SOLAR MAGNETIC FIELDS: CREATION AND EVOLUTION

Transcription

1 THE MULTIDISCIPLINARY UNIVERSE - A CONFERENCE IN HONOR OF JORGE DIAS DE DEUS Lisbon, Portugal, July 14 15, 2011 SOLAR MAGNETIC FIELDS: CREATION AND EVOLUTION Dário Passos CENTRA-IST, University of Évora and GRPS-University of Montreal dariopassos@ist.utl.pt Keywords: Sun, Solar cycle, Solar Dynamo Model, Solar Magnetic Fields, Solar Variability Abstract. This work starts with a small review about the characteristics of the solar magnetic field and how this field might influence our daily life. Then the solar dynamo theory is presented in a compact/resumed way showing its main characteristics and its different implementations in solar dynamo models. Based in one of these dynamo types, the flux transport dynamo, we develop a simplified version of it, i.e. a low order dynamo model (LODM) that retains the majority of the physical mechanisms necessary for the dynamo operation. Furthermore several applications for the LODM working in the kinematic regime are proposed and some are presented in a more detailed way. By pairing the LODM with a proxy of magnetic activity in the Sun, dynamo parameter s evolution and reconstructions back in time past are presented for some physical quantities. In the final section we explore the non-kinematic dynamo regime where the Lorentz force can not be neglected. This is done by expanding the LODM with a term that takes into account the backreaction of the magnetic field into the meridional circulation. Preliminary results show that this backreaction is indeed important in the long term evolution of the dynamo and that it should be included in other dynamo models.

2 1 INTRODUCTION It has passed just a couple of decades since we became aware of the importance that the Sun has in regulating the space environment around our planet. Taking into consideration that our present way of life relies heavily on technological systems that depend directly of satellite operations and energy and communication networks, this subject becomes even more interesting. All the aforementioned infrastructures can be endangered by extreme space climate events originated by strong solar activity. Mostly known for the cyclic appearance of dark spots in the Sun s surface (sunspots), the solar cycle, with its eleven years periodicity, is a consequence of the underlying 22 year solar magnetic cycle. Noteworthy is the fact that the peak amplitude or strength of this cyclic activity changes from cycle to cycle, and its most of the times referred as solar variability. Around solar cycle maximum, explosions in the solar photosphere, flares, release copious amounts of radiation into space. Some times these events are accompanied by bursts of solar plasma, coronal mass ejections (CMEs), that travel at several hundred kilometers per second into space. It is the violent combination of these two phenomena that can create solar magnetic storms. Also, the stronger the cycle is, the higher is the probability of occurrence of these events. Constantly streamed by the solar wind, the interplanetary space in the solar system is prone to all kinds of solar stimuli and modulations creating what is nowadays called Space Weather [1]. In occasions, when the Sun bombards the Earth s magnetosphere (the Earth s magnetic field) with radiation and CME s the conditions of space around our planet change severely. The radiation can increase to harmful levels endangering astronauts in the International Space Station or during space walks; the electronics of satellites can also be affected and even high latitude plane travelers can be exposed to high doses of radiation. This is also an issue for the impending industry of space tourism. A strong radiation burst can also increase the temperature of the ionosphere, the upper layer of our atmosphere making it expand. This creates an additional problem for low orbit satellites since the expanded ionosphere will increase the drag forces that act upon the spacecrafts and shorten its life cycle resulting in losses of millions of euros [2]. Also, the ionized plasma carried by CMEs and the solar wind, can originate atmospheric currents at several altitudes that can again damage the electronics of satellites or even create power surges that disrupt power and communication distribution networks. Besides the possible technological problems that might advent from magnetic storms, Mankind also faces the challenge of understanding the Sun s influence in the Earth s climate [3]. Although the International Panel for Climate Change (IPCC) has attributed small weight to the Sun n influence in the climate, there are strong evidences that in the past the Sun was the major driver behind the variations in the Earth s climate. Strong correlations exist between the Earth s temperature and solar grand minima. These grand minima episodes correspond to extended periods (a few decades) where very low or no solar activity occurred. During theses periods no sunspots (or very few) are observed in the solar photosphere and it is believed that other solar phenomena also exhibit low levels of activity. The most famous grand minima that has been registered is the Maunder

