tokens: Rincon, Ogilvie Proctor ( 2007 ), where a subcritical dynamo generated by the interaction between (fluctuating) non-axisymmetric and (mean) axisymmetric toroidal magnetic field leads to ultimate saturation in a bounded domain with zero initial magnetic field. Indeed, in our theory the domain-integrated helicities of the flow and the magnetic field are both zero, while the domain-integrated cross-helicity vanishes because and are out of phase. The theory also allows us to investigate the astrophysically relevant weakly dissipative regime. We do this by showing that the inviscid limit of the theory, Re ≫ 1, is well behaved, and then examine the further limit of nearly ideal MHD by considering large values of Re m . Although the zonal magnetic field fluctuation becomes very large in this regime the angular momentum flux remains well behaved for all wavenumbers and decreases as | m | −1 (equation 34 ). A simple prediction for the magnitude of this flux follows from the assumption that the dominant wavenumber m corresponds to the optimum wavenumber from linear theory that yields the fastest MRI growth rate. The resulting expression for the angular momentum flux, , resembles the prescription of the α-model of Shakura Sunyaev ( 1973 ) but with the sound speed replaced by the Alfvén speed. A similar dependence of the momentum flux on the square of the magnetic field was observed in the simulations of a compressible but magnetic field dominated disc by Hawley et al. ( 1995 ), although subsequent computations by Sano et al. ( 2004 ) and Pessah, Chan Psaltis ( 2007 ) suggest a weaker dependence on v A . We emphasize that throughout this paper the symbols Re and Re m refer to scaled values of the Reynolds number and the magnetic Reynolds number, i.e. the condition Re m ≫ 1, for example, refers to Re m * ≫ ε −1 while the condition Re m ≪ 1 refers to Re m * ≪ ε −1 . The latter condition can therefore be satisfied for the large Re m * expected of astrophysical systems provided only that these are small compared to ε −1 , a quantity defined here in terms of the shear rate. Our approach has several attractive features, one of which is the fact that angular momentum is inevitably transported outward. In this connection we mention that Lesur Longaretti ( 2007 ) find that the transport coefficient F v /|σ| in a turbulent disc increases with the magnetic Prandtl number Re m / Re . In our (non-turbulent but fully non-linear) study we find a similar tendency when m is fixed (cf. Fig. 7 a): in the supercritical regime, above the marginal stability curve, F v decreases as Re increases for fixed Re m ; when Re is fixed, F v increases as Re m increases. Possible concerns include neglecting the possibility that saturation is accompanied by net outflow in the vertical direction, as measured by the quantity although this is a consistent assumption that follows from the relation obtained from equation ( 1a ). This relation is identically satisfied by . A further concern is
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