319. Turbulence. Dynamical approach

When we study a turbulent flow, whether turbulence is realized in fluid, plasma, etc., one of the most interesting and complicated questions is the one about transition to turbulence: how exactly the smooth motion of the field becomes turbulent, chaotic, independent of external noise?

It seems to be impossible to answer to this question using statistical approach to turbulence, although I hope to discuss some recent ideas involving diagrammatic methods some day.

The first attempts to develop another approach were made by L. Landau in 1944 and E. Hopf in 1948. In their models, behaviour of a fluid is getting more chaotic (i.e., turbulent) since instabilities (hierarchy of instabilities, actually) with incommensurable time scales develop for the flow. The velocity field becomes more disordered if more and more excitations with incommensurable time scales become present in the flow. In the latter case, velocity autocorrelation function rapidly falls off with time, although it is possible to observe some regularity pattern in the flow if the observation process takes longer than the Poincare recurrence time

,

where is a constant of the order 1, while is the number of excitations with incommensurable time scales. Although this model seems to be reasonable, Ruelle and Takens have shown later that the corresponding attractor is actually unstable, so this very complicated quasi-periodic motion can be really realized in Nature and has definitely nothing to do with the generic situation.

We have got better understanding of the transition phenomenon after discovery of dynamical chaos – randomess of dynamics in deterministic systems. One famous result in chaos theory is existence of strange attractors. Strange attractor is an attracting set of trajectories in the phase space of the system such that (almost) all trajectories in the set correspond to saddle points and are therefore unstable. As it turns out, many rather simple (as well as complex) systems get attracted to this set after a small number of bifurcations.

It is currently believed that a strange attractor should exist in the phase space of the Navier-Stokes equation. Once the system gets attracted to it, the turbulence pattern is fully developed (and regime of turbulence remains stationary in the statistical sense).

Dynamical approach to turbulence and chaos theory is especially useful if we consider transition from the laminar regime to the regime chaotic in time. The simplest scenarios (for the Taylor-Quette flow for example) of such transition include intermittency, infinite sequence of period doubling and breakdown of quasi-periodic motion. There are many much more complicated scenarios observed in Nature, but those three are a kind of canonical ones. The same scenarios are observed in lattice simulations of Navier-Stokes hydrodynamics on large lattices at .

It is also possible to explain spatial chaos using these considerations. For that, we have to consider Lagrangian formulation of hydrodynamics. Introducing Lagrangian markers, we rewrite the Navier-Stokes equation on the form

,

where is velocity of the flow satisfying to the Navier-Stokes equation. The point is that even if the dynamics of is regular, dynamics of can be chaotic. Such regime is called the regime of Largangian turbulence. It seems that Lagrangian turbulence can be developed into usual turbulence of the velocity field, but I don’t think I can say more about that at this point