3 Minimum which occurred between the years of 1645 and 1715 [4]. Interestingly enough, during the same period, the northern hemisphere was subjected to very long and cold winters and short mild summers. This period is also known as Little Ice Age. In the last decades, fueled in part by the global warming problem, much was learned about the interaction mechanisms between the Sun and the Earth. Cosmic ray modulation, magnetospheric coupling and irradiance variations are just a few examples of active research fields that make this a hot topic heavily debated in many scientific circles. 2 THE SOLAR DYNAMO The heart of most of the phenomena that contribute to the presented issues, is the solar magnetic field. Observations of strong magnetic fields inside sunspots by Hale in the beginning of the XX century [5] seeded the need for a theory that explained its origin. Due to the works of Parker in the 1950 and Krauss & Rädler short after the concept of solar dynamo was born [6] [7]. Nowadays it is widely accepted that the Sun produces magnetic fields and that these evolve according to the dynamo concept. The idea behind the solar dynamo theory is that the kinetic energy from turbulent plasmas motions and solar rotation is transformed into magnetic energy [6]. Mathematically speaking this translates in solving the magnetohydrodynamic (MHD) induction equation B t = (u B λ B), (1) where B is the magnetic field, u is the velocity field of the solar plasma and λ is the microscopic magnetic diffusivity. The plasma characteristics in the solar convection zone are such that the magnetic is said to be frozen to the plasma flows, i.e., if the plasma moves, the field moves along [8]. Current solar dynamo models rely heavily on the mean field approximation [7]. In this theoretical framework a scale separation between turbulent small scale phenomena and large coherent phenomena is assumed. Equation 1 is usually solved for the large scale (laminar) fields and flows while the small scale phenomena related to turbulence is usually subsumed into constant coefficients that will appear in the expansion of this equation. For the Sun, due to a relatively strong rotation, the axisymmetric approximation can be used very efficiently. Working in spherical polar coordinates (r, θ, φ) field and flows are separated into two components: a toroidal component in the direction of rotation, φ and a poloidal component in the (r, θ) plane u(r, θ) = v p (r, θ) + rω(r, θ)ê φ, (2) B(r, θ, t) = (A(r, θ, t)ê φ ) + B(r, θ, t)ê φ ), (3) where r = r sin(θ), v p is the plasma velocity component in the (r, θ) plane also called meridional circulation and Ω is the angular velocity of rotation, B φ is the toroidal magnetic field component and A p is the vector potential that describes the poloidal magnetic field. The solar differential rotation profile (strong flow) is well mapped for the solar convection zone (SCZ) by Helioseismology and its inclusion on dynamo models it s

4 mandatory. Of crucial importance is a thin layer in the base of the SCZ, the tachocline, where the differential rotation presents a strong radial shear. This is the region where B φ is thought be generated. Nevertheless the meridional flow (weak flow) is only mapped for the outer 2% of the solar radius. This lack of information about this flow component and the fact that it is considered a weak flow, made the first dynamo models skip it in their mechanics. Also note that equation 2 is not time dependent, which is the essence of the kinematic approximation. In this kinematic regime it is assumed that the large scale flows are strong enough to resist the backreaction of the magnetic field, i.e., a steady flow configuration will determine the evolution of B. Substituting equations 2 and 3 into 1 and separating them yields the two field component evolution equations given by B φ t A p t = η ( 2 1 r ) 2 B φ + 1 r ( ) ( rb φ ) η r r rv Bφ p B φ v p r + r [ (A p ê φ )] Ω (4) = η ( 2 1 r ) A p v p r ( ra p) + αbφ, (5) In order to bypass Cowling s anti-dynamo theorem and achieve dynamo action, the term α B φ was added to the r.h.s. of equation 5. This term, also called α-effect acts as a source term for the poloidal component of the field and its extracted from the average properties of turbulence. Depending on the inclusion of meridional circulation and on the nature of this α source term term (e.g. helical turbulence, decay of active regions at the surface (Babcock- Leighton mechanism), etc.) we define different dynamo types. Also note that λ, the microscopic magnetic diffusivity, has now been substituted by η. This is due to the fact that when we resort to the mean properties of turbulence to extract α we get an additional term that behaves exactly like λ, i.e. we also get magnetic diffusivity though turbulence. Usually this turbulent diffusivity is much more efficient then its microscopic analog and it is a current practice to use this symbol to represent the magnetic diffusivity of the system. The pictorial idea is that an initial aligned poloidal field A p (in dipolar configuration) is stretched and winded up around the Sun in the φ direction by the differential rotation (it acts as source for B φ ). This stretching process happens in the tachocline and increases the magnetic field until buoyancy instabilities appear. After reaching a certain threshold, B φ becomes buoyant and rises through the SCZ piercing the photosphere and originating sunspots. During the rising process and/or on the photosphere, B φ will the acted by the α-effect transforming it again in A p and closing the cycle. Different prescriptions for the α-effect, with or without quenching by the magnetic field and at different locations in convection zone and also different parameterizations for the magnetic diffusivity and large scale flows originate a vast array of models. These models usually solve the previous equations in 2 dimensional spherical axisymmetric grid and return information about the field spatial distribution and evolution in time. The main goal of any dynamo model is to reproduce the observed solar cycle features and this is what conditions their success.

