In ̄ation and twist of a stellar magnetic ®eld in the presence of a turbulent accretion disc

Accretion discs are ubiquitous in astrophysics and in particular in accreting objects like cataclysmic variables, X-ray binaries or young stars. For many years magnetic ®elds have been included in models of such objects to be able to generate out ̄ows or jets or to explain accretion ofmatter on to the poles of the central objects. The interest in magnetic ®elds in accretion systems has been recently strengthened. On the one hand, small-scale magnetic ®elds are now believed to play an important role in the angular momentum transport in discs. On the other hand, the signi®cant part played by large-scale magnetic ®elds is revealed by recent observations of the spin period of the central objects. Without magnetic ®elds, the accretion of matter always spins-up the central objects. However, observations show the existence of spin-down torques both in X-ray binaries (Bildsten et al. 1997; Bildsten 1998) and in T Tauri stars (Bouvier et al. 1995). Magnetic ®elds in the corona (the region above the disc) establish a connection between the disc and the central object, and are thought to be responsible for such a torque. Moreover, quick switches from the spin-up state to the spin-down state are now observed in several X-ray binaries, and cannot be explained by the standard model for the interaction between the magnetic ®eld of the central object and the disc (Ghosh & Lamb 1978). Thus the study of this interaction must be revisited, as it plays a crucial role in accreting systems determining, in particular, the net torque exerted on the central objects. Concerning accretion discs, the main problem is to model how matter loses angular momentum (Papaloizou & Lin 1995). We assume here the presence of turbulence in the disc which generates an effective turbulent viscosity. This idea is nowadays well supported by two facts. First, ̄ows in accretion discs have high Reynolds numbers. Second, instabilities developing in accretion discs and giving rise to turbulence are now quite well identi®ed and studied, especially the magneto-rotational one (Stone et al. 1996). Nevertheless, the whole process from the development of the instability to a global description of the effective viscosity is not yet fully understood. Prescriptions for viscosity are thus still very useful. The most well-known is the Shakura±Sunyaev prescription (Shakura & Sunyaev 1973), which is a parametrization