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

Abstract

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 stat...