Onset of chaos in finite-dimensional models of turbulence

Abstract

A review of the approach to the study of turbulence (spatiotemporal chaos) in a finite spatial region, based on decomposition of the underlying partial differential equations over an infinite set of modes and subsequent truncation of the infinite chain of the corresponding ordinary differential equations (ODEs), is given. The review starts with a general survey of this approach. Then, to illustrate it in terms of real systems, two recently studied models are considered in some detail. The first model is a system of six ODEs for amplitudes of eigenmodes of the curl operator, which is a simplest nontrivial truncation of the three-dimensional (3D) Euler equations; the wave vectors of the six modes are closed into a simplex (tetrahedron) in the 3D space, which makes this truncation basis very natural. The resultant dynamical system (DS) has three integrals of motion, that can be interpreted as the energy, helicity, and squared angular momentum of the flow. Depending on values of these conserved quantities, numerical simulations reveal both regions of purely chaotic (ergodic) motion, and mixed regions where chaotic and regular motions coexist. The second model is produced by truncation of the two-dimensional (2D) complex Ginzburg-Landau (CGL) equation in a rectangular region, the spatiotemporal chaos being triggered by the modulational instability (MI) of a spatially uniform solution. Close to the MI threshold, the instability is dominated by few spatial modes, which makes it possible to perform the truncation in a consistent way. This leads to a fifth-order DS, that admits two different invariant reductions to third-order DSs. One reduction corresponds to a particular case of the 1D CGL equation, and it this case the system can be further transformed into the famous Lorenz system or its modified form. The other reduction produces DS that has coexisting exploding (singular) solutions and dynamical chaos accounted for by a compact strange attractor. Both types of the solutions are meaningful, the singular ones corresponding to the wave collapse in the 2D CGL equation.