Passive Advection and the Degenerate Elliptic Operators M n

Hakulinen, Ville

doi:10.1007/s00220-002-0778-0

Cited by 10 publications

(8 citation statements)

References 18 publications

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“…Note the multibody structure of the operators in ( 69) and ( 70). The limiting PDF's obtained for κ → 0 define the heat kernels of the κ = 0 version of the operators that are singular elliptic and require some care in handling (Hakulinen, 2000).…”

Section: Multiparticle Motion In Kraichnan Velocitiesmentioning

confidence: 99%

“…The κ → 0 limit of the stationary correlation functions does exist and the κ-dependence drops out of the expressions in the convective interval, as for the pair correlator. The limit is given by the formulae (154) with the κ = 0 versions of G N (Hakulinen, 2000). Note in passing that the advection preserves any power of the scalar so that dissipative anomalies are present also for orders higher than the second.…”

Section: Passive Fields In the Inertial Interval Of Turbulencementioning

confidence: 99%

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Particles and fields in fluid turbulence

2001

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The understanding of fluid turbulence has considerably progressed in recent years. The application of the methods of statistical mechanics to the description of the motion of fluid particles, i.e. to the Lagrangian dynamics, has led to a new quantitative theory of intermittency in turbulent transport. The first analytical description of anomalous scaling laws in turbulence has been obtained. The underlying physical mechanism reveals the role of statistical integrals of motion in non-equilibrium systems. For turbulent transport, the statistical conservation laws are hidden in the evolution of groups of fluid particles and arise from the competition between the expansion of a group and the change of its geometry. By breaking the scale-invariance symmetry, the statistically conserved quantities lead to the observed anomalous scaling of transported fields. Lagrangian methods also shed new light on some practical issues, such as mixing and turbulent magnetic dynamo.Comment: 165 pages, review article for Rev. Mod. Phy

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Section: Multiparticle Motion In Kraichnan Velocitiesmentioning

confidence: 99%

Section: Passive Fields In the Inertial Interval Of Turbulencementioning

confidence: 99%

Particles and fields in fluid turbulence

2001

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“…In the third case, we notice that v(k) k may be written as Proof. Go back to the framework of Section 2 and recall (16) and (17). Existence and uniqueness in Condition 3 follow from Theorem 22 below.…”

Section: Proposition 18 If There Exists a Borel Setmentioning

confidence: 99%

Noise Prevents Collapse of Vlasov‐Poisson Point Charges

Delarue

Flandoli

Vincenzi

2013

Comm Pure Appl Math

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We elucidate the effect of noise on the dynamics of N point charges in a Vlasov-Poisson model driven by a singular bounded interaction force. A too simple noise does not impact the structure inherited from the deterministic case and, in particular, cannot prevent coalescence. Inspired by the theory of random transport in passive scalars, we identify a class of random fields which generate random pulses that are chaotic enough to disorganize the deterministic structure and prevent any collapse of the particles. We obtain strong unique solvability of the stochastic model for any initial configuration of different point charges. In the case where there are exactly two particles, we implement the "vanishing noise method" for determining the continuation of the deterministic model after collapse.

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“…The nullspace of M (ξ) n can be thought as consisting of local martingales of an effective purely multiplicative stochastic process for each value of n. The relevance of these quantities for the unique solution [20] in L 2 (R dn ) of (4) is discussed in details in [5,13]. The limit ξ ↓ 0 illustrates the situation.…”

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confidence: 99%

Scaling Properties of the Two-Dimensional Randomly Stirred Navier-Stokes Equation

2007

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We show that the statistics of a turbulent passive scalar at scales larger than the pumping may exhibit multiscaling due to a weaker mechanism than the presence of statistical conservation laws. We develop a general formalism to give explicit predictions for the large scale scaling exponents in the case of the Kraichnan model and discuss their geometric origin at small and large scale.PACS numbers: 47.27Gs, 05.10GgTurbulent transport poses challenges for fundamental research with important implications for many environmental (e.g. impact of natural and anthropogenic pollutants on climate) and industrial (e.g. design of effective mixers of chemical products) applications. During the last fifteen years, the field has seen major developments [14]. The study of an analytical tractable model, the Kraichnan model of passive advection [22,23], permitted for the first time [9,18] to prove that the statistics of a turbulent passive field (e.g. the temperature) is intrinsically not self-similar in the inertial range (fine scales of fluid motion not affected by thermal dissipation). More importantly, drawing on concepts and methods from stochastic analysis [5,19] pointed out a general mechanism accounting for the experimentally and numerically observed multiscaling (see e.g. [21,30]) of inertial range statistical indicators. Accordingly, the statistics of equal time correlation functions is dominated by global statistical invariants of the Lagrangian dynamics [5,13]. Although this picture can be established in a mathematically controlled way only for the Kraichnan model, numerical investigations of passive scalar advected by the Navier-Stokes equations [8] together with experiments [26,30] give strong evidences of the generality of the mechanism. In the unfolding of these developments, thoroughly summarized in [13], much attention has been devoted to the turbulent inertial range. However, in many physical contexts (e.g. the study of the large scale structures in cosmology [27]) it is important to understand the defining properties of statistical indicators of fluid tracers at scales larger than the typical energy source. As the energy of tracers transported by an incompressible velocity field is expected to "cascade" towards finer-scale, one might be tempted to infer from the absence of a "constant-flux" solution of the type predicted by Komogorov's 1941 theory [16] the onset of a thermodynamical equilibrium with Gaussian statistics and equipartition of scalar variance. However it was recently shown analytically [12] and numerically [6,7] that the presence of an equipartition-like scalar powerspectrum may well co-exist with higher order correlation functions exhibiting breakdown of self-similarity and multiscaling. Underlying these results is the existence, predicted in [5] for the Kraichnan model, of an asymptotic zero-mode expansion of correlation functions also at scales larger than the pumping. Here, we device a formalism to calculate (perturbatively) the scaling dimensions of the large scale zero modes. We show that l...

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Passive Advection and the Degenerate Elliptic Operators M n

Cited by 10 publications

References 18 publications

Particles and fields in fluid turbulence

Particles and fields in fluid turbulence

Noise Prevents Collapse of Vlasov‐Poisson Point Charges

Scaling Properties of the Two-Dimensional Randomly Stirred Navier-Stokes Equation

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