On the fine-scale structure of vector fields convected by a turbulent fluid

Abstract

This paper is a contribution to the study of statistically homogeneous, dynamically passive vector fields convected by a turbulent fluid and subject to a molecular diffusivity λ that is small compared with the kinematic viscosity v. Two types are considered: the first, denoted by F, has the property that the total flux across a material surface moving with the fluid is conserved if λ = 0 (e.g. magnetic field); and the second, denoted by G, is the gradient of a conserved scalar quantity θ (e.g. temperature gradient). Attention is focused on small-scale variations with length-scale less than $(v^3|\epsilon)^{\frac {1}{4}}$. A theory of Batchelor's in terms of Eulerian correlations for the distribution of θ for the case when λ [Lt ] v is extended and applied to the vector fields, thereby giving equations for the covariance tensors of F and G appropriate for separations less than $(v^3|\epsilon)^{\frac {1}{4}}$. According to these equations, the effect of convection on small-scale components of the fields is to amplify and also to distort by reducing the scale; the ratio of these two effects is measured by a parameter σ. It is shown that if $\sigma \textless {\frac{5}{2}$, the small-scale structure is stable against perturbations however small λ/v may be, the amplification being eventually balanced by the dissipation which is increased by the reduction of scale. In the case of the quantity G, σ = 1. The value of σ for the case of F is not known, but reasons are given for believing that it is less than one, and it is concluded that the behaviour of $\overline{\bf F^2}$ and $\overline{\bf G^2}$ in a field of homogeneous turbulence is qualitatively the same. In particular, $\overline{\bf F^2}$ does not grow indefinitely with time as predicted by previous arguments. The correlation functions for small separations and the corresponding spectrum functions for a statistically steady state are obtained. The relation between this analysis and that for random vector fields in a uniform straining motion of infinite extent is considered in detail, for Pearson has shown that, if the strain is an irrotational distortion, then $\overline{\bf F^2} \rightarrow \infty$ with time. It is shown that this divergence is due to the amplification of components with very small wave-numbers or, equivalently, of very large scale, and it is therefore not considered relevant to a study of homogeneous turbulence.

The particular case of the magnetic field in a good conductor is considered. If the Lorentz forces are unimportant, it is estimated that the magnetic energy of a weak seed field will be in general amplified by the turbulence by a factor lying somewhere between the Reynolds and magnetic Reynolds numbers of the turbulence before ohmic dissipation as increased by the reduction of scale limits the growth, and it is suggested further that the magnetic field will eventually decay to zero in the absence of external electromotive forces.

In an appendix, the theory is applied tentatively to the turbulent vorticity (which satisfies the same equation as F if λ = v) and an expression for the energy spectrum function for very large wave-numbers is deduced. This is compared with an expression given by Townsend, and is found to have a similar qualitative behaviour but gives values about one-half as large.