Evolution of magnetic field curvature in the Kulsrud-Anderson dynamo theory

We find that in the kinematic limit the ensemble averaged square of the curvature of magnetic field lines is exponentially amplified in time by the turbulent motions in a highly conductive plasma. At the same time, the ensemble averaged curvature vector exponentially decays to zero. Thus, independently of the initial conditions, the fluctuation field becomes very curved, and the curvature vector becomes highly isotropic. Subject headings: ISM: magnetic fields — MHD — turbulence — methods: analytical It was shown by Kulsrud and Anderson (1992) that MHD turbulent dynamo action builds magnetic field energy primarily on scales smaller than the smallest turbulent eddy size (which is the viscosity scale), but still larger than the resistive scale (provided the magnetic Prandtl number is large). This result was found under assumption that the “kinematic” approximation is valid, i.e. the field is weak enough that it does not affect the turbulent motions. In this paper we calculate the evolution of magnetic field curvature within the framework of the Kulsrud-Anderson kinematic dynamo theory. Such calculations are of great interest because a rapid built up of curvature may quickly break down the kinematic approximation and change the dynamo action on very small scales considerably. Following Kulsrud and Anderson (1992) we make the following assumptions. We use the “kinematic” approximation. We neglect resistivity (infinite magnetic Prandtl number limit). We assume that the turbulence is incompressible, homogeneous, isotropic and static, and we use zero correlation time approximation for the turbulent motions: Vα(t, r) = 1 (2π)3 ∫ Vα(t,k) e ik·r dk, (1) 〈Vα(t,k)〉 = 0, (2) 〈V ∗ α (t ,k)Vβ(t,k)〉 = [Jk(δαβ − k̂αk̂β) + iJ̄kεαγβkγ ]δ(k ′ − k)δ(t − t). (3) Here and below 〈...〉 means ensemble average, δαβ is the Kronecker symbol, εαβγ is the unit antisymmetric tensor, δ(t − t) and δ(k − k) are the Dirac δ-functions, k̂ = k/k is a unit vector, and we always assume summation over repeated indices. Functions Jk and J̄k are the normal and the helical parts of the turbulence, they depend only on the absolute value of k. We make no assumptions about the statistics of initial magnetic field. Using equations (1) and (3), it is straightforward to calculate the correlation tensors between velocities and their spatial derivatives, taken at the same point of space r but at different time, t and t: 〈Vα(t , r)Vβ(t, r)〉 = (ηT /2π)δαβδ(t ′ − t), 〈VαVβ;γ〉 = αεαβγδ(t ′ − t), 〈Vα;βVγ;δ〉 = (γ/5)(5δαγδβδ − δαβγδ)δ(t ′ − t), 〈VαVβ;γδ〉 = −〈Vα;δVβ;γ〉 , 〈Vα;βVγ;δτ 〉 = ωεαγηδηβδτ δ(t ′ − t), (4) 〈VαVβ;γδτ 〉 = −〈Vα;τVβ;γδ〉 , 〈Vα;βγVδ;τη〉 = (λ/15)(7δαδδβγτη − δαβγδτη)δ(t ′ − t). Here and below, in order to shorten notations, spa-