The early concept of the origin of sunspots (Babcock 1961 , Parker 1979) was based on the idea of a magnetic loop emerging from the convective zone to the solar surface due to buoyancy. The magnetic loops were assumed to be caused by instabilities of the largescale, mean magnetic field generated by the joint action of the differential rotation and mean helicity of the convection (dynamo). The mean field itself had the form of waves propagating from the poles to the equator in 11-years. This concept explained the solar sunspot cycle in its basic manifestations: the Maunder butterfly diagram and Hale's law of the field polarities. The idea of the mean field generation and its evolution has been justified and developed in numerous studies (c.f. Moffatt 1978 , Parker 1979 , Krause & Rädler 1981 , Zeldovich et al. 1983). Theories of instabilities that could lead to the formation of emerging magnetic loops were less advanced however.
Recently a model of storage, instability and dynamical eruption of magnetic flux tubes in the convection zone has been developed (Schüssler et al. 1994 , Caligari 1995). According to this model, a toroidal flux tube stored at the core/convective zone overshoot layer becomes unstable and erupts to the surface of the Sun when its field strength exceeds G. Weaker fields do not erupt. The model is in a good agreement with the basic observational facts related to sunspot magnetic fields in that the flux tubes emerge at low heliolatitudes and have the correct inclination and asymmetry with respect to the east-west direction. However, there is still a problem to resolve. Conventional dynamo theories do not predict such a strong magnetic field. The predicted field does not exceed the equipartition field which is only about G (Schüssler et al. 1994).
This paper suggests that the random, fluctuating magnetic fields in the solar convection zone play a central role in flux eruption. Although there are regularities in sunspot behavior, within these regularities each sunspot appears at a random time and at a random place. The number of sunspots observed in a given cycle varies from cycle to cycle. In addition, the simple question of why sunspots occupy so small an area on the solar surface (less than or about 1%) becomes a problem if we relate the sunspot origin to only the mean, largescale magnetic field produced by the dynamo.
The generation of fluctuating fields and their role in the generation of the mean field has been widely discussed: for early studies see for example Krause & Rädler (1981), Zeldovich et al. (1983), for numerical simulations see for example Meneguzzi et al. (1981), Brandenburg et al. (1990). Recently Ossendrijver et al. (1996) developed a solar dynamo model with a stochastic kinetic helicity. The fluctuations of the kinetic helicity excite overtones of the basic mode of the mean magnetic field. Schmitt et al. (1996) used magnetic fluctuations as a stochastic forcing control of the dynamo leading to grand minima in solar activity. The importance of fluctuating fields is indicated by sunspots observations. It has been noted that no single large flux tube emerges when sunspots are formed. Instead, the sunspot magnetic field is assembled over a period of hours and days through the progressive gathering of many flux tubes (Zwaan 1978). In accord with these observations the original concept of sunspot formation through flux tube emergence had been modified in such a way that the sunspot appears as a dynamical clustering of many separate flux tubes (Parker 1979). However, in this modification each flux tube was treated as regular and non-random.
In the model suggested here the randomness of the sunspots is considered to be a fundamental attribute. Also central for the model is the concept that noise plays a constructive role in the detection of weak periodic signals. This concept recently developed in the study of dynamical systems (see for example Wiesenfeld & Moss 1995) is often called "stochastic resonance" although it has been recognized that it is not a true resonance phenomenon. The present paper is restricted to the simplest model of the noise-periodic signal interaction.
© European Southern Observatory (ESO) 1997
Online publication: July 3, 1998