5 For a complete review on the several existing dynamo model c.f. [9]. Currently the most successful models are the so called flux transport dynamos that include the meridional circulation. These type of models overcome theoretical constraints of other types of models related with the efficiency of the α mechanism. In these models the meridional circulations acts as a conveyor belt transporting field at the solar surface from the lower latitudes to the poles. By the time these fields that come from the decay of active regions reach the pole they are in the poloidal configuration. At high latitudes, near the poles, this circulations sinks down transporting the poloidal field down through the convection zone. Upon reaching the tachocline these fields are acted upon by the shearing action of rotation, transformed and stretched into toroidal fields and buoyancy take care of closing the cycle. In this type of models the meridional flow control not only the period of the cycle but also its amplitude. 3 A LOW ORDER DYNAMO MODEL The dynamo origin of the solar magnetic field is solid enough that even a simplified model of the solar cycle can capture most of features observed. That is the case of the 1D low order dynamo model (LODM) developed by [10], [11] that focuses on the evolution of the amplitude of the toroidal component of the solar magnetic field, B φ. This component was chosen for modelling because it can be directly related to the sunspot number, the longest time series of solar activity available. It is important to stress out that when we are dealing with low-order models, care must be taken when comparing their physical quantities with their typical parameterizations found in 2.5D or 3D models of the Sun. For example in this LODM the meridional circulation acts as a source term and not like a pure transport term found in higher dimension dynamo models. Also to be taken into account is the inclusion of flux removal by the process of magnetic buoyancy which has an important effect in the evolution of this dynamical system. All these features might induce different dynamics from the ones that are seen in 2D spatially resolved dynamo models. Parallels can be made but the pointed differences must be taken in mind. Nevertheless the long term evolution dynamics of the system remains basically the same and that is what this model aims to capture. We start from equations 4 and 5 and introduce the following simplifications: assume an average magnetic diffusivity for the entire convection zone ( η/ r = 0), plasma incompressibility and that magnetic buoyancy is actively acting in the system. After that we introduce a dimensional truncation as suggested in [12] to collapse all spatial coordinates by substituting 1/l 0, where l 0 is a specific length of interaction for the large scale magnetic fields, usually taken in the range l 0 [0.01R, 0.3R ]. The model base equations are now db φ dt da p dt = c 1 B φ + c 2 A p c 3 B 3 φ, (6) = c 1 A p + αb φ. (7)

6 with the structural coefficients, c n, defined as ( 1 c 1 = η 1 r ) 2 v p, (8) l 0 l 2 0 c 2 = rω, (9) c 3 = l 2 0 γ 8πρ, (10) where η now represents an average turbulent magnetic diffusivity for the bulk of the solar convection zone (SCZ), v p represents the average amplitude of the meridional circulation, r and l 0 are scaling terms of the system (related to the size of the solar convection zone), Ω corresponds to the amplitude of the solar differential rotation and α (with α 0) represents the regeneration mechanism(s) between toroidal and poloidal components (the α-effect). The coefficient c 3 is related to the flux removal by magnetic buoyancy that as suggested by [13] that depends on a removal rate, γ, and an average density value for the SCZ, ρ. This coefficient will act in the same way as the diffusivity, i.e., it stops the field from growing indefinitely when dynamo action is achieved. In the parameter regime presented in this work this is the main limiting mechanism to dynamo growth. In dynamical system s, these structural coefficients control several aspects of the solution of system, namely c 1 and c 3 control the field amplitude, c 1 also controls the cycle s rise and decay asymmetry while the period is controlled mainly by c 2 (c 1 also appears to have a small influence). A typical solution is presented in Figure 1, by solving equations 6 and 7 for an small seed field (i.e. initial conditions B(0) = 0.1 and A p (0) = 0.1). (A) f Figure 1: Typical solution of the LODM after stabilization which takes place after 30 years approximately (to evolve from the used initial conditions). In black we have B 2 φ, the usual proxy for the solar cycle and in red A 2 p. The solution were generated using c 1 = 0.1, c 2 = 0.95, c 3 = and α = 0.1 We pay special attention to the B φ component as it will be more useful for comparison with solar data later on. By deriving equation 6 in order to the time, and making the

