Passive Fields and Particles in Chaotic Flows

Abstract

Two examples for the interplay between chaotic dynamics and stochastic forces within hydrodynamical systems are considered. The first case concerns the relaxation to equilibrium of a concentration field subject to both chaotic advection and molecular diffusion. The concentration field develops filamentary structures and the decay rate depends nonmonotonically on the diffusion strength. The second example concerns polymers, modelled as particles with an internal degree of freedom, in a chaotic flow. The length distribution of the polymers turns out to follow a power law with an exponent that depends on the difference between Lyapunov exponent and internal relaxation rate.

Advected fields in maps—III. Passive scalar decay in baker's maps

“…In the latter case oscillations will occur in the approach to the limiting decay rate as " ! 0, as is found by [4] and also in related dynamo problems [13]. By contrast, in [34] a regular perturbation theory in integer powers of " is developed for one-dimensional mappings with noise that tends to zero at the boundaries.…”

“…0. This surprising feature has been noted in a related model [4]. The final section 8 offers concluding discussion.…”

“…A similar non-monotonic approach of to 0 as " ! 0 is seen in a closely related model [4] (also involving a fold in the mapping).…”

Deceleration of one-dimensional mixing by discontinuous mappings

“…[25] observed similar behaviour for permutations composed with diffusion, however non-monotonicity was not reported. Examples of maps in which non-monotonicity has been seen include an expanding map with three branches [33], and the non-uniform inverted baker's map with a no-flux boundary condition, where a power law relation had oscillatory non-monotonic behaviour [34]. Both maps contain points that are non-differentiable.…”

Rates of mixing in models of fluid devices with discontinuities

“…However, this approach to the mixing rate in the zero-diffusivity limit is nonmonotonic in many cases, predicting a deceleration with increasing diffusivity coefficient, which is counter-intuitive. Non-monotonicity in profiles of |λ 2 | with κ have been observed in one-dimensional maps before; for an expanding map with three branches where one branch is inverted Eckhardt et al (2003), and a non-uniform inverted baker's transformations with a no-flux boundary condition Gilbert (2006). However the oscillations in both of these maps occur as |λ 2 | converges to τ , the latter shown to have a polynomial power-law on average, in line with (5.30).…”