Random stretching December 7 , 2000 1 Overview
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چکیده
1 Overview In the previous lecture we emphasized that the destruction of tracer variance by molecular diffusivity relies on the increase of ∇c by stirring. Thus the term κ∇c ·∇c in the variance budget eventually becomes important, even though the molecular diffusivity κ is very small. One goal of this lecture is to understand in more detail how tracer gradients in a moving fluid are amplified by simple velocity fields. We will assume that κ = 0 so that there stirring without mixing. This is a good approximation provided that the smallest scale in the tracer field is much greater than the length = κ/α which we identified in lecture 1. Gradient amplification is closely related to the stretching of material lines, a subject which was opened by Batchelor in 1952. A material line is a curve which consists always of the same fluid particles. Batchelor's main conclusion is that there is a timescale governing the ultimate growth of an infinitesimal line element, but no length scale other than that of the element itself. These dimensional considerations force the conclusion that the element grows exponentially, = 0 e γt , (1) where γ is a constant. Just as some close particle pairs separate exponentially, other pairs starting at distant points are brought close together. This might seem paradoxical until one recalls the folded tracer patterns evident in Welander's 1955 experiments (see the final figures in lecture 1). If two closely approaching particles are carrying different values of c then the gradient ∇c will be amplified.
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