7 adequate substitutions of equation 7 one can rewrite this dynamo system as d 2 B φ dt 2 = ( 2c 1 3c 3 Bφ 2 ) db φ ( c 2 ) 1 c 2 Bφ + c 3 c 1 Bφ 3, (11) dt The main solar cycle characteristics that this model successfully reproduces are: dynamo action producing coherent magnetic field with pole reversals at a period of approximately 11 years, correct amplitude and phase difference between B 2 φ and A2 p and cycle s rising time shorter then decaying time. 4 KINEMATIC LODM APPLICATIONS 4.1 Example of application 1: evolution of dynamo parameters A good example of an application for this LODM is the inference of dynamo parameters evolution through time by pairing the model with observational data [10] [15] [16]. The following strategy is used. As a first approximation we assume that the structural coefficients can change only discretely in time, more specifically from cycle to cycle while the magnetic field evolves continuously. Different coefficients generate theoretical solutions with different amplitudes, periods and eigen-shapes. By comparing these different solutions with the observed variations in the solar magnetic field, we are able to infer information about the physical mechanisms associated with the coefficients. The second step is to look for a solar observable that could translate in a good proxy for the magnetic field, e.g., F10.7 radio flux, aa index, etc. Since we are interested in a signature of the toroidal field component we use the sunspot number record for this purpose. Taking into account that sunspots are produced by the toroidal field that erupts at the photosphere we follow the suggestions of [17] and create a proxy for B φ based on the sunspot number by assuming that B φ SSN, i.e., the sunspot number is proportional to magnetic energy. For simplifications purposes we take this proportionality as 1. The International monthly averaged SSN since 1750 to the present is used to construct this proxy. The time series is smoothed with the application of an FFT filter low pass filter (τ = 24 months), which retains the main features of the cycles. We also impose a sign change every cycle to emulate field polarity reversals (this sign change takes place at solar minima). The result of this procedure is shown in Figure 2. With the proxy built we separate the data into individual cycles and fit equation 11 to each one individually, considering that the buoyancy properties of the system are immutable, i.e. c 3 is constant throughout the time series. This means that when we fit equation 11 to solar cycle N, we will retrieve the set of c nn coefficients that best describe that cycle. This allows to probe how these coefficients vary and consequently how the physical mechanisms associated with them evolve in time [16]. The result of the fit procedure is shown in Figure 3.

8 Figure 2: Black dashed line represents the built proxy for the toroidal field. B φ is obtained by calculating SSN, changing the sign of alternate cycles (represented in gray), and smoothing it down. Vertical thin dotted lines represent solar cycle minima. c 2 c 1 0,1 6 0,1 2 0,0 8 0,0 4 0,0 0-0,0 4-0,0 6-0,0 8-0,1 0-0,1 2-0, C y c le N Figure 3: Values of c 1 and c 2 found by the fitting procedure (c 3 = 0.002). The vertical error bars represent the standard deviation of each sample Modelling the evolution of solar activity Since the fitting process involves errors and approximations, we perform a test to see if we can retrieve the same evolution pattern found for our initial proxy data (Figure 2). This is done solving equation 11 by forcing it at each cycle with the fit determined values of c n. We used the numerical differential solver in the Mathematica software package in the following way. Equation 11 is solved from t=1749 to t=2010, with c 1, c 2 and c 3 inputted as a step function defined between consecutive cycle minima. The result of this procedure is shown in the top panel of Figure (4). The observed phase shift between

