Scale of turbulence and turbulent diffusion

A discussion on the scale of turbulence and turbulent diffusion that can be derived from the newly defined ESD is given as follows. 

For t0, this double correlation can be expressed by a parabolic curve as 

According to the expression of R(t) described above, the micro time and macro time scales can be determined as 

From these equations, it is clarified that both the scales are strongly affected by the smaller frequency fluctuation, that is, by the large-scale eddy. 

When it is assumed that the double correlation R(t) is equal to the Lagrangean double correlation RL(t), the variance that shows the degree of turbulent diffusion is expressed as 

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