Chaotic advection

Studying the advection of a passive scalar field is the basic approach for understanding the general problem of mixing in fluids.

The evolution of a scalar field $\theta ({\mbox{\boldmath$x$}},t)$ diffusing and transported by a given velocity field ${\mbox{\boldmath$u$}}({\mbox{\boldmath$x$}},t)$ is given by the advection-diffusion equation

$${\partial \theta \over \partial t} + {\mbox{\boldmath$u$}} \cdot \nabla \theta = \kappa \nabla^2 \theta \;.$$ (2.8)

In the passive approach one assumes that it is possible to neglect all the possible kind of feedback of the transported field on the flow, e.g. the buoyancy force in the case of temperature field. Moreover Equation (2.8) is obtained assuming that the trajectories of the advected field coincide with those of the fluid particle, i.e. neglecting inertial effects [37] which are originated by difference of density between the solution and the solute or by finite size of the particle in suspension.

This clearly means that one is just giving an approximate description of fluid transport, but on the other side this approach has the great advantage that the behavior of a blob of passive scalar injected in the fluid can be understood in term of the statistical properties of fluid trajectories, which is the essence of the Lagrangian description of transport.

Fluid trajectories in realistic flows are typically chaotic, i.e they show a sensible dependence on initial conditions, thus also the advection of a scalar field will be chaotic. Chaotic advection contains the essence of the mixing properties of fluid transport, that is the characteristic mechanism of stretching and folding of material elements.

Here we will focus on chaotic advection by a smooth, two-dimensional, time-dependent, incompressible velocity field, which is the case of the velocity field in the direct enstrophy cascade in presence of friction, as discussed in the previous section.

Because of the presence of chaos, nearby fluid trajectories typically separate with an exponential rate given by the leading Lyapunov exponent [38]:

$$\lambda = \lim_{t \to \infty} \lim_{\delta_0 \to 0} {1 \over t} \ln {\delta(t) \over \delta_0}$$ (2.9)

where $\delta_0$ and $\delta(t)$ are respectively the initial separation and the separation at time $t$ of the two Lagrangian trajectories: $\delta(t) = \vert {\mbox{\boldmath$x$}}^{(1)}(t) - {\mbox{\boldmath$x$}}^{(2)}(t) \vert$. As a consequence an element of fluid is elongated exponentially in the direction of the leading Lyapunov vector, resulting in a long filament, which is folded on itself several times. At the same time, because of incompressibility, the area of the two-dimensional fluid element is preserved, and so the thickness in the transverse direction must decrease exponentially fast with the same rate.

Injecting in the fluid a blob of passive tracer with a typical size $\ell_0$ it will follow the same evolution, and in a time $t$ it will became a filament of length $L=\ell_0 \exp(\lambda t)$ and thickness $r=\ell_0 \exp(-\lambda t)$. The thinning process is finally stopped by the diffusion process on the diffusive scale $\sqrt{\lambda \kappa}$

This means that the original fluctuation of passive scalar which was injected at the scale $\ell_0$ in a time $t$ has been transported down to the small scale $r$. This phenomenon is known as the direct cascade of passive scalar, in which large scale inhomogeneities of the concentration field are transported by the flow to the small scales where they are omogeneized by molecular diffusivity.

Chaotic advection provides a mechanism to transport quickly a large scale fluctuation to small scale, originating the typical filamental pattern which are observed in transported fields. The roughness of the concentration field can be characterized by a Hölder exponent $h$:

$$\vert\delta \theta({\mbox{\boldmath$x$}},{\mbox{\boldmath$r$... ...{\boldmath$r$}}\vert^h, \vert{\mbox{\boldmath$r$}}\vert \to 0$$ (2.10)

For a smooth field (i.e differentiable) at ${\mbox{\boldmath$x$}}$ we have $h = 1$ while the range $0 < h < 1$ correspond to an irregular (e.g. filamental) field.