Chaotic Fluid Convection and the Fractal Nature of Passive Scalar Gradients

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Abstract
The chaotic convection of passive scalars by an incompressible fluid is considered. (The convection is said to be chaotic if nearby fluid elements typically diverge from each other exponentially in time.) It is shown that during the time evolution the square of the gradient of chaotically convected passive scalars typically concentrates on a fractal set. Considerations of the local stretching properties of the flow lead to a partition function which yields the dimension spectra of the resulting fractal measure. Fractal structure is a result of the nonuniform stretching of typical chaotic flows.