TRANSPORT AND DIFFUSION




mixing



Transport processes play a crucial role in many natural systems. Among the many possible examples, we mention the problem of particle transport in geophysical flows which is of obvious interest for atmospheric and oceanic problems. The most natural framework for investigating such phenomena is to adopt a Lagrangian viewpoint in which the particles are advected by a given Eulerian velocity field u(x,t) according to the differential equation
dx / dt = u(x,t) = v(t),
where v(t), is, by definition, the Lagrangian particle velocity.

Despite its apparent simplicity the problem of connecting the Eulerian property of v, to the Lagrangian properties of the trajectories x(t), is a very difficult task. In the last 20-30 years the situation has become even more complex by the recognition of the ubiquity of Lagrangian chaos (chaotic advection). Even very simple Eulerian fields can generate very complex Lagrangian trajectories which are indistinguishable from those obtained in a complex, turbulent, flow (see the figure).

This fact is of great importance to mixing, transport and diffusion in fluids. The use of techniques of dynamical systems allows one, e.g., to determine the scaling range of the Batchelor law for the passive scalar fluctuations at small scales or the connection between variations of the effective Lyapunov exponent and the strong spatial fluctuations of the magnetic field in the dynamo problem. This leads naturally to the problem of chaotic diffusion.

One of the main issues is: what does the knowledge of structure of the velocity field tell us about the diffusion of fluid particles e.g. whether it is anomalous and, if it is normal, how to compute the diffusion coefficient. Let us mention the fact that even in presence of simple Eulerian fiels, e.g. laminar and periodic in time, the diffusion coefficients as function of the parameters of the velocity field can display a rather non trivial behaviour.

In addition in many cases, e.g. closed basins and open systems, the diffusion coefficents are not able to give a complete description of many interersting phenomena, e.g. the spreading of polluttant initially confined in a small region. So it is necessary to take into account, using methods and techniques of the dynamical systems theory, the effects of the boundaries. This can be done in terms of the finite size Lyapunov exponent and exit time approach.

An important open problem, in fully developed turbulence, is the relationship between Eulerian and Lagrangian properties. Up to now there are not clear results, in particular the effects of intermittency corrections on the Lagrangian statistics of advected particles are not well understood. Even in a context of simple phenomenological models, e.g. multifractal one, it is not trivial to generalize the anomalous scalings of the Richardson law for relative dispersion which is valid for the K41 limit.

A first way to address the problem of Lagrangian/Eulerian intermittency relationship for particle dispersion in fully developed turbulence is to investigate the transport in velocity fields generated by simplified turbulence models (shell models) or in synthetic turbulent fields.