9 theoretical (black dashed line) and the original data (gray solid line) is a consequence of a systematic fitting error in c 2. To overcome this we add a correction to the values of c 2 by defining a corrected coefficient, c 2 = c 2 + δc 2. This compensates for the phase lag between the theoretical solution and the original data. We would like to point out that the theoretical curves shown in Figure 4 (black dashed line) represents the dynamical evolution of equation 11) as a response to the changing background structure. So although the structural coefficients are assumed to vary only at solar minima, the magnetic field changes continuously and reacts to these variations (kinematic regime). The initial conditions introduced for solving equation 11) are introduced only at t = 0 and are not corrected for every cycle when the coefficients are forced. This implies that a final state of a cycle will influence the solution for the following cycle. When the squared B φ is plotted and compared with the SSN (bottom panel of Figure 4), the differences also get amplified and it easier to see where the model is less efficient in reproducing the time series. Nevertheless the correlation coefficient for the two curves in the bottom panel of figure (4) is high with r = The fact that such a simplified dynamo model can get this degree of resemblance with the observed data just by controlling two parameters is an indication that it capture the most important physical processes occurring in the Sun Reconstructing variations in the meridional flow From equation 8 we can see that c 1 depends on the strength of the meridional flow and the average magnetic diffusivity in the SCZ. The magnetic diffusivity depends on the microscopic properties of the solar plasma and the turbulence power spectrum in the SCZ and it is believed to be constant over time. Nevertheless complex phenomena such as magnetic quenching of turbulent diffusivity might occur [18] but this is beyond the scope of this modelling methodology. The only way of explaining the variations in the coefficient c 1 is to admit variations in the strength of the meridional flow, v p. This is a very plausible scenario given that variations in the surface meridional flow have been reported by [19], [20] and more recently by [21] and [22]. The methodology presented here is only sensible to variations of the average meridional flow from cycle to cycle that can be interpreted by variations in c 1. In Figure 5 it is presented the variation of c 1, which we believe that translates in variations of the average strength of the meridional flow, superimposed on the smoothed sunspot number used (or B 2 φ ). The features of this figure are comparable with the 2D dynamo simulations with varying meridional circulation of [14] and [23]. This provides another indication that this low order model is reproducing qualitatively the same results as more complex spatially resolved models. Another application derived from a similar study is the explanation of grand minima episodes. It was found in [11], using a this LODM, that a steep decrease in the meridional flow amplitude can lead to grand minima episodes like the Maunder minimum referred in 1. This result was later confirmed by [23], again using a more complex numerical flux transport model.

10 f f f time (yr) time (yr) time (yr) Figure 4: Gray solid line represents the built proxy for B φ and the black dashed line is the solution of equation (11) forced for different values of c 1 and c 2 (c 3 constant) between consecutive minima. In the middle panel, the same simulation is given with c 2 corrected for the systematic delay, c 2 -δc 2, with δc 2 = In the bottom panel, the smoothed SSN is compared with B 2 φ. Other applications of this LODM are being studied at this moment. A tempting scenario is its application is the predictability of future solar cycles amplitudes. The first step toward this objective is presented in [16]. In this work the we study the correlations

11 c 1 0,2 5 0,2 0 0,1 5 0,1 0 0,0 5 0,0 0-0,0 5-0, Y e a r S u n s p o t n u m b e r Figure 5: Smoothed SSN used for the fit (dotted line). The value of c 1 obtained for each cycle is superimposed with a black solid line. between the LODM structural coefficients and cycle s characteristics (amplitude, period and rising time) and we find useful relationships between this quantities for consecutive cycles. These relationships are then used to predict the behavior of solar cycle 24. The obtained results indicate that solar cycle 24 will have an annual mean sunspot number around 84 and will peak around June Another possible application for this model is the inference of stellar rotation periods. Using information about the magnetic activity on other starts, this model could be used in a similar way as presented here to infer rotation rates of solar analogs. This front will be explored soon... 5 EXTENDING THE LODM FOR THE NON-KINEMATIC REGIME In the previous sections we followed the traditional assumption that the solar dynamo can be correctly modeled in the kinematic regime, where the flows are not influenced in any way by the magnetic field. Nevertheless in the last couple of years evidence started to appear supporting the claim that this kinematic regime might be overlooking important physical mechanisms for the evolution of the dynamo. Observations of the surface meridional flow by [22] and [21] indicate that the observed variation in this flow is highly correlated with the levels of magnetic activity. This hints for a close relation between field and flows but the answer to the question: who is driving who, the field or the flow? still eludes us. The answer is starting to appear from full MHD simulations of the Sun. The analysis of the the output of one of the large-eddy global MHD simulations of the solar convection zone produced by [24] shows interesting clues to solve this mystery. These simulations solve the full set of MHD equations in the anelastic regime, in a broad, thermally-forced stratified plasma shell mimicking the SCZ and are fully dynamical on all spatiotemporally-resolved scales. A posterior analysis of the interaction between the toroidal magnetic field and the meridional flow in the base of the convection zone indicates that the magnetic field is indeed acting on the θ component of this flow accelerating

