Fractal Measures of Passively Convected Vector Fields and Scalar Gradients in Chaotic Fluid Flows

The passive convection of vector fields and scalar functions by a prescribed incompressible fluid flow v(x,t) is considered for the case where v(x,t) is chaotic. By chaotic v(x,t) it is meant that typical nearby fluid elements diverge from each other exponentially in time. It is shown that in such cases, as time increases, a convected vector field and the gradient of a convected scalar will generally concentrate on a set which is fractal. The present paper relates the stretching properties of the flow to the resulting fractal dimension spectrum. Motivation for these considerations is provided by the kinematic magnetic dynamo problem (in the vector case) and (in the scalar case) by recent experiments which demonstrate the possibility of measuring the fractal dimension of the gradient squared of convected passive scalars.