Chaotic advection of irrotational flows and of waves in fluids

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Abstract
This paper treats the kinematics of particles advected passively by flow of an incompressible fluid. It is shown that for steady irrotational flow without circulation, and for many monochromatic waves in a fluid the particle paths are not chaotic, i.e. do not depend sensitively on initial conditions. However, if the flow is a time-periodic potential flow or a superposition of waves then the particle paths may be chaotic. This is shown by the application of the theory of Melnikov to the breakup of a heteroclinic orbit (which connects two stagnation points and may bound a region of closed streamlines) and the onset of chaos in two examples of two-dimensional flow. The first example is a simple unbounded irrotational flow comprising a steady flow with two stagnation points which has a time-periodic perturbation. The second example is of two Rossby waves with a mean zonal flow; the particle paths are examined geometrically and numerically, and consequences for pollutant dispersion are discussed in physical terms. Also the combination of the effects of chaotic advection and molecular diffusion on the transport of a solute are examined.