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

Abstract

We have recently proposed that the statistics of active fields (which affect the velocity field itself) in well-developed turbulence are also dominated by the Statistically Preserved Structures of auxiliary passive fields which are advected by the same velocity field. The Statistically Preserved Structures are eigenmodes of eigenvalue 1 of an appropriate propagator of the decaying (unforced) passive field, or equivalently, the zero modes of a related operator. In this paper we investigate further this surprising finding via two examples, one akin to turbulent convection in which the temperature is the active scalar, and the other akin to magneto-hydrodynamics in which the magnetic field is the active vector. In the first example, all the even correlation functions of the active and passive fields exhibit identical scaling behavior. The second example appears at first sight to be a counter-example: the statistical objects of the active and passive fields have entirely different scaling exponents. We demonstrate nevertheless that the Statistically Preserved Structures of the passive vector dominate again the statistics of the active field, except that due to a dynamical conservation law the amplitude of the leading zero mode cancels exactly. The active vector is then dominated by the sub-leading zero mode of the passive vector. Our work thus suggests that the statistical properties of active fields in turbulence can be understood with the same generality as those of passive fields.

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).…”

Statistics of active versus passive advections in magnetohydrodynamic turbulence

“…This is indeed what has been measured for similar models [11], where both fields appear to display the same anomalies. Further, as shown in [6], the conservation of the third invariant allows for another scaling, |b n | 2 ∼ k α−2/3 n , for which the magnetic helicity flux is constant. But since this scaling is incompatible with the conservation of the two other invariants, it is not relevant to the statistics of the magnetic field.…”

“…The two fields have different scaling properties. In [6], despite this difference, the claim was made that the analogy does hold in the sense that there exist subleading zero modes of the propagator of the correlation functions of the auxiliary passive field with the scaling of the correlation functions of the active field.The purpose of this note is to show that this is actually not the case. To this end, we will limit our investigation…”

“…Only the two-dimensional model can sustain a stationary state and we will limit ourselves to this case. As dimensional analysis shows [6], the conservation of the first two invariants implies that u n and b n must both have Kolmogorov scalings,…”

“…In [4] and [5,6], the case of thermal convection in the Boussinesq approximation was studied. There it was shown that the scaling of the active field is also dominated by the statistically preserved structures of auxiliary passive fields.…”