12 it [25]. The authors also show that this acceleration is due to the work that the Lorentz force makes on the flow. This observed relationship runs contrary to the usually assumed kinematic approximation. In order to check if this non-kinematic regime has any impact in the long scale evolution of the dynamo, we turn again to our LODM. By implementing a term that accounts for the Lorentz, the LODM allows us to fully isolate the global aspects of the dynamical interactions between the meridional flow and magnetic field in a simplified way. We now assume that the large-scale meridional circulation, v p, is divided into a kinematic constant part, v 0 (due to angular momentum distribution) and a time dependent part, v(t), that encompasses the Lorentz feedback of the magnetic field. Since there is no way to account for the complex vectorial relations present in the definition of the Lorentz force in the LODM, we simply assume that this feedback depends on both components of the magnetic field. Therefore we redefine v p as v p (t) = v 0 + v(t) where the time dependent part evolves according to dv(t) dt = a B φ A p b v(t), (12) The first term is a magnetic nonlinearity representing the Lorentz force and the second is a newtonian drag that mimics the natural resistance of the flow to outside kinematic perturbation and ensures an exponentially decay of v(t) towards its imposed kinematic value v 0 in the absence of magnetic fields. In other words we are assuming that the Lorentz force associated with the cyclic large-scale magnetic field acts as a perturbation on the otherwise dominant kinematic meridional flow. This approach has already been used to model magnetically-mediated variations of differential rotation in mean-field dynamo models [26] [27] [28]. The modified LODM equation are now db φ dt da p dt = = ( c 1 v ) p(t) B φ + c 2 A p c 3 Bφ 3, (13) l 0 ( c 1 v ) p(t) A p + αb φ. (14) l 0 η R 2, while the other coefficients remain the same. A with c 1 now defined as c 1 = η l 2 0 typical solution for this system depicting B φ and the components of the meridional flow is presented in Figure 6. The formulation proposed for v p satisfies the requirements at hand, i.e., the flow component influenced by the Lorentz force, v(t) varies in phase with the toroidal field (like it is observed in the 3D MHD simulation) while the complete meridional flow v p follows the observational behavior and varies in anti-phase with B φ. The system presents a wide range of solutions, according to the values chosen for the various numerical coefficients. The values used for the structural coefficients, are mean values derived in section 4. The parameters associated with the meridional flow evolution, a, b and v 0 deserved now the attention. These parameters have an important

13 BΦ t 2, 5 v t, 5 vp t, 5v Figure 6: Comparison between the toroidal field (black thick line), v p (dashed blue line), v(t) (purple dotted line) and v 0 (gray dot-dashed line) for a standard solution. Here the values of the flow are scaled 5 times to be visible in the scale of B 2 φ. role in the evolution of the solution space. The behavior observed in the solutions range from fixed-amplitude oscillations closely resembling kinematic solutions, multiperiodic solutions, and even chaotic solutions. This is easier to visualize in Figure 7 where we present analogs of classical bifurcation diagrams by plotting successive peak values of cycle amplitudes, for solutions with fixed (a, v 0 ) combinations but spanning through values of b. Transition to chaos through bifurcations is also observed when holding b fixed and varying a instead. 5.1 Fluctuations and intermittency To simulate the fact that the field s feedback into the flow happens in a very turbulent environment we assume that parameter a, the one that controls the influence of the Lorentz force, is subjected to some sort of stochastic fluctuations. This is done by adding noise to a which is extracted from a zero-mean bounded uniform distribution at a defined coherence time. We choose this coherence time to be one year, i.e. much smaller than the 11 years length of the cycle. Afterwards the LODM is solved under the forcing of the variations of a in a procedure analog at the one used in 3. As a result, and depending on the range of fluctuations, we see that the short term stochastic kicks in the Lorentz force amplitude create long term modulations in the amplitude of the cycles (hundreds of years) and even episodes where the field decays to near zero values analog the previously mentioned grand minima. The duration and frequency of these long quiescent phases, where the magnetic field decays to very low values, is determined by the level of fluctuations of a and the value of b. The stronger this drag term b is, the

