Introduction

As early as in 1941 Kolmogorov[1] predicted some universal features of fluid turbulence which have been confirmed by experiments in later years. It is rather surprising that they are derived from dimensional analysis based on the simple assumption of `local homogeneity' for small-scale turbulence. More surprising is the fact that those have little to do with the equation of dynamics of fluid. It is addressed in this paper to locate this `missing link' through rederivation of the -5/3 power law of the spectrum for the inertial subrange as the universal law using the dynamical equation proposed in ref.2. The key issue for this task to work is that equation on which to describe the features of small eddies as independent of individual flow geometry. To meet this purpose a six-dimensional Navier-Stokes equation is employed, where additional 3D space having a length dimension , corresponding to eddy size, is introduced. Originally such an equation has been derived elsewhere[2], using non-equilibrium statistical mechanics starting from Liouville's equation. In this formalism the 3D `eddy' space has been introduced as natural consequence of a mathematical procedure, namely, the separation of variables of turbulent fluctuation-correlation equation. The equation stands as 6D generalization of the Navier-Stokes equation which in 3D physical space degenerates to the classical equation. In what follows, however, it is intended to rederive the same equation using phenomenologies alone on the basis same as the classical theory of Kármán and Howarth[3].　Then,　the wave-number space is introduced　for separating variables (Sec.2). It is then Fourier-transformed into eddy space, thereby the 6D Navier-Stokes equation, the basis to all what follows is established (Sec.3). The equation gives a novel expression for turbulent dissipation, enabling to predict existence of a pseudo-singularity surrounding the dissipation region (Sec.4). In inertial subrange, this singularity turns out to be a couple of dipoles of opposite sense, separated by the order of several tens to hundreds the Kolmogorov length (Sec.5). Power spectrum of this locally homogeneous turbulence is calculated, showing a wave-number dependence close to -5/3 power law of the Kolmogorov theory.(Sec.6)