycle). 4. Discussion and Conclusions A simple conventional analysis of the solar cycle (represented by the sunspot number data) points to the existence of an asymmetry between the rising and falling phases of the cycle. After relating the sunspot number with surface magnetic fields, we present a statistical analysis of the solar cycle based on the phase space of the toroidal component of the magnetic field at the surface. This method, which we used to compute the most probable cycle, has the advantage of being time independent and can be used to infer properties of the magnetic structure from which the cycle originates. This leads us to believe that the solar cycle can be described as some type of nonlinear oscillator that has an attractor as solution.

8 410 D. Passos, I. Lopes Following the suggestions of other authors we positively matched this average cycle as the solution of a van der Pol oscillator. If we consider the solar magnetic cycle as a dynamical system, the average cycle found here is our best estimative for the equilibrium solution. We also believe that this oscillator is a signature of the dynamo that operates within the Sun. The variable trajectories in the phase space indicates the existence of a physical mechanism that induces variability into the dynamo process. This variability can come from several sources: solar differential rotation, meridional circulation, oscillations in the tachocline, or even other phenomena. We believe that the large-scale dynamo action is responsible for the low order dynamics that originates the solar cycle but not for the fluctuations that it presents. Also, in the future, after introducing more realistic models, these results might allow constraints to be imposed on solar dynamo theories. This analysis also returns us the empirical Gnevyshev Ohl s rule between the first and second half of the magnetic cycle. This indicates that the mechanism responsible for the variability has a more active role in one half of the magnetic cycle. As this seems to be a characteristic of the magnetic cycle over time, dynamo models should try to emulate this behavior. Another idea to implement in the future is to feed the trajectories of each cycle in the phase space into a neural network and see whether it can learn some characteristic behaviors, leading ultimately to a prediction technique. In this case the level of smoothing applied to SSN data will play an important role in the neural network learning curve. Acknowledgements We would like to thank and acknowledge the useful comments of the anonymous referee, which allowed us to improve and widen the scope of our work. This work was done under the financial support of Fundação para a Ciência e Tecnologia and Fundação Calouste Gulbenkian. References Berhanu, M., Monchaux, R., Fauve, S., Mordant, N., Pétrélis, F., Chiffaudel, A., Daviaud, F., Dubrulle, B., Marié, L., Ravelet, F., Bourgoin, M., Odier, Ph., Pinton, J.-F., Volk, R.: 2007, Eur. Phys. Lett. 77, Bracewell, R.N.: 1953, Nature 133, 512. Chaplin, W.J., Elsworth, Y., Miller, B.A., Verner, G.A., New, R.: 2007, Astrophys. J. 659, Dubrulle, B., Blaineau, P., Mafra Lopes, O., Daviaud, F., Laval, J.-P., Dolganov, R.: 2007, New J. Phys. 9, 308. Houdek, G., Chaplin, W.J., Appourchaux, T., Christensen-Dalsgaard, J., Däppen, W., Elsworth, Y., Gough, D.O., Isaak, G.R., New, R., Rabello-Soares, M.C.: 2001, Mon. Not. Roy. Astron. Soc. 327, 483. Libbrecht, K.G., Woodard, M.F.: 1990, Nature 345, 779. Marsh, N., Svensmark, H.: 2003, Space Sci. Rev. 107, 317. Mininni, P.D., Gomez, D.O., Mindlin, G.B.: 2001, Solar Phys. 201, 203. Parker, E.N.: 1955, Astrophys. J. 121, 491. Polygiannakis, J.M., Moussas, X.S.: 1996, Solar Phys. 163, 193. Pontieri, A., Lepreti, F., Sorriso-Valvo, L., Vecchio, A., Carbone, V.: 2003, Solar Phys. 213, 195. Priest, E.R.: 1984, Solar Magnetohydrodynamics, D. Reidel, Dordrecht. Reid, G.C.: 2000, Adv. Space Res. 94(1/2), 1. Rogers, L.M., Richards, M.T., Richards, D.St.P.: 2006, astro-ph/ Schatten, K.H.: 2003, Adv. Space Res. 32(4), 451. Solanki, S.K.: 2003, Astron. Astrophys. Rev. 11, 153. Solanki, S.K., Krivova, N.A.: 2005, Solar Phys. 224, 197. Usoskin, I.G., Mursula, K.: 2003, Solar Phys. 218, 319. Walther, G.: 1999, Astrophys. J. 513, 990.