The properties of a stochastic model with non-Gaussian random noise describing turbulent dispersion have been investigated, with reference to its Mathematical structure and to its behaviour simulating the inertial subrange. The process is Markovian, mean-square continuous and with correlated increments. The model is influenced by the turbulence inhomogeneities also at the smallest scales, that is, it does not correctly simulate the existence of a well-developed inertial subrange. Some numerical computations have been performed confirming the theoretical results

Cesari, R.

Tampieri, F.

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IL NUOVO CIMENTO VOL. 20 C, N. 3 Maggio-Giugno 1997On some properties of Lagrangian dispersion modelswith non-Gaussian noise()()R. CESARI(1) and F. TAMPIERI(2)(1) IMGA/CNR, Modena, Italy(2) FISBAT/CNR, Bologna, Italy(ricevuto il 10 Gennaio 1996; approvato il 19 Luglio 1996)Summary. — The properties of a stochastic model with non-Gaussian random noisedescribing turbulent dispersion have been investigated, with reference to its mathe-matical structure and to its behaviour simulating the inertial subrange. The processis Markovian, mean-square continuous and with -correlated increments. The modelis influenced by the turbulence inhomogeneities also at the smallest scales, that is, itdoes not correctly simulate the existence of a well-developed inertial subrange. Somenumerical computations have been performed confirming the theoretical results.PACS 92.60.Ek – Convection, turbulence, and diffusion.PACS 47.27.Qb – Turbulent diffusion.PACS 01.30.Cc – Conference proceedings.1. – IntroductionThe Lagrangian description of turbulent dispersion by means of stochastic pro-cesses was initiated by Taylor (1921) [1] with reference to homogeneous turbulence.Obukhov (1959) [2] made explicit use of a stochastic differential (Langevin) equationto deal with the same problem, whereas more recently Thomson [3, 4] generalized theLangevin equation in the case of inhomogeneous turbulence.Thomson [4] outlined some general conditions that have to be satisfied by a stochasticmodel for describing the turbulent dispersion of marked fluid particles, that is:1. At small time scale, the accelerations of the fluid particles are independent of eachother [5], so that the stochastic process describing the velocity is a Markov process.() Paper presented at EUROMECH Colloquium 338 “Atmospheric Turbulence and Dispersion inComplex Terrain” and ERCOFTAC Workshop “Data on Turbulence and Dispersion in ComplexAtmospheric Flows”, Bologna, 4-7 September 1995.() The authors of this paper have agreed to not receive the proofs for correction.c Societa` Italiana di Fisica 425426 R. CESARI and F. TAMPIERI2. The small time behaviour of the statistics of the particle velocity has to be consis-tent with the Kolmogorov hypothesis referring to the inertial subrange, that is, thecorrelation of the velocity increments has to be independent of the initial conditionsand grow linearly with time [5, 6].3. A true fluid particle is characterized by a continuous trajectory, so that it seemsreasonable to require the stochastic process to be continuous (this is the assumptionleading to the model choice made by Thomson [4] and other authors).4. If the particles of tracer are initially well mixed, they will remain so. This criterionis referred to as well-mixed condition and applies to general, unsteady situations.A weaker condition was used in [3], that is, for times larger than the time scale ofthe inertial subrange the unconditioned statistics of the particle velocities has to bethe same as the Eulerian statistics of the fluid velocity.A detailed analysis of the previous points has been performed by [4] and by [6] referringto stochastic models with noise described by a Wiener process. In particular, [4] proposeda model based on the previous assumptions, which has been used to investigate dispersionin complex flows (see, for istance, Luhar and Britter [7] and Na¨slund et al. [8]). Themain shortcomings of this type of model have been also discussed in details, for istance bySawford [9]. Namely, in the 3D case, the equation leading to define the drift term in theLangevin equation does not have in general a unique solution.On the other hand, the model described by Thomson [3], characterized by a non-Gaussian noise, appears to be of easy application, although there are some problems withrespect to the conditions outlined above. In particular, the trajectories of the stochasticprocess do not display sample-path continuity (a strong continuity condition, as will bediscussed after), thus lacking to satisfy condition 3). In spite of this, implementations ofthis model turn out to describe quite well dispersion experiments in perturbed boundarylayers (see Thomson [10], Tinarelli et al. [11], Brusasca et al. [12], Tampieri et al. [13]).In this paper we shall discuss the structure of the stochastic process with non-Gaussian noise, as derived by Thomson [3], in order to discuss the applicability conditionof the model.2. – An outline of the modelAccording to the model proposed by Thomson [3], and referring to a monodimen-sional case, the motion of each particle in the phase space (x; v) is described by a Langevinequation, as follows:8<:dv =  vTdt+ d(t) ;dx = vdt ;(2.1)where T has the meaning of a decorrelation time (in homogeneous turbulence T is theLagrangian correlation time) and  is a random function withh(ti)(tj)i = (ti  tj) ;(2.2)where hi indicates the ensemble average.Information about the structure of turbulence is put into the explicit form of T andof the probability distribution (PDF) of , so that in general they will be function of posi-tion x.ON SOME PROPERTIES OF LAGRANGIAN DISPERSION MODELS ETC. 427In the following we shall examine the properties of the discretized form of themodel (2.1), that is:8><>:vi+1=1 tTi+1vi+ i+1;xi+1= vit+ xi;(2.3)where now kis a discrete random process whose PDF is determined, according to Thom-son [3], imposing that the unconditioned steady-state distribution of the particles in the(x; v) space is the same as the Eulerian distribution of the fluid.The expressions for the moments i are reported in [3], (eq. 10), at first order in T ,and in appendix A at the second order in t following [14] (eq. 3.7).To generate the realizations of the process kwe shall use random numbers sampledfrom a distribution with the first four moments i; i = 1; : : : ; 4 given by (A.1). We note herethat only using the moments of the random forcing expressed at order (t)2 the Gaussiancase can be recovered exactly (because 4 turns to be equal to 3(2)2). Moreover theSchwartz inequality 24 > (3)2 is satisfied, at least as far as the order of magnitudes areconcerned, for all but the case of Gaussian inhomogeneous turbulence. In this respect, wenote that this is a quite unphysical case, and does not describe the atmospheric boundarylayer [15–17].Condition (2.2) ensures that the process described by (2.1) is Markovian [18].A few comments on the local structure of the increments of the process are in order.For small times it T , being T the scale of the decorrelation times Ti, there resultsvi+1  v0=i+1Xk=1k  v0ti+1Xk=11Tk+O(t2) :(2.4)The correlation of the velocity increments results, for i j + 1h(vi+1  vi)(vj+1  vj)i =t2Ti+1Tj + 1hvivji+ hi+1j+1i  tTj+1j;i+1hi+1ji;(2.5)but hi+1j+1i = 0 by (2.2) and hvivji = 0 by Markovianity. Thus, the process is -correlated.3. – Continuity of the trajectoriesThe continuity of the trajectories of the particles in the phase space is a physical re-quirement, if real trajectories have to be simulated. Because the stochastic model is usedto describe the average properties over an ensemble of trajectories, we are not interestedin the behaviour of the single trajectory, but only in the average behaviour of the ensem-ble, thus we argue that the relevant continuity condition for the process is a conditionrequiring the continuity in probability.An appropriate continuity condition is a criterion [19] which readslimt!0h(vi+1  vi)2i = 0 :(3.1)In fact there results(vi+1  vi)2=t2T2i+1v2i+ 2i+1  2tTi+1vii+1(3.2)428 R. CESARI and F. TAMPIERIand thush(vi+1  vi)2i = t2v2iT2i+1+ h2i+1i ;(3.3)solimt!0h(vi+1  vi)2i = 0:(3.4)Thus, the present process is mean-square continuos, a weaker condition with respect tothe sample path continuity, which could be expressed following a condition given by Kol-mogorov [20]. This criterion implies the existence of positive constants p; r; L such thath(vi+1  vi)pi Ljtjjlg2jtjj1+r; p < r ;(3.5)where vi+1, viare the velocities at times (i+ 1)t and it. This stronger continuity con-dition is verified in the Gaussian case, and in general by the Thomson [4] model, becausethe noise is a Wiener process and the stochastic process has locally the same form as aprocess with independent increments.4. – The small time behaviourThe existence of the inertial subrange has some implications on the behaviour of thetrajectories of the stochastic model (as outlined by [6]). The Kolmogorov hypothesis statesthat in the inertial subrange the structure of the turbulence is locally homogeneous andisotropic. Because the accelerations are shown to be almost independent of each other [6],any process describing the trajectories has to be a Markovian process with -correlatedincrements.In the inertial subrange the correlation of the velocity increments is given byh(vi  v0)2; v0i = c0it ;(4.1)where  is the dissipation rate, and the constant c0takes a value between 2 and 7, accord-ing to the literature.In general, we haveh(vi  v0)q; v0i = (c0it)q2;(4.2)but the odd moments have to be zero to the leading order due to isotropy.In particular, we must note that the moments are independent of the value of theinitial velocity v0.The present model is characterized by a non-Gaussian random noise, and then itsPDF does not satisfy a Fokker-Planck equation (at variance with models with Gaussiannoise). The relevant equation is a Kramers-Moyal equation, which does not give an ex-pression for the PDF that can be used for the present purposes. More simply, it is possibleto compute the first moments of the velocity increments directly from (2.3). The velocityincrement may be written asvi+1  v0=  v0t0@i+1Xj=11Tj1A+i+1Xj=1j+O(t2) :(4.3)ON SOME PROPERTIES OF LAGRANGIAN DISPERSION MODELS ETC. 429Fig. 1. – h(vi+1  v0)2i as a function of time  , for   T , in the case of homogeneous turbulence.Continuous line: first term of eq. (4.6); stars: numerical simulation average over 104 trajectories.In general, the value of the decorrelation time T and the PDF of the noise vary withthe position along the trajectory. However, for small times (kt  T ) we assume thaton average the total distance covered by each particle is small with respect to the spatialscale of variation of the turbulence. Consequently, in the following computations we willuse the values of the time scale T and of the moments of the noise computed in the initialpoints of the trajectories. Obviously this is not a restriction in homogeneous turbulence.Averaging over the ensemble of trajectories we haveh(vi+1  v0)2i = G1  G2t +v20T2 2v0A1T2+O(t)2)(4.4)andh(vi+1  v0)3i = H1  H2t   3v0G1T2+ 3v0G2Tt2+(4.5)+3v20A1T2+ 3v0G1T23+ 3v20A2T2  3v0G2T2t3+O(t2)430 R. CESARI and F. TAMPIERIFig. 2. – As in fig. 1, but for the third moment h(vi+1  v0)3i, eq. (4.7).where  = (i+ 1)t.The expressions for the coefficients are reported in appendix A.In homogeneous, non-Gaussian turbulence the previous expression reduces toh(vi+1  v0)2i = 2k2T  k2T2t +v20T22+O(t2)(4.6)andh(vi+1  v0)3i = 3k3T   3k3T2t   6v0k2T22+ 3v0k2T3t2+ 6v0k2T33 (4.7) 3v0k2T4t3+O(t2) :ON SOME PROPERTIES OF LAGRANGIAN DISPERSION MODELS ETC. 431Fig. 3. – As in fig. 1, but in the case of inhomogeneous turbulence, eq. (4.4). The initial positionof the trajectories is such that turbulence in the mixed region is strongly inhomogeneous and non-Gaussian.At order  the second moment is consistent with (4.2), holding in the inertial subrange,that is, it does not depend on the initial velocity v0and the coefficient of  is exactly c0.The dependence on the initial velocity appears at second order in  . The independencefrom v0at first order has a relation with the mean-square continuity discussed above. Infact Borgas and Sawford [6] showed that in a discontinuos process (with jump) this depen-dence appears at first order in  ; the weak continuity condition of the present process isthen consistent with a good behaviour of the velocity correlation.The third moment is different from zero at leading order, contradicting the aboveconsiderations on the odd moments, and showing that in this model the physical informa-tion derived from the large scale structure of the turbulence has an effect on the smalltime behaviour, being present in the structure of the noise. Consistently, in inhomoge-neous turbulence, the moments of the velocity increments depend at first order in  onthe derivatives of the moments of the random forcing.We notice that (4.5) and (4.7) highlight a contradiction with the inertial subrange be-432 R. CESARI and F. TAMPIERIFig. 4. – As in fig. 3, but for the third moment h(vi+1  v0)3i, eq. (4.5).haviour condition, namely (4.2) above. However, if the model is applied to describe disper-sion over scales larger than the integral turbulence scale (or over times larger than T ), wecan neglect the small time behaviour of the dispersion. Thus, the model should be appliedproperly under these conditions.The results obtained for the behaviour of the model at small times have been testednumerically using data on turbulence in a mixing region. The turbulent mixing regionthat develops when a high-velocity flow mixes with a low-velocity one presents a case ofhighly inhomogeneous turbulence. Detailed measurements of high-order moments and ofmany other quantities are available from laboratory experiments using high-pressure airinflated trough a nozzle in a chamber with still air [21].For these experiments the PDF of the random forcing has been assumed to be givenby the sum of two Gaussian distributions. Given only the first four moments the problemis not closed; we have adopted a closure presented by [14] wich minimizes the differencebetween the amplitude of the two Gaussian distributions. Because the time step was cho-sen to be quite small, the PDF of the random noise turns out to be quite skewned. ThisON SOME PROPERTIES OF LAGRANGIAN DISPERSION MODELS ETC. 433fact leads to random numbers chosen more frequently from one Gaussian curve than fromthe other one. On the other hand, small time steps are necessary to obtain stable numer-ical results in the integration of the Langevin equation. Comparing the statistics of theparticle velocity with those prescribed for the fluid, a good agreement results, showingthat the ensemble average is made on a sample large enough to ensure stability of thecomputed moments [14].Equations (4.4) and (4.5) have been verified using different values of the initial veloc-ity v0. The numerical tests have shown the independence of velocity correlation on v0.In figs. 1-4 the behaviours of h(vi+1 v0)2i and h(vi+1 v0)3i are reported for v0= 0 asa function of time: there results that the theoretical formula at order t, i.e. the first termof eqs. (4.4)-(4.7) (continuous line), agrees with the numerical estimate (stars) within 10%.5. – ConclusionsSome properties of a stochastic model for turbulent dispersion based on the Langevinequation with non-Gaussian noise have been examined.The model turns out to be Markovian, with -correlated increments. Moreover, acontinuity condition for the trajectories has been verified.The main difference with models characterized by a Gaussian noise lies in the be-haviour in the inertial subrange. The non-Gaussian model shows odd moments of thevelocity increments which account for the inhomogeneous structure of turbulence. Thusthis model does not describe in a proper way the existence of the inertial subrange, butcan be applied to dispersion problems with time scale larger than that of the inertial sub-range. This was, in fact, done in the practical implementations of the model, already citedin the introduction. Moreover, the possibility to use such models in conditions when theinertial subrange is not developed, or is not explicitly described by the velocity field, hasto be investigated.APPENDIX A.In this appendix we consider the explicit expression of the first four moments of thePDF of the random forcing used in (2.3). According to [14] (par. 3), we have8>>>>><>>>>>>:1= tA1  (t)2A2;2= tG1  (t)2G2;3= tH1  (t)2H2;4= tL1  (t)2L2;(A.1)with434 R. CESARI and F. TAMPIERI8>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>><>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>:A1=dk2dx;A2=dk2dx1T+12d2k3dx2;G1=2k2T+dk3dx;G2="k2T2+2Tdk3dx+52k2d2k2dx2+dk2dx22+12d2k4dx2#;H1=3k3T+dk4dx+ 3k2dk2dx;H2=3k3T2+3Tdk4dx+3Tk2dk2dx+72k2d2k3dx2+92k3d2k2dx2+ 4dk2dxdk3dx+12d2k5dx2;L1=4Tk4+ 6k3dk2dx+dk5dx+ 4k2dk3dx;L2=" 12T2k22+6k4T2+4Tdk5dx+ k2dk3dx+ 3k3dk2dx+ 7k4d2k2dx2+ 8k3d2k3dx2++92k2d2k4dx2+ 6k22d2k2dx2+ 3k2dk2dx2+ 4dk3dx2+ 7dk2dxdk4dx#;(A.2)where ki; i = 1; : : : ; 5 are the cumulants of the flow velocity, and we choose k1= 0 accord-ing to the data [10].REFERENCES[1] TAYLOR G. I., Proc. London Math. Soc., 20 (1921) 196.[2] OBUKHOV A. M., Adv. Geophys., 6 (1959) 113.[3] THOMSON D. J., Q. J. R. Meteorol. Soc., 110 (1984) 1107.[4] THOMSON D. J., J. Fluid Mech., 180 (1987) 529.[5] MONIN A. S. and YAGLOM A. M., Statistical Fluid Mechanics of Turbulence, Vol. 2 (MITPress, Cambridge, MA) 1975.[6] SAWFORD B. L. and BORGAS M. S., Physica D, 76 (1994) 297.[7] LUHAR A. K. and BRITTER R. E., Atmos. Environ., 23 (1989) 1911.[8] NA¨SLUND E. et al., Boundary-Layer Meteorol., 67 (1994) 369.[9] SAWFORD B. L., Boundary-Layer Meteorol., 62 (1993) 197.[10] THOMSON D. J., Q. J. R. Meteorol. Soc., 112 (1986) 511.[11] TINARELLI G. et al., J. Appl. Meteorol., 33 (1994) 744.[12] ANFOSSI D. et al., Nuovo Cimento C, 15 (1992) 139.[13] TAMPIERI F. et al., Ann. Geophys., 10 (1992) 749.[14] TAMPIERI F., SCARANI C. and GIOSTRA U., in Stably Stratified Flows, edited by I. P. CAS-TRO and N. J. ROCKLIFF (Clarendon Press, Oxford) 1994, p. 275.[15] CHIBA O., J. Meteorol. Soc. Jpn., 56 (1978) 140.[16] MOENG C. H. and ROTUNNO R., J. Atmos. Sci., 47 (1990) 1149.[17] TOWNSEND A. A., The Structure of Turbulent Shear Flow (Cambridge Univ. Press) 1976.[18] SANCHO J. M. and SAN MIGUEL M., in Noise in nonlinear dynamical systems, edited byF. MOSS and P. V. E. MCCLINTOCH, Vol. 1 (Cambridge Univ. Press) 1989, pp. 72-109.[19] KLOEDEN P. E. and PLATEN E., Numerical Solution of Stochastic Differential Equations(Springer-Verlag) 1992.[20] GIHMAN I. I. and SKOROHOD A. V., The Theory of Stochastic Processes, Vol. 1 (Springer)1974.[21] CHAMPAGNE F. H., PAO Y. H. and WYGNANSKY I. J., J. Fluid Mech., 74 (1976) 209.

On some properties of Lagrangian dispersion models with 

non-Gaussian noise

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On some properties of Lagrangian dispersion models with non-Gaussian noise

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