1 Introduction

Many late-type stars in close binary systems, e.g. the RS CVn group, have manifestations of magnetic activity which seems to be persistently nonaxisymmetric (e.g. Piskunov et al. 1990; Henry et al. 1995; Olah et al. 1997). The non-axisymmetric nature of stellar activity in close binaries is much more pronounced than that discussed in connection with the possible existence of active longitudes for the solar activity (see e.g. Jetsu et al. 1997). The magnetic activity of late-type stars is thought to be connected with dynamo action. The aim of this paper is to discuss the nonaxisymmetric nature of the large scale magnetic fields of close binaries as a result of nonaxisymmetric dynamo action; we base our analysis on the example of an active RS CVn type binary ER Vulpeculae. The system consists of two late-type main sequence stars ( $T_{\rm eff} = 5800$ and 5500 K, R_* = 1.32 and 1.27 $R_\odot$), separated by 4.2 $R_\odot$, the orbital period is 0.698 d and the system appears to be rotationally synchronized (Gunn & Doyle 1997). Doppler images of the system have been obtained by Piskunov (1996) and Piskunov et al. (2001) for two observational seasons (1994 and 1996); see Sokoloff & Piskunov (2002) for a discussion of the observational situation from the viewpoint of dynamo theory.

The distribution of spots on ER Vulpeculae appears to be remarkably nonaxisymmetric. Dark spots, which are thought to be connected with magnetic activity, seem to be anchored to the bright spot associated with the reflection effect. The dark spots are located in polar and subpolar latitudes, and their form appears to be quite variable with time, so it would be natural to associate them with an oscillating dynamo solution, with magnetic field concentrated in the subpolar regions. This form of spot structure seems to be intrinsically connected with the tidal interaction between two members of the close binary system. When considering similar systems, there appears to be a gradual reduction of the apparent degree of nonaxisymmetry with increasing stellar separation. Long-term monitoring of active regions on ER Vul with Doppler Imaging techniques (Piskunov et al. 2001), HR 1099 (Vogt et al. 1999) and II Peg (Berdyugina et al. 1999) shows a progressive increase in drift velocity of the active regions with respect to the line connecting the binary components. ER Vul which has the smallest separation between its components shows no such drift. On the other hand, the separation of the components of ER Vulpeculae is large enough to exclude direct magnetic flux exchange between the primary and secondary components.

This conclusion, being based on only three examples, obviously needs additional confirmation. Hall (1991) studied the connection between the Roche-lobe filling factor and a differential rotation coefficient, and found that differential rotation decreases as the stellar separation decreases. Such a variation in differential rotation is certainly consistent with our idea. We stress that additional observations to clarify the relations between stellar separation, differential rotation and nonaxisymmetric structures would be more than welcome.

Because a binary system is intrinsically nonaxisymmetric the idea of nonaxisymmetric dynamo action in such a system prima facie appears quite natural. Of course, heating by the reflection effect as well as by tidal forces can result in a nonaxisymmetric distribution of the dynamo control parameters (for example, the coefficient $\alpha$, which is proportional to the helicity). We present the relevant estimates in Sect. 2. We demonstrate in Sect. 4 that the associated magnetic field distribution has a substantial nonaxisymmetry, whose geometrical shape is quite similar to that of the observed dark spots. This result persists even in the presence of a degree of differential rotation; we allow the $\alpha$-spot to corotate with the stellar envelope at various depths below the surface, so that the spot moves with respect to the stellar surface. The timescale connected with this motion is much longer than the orbital period (about 1 day) of these binary systems, because the system is synchronized by tidal interaction. However the dynamo timescale (of order 10 years) is in turn much longer than the orbital timescale. So an obvious question to be clarified is to what extent the differential rotation can coexist with a nonaxisymmetric structure generated by an $\alpha$-spot (see Sect. 5).

The following more delicate problem is connected with the temporal behaviour of a nonaxisymmetric solution. The form of the observed dark spot appears to be consistent with the idea of a nonoscillatory dynamo solution, and a standard dynamo for a basically rigid rotating body, i.e. the so-called $\alpha ^2$-dynamo, usually does give nonoscillatory solutions. Sokoloff & Piskunov (2002) suggested a swing exitation between a slowly rotating $\alpha$-spot and an oscillating $\alpha ^2\omega$-dynamo mode, driven by the joint action of helicity and differential rotation, and which is subcritical in a nonresonant case. We demonstrate in Sect. 6 that for some rotation laws a resonant excitation of oscillating magnetic structure occurs, and the geometrical shape of such "spots'' remains similar to that of the observed spot structure.