Statistical Geometry in Scalar Turbulence

Abstract

A general link between geometry and intermittency in passive scalar turbulence is established. Intermittency is qualitatively traced back to events where tracer particles stay for anomalousy long times in degenerate geometries characterized by strong clustering. The quantitative counterpart is the existence of special functions of particle configurations which are statistically invariant under the flow. These are the statistical integrals of motion controlling the scalar statistics at small scales and responsible for the breaking of scale invariance associated to intermittency. PACS number(s) : 47.10.+g, 47.27.-i, 05.40.+j Scalar fields transported by turbulent flow occur in many physical situations, ranging from the dynamics of the atmosphere and the ocean to chemical engineering (see, e.g., Ref.[1]). Specific examples are provided by pollutant density, temperature or humidity fields and the concentration of chemical or biological species. The advection-diffusion equation governing the transport of the scalar field θ is :where v(r, t) is the incompressible advecting flow and κ is the molecular diffusivity. Two broad cases are distinguished : active scalars, where v depends on θ, e.g. by an explicit relation v = v (θ), and passive scalars, where the statistics of v is independent of θ. Here, we shall be concerned with the latter, although we conjecture that the physical mechanisms presented in the following are quite general and relevant also for the active cases. The Fokker-Planck equation (1) is associated to the Lagrangian dynamics of tracer particles whose position ρ(t) obeys dρ(t) = v(ρ(t), t) dt + √ 2κ dβ(t), where β(t) is the isotropic Brownian motion [2]. The equation (1) governs the evolution of the probability density of particles at position r and time t.Scalar turbulence is typically generated by maintaining a mean scalar gradient θ = g·r, e.g. by heating/cooling devices in temperature field experiments. The notation • denotes the average with respect to the velocity statistics, which is in principle arbitrary. We shall be interested in flows with correlations having a nontrivial power law behavior in the inertial range of scales r ≪ L, where L is the velocity correlation length. Examples are provided by the two and three dimensional Navier-Stokes turbulent flow (see, e.g., Ref.[3]). A very robust feature of scalar turbulence is its strong intermittency : rare strong events (such as the sharp cliffs observed in Fig. 1) dominate the scalar statistical properties. More quantitatively, intermittency reflects in the anomalous scaling of the correlations. In the inertial range, the scalar structure functions S n (r) = (θ(r, t) − θ(0, t)) n take the form S n (r) ∝ r ζ dim n

Turbulence with pressure: Anomalous scaling of a passive vector field

“…An important step is breaking the artificial assumption of the time decorrelation of the advecting velocity field; see the remarks in Refs. [21,22].…”

“…The concept of statistical conservation laws appears rather general, being also confirmed by numerical simulations of Refs. [21,22], where the passive advection in the two-dimensional Navier-Stokes (NS) velocity field [21] and a shell model of a passive scalar [22] were studied. This observation is rather intriguing because in those models no closed equations for equal-time quantities can be derived due to the fact that the advecting velocity has a finite correlation time (for a passive field advected by a velocity with given statistics, closed equations can be derived only for different-time correlation functions, and they involve infinite diagrammatic series).…”

Strong universality in forced and decaying turbulence in a shell model

“…The reason is that the riddle of anomalous scaling of correlation and structure functions in forced turbulent advection (passive and active) had been solved recently. First in the context of the non-generic Kraichnan model of passive scalar advection [18], and then, in steps, for passive vectors [19,20], generic passive scalars and vectors [21,22,23] and finally for generic active scalar and vectors [24,25,26]. The common thread of this advance is that anomalous scaling is discussed in the context of the decaying (unforced problem), in which one shows that there exist Statistically Preserved Structures (eigenfunctions of eigenvalue 1 of the appropriate propagator of the decaying correlation functions).…”

Active and passive fields in turbulent transport: The role of statistically preserved structures

“…To understand the progress made in the context of passive fields [4,5], note that the passive fields satisfy a linear equation of motion that can be written as…”