K. Petrovay

Solar Cycle Prediction

A review of solar cycle prediction methods and their performance is given, including forecasts for cycle 24. The review focuses on those aspects of the solar cycle prediction problem that have a bearing on dynamo theory. The scope of the review is further restricted to the issue of predicting the amplitude (and optionally the epoch) of an upcoming solar maximum no later than right after the start of the given cycle.Prediction methods form three main groups. Precursor methods rely on the value of some measure of solar activity or magnetism at a specified time to predict the amplitude of the following solar maximum. Their implicit assumption is that each numbered solar cycle is a consistent unit in itself, while solar activity seems to consist of a series of much less tightly intercorrelated individual cycles. Extrapolation methods, in contrast, are based on the premise that the physical process giving rise to the sunspot number record is statistically homogeneous, i.e., the mathematical regularities underlying its variations are the same at any point of time and, therefore, it lends itself to analysis and forecasting by time series methods. Finally, instead of an analysis of observational data alone, model based predictions use physically (more or less) consistent dynamo models in their attempts to predict solar activity.In their overall performance during the course of the last few solar cycles, precursor methods have clearly been superior to extrapolation methods. Nevertheless, most precursor methods overpredicted cycle 23, while some extrapolation methods may still be worth further study. Model based forecasts have not yet had a chance to prove their skills. One method that has yielded predictions consistently in the right range during the past few solar cycles is that of K. Schatten et al., whose approach is mainly based on the polar field precursor.The incipient cycle 24 will probably mark the end of the Modern Maximum, with the Sun switching to a state of less strong activity. It will therefore be an important testbed for cycle prediction methods and, by inference, for our understanding of the solar dynamo.

Making sense of sunspot decay I. Parabolic Decay Law and Gnevyshev-Waldmeier Relation

In a statistical study of the decay of individual sunspots based on DPR data we find that the mean instantaneous area decay rate is related to the spot radius ro and the maximum radius ro as D = CD r/ro, CD = 32.0±0.26 MSH day -1. This implies that sunspots on the mean follow a parabolic decay law; the traditional linear decay law is excluded by the data. The validity of the Gnevyshev–Waldmeier relationship between the maximum area A 0 and lifetime T of a spot group, A0/T ≃10 MSH day-1, is also demonstrated for individual sunspots. No evidence is found for a supposed supergranular ‘quantization’ of sunspot areas. Our results strongly support the recent turbulent erosion model of sunspot decay while all other models are excluded.

ASYMMETRIC FLUX LOOPS IN ACTIVE REGIONS, II

We propose that magnetic flux loops in the subphotospheric layers of the Sun are seriously asymmetrical as a consequence of the drag force exerted on them because of the different rotational rate of the surrounding plasma. In numerical models of stationary slender flux loops in the plane parallel approximation we show that a serious tilt is both possible and probable. Observational facts (see van Driel-Gesztelyi and Petrovay, 1989; Paper I) strongly support the case for high asymmetry. The different stability of p and f spots may also be related to such an asymmetry.The tilts are very sensitive to the rotational profile and to the magnetic field structure. Nevertheless the characteristic maximal tilts can be tentatively estimated to be 20° for thin flux tubes and 5° for thick tubes.For two of the five observational consequences of such a tilt (described in detail in Paper I) order-of-magnitude estimates of the effects are given. The estimates are in reasonable accord with observations.We also explore the possibilities of an inverse treatment of the problem whereby subphotospheric rotation and/or flux tube shapes can be inferred from observations of velocities of photospheric spot motions. In particular we demonstrate how analytic inverse solutions can be obtained in special cases.

Turbulent erosion of magnetic flux tubes

Results from a numerical and analytical investigation of the solution of a nonlinear axisymmetric diffusion equation for the magnetic field are presented for the case when the nonlinear dependence of the diffusivity ν(B) on the magnetic field satisfies basic physical requirements. We find that for sufficiently strong nonlinearity (i.e., for sufficiently strong reduction of ν inside the tube) a current sheet is spontaneously formed around the tube within one diffusion timescale. This sheet propagates inward with a velocity inversely proportional to the ratio of the field strength just inside the current sheet to the equipartition field strength B0/Be, so the lifetime of a tube with constant internal flux density is increased approximately by a factor not exceeding B0/Be, even for infinitely effective inhibition of turbulence inside the tube. Among the applications of these results, we point out that toroidal flux tubes in the solar convective zone are subject to significant flux loss owing to turbulent erosion on a timescale of ~1 month and that turbulent erosion may be responsible for the formation of a current sheet around a sunspot. It is further proposed that, despite the simplifying assumptions involved, our solutions correctly reflect the essential features of the sunspot decay process.

Turbulence in the solar photosphere

The precise nature of photospheric flows, and of the transport effects they give rise to, has been the subject of intense debate in the last decade. Here we attempt to give a brief review of the subject emphasizing interdisciplinary (solar physics–turbulence theory) aspects, key open questions, and recent developments.

Making sense of sunspot decay – II. Deviations from the Mean Law and Plage Effects

AbstractIn a statistical analysis of Debrecen Photoheliographic Results sunspot area data we find that the logarithmic deviation (log D)′ of the area decay rate D from the parabolic mean decay law (derived in the first paper in this series) follows a Gaussian probability distribution. As a consequence, the actual decay rate D and the time-averaged decay rate

Oscillator Models of the Solar Cycle

\bar D

THE ROLE OF ACTIVE REGIONS IN THE GENERATION OF TORSIONAL OSCILLATIONS

are also characterized by approximately lognormal distributions, as found in an earlier work. The correlation time of (log D)′ is about 3 days. We find a significant physical anticorrelation between (log D)′ and the amount of plage magnetic flux of the same polarity in an annulus around the spot on Kitt Peak magnetograms. The anticorrelation is interpreted in terms of a generalization of the turbulent erosion model of sunspot decay to the case when the flux tube is embedded in a preexisting homogeneous ‘plage’ field. The decay rate is found to depend inversely on the value of this plage field, the relation being very close to logarithmic, i.e., the plage field acts as multiplicative noise in the decay process. A Gaussian probability distribution of the field strength in the surrounding plage will then naturally lead to a lognormal distribution of the decay rates, as observed. It is thus suggested that, beside other multiplicative noise sources, the environmental effect of surrounding plage fields is a major factor in the origin of lognormally distributed large random deviations from the mean law in the sunspot decay rates.