14 Figure 7: Bifurcation maps for maximum amplitude of the toroidal field (equivalent to solar cycle maximum) obtained by varying b between 10 4 and 1 for different a and v 0. (A) Single period regime, v 0 = 0.1, a = (B) Appearance of period doubling, v 0 = 0.1, a = 0.1. Panels (C) and (D) show signatures of chaotic regimes with multiple attractors and windows, obtained with v 0 = 0.13, a = 0.05 and v 0 = 0.13, a = 0.2 correspondingly. shorter the minima are and the higher the level of fluctuation of a, the more common these intermittency episodes become. Figure 8 shows a solution example computed over years that presents all the behaviors described before. In this specific example we used a [0.01, 0.03] which corresponds to a 100% fluctuation level relative to its mean value (0.02), b = 0.05 and v 0 = It is important to note that in this parameter regime, the solution without stochastic forcing is well behaved in the sense that it presents a single period regime. Therefore, the fluctuations observed in this solution are a direct consequence of the stochastic forcing of the Lorentz force and not from a chaotic regime of the solution s space. We believe that even for lower fluctuations threshold the system will display these grand minima periods specially if we use value of a, b and v 0 that lead to corners of the solution space where chaos or period doubling lure. To understand how the grand minima episodes arise we find useful to resort to phase space diagrams of {B φ, A p, v p }. This allows us to see how these quantities vary in

15 (A) f f f (B) (C) Figure 8: Simulation result fluctuating a [0.01, 0.03], b = 0.05 and v 0 = All other model parameters are the same as in the reference solution. Panel (A) shows B φ (t) 2 (black) and A p (t) 2 (red) for years simulation. Panel (B) is a zoom in to show a section of the simulation where the long term modulation can be seen. In black is Bφ 2(t), red A2 p(t) and blue a scaled version of the meridional flow, in this case 5v p (t). In panel (C) the same quantities but this time zooming in into a grand minimum (off phase) period. relation to each other and try to understand the chain of events that trigger a grand minimum. Using the solution presented in Figure 8 we built several phase spaces shown in Figure 9. The standard solution for the LODM without stochastic forcing, i.e. with a fixed at the mean value of the random number distribution used, is a limit cycle attractor,

16 i.e., a closed trajectory in the {B φ, A p } phase space. This curve is represented as a black dashed trajectory in the panels of Figure 9. The gray points in this figure are the stochastic forced solution values sampled at 1 year interval. These points scatter around the attractor representing the variations in amplitude of the solution. Occasionally we see that the trajectories defined by these points collapse to the center of the phase space (the point {0,0} is also another natural attractor of the system) indicating a decrease in amplitude of the cycle, i.e. a grand minimum. The colored trajectory evolving in time from purple to red represents one of those grand minimum. This happens when the solution is at a critical distance from the limit cycle attractor and gets a random kick further away from it. This kick makes the field grow rapidly. In turn, since the amplitude of the field grows fast, the Lorentz force will induce a similar growth in v(t) eventually making v p change sign. When this occurs, v p behaves as a sink term quenching the field growth very efficiently. This behavior is seen in the two bottom panels of Figure (9) where we can see v p decaying to it imposed kinematic value v 0 after the fields decay. After this collapse of v p to v 0 it starts behaving has a source term again and the cyclic activity proceeds. The fact that this feedback mechanism provides an alternative mechanism for limiting the growth of the magnetic field is important and should be further investigated. 6 CONCLUDING REMARKS As we have seen in a resume way throughout the previous sections, the applications of a low order dynamo model are vast and yield interesting results. This is only possible due the the strong nature of the solar dynamo process and our current knowledge about it. In the past this type of simplified models where many times used in the context of chaos and its existence in the solar cycle but their lack of physical context deemed them to a secondary plane. The new methodology reviewed here where much of the physics is captured by the model makes these models more viable. Nevertheless one should always keep in mind that this is a simplified model and although it provides useful results it has its limitations. We showed how the LODM can be applied to reconstruct the behavior and evolution of dynamo parameters by pairing it with observational time series. Although possible, this is much more difficult to do with spatially resolved dynamo models. Nevertheless the results obtained here and in the cited publications provided valuable contributions to the improvement of more sophisticated numerical models. When we explored applications in the kinematic regime applied to the sunspot number time series, we found interesting variations patterns in the meridional flow which motivated other works that explored further this fact with flux transport dynamo models. Other interesting application areas for this model are its use as a prediction tool or as a way of determine the rotation rate of solar-like stars (study underway). The model also proved to be useful to explore dynamo solutions in the non-kinematic. The results obtained with the modified LODM presented in 5 indicate that indeed the Lorentz force feedback of the field into the flow has an important effect in the evolution

17 Figure 9: Phase spaces of the solution with stochastic fluctuations. The gray dots represent 1 year intervals between t=35000 and t= The colored line shows the trajectory of a grand minima (starting from purple, t=27300 and ending in red, t= The black dashed line represents the unperturbed solution with a = of the solar dynamo. It is expected that results obtained from full MHD simulations backed up by this exploratory analysis may lead to the upgrade of many flux transport dynamo models to work in the non-kinematic regime. REFERENCES [1] Tuija Pulkkinen, Living Rev. Solar Phys. 4, 1. URL (cited on 010/2011):

18 [2] Severe Space Weather Events - Understanding Societal and Economic Impacts: A Workshop Report, Committee on the Societal and Economic Impacts of Severe Space Weather Events: A Workshop, National Research Council, URL: edu/catalog/12507.html, [3] Joanna D. Haigh, Living Rev. Solar Phys. 4, 2. URL (cited on 10/2011): [4] Eddy, J.A., Science, 192, , [5] Hale, G.E., ApJ, 28, 315, [6] Parker, E., ApJ, 121, 491, [7] Krause, F., Rädler, H.-H., Mean-Field Magnetohydrodynamics and Dynamo Theory, Pergamon Press, [8] Alfvén, H., Ark. f. Mat. Astr. o Fysik, 29B, 2, [9] Charbonneau, P., Living Rev. Solar Phys. 7, 3. URL (cited on 11/2011): [10] Passos, D., Lopes, I.P., ApJ, Vol. 686, 2, 1420, [11] Passos, D., Lopes, I.P., J. of Atmosph. Sol-Ter. Phys. 73, , [12] Mininni, P.D., Gomez, D.O., Mindlin, G.B., Sol. Phys. 201, , [13] Pontieri, A., Lepreti, F., Sorriso-Valvo, L., Vecchio, A., Carbone, V., Sol. Phys., 213, , [14] Lopes, I., and Passos, D., Sol. Phys., 257, 1, [15] Lopes, I., and Passos, D., MNRAS, 397, , [16] Passos, D., ApJ, Accepted Oct. 2011, in press, [17] Mininni, P.D., Gomez, D.O., Mindlin, G.B., PRL, 85, , [18] Muoz-Jaramillo, A., Nandy, D., Martens, P.C.H., ApJ, 727, L23, [19] Komm, R.W., Howard, R.F., Harvey, J.W., Sol. Phys., 147, 207, [20] Basu, S., Antia, H., ApJ, 585, 553, [21] Basu, S., Antia, H. M., ApJ, 717, 488, [22] Hathaway, D., Rightmire, L., ApJ, 729, 80, [23] Karak, B., ApJ, 724, 2, 1021, 2010.

19 [24] Ghizaru, M., Charbonneau, P. and Smolarkiewicz, P.K., ApJL, 715, 133, [25] Passos, D., Charbonneau, P., Beaudoin, P., An exploration of non-kinematic effects in flux transport dynamos, Sol. Phys., submitted Nov. 2011, [26] Tobias, S.M., ApJ, 467, 870, [27] Moss, D., Brooke, J.M., MNRAS, 315, 521, [28] Bushby, P.J., MNRAS, 371, 772, 2006.