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

Inertial Range Scaling, Karman-Howarth Theorem and Intermittency for Forced and Decaying Lagrangian Averaged MHD in 2D

We present an extension of the Karman-Howarth theorem to the Lagrangian averaged magnetohydrodynamic (LAMHD-alpha) equations. The scaling laws resulting as a corollary of this theorem are studied in numerical simulations, as well as the scaling of the longitudinal structure function exponents indicative of intermittency. Numerical simulations for a magnetic Prandtl number equal to unity are presented both for freely decaying and for forced two dimensional MHD turbulence, solving directly the MHD equations, and employing the LAMHD-alpha equations at 1/2 and 1/4 resolution. Linear scaling of the third-order structure function with length is observed. The LAMHD-alpha equations also capture the anomalous scaling of the longitudinal structure function exponents up to order 8.Comment: 34 pages, 7 figures author institution addresses added magnetic Prandtl number stated clearl

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

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

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&

Solar Grand Minima and random fluctuations in dynamo parameters

We consider to what extent the long-term dynamics of cyclic solar activity in the form of Grand Minima can be associated with random fluctuations of the parameters governing the solar dynamo. We consider fluctuations of the alpha-coefficient in the conventional Parker migratory dynamo, and also in slightly more sophisticated dynamo models, and demonstrate that they can mimic the gross features of the phenomenon of the occurrence of Grand Minima over a suitable parameter range. The temporal distribution of these Grand Minima appears chaotic, with a more or less exponential waiting time distribution, typical of Poisson processes. In contrast however, the available reconstruction of Grand Minima statistics based on cosmogenic isotope data demonstrates substantial deviations from this exponential law. We were unable to reproduce the non-Poissonic tail of the waiting time distribution either in the framework of a simple alpha-quenched Parker model, or in its straightforward generalization, nor in simple models with feedback on the differential rotation. We suggest that the disagreement may only be apparent and is plausibly related to the limited observational data, and that the observations and results of numerical modeling can be consistent and represent physically similar dynamo regimes.Comment: Solar Physics, in prin

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

The case for a distributed solar dynamo shaped by near-surface shear

Arguments for and against the widely accepted picture of a solar dynamo being seated in the tachocline are reviewed and alternative ideas concerning dynamos operating in the bulk of the convection zone, or perhaps even in the near-surface shear layer, are discussed. Based on the angular velocities of magnetic tracers it is argued that the observations are compatible with a distributed dynamo that may be strongly shaped by the near-surface shear layer. Direct simulations of dynamo action in a slab with turbulence and shear are presented to discuss filling factor and tilt angles of bipolar regions in such a model.Comment: 10 pages, 6 figures, Astrophys. J. 625 (scheduled for the 1 June 2005 issue

The Origin of T Tauri X-ray Emission: New Insights from the Chandra Orion Ultradeep Project

We use the data of the Chandra Orion Ultradeep Project (COUP) to study the nearly 600 X-ray sources that can be reliably identified with optically well characterized T Tauri stars (TTS) in the Orion Nebula Cluster. We detect X-ray emission from more than 97% of the optically visible late-type (spectral types F to M) cluster stars. This proofs that there is no ``X-ray quiet'' population of late-type stars with suppressed magnetic activity. All TTS with known rotation periods lie in the saturated or super-saturated regime of the relation between activity and Rossby numbers seen for main-sequence (MS) stars, but the TTS show a much larger scatter in X-ray activity than seen for the MS stars. Strong near-linear relations between X-ray luminosities, bolometric luminosities and mass are present. We also find that the fractional X-ray luminosity rises slowly with mass over the 0.1 - 2 M_sun range. The plasma temperatures determined from the X-ray spectra of the TTS are much hotter than in MS stars, but seem to follow a general solar-stellar correlation between plasma temperature and activity level. The large scatter about the relations between X-ray activity and stellar parameters seems to be related to the influence of accretion on the X-ray emission. While the X-ray activity of the non-accreting TTS is consistent with that of rapidly rotating MS stars, the accreting stars are less X-ray active (by a factor of ~2-3 on average) and produce much less well defined correlations than the non-accretors. We discuss possible reasons for the suppression of X-ray emission by accretion and the implications of our findings on long-standing questions related to the origin of the X-ray emission from young stars.Comment: accepted for ApJS, COUP Special Issu