Influence of Eulerian and Lagrangian scales on the relative dispersion properties in Lagrangian stochastic models of turbulence A. Maurizi, 1 G. Pagnini, 1,2 and F. Tampieri 1 1 ISAC-CNR, Istituto di Scienze dell’Atmosfera e del Clima del CNR, via Gobetti 101, I-40129 Bologna, Italy 2 Facolta ` di Scienze Ambientali, Universita ` di Urbino, Campus Scientifico Sogesta, I-61029 Urbino, Italy ~Received 22 September 2003; published 30 March 2004! The influence of Eulerian and Lagrangian scales on the turbulent relative dispersion is investigated through a three-dimensional Eulerian consistent Lagrangian stochastic model. As a general property of this class of models, it is found to depend solely on a parameter b based on the Kolmogorov constants C K and C 0 . This parameter represents the ratio between the Lagrangian and Eulerian scales and is related to the intrinsic inhomogeneity of the relative dispersion process. In particular, the quantity g * 5g / C 0 ~where g is the Rich- ardson constant! and the temporal extension of the t 3 regime are found to be strongly dependent on the value adopted for b . DOI: 10.1103/PhysRevE.69.037301 PACS number~s!: 47.27.Qb In the frame of the Eulerian description of turbulence, the Kolmogorov @1,2# theories are based on spatial separation, while the natural description in the Lagrangian approach is in terms of time elapsing from an initial condition ~see, e.g., Ref. @3#!. Each description naturally leads to the definition of two characteristic scales, say l and t, for space and time, respectively. The effects of the two scales on turbulence dy- namics are particularly evident in the relative dispersion pro- cess. When the particle separation lies in the inertial sub- range, the spatial structure ~Eulerian length scale! affects the dispersion features ~Lagrangian property!@4#. The relative dispersion in the inertial subrange regime is characterized by the Richardson law @5# and the nondimensional constant g which should be considered universal even though measured values range from 0.06 to 6 @6–9#. Lagrangian stochastic models ~LSMs!, along with their utility in dispersion studies, provide a powerful tool for in- vestigating some properties of turbulence, as shown in Refs. @10–13#. Its formulation naturally connects Lagrangian and Eulerian statistics through the requirement of statistical Eu- lerian consistency ~well mixed condition, hereinafter WMC, @14,15#! of Lagrangian particle trajectories. Accordingly, model results should be dependent on the values of l and t. The aim of this Brief Report is to investigate the properties of WMC formulation of LSMs with respect to the parameters determining changes in the duration and the value of g of the Richardson t 3 regime, as the Lagrangian and Eulerian scales are modified. The basic assumption of the LSM relies on the Mark- ovianity of the velocity process. In fact, in the inertial sub- range the Lagrangian acceleration autocorrelation scale is of the order of the Kolmogorov time scale t h , which decreases with increasing Reynolds number, as predicted by the Heisenberg-Yaglom formula based on the Kolmogorov theory @4#. This prediction was experimentally confirmed in Ref. @16#. Another experimental support for the LSM, in par- ticular to the WMC, was given in Refs. @17,18#, where the Eulerian probability density function ~PDF! is shown to fulfil the Chapman-Kolmogorov equation underlying the Markov- ian assumption. Intermittency effects are not considered be- cause they are found to be negligible in the LSM @19#. Thus, according to Ref. @15#, the particle separation d x and velocity difference d v are represented by a stochastic differential equation ~SDE! of the Langevin type. Using the above defined scales and a velocity scale y, to make the model nondimensional, the Langevin equation turns out to be d d x i 5b d v i dt , d d v i 5a i ~ d v, d x, t ! dt 1b ij ~ d x, t ! dW j , ~1! where dW j is a component of a three-dimensional Wiener process and b 5yt l 21 is recognized as the well known Lagrangian-to-Eulerian scale ratio. In Ref. @14# the SDE co- efficients a i and b ij are called the drift and diffusion coeffi- cients, respectively. The associated nondimensional Fokker- Planck equation turns out to be ] p ] t 52b ] ] d x i ~ d v i p ! 2 ] ] d v i ~ a i p ! 1b ij ] 2 p ] d v i ] d v i , ~2! where p is the Lagrangian PDF of the process ~d x,d v!. To avoid physical inconsistencies caused by different Ito and Stratonovich interpretations of the stochastic calculus, the tensor b ij must be independent of d v @20#. For consistency with the Lagrangian structure function of the second order, b ij 5A 2 d ij . The drift term is retrieved by imposing the Eu- lerian statistical consistency ~WMC!@14#, which is obtained by the Novikov relation between Eulerian and Lagrangian PDFs @21# in combination with the Fokker-Planck equation. Under this assumption, it turns out that a i 5 ] ln p E ] d v i 1F i , ~3! where p E is the Eulerian PDF and ] ~ F i p E ! ] d v i 52 ] p E ] t 2b ] ~ d v i p E ! ] d x i , ~4! PHYSICAL REVIEW E 69, 037301 ~2004! 1063-651X/2004/69~3!/037301~4!/$22.50 ©2004 The American Physical Society 69 037301-1