Cluster analysis of the space-time distribution of sunspot groups during solar cycle no. 20

Cluster analysis (a Bayesian iteration procedure) was used to study the space-time distribution of sunspot groups in the time interval from 1965 to 1977. (Data were taken from the Greenwich and Debrecen Heliographic Results.) The distribution proved to be significantly non-random for the 8–10 groups cluster^–1 (gr cl^–1) level of clustering. Convincing evidence also favours non-random behaviour for other levels of clustering from the lowest (3–4 gr cl^–1) up to the highest ( 150 gr cl^–1) level. The rotation rate of the non-random pattern is generally slightly lower than the Carrington rate.The 8–10 gr cl^–1 level, crudely corresponding to the sunspot nests investigated earlier, was studied in more detail. The cycle- and latitude-averaged rotational rate of the nests is slightly ( 1%) but significantly lower than the Carrington rate. Their differential rotation is strongly reduced: the cycle-averaged rotational rate varies only by 2–3% within the sunspot belt. A slight but significant bimodality is seen in the differential rotation curve: the intermediate latitudes ( 10°–20°) show a somewhat slower rotation than both the equatorial and the higher latitude regions. This might be explained by a time-dependence of the rotation rate coupled with the butterfly diagram.     

On the properties of toroidal flux tubes in the solar dynamo

We consider a hypothesis that nondissipating wave-bound beam structures may be created in plasma due to interaction between energetic charged particles and undamped Van Kampen (1955) waves. This hypothesis, if correct, might account for several phenomena observed in the Sun. For instance, on entering the atmosphere and then decaying, the structures may lead to flares. Furthermore, the idea offers propitious prospects of solving the well-known problem of the solar neutrino deficit. In addition, a relation between phenomena occuring within the Sun's core and those on its surface becomes possible.     

Properties of a spherical galaxy with exponential energy distribution

Some analytical relations for the phase space functions of a self-consistent spherical stellar system are derived. The integral constraints on the distribution function by imposing a given (r) density distribution andN(E) fractional energy distribution are determined. For the case of radially-anisotropic velocity distribution in theE0 limit the constraint by an exponentialN(E) implies thatf(E, J ²) tends to zero in the order (–E)^3/2. This lends analytical support to the use of the Stiavelli and Bertin (1985) distribution function for modeling elliptical galaxies. Maximum phase space density constraint confirms the necessity of high collapse factors to produce such a distribution function. Limits on the steepness of an exponentialN(E) for the case when (r) resembles the emissivity law of ellipticals are also derived.     

The Role of Active Regions in the Generation of Torsional Oscillations

We present a model for torsional oscillations where the inhibiting effect of active region magnetic fields on turbulence locally reduces turbulent viscous torques, leading to a cycle- and latitude-dependent modulation of the differential rotation. The observed depth dependence of torsional oscillations as well as their phase relationship with the sunspot butterfly diagram are reproduced quite naturally in this model. The resulting oscillation amplitudes are significantly smaller than observed, though they depend rather sensitively on model details. Meridional circulation is found to have only a weak effect on the oscillation pattern.     

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.     

On the validity of quasi-linear kinematic mean-field electrodynamics in astrophysical flows

Mean-field theory in its kinematic form with the quasi-linear approximation is widely used for the modelling of the transport of weak magnetic fields in turbulent media. The validity of this approach to real astrophysical flows is discussed. Numerically evaluating the turbulent electromotive force using Lagrangian analysis for a set of simple, prescribed 2D flow patterns with a wide range of parameters, we find that quasi-linear expressions for the turbulent diffusivities and for the pumping velocities are correct within a factor of 2 for a wide variety of flow types with order of unity (or even higher) effective Strouhal numbers. The degree of the non-linear quenching of turbulent transport by a weak magnetic field is also discussed. We argue that, owing to the intermittency and small filling factors of magnetic fields in realistic astrophysical media, diffusivity and pumping effects are not quenched to order of magnitude, while a more moderate quenching of order 10 per cent is still present.     

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 r_o and the maximum radius r_o as D = C_D r/r_o, C_D = 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.     

On the Compatibility of a Flux Transport Dynamo with a Fast Tachocline Scenario

The compatibility of the fast-tachocline scenario with a flux-transport dynamo model is explored. We employ a flux-transport dynamo model coupled with simple feedback formulae relating the thickness of the tachocline to the amplitude of the magnetic field or to the Maxwell stress. The dynamo model is found to be robust against the nonlinearity introduced by this simplified fast-tachocline mechanism. Solar-like butterfly diagrams are found to persist and, even without any parameter fitting, the overall thickness of the tachocline is well within the range admitted by helioseismic constraints. In the most realistic case of a time- and latitude-dependent tachocline thickness linked to the value of the Maxwell stress, both the thickness and its latitudinal dependence are in excellent agreement with seismic results. In nonparametric models, cycle-related temporal variations in tachocline thickness are somewhat larger than admitted by helioseismic constraints; we find, however, that introducing a further parameter into our feedback formula readily allows further fine tuning of the thickness variations.     

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

In a statistical analysis of Debrecen Photoheliographic Results sunspot area data we find that the logarithmic deviation (logD) 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 are also characterized by approximately lognormal distributions, as found in an earlier work. The correlation time of (logD) is about 3 days. We find a significant physical anticorrelation between (logD) 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.