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

Abstract

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...

Nonperturbative renormalization group study of the stochastic Navier-Stokes equation

“…On the other hand, the asymptotic solution of the Kármán-Howarth-Monin equation [22] shows that (4) is always subdominant with respect to the inverse energy cascade spectrum E(p) ∝ p −5/3 , for ε 2, i.e., even in the regime where renormalization group analysis should apply. Direct numerical simulations up to 2048 2 resolution give clear evidence of the inverse cascade [21,22]. A scenario reconciling these findings may be that the Kraichnan-Kolmogorov inverse cascade corresponds to a renormalization group nonperturbative fixed point which does not bifurcate from the Gaussian fixed point at marginality.…”

“…Direct numerical simulations [19,20] exhibited, within a 512 3 -lattice accuracy, a transition in the ε dependence of η 2 which is consistent with the freezing scenario. However, the situation completely differs in two dimensions [21,22]. On the one hand, perturbative renormalization group analysis [23] upholds the validity of (4) for any ε.…”

“…An alternative way to derive the results of this appendix is based on the Janssen-De Dominicis [53,54] path integral representation of (21). We refer to Ref.…”

Nonlinear Superposition of Direct and Inverse Cascades in Two-Dimensional Turbulence Forced at Large and Small Scales

“…The decomposition in terms of auxiliary velocity fields here proposed can be straightforwardly generalized to 2d Navier-Stokes equations with an energy input distributed over different scales. In this perspective, the cascade overlap of the two sources model, here investigated, evinces the physical mechanism for why Kraichnan theory applies also in the presence of power-law sources and, consequently, for the inability of renormalization group approach to correctly predict Navier-Stokes energy spectra [30], even in what may seem a priori a perturbative regime.…”

“…Using these hypotheses, a careful analysis along the lines of [10,28,30] justifies in the range ℓ ≪ r ≪ L, the expansion…”