On Predicting the Solar Cycle using Mean-Field Models

We discuss the difficulties of predicting the solar cycle using mean-field models. Here we argue that these difficulties arise owing to the significant modulation of the solar activity cycle, and that this modulation arises owing to either stochastic or deterministic processes. We analyse the implications for predictability in both of these situations by considering two separate solar dynamo models. The first model represents a stochastically-perturbed flux transport dynamo. Here even very weak stochastic perturbations can give rise to significant modulation in the activity cycle. This modulation leads to a loss of predictability. In the second model, we neglect stochastic effects and assume that generation of magnetic field in the Sun can be described by a fully deterministic nonlinear mean-field model -- this is a best case scenario for prediction. We designate the output from this deterministic model (with parameters chosen to produce chaotically modulated cycles) as a target timeseries that subsequent deterministic mean-field models are required to predict. Long-term prediction is impossible even if a model that is correct in all details is utilised in the prediction. Furthermore, we show that even short-term prediction is impossible if there is a small discrepancy in the input parameters from the fiducial model. This is the case even if the predicting model has been tuned to reproduce the output of previous cycles. Given the inherent uncertainties in determining the transport coefficients and nonlinear responses for mean-field models, we argue that this makes predicting the solar cycle using the output from such models impossible.Comment: 22 Pages, 5 Figures, Preprint accepted for publication in Ap

In--out intermittency in PDE and ODE models

We find concrete evidence for a recently discovered form of intermittency, referred to as in--out intermittency, in both PDE and ODE models of mean field dynamos. This type of intermittency (introduced in Ashwin et al 1999) occurs in systems with invariant submanifolds and, as opposed to on--off intermittency which can also occur in skew product systems, it requires an absence of skew product structure. By this we mean that the dynamics on the attractor intermittent to the invariant manifold cannot be expressed simply as the dynamics on the invariant subspace forcing the transverse dynamics; the transverse dynamics will alter that tangential to the invariant subspace when one is far enough away from the invariant manifold. Since general systems with invariant submanifolds are not likely to have skew product structure, this type of behaviour may be of physical relevance in a variety of dynamical settings. The models employed here to demonstrate in--out intermittency are axisymmetric mean--field dynamo models which are often used to study the observed large scale magnetic variability in the Sun and solar-type stars. The occurrence of this type of intermittency in such models may be of interest in understanding some aspects of such variabilities.Comment: To be published in Chaos, June 2001, also available at http://www.eurico.web.co

Solar-Cycle Characteristics Examined in Separate Hemispheres: Phase, Gnevyshev Gap, and Length of Minimum

Research results from solar-dynamo models show the northern and southern hemispheres may evolve separately throughout the solar cycle. The observed phase lag between the hemispheres provides information regarding the strength of hemispheric coupling. Using hemispheric sunspot-area and sunspot-number data from Cycles 12 - 23, we determine how out of phase the separate hemispheres are during the rising, maximum, and declining period of each solar cycle. Hemispheric phase differences range from 0 - 11, 0 - 14, and 2 - 19 months for the rising, maximum, and declining periods, respectively. The phases appear randomly distributed between zero months (in phase) and half of the rise (or decline) time of the solar cycle. An analysis of the Gnevyshev gap is conducted to determine if the double-peak is caused by the averaging of two hemispheres that are out of phase. We confirm previous findings that the Gnevyshev gap is a phenomenon that occurs in the separate hemispheres and is not due to a superposition of sunspot indices from hemispheres slightly out of phase. Cross hemispheric coupling could be strongest at solar minimum, when there are large quantities of magnetic flux at the Equator. We search for a correlation between the hemispheric phase difference near the end of the solar cycle and the length of solar-cycle minimum, but found none. Because magnetic flux diffusion across the Equator is a mechanism by which the hemispheres couple, we measured the magnetic flux crossing the Equator by examining magnetograms for Solar Cycles 21 - 23. We find, on average, a surplus of northern hemisphere magnetic flux crossing during the mid-declining phase of each solar cycle. However, we find no correlation between magnitude of magnetic flux crossing the Equator, length of solar minima, and phase lag between the hemispheres.Comment: 15 pages, 7 figure

Global-Scale Turbulent Convection and Magnetic Dynamo Action in the Solar Envelope

The operation of the solar global dynamo appears to involve many dynamical elements. Self-consistent MHD simulations which realistically incorporate all of these processes are not yet computationally feasible, though some elements can now be studied with reasonable fidelity. Here we consider the manner in which turbulent compressible convection within the bulk of the solar convection zone can generate large-scale magnetic fields through dynamo action. We accomplish this through a series of three-dimensional numerical simulations of MHD convection within rotating spherical shells using our ASH code on massively parallel supercomputers. Since differential rotation is a key ingredient in all dynamo models, we also examine here the nature of the rotation profiles that can be sustained within the deep convection zone as strong magnetic fields are built and maintained. We find that the convection is able to maintain a solar-like angular velocity profile despite the influence of Maxwell stresses which tend to oppose Reynolds stresses and thus reduce the latitudinal angular velocity contrast throughout the convection zone. The dynamo-generated magnetic fields exhibit a complex structure and evolution, with radial fields concentrated in downflow lanes and toroidal fields organized into twisted ribbons which are extended in longitude and which achieve field strengths of up to 5000 G. The flows and fields exhibit substantial kinetic and magnetic helicity although systematic hemispherical patterns are only apparent in the former. Fluctuating fields dominate the magnetic energy and account for most of the back-reaction on the flow via Lorentz forces. Mean fields are relatively weak and do not exhibit systematic latitudinal propagation or periodic polarity reversals as in the sun. This may be attributed to the absence of a tachocline.Comment: 55 pages (ApJ refereeing format), 15 figures (low res), published by ApJ on October 2004 (abstract slightly reduced in order to fit in 24 lines limit) see also Browning, Miesch, Brun & Toomre 2006, ApJL, 648, 157 (astro-ph/0609153) for the effect of a tachocline in organizing the mean field

Large-scale dynamos in turbulent convection with shear

(abridged) Aims: Three-dimensional numerical simulations of penetrative compressible convection with uniform horizontal shear are used to study dynamo action and the generation of large-scale magnetic fields. Methods: We consider cases where the magnetic Reynolds number is either marginal or moderately supercritical with respect to small-scale dynamo action in the absence of shear and rotation. The effects of magnetic helicity fluxes are studied by comparing results for the magnetic field with open and closed boundaries. Results: Without shear no large-scale dynamos are found even if the ingredients necessary for the alpha-effect (rotation and stratification) are present in the system. When uniform horizontal shear is added, a large-scale magnetic field develops, provided the boundaries are open. In this case the mean magnetic field contains a significant fraction of the total field. For those runs where the magnetic Reynolds number is between 60 and 250, small-scale dynamo is expected to be excited, but the field distribution is found to be similar to cases with smaller magnetic Reynolds number where the small-scale dynamo is not excited. In the case of closed (perfectly conducting) boundaries, magnetic helicity fluxes are suppressed and no large-scale fields are found. Similarly, poor large-scale field development is seen when vertical shear is used in combination with periodic boundary conditions in the horizontal directions. If, however, open (normal-field) boundary conditions are used in the x-direction, a large-scale field develops. These results support the notion that shear not only helps to generate the field, but it also plays a crucial role in driving magnetic helicity fluxes out of the system along the isocontours of shear, thereby allowing efficient dynamo action.Comment: 10 pages, 19 figures, accepted for publication in Astron. Astrophy

Alpha effect and turbulent diffusion from convection

(abridged) Aims: To study turbulent transport coefficients that describe the evolution of large-scale magnetic fields in turbulent convection. Methods: We use the test field method together with 3D numerical simulations of turbulent convection with shear and rotation to compute turbulent transport coefficients describing the evolution of large-scale magnetic fields in mean-field theory in the kinematic regime. 1D mean-field models are used with the derived turbulent transport coefficients to compare with direct simulations. Results: The alpha-effect increases monotonically as rotation increases. Turbulent diffusivity, eta_t, is proportional to the square of the turbulent vertical velocity. Whereas eta_t decreases approximately inversely proportional to the wavenumber of the field, the alpha-effect and turbulent pumping show a more complex behaviour. In the presence of shear and no rotation a small alpha-effect is induced which does not seem to show any consistent trend as a function of shear. If the shear is large enough, this small alpha is able to excite a dynamo in the mean-field model. The coefficient responsible for driving the shear-current effect shows several sign changes as a function of depth but is also able to contribute to dynamo action in the mean-field model. The growth rates in these cases are well below those in direct simulations suggesting that an incoherent alpha-shear dynamo may also act in them. If both rotation and shear are present, the alpha-effect is more pronounced. The combination of the shear-current and Omega x J-effects is also stronger than in the case of shear only, but subdominant to the alpha-shear dynamo. The results of direct simulations are consistent with mean-field models where all of these effects are taken into account without the need to invoke incoherent effects.Comment: 14 pages, 14 figures, minor changes to match with the published versio

MOST detects variability on tau Bootis possibly induced by its planetary companion

(abridged) There is considerable interest in the possible interaction between parent stars and giant planetary companions in 51 Peg-type systems. We demonstrate from MOST satellite photometry and Ca II K line emission that there has been a persistent, variable region on the surface of tau Boo A which tracked its giant planetary companion for some 440 planetary revolutions and lies ~68deg (phi=0.8) in advance of the sub-planetary point. The light curves are folded on a range of periods centered on the planetary orbital period and phase dependent variability is quantified by Fourier methods and by the mean absolute deviation (MAD) of the folded data for both the photometry and the Ca II K line reversals. The region varies in brightness on the time scale of a rotation by ~1 mmag. In 2004 it resembled a dark spot of variable depth, while in 2005 it varied between bright and dark. Over the 123 planetary orbits spanned by the photometry the variable region detected in 2004 and in 2005 are synchronised to the planetary orbital period within 0.0015 d. The Ca II K line in 2001, 2002 and 2003 also shows enhanced K-line variability centered on phi=0.8, extending coverage to some 440 planetary revolutions. The apparently constant rotation period of the variable region and its rapid variation make an explanation in terms of conventional star spots unlikely. The lack of complementary variability at phi=0.3 and the detection of the variable region so far in advance of the sub-planetary point excludes tidal excitation, but the combined photometric and Ca II K line reversal results make a good case for an active region induced magnetically on the surface of tau Boo A by its planetary companion.Comment: 7 pages, 7 figures; accepted for publication in A&

Hemispheric Sunspot Numbers R_n and R_s: Catalogue and N-S asymmetry analysis

Sunspot drawings are provided on a regular basis at the Kanzelhoehe Solar Observatory, Austria, and the derived relative sunspot numbers are reported to the Sunspot Index Data Center in Brussels. From the daily sunspot drawings, we derived the northern, R_n, and southern, R_s, relative sunspot numbers for the time span 1975-2000. In order to accord with the International Sunspot Numbers R_i, the R_n and R_s have been normalized to the R_i, which ensures that the relation R_n + R_s = R_i is fulfilled. For validation, the derived R_n and R_s are compared to the international northern and southern relative sunspot numbers, which are available from 1992. The regression analysis performed for the period 1992-2000 reveals good agreement with the International hemispheric Sunspot Numbers. The monthly mean and the smoothed monthly mean hemispheric Sunspot Numbers are compiled into a catalogue. Based on the derived hemispheric Sunspot Numbers, we study the significance of N-S asymmetries and the rotational behavior separately for both hemispheres. We obtain that about 60% of the monthly N-S asymmetries are significant at a 95% level, whereas the relative contributions of the northern and southern hemisphere are different for different cycles. From the analysis of power spectra and autocorrelation functions, we derive a rigid rotation with about 27 days for the northern hemisphere, which can be followed for up to 15 periods. Contrary to that, the southern hemisphere reveals a dominant period of about 28 days, whereas the autocorrelation is strongly attenuated after 3 periods. These findings suggest that the activity of the northern hemisphere is dominated by an active zone, whereas the southern activity is mainly dominated by individual long-lived sunspot groups.Comment: 9 pages, 8 figures, data catalogue online available at http://cdsweb.u-strasbg.fr/cgi-bin/qcat?J/A+A/390/70

Generalized Boltzmann Equation for Lattice Gas Automata

In this paper, for the first time a theory is formulated that predicts velocity and spatial correlations between occupation numbers that occur in lattice gas automata violating semi-detailed balance. Starting from a coupled BBGKY hierarchy for the $n$-particle distribution functions, cluster expansion techniques are used to derive approximate kinetic equations. In zeroth approximation the standard nonlinear Boltzmann equation is obtained; the next approximation yields the ring kinetic equation, similar to that for hard sphere systems, describing the time evolution of pair correlations. As a quantitative test we calculate equal time correlation functions in equilibrium for two models that violate semi-detailed balance. One is a model of interacting random walkers on a line, the other one is a two-dimensional fluid type model on a triangular lattice. The numerical predictions agree very well with computer simulations.Comment: 31 pages LaTeX, 12 uuencoded tar-compressed Encapsulated PostScript figures (`psfig' macro), hardcopies available on request, 78kb + 52k

The Differential Rotation of Kappa1 Ceti as Observed by MOST

We first reported evidence for differential rotation of Kappa1 Ceti in Paper I. In this paper we demonstrate that the differential rotation pattern closely matches that for the Sun. This result is based on additional MOST (Microvariability & Oscillations of STars) observations in 2004 and 2005, to complement the 2003 observations discussed in Paper I. Using StarSpotz, a program developed specifically to analyze MOST photometry, we have solved for k, the differential rotation coefficient, and P_{EQ}, the equatorial rotation period using the light curves from all three years. The spots range in latitude from 10 to 75 degrees and k = 0.090^{+0.006}_{-0.005} -- less than the solar value but consistent with the younger age of the star. k is also well constrained by the independent spectroscopic estimate of vsini. We demonstrate independently that the pattern of differential rotation with latitude in fact conforms to solar. Details are given of the parallel tempering formalism used in finding the most robust solution which gives P_{EQ} = 8.77^{+0.03}_{-0.04} days -- smaller than that usually adopted, implying an age < 750 My. Our values of P_{EQ} and k can explain the range of rotation periods determined by others by spots or activity at a variety of latitudes. Historically, Ca II activity seems to occur consistently between latitudes 50 and 60 degrees which might indicate a permanent magnetic feature. Knowledge of k and P_{EQ} are key to understanding the dynamo mechanism and rotation structure in the convective zone as well assessing age for solar-type stars. We recently published values of k and P_{EQ} for epsilon Eri based on MOST photometry and expect to analyze MOST light curves for several more spotted, solar-type stars.Comment: 16 pages, 7 Figures, published in Ap