Turbulent transport of magnetic fields. V. Distribution of magnetic energy in a simple α2-dynamo

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Abstract
In this paper we analyse the stationary mean energy density tensor Tij = BiBj for the x 2-sphere. This model is one of the simplest possible turbulent dynamos, originally due to Krause and Steenbeck (1967): a conducting sphere of radius R with homogeneous, isotropic and stationary turbulent convection, no differential rotation and negligible resistivity. The stationary solution of the (linear) equation for Tij is found analytically. Only Trr , T θθ and T φφ are unequal to zero, and we present their dependence on the radial distance r. The stationary solution depends on two coefficients describing the turbulent state: the diffusion coefficient β≈⟨u2⟩τc/3 and the vorticity coefficient γ ≈ ⟨|▿×u|2⟩τc/3 where u(r, t) is the turbulent velocity and c its correlation time. But the solution is independent of the dynamo coefficient α≈−⟨u·▿×u⟩τc/3 although α does occur in the equation for Tij . This result confirms earlier conclusions that helicity is not required for magnetic field generation. In the stationary state, magnetic energy is generated by the vorticity and transported to the boundary, where it escapes at the same rate. The solution presented contains one free parameter that is connected with the distribution of B over spatial scales at the boundary, about which Tij gives no information. We regard this investigation as a first step towards the analysis of more complicated, solar-type dynamos.

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