Scaling Theory of Relative Diffusion in Chaos and Turbulence

Abstract

The characteristic feature of chaos and turbulence is studied in the present paper from a general point of view of the scaling theory of transient phenomena proposed by the present author [1∿5]. For this purpose, we investigate here the relative diffusion in chaos and turbulence, namely

$$y\left( t \right) = < {\left( {{x_1}\left( t \right) - {x_2}\left( t \right)} \right)^2} > ,$$

(1.1)

where x_j(t) denotes the trajectory of the j-th particle. The essence of chaos and turbulence is characterized by the instability of trajectories and consequently by the exponential growing of the relative diffusion y(t) in the initial time region, for a small initial deviation y(o); namely

$$y\left( t \right) \simeq y\left( 0 \right) {e^{\gamma t}};\gamma > 0.$$

(1.2)