Theory of stochastic models for turbulent dispersion. Francesco Tampieri ISAC CNR, Bologna, Italy

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1 Theory of stochastc models for turbulent dsperson Francesco Tamper ISAC CNR, Bologna, Italy September 1, 26

2 1 Introducton: the transport problem The behavour of a substance advected by a turbulent flow shows complex, chaotc features. The substance may be a pollutant, may be temperature, may be a mcroorgansm lvng n the sea. Some substances are characterzed by an almost nfnte lvng tme: they do not react wth the envronment; others combne accordng to the rules of chemcal reactons. Some of them behave lke the flud molecules, n the sense that ther velocty s at any tme the same of that of the flow; others are subject to the gravty acceleraton, lke aerosols, or exbt autonomous movements, lke flagellates. Last but not least, some substances nteract wth the flow changng the velocty feld, for nstance when temperature fluctuatons occurr. In ths lecture we shall address the smplest case of a non reactng passve tracer: an deal partcle, whch can be dstngushed from the flud ones by some markng, but whose velocty v(t) when passng at the pont x s the same of that of the flow u(x, t): v(t) = u(x, t) (1) Let we consder flows wth large Reynolds numbers R e >> 1 3. The transport problem for ths tracer can be tackled as the source s defned, namely the ntal poston of the partcle and the mmsson rate n the flow are known. Agan for reducng to the smplest case, let we magne a pont source n x and an nstantaneous release at the tme t. The formal soluton of the transport problem s thus expressed by the ntegral x(t) = x(t ) + v(τ) dτ t (2) whch gves the poston of the tracer partcle as a functon of tme. If the velocty feld s completely determned, Eq. 2 allows to compute the poston of the partcle released from the source: dsperson for a set of partcles s explctly computed. In hgh Reynolds number flows,.e. n geophyscal flows, ths approach cannot be utlzed. At frst, wth the measurements we have access only to some statstcs of the flud tself. Moreover, apart from Drect Numercal Smulatons whch can be performed n a restrcted varety of condtons, any other numercal approach gves only a partal descrpton of the velocty feld. 1

3 Thus, we shall adopt a stochastc approach, namely, we shall look for solutons n terms of statstcal propertes of x resultng from Eq. 2. The stochastc approach allows to model dsperson accordng to a descrpton of the velocty feld as a random feld wth known statstcal propertes. Qute obvously, detals n the results wll ncrease as much nformaton we shall put nto the model. On the other hand, ths approach wll be lmted by the fact that the smulated trajectores wll have the same statstcal propertes of the real ones, but nothng more. 2 Absolute dsperson and meanderng Let us consder a set of N partcles, wth postons x (n) (t), = 1, 2, 3, n = 1,..., N. Averagng over N s dentfed by.. Mean poston x = 1 N N n=1 x (n) (3) Relatve postons y (n) = x (n) x. Obvously, y =. The second order moments of the relatve poston are: y y j = x x j x x j (4) Let us consder now M sets of N partcles each. For each set out of M, moments are dentfed by (m), wth m = 1,..., M. Averagng over M s dentfed by. The mean poston s x = 1 NM NM k=1 x (k) = 1 M ( M 1 N N m=1 The second order moments are n=1 x (n+(m 1)N) ) = 1 M M x (m) (5) m=1 x x j = y y j + 1 M M x (m) x j (m) = y y j + x x j (6) m=1 whch means that the covarance of the absolute postons s made by the sum of the covarances of the relatve postons (the relatve dsperson about the mean poston, averaged over the M sets), plus the covarance of the mean postons (the meanderng). 2

4 Let us consder the separaton between two partcles, for each m set: l = x (j) x (l). The varance of the separaton s: l 2 1 N 2 N N j=1 l=1 [x (j) (the cases wth j = l should be accounted for). Because of Eq. 4 t results x (l) ] 2 = 2 x 2 2 x 2 (7) l 2 2 y 2 (8) namely, the varance of the dstance between par of partcles s two tmes the dsperson about the mean poston. 3 A basc paradgm Inertal subrange theory (Kolmogorov, 1941) gves the nterpretaton paradgm for velocty spectra, assumng the exstence of the (unversal) nertal subrange: Let consder the velocty fluctuatons about the mean: u = u u. Averagng s n prncple ensemble averagng; f homogenety and/or steadyness occurr, volume and/or tme averagng s used (mostly tme averagng n geophyscal applcatons). The two ponts x and x+r velocty correlaton tensor, at the same tme, reads R j (r, x) = u (x)u j (x + r) (9) The two tme t and t + τ velocty correlaton, n the pont x, s: R j (τ, t, x) = u (x, t)u j (x, t + τ) (1) We assume homogenety and steadness condtons. The ntegral space and tme scales are defnes as the ntegrals converge: L (k) j = 1 u u j T j = 1 u u j R j (r)dr k (11) R j (τ)dτ (12) 3

5 Correlaton functons and spectra n space doman are related by: and n tme: From Eq. 15: E(k) = 2 π R(r) = E(ω) = 2 π R(t) = R(r) cos(kr)dr (13) E(k) cos(kr)dk (14) R(t) cos(ωt)dt (15) E(ω) cos(ωt)dω (16) E() = 2 π R(t) dt = 2 π T u 2 (17) The second order structure functons: ( ) D (2) (r) = [u(r) u()] 2 = 2 u 2 R(r) = C K ε 2/3 r 2/3 (18) ( ) D (2) (t) = [v(t) v()] 2 = 2 u 2 R(t) = C εt (19) where v(τ) 2 = u 2 for any τ (thanks to homogenety and steadyness). Eq. 18 s an Euleran relatonshp, Eq. 19 a Lagrangan one. A model for the Lagrangan correlaton (Gfford, 1982) s R(t) = u 2 exp( t/t ) (2) For t << T, n the nertal subrange, R(t) u 2 (1 t/t ) and D (2) (t) 2u 2 t/t. Comparng wth 19 we obtan an expresson for the dsspaton as a functon of the ntegral quanttes characterzng the flow (Tennekes, 1982): ε = 2u2 C T (21) In the nertal subrange the spectrum for each velocty component depends on dsspaton and on space scale l (.e. from wavenumber k l 1 ). From dmensonal arguments [E(k)dk] = U 2 l (22) 4

6 and E(k) ε 2/3 l 2/3 k 1 ε 2/3 k 5/3 (23) Smlarly: and [E(ω)dω] = V 2 t (24) E(ω) εtω 1 εω 2 (25) 4 Dsperson by contnuous movements: Taylor (1921) Let we study the statstcs of flud partcle poston (the same as Eqs. 2 and 1) : x (t) = v (t 1 ) dt 1 + x () (26) wth v (t) u (x(t), t). The poston varance can be computed. Assumng u =, x () = we get x x j = dt 1 dt 2 v (t 1 )v j (t 2 ) (27) The Lagrangan correlaton v (t 1 )v j (t 2 ) = R j (t 2 t 1 ) s computed along the partcle trajectory, not n a fxed pont: we need homogenety and steadyness (at least locally). The decorrelaton tme defnton s smlar to Eq. 12 T j = 1 v v j R j (τ)dτ (28) For t << T j v (t 1 )v j (t 2 ) v ()v j () the ballstc regme occurs: x x j = v ()v j ()t 2 (29) and f the partcles are nserted n equlbrum wth the flow v ()v j () = u u j. 5

7 For t >> T j dffuson occurs: x x j 2T j u u j t (3) and the eddy dffuson coeffcent (computed as a lmt) s D j = u u j T j. 4.1 Correlaton, spectra and dffuson From we get R j (t) = E j (ω) cos(ωt)dω (31) x x j = 2 R j (s)(t s)ds = 4 For t T the ntegral may be approxmated as: E j (ω) sn2 (ωt/2) ω 2 dω (32) sn 2 (k) x x j 2tE j () k 2 dk = πte j () (33) (for the computaton Gradshteyn and Ryzhk (198, pag. 446(9))). Note that for long tmes only the large scales contrbute to dsperson. The process s dffusve, usng Eq. 17 we get Eq. 3 agan. 5 Formulaton of stochastc models of dsperson: prelmnary observatons The equatons descrbng poston of a partcle, Eq. 2, s a Lagrangan expressons of the transport problem. On the other hand, t s customary n the dsperson problems to utlze the concentraton of a tracer (a pollutant) and the approprate Euleran equaton, namely the adecton-dffuson equaton. The lnk between the two descrptons arses from a well known mathematcal result (see for nstance Rsken (1989); n partcular for an ntroductory dscusson, Par ). Let z the vector of dmensons N defnng the state of the partcles: for nstance, f we dentfy z wth the poston of ndependent partcles then N = 3, whereas N = 6 needs as postons and veloctes have to be accounted for. If we want to nvestgate relatve 6

8 dsperson, we neet to defne the state of a par of partcles, so N = 12 s necessary to descrbe poston and velocty of partcle pars. Let p the probablty densty functon (pdf) of z: p(z; t z, t ) = N z 1 z 2... z N P (y < z z(t ) = z ) (34) where the ntal state of the system (at t = t ) s gven by z and P s the probablty. We assume that the evoluton of the state of the system s descrbed by the Langevn equaton (LE): dz = α (z, t) dt + β j (z, t) dw j (t) (35) where dw j (t) s a Wener process such that dw (t) =, dw (t) dw j (s) = δ j δ(t s) ds dt (36) then the pdf of z satsfes the Fokker-Planck equaton (FPE): ( ) p t = (α p) + 2 β 2 j z z z j 2 p (37) Then, ths result gves us the lnk between the Lagrangan descrpton of the evoluton of the state of each partcle, Eq. 35 and the probablty densty feld of fndng partcles wth state z at tme t, gven that the ntal state s z at t, Eq The Thomson (1987) formulaton for absolute dsperson modellng 6.1 The dervaton of the LE equaton terms The prevous consderatons can be appled to model absolute dsperson n very hgh Reynolds number flow (so that the nertal subrange extends to very small spatal scales and vscosty effects wll be neglected). A flud partcle trajectory s gven by dx = v dt (38) where v (t) u (x, t), beng u the Euleran velocty of the flow (.e. the soluton of the Naver Stokes equatons). As far as u s unknown, a model for the velocty v s necessary: 7

9 dv = a (x, v, t) dt + b (x, t) dw (39) The model formulaton must be made n such a way that the the statstcal propertes (at least, some statstcal propertes) of the vector z are the same as for true partcles, whose moton s exactly determned by the (unknown) soluton of the Naver-Stokes equatons. Of course, each realsaton do not correspond to an actual trajectory (note also that the trajectory estmated from the stochastc equaton s contnuous but not dfferentable). The challenge s to obtan the correct forms for a and b. A couple of remarks about 39 has to be made. To avod ambgutes (see van Kampen (1981)), we chose the tensor b j ndependent on v. Secondly, n general β j s non-dagonal, as n 35, whereas we have chosen to have only one component of nose for each velocty component; ths choce wll be shown to be consstent wth the requrements of Kolmogorov (1941) theory. The second order moment of the velocty ncrements can be derved from 39: dv dv j = b b j dw dw j + O( dt 3/2 ) b b j δ j dt (4) for dt small wth respect to a tme scale approprate for the equaton: at frst glance, such scale should be O( v / a ). In order to be consstent wth the nertal subrange relaton for Lagrangan velocty ncrements: t must be (v (t) v (t ))(v j (t) v j (t )) = C ε(t t )δ j (41) b 2 = C ε (42) Ths suggests also that dt must belong to the nertal subrange. Thus 38 and 39 are canddate for modellng flud partcle trajectores, consstently wth the nertal subrange propertes as far 42 s satsfed. ε s usually estmated through macroscopc varables as n Eq. 21 (Tennekes, 1982) ε = 2u 2 (43) C T where u 2 s the varance of the -th component of the turbulent feld (f sotropc, two-thrds of TKE) and T s the Lagrangan decorrelaton tme 8

10 scale, whch does not correspond n general to the Lagrangan ntegral tme (Maurz and Lorenzan, 21). To obtan a the FPE s consdered. In ths case we have N = 6 and z = x (44) z +3 = v (45) α = v (46) α +3 = a (47) β +3,+3 = (C ε) 1/2, = 1, 2, 3 (48) whereas for the other ndexes β j =. Thus the FPE, 37, becomes p t = (v p) (a p) + C ε x v 2 2 p (49) v v Eqs. 38 and 39 descrbe the evoluton of the state of a partcle, from the ntal pont (x, v ) at tme t to (x, v) at tme t. From the transton probablty, the concentraton of tracer partcles from a source can be computed. If S(x, v ; t ) s the number of marked partcles n the ntal poston (n the state space) for tme nterval, volume and velocty range (namely, the source) then n(x, v; t) = d 3 x d 3 v t <t dt p(x, v; t x, v ; t )S(x, v ; t ) (5) s the number of tracer partcles for volume and velocty range found n the fnal poston. It must be notced that Eq. 37 does not depend on the ntal state, so t s satsfed by n as well as by p. Then, ntegratng n over velocty gves the number of partcles for volume range,.e. the average concentraton: c(x; t) = d 3 vn(x, v; t) (51) Now we defne the source to be equal to the mean densty of the flud ρ (whch s the probablty of fndng flud partcles n a gven volume, multpled by the mass of the partcles themselves: n what follows, we shall avod to ntroduce the mass just defnng densty as the number of partcles for 9

11 unt volume) tmes the Euleran probablty densty functon of the velocty p E : S = ρ(x, t )p E (v ; x, t )δ(t ) (52) Accordng to ths defnton, the source s made by the flud partcles leavng all the possble ntal poston, wth ntal velocty dstrbuted as the flow velocty (namely, the Euleran velocty dstrbuton), and the ntegraton n 5 s made over the entre space and velocty doman. Then n must be the number (for volume and velocty range) of the flud partcles n the fnal poston,.e. n = ρ(x, t)p E (v; x, t). In a slghtly dfferent form, ths pont was formulated by Novkov (1986) and Thomson (1987). In a constant densty flud, substtutng 52 n Eq. 5 (namely, ntegratng over the ntal tme and over the entre space and velocty doman), we get p E (v; x, t) = d 3 x d 3 v p(x, v; t x, v ; t )p E (v ; x, t ) (53) Thomson (1987) observed that f p s a soluton of 49 also p E ρ must be a soluton. Ths s the so called Well Mxed Condton (WMC): partcles mxed at a gven tme must reman mxed at any later tme. Then 49 may be wrtten for p E ρ, for a constant densty flow ρ can be smplfed and because the Euleran pdf s assumed to be known, t becomes an equaton for a : (a p E ) = C ε 2 p E p E u 2 u u t u p E x (54) where the effects of unsteadyness and of nhomogenety of the flow on the drft coeffcent are put nto evdence by the presence of dervatves of the velocty pdf. Eq. 54 may be wrtten as: a p E = C ε p E + Φ (x, u, t) (55) 2 u Φ = p E u t u p E x (56) wth the condton Φ as u. Note that any solenodal feld decayng fast enough can be added to Φ (non unqueness of the soluton). In the one-dmensonal case, ths ndetermnacy does not arse. 1

12 6.2 Comments and shortcomngs The use of FPE mples the knowledge of the Euleran pdf feld. We know at best some moments, possbly measured n a few ponts. In general a model for the p E must be adopted: see Maurz and Tamper (1999); Tamper et al. (21). A dfferent possblty s based on the moment approxmaton, exploted by Kaplan and Dnar (1993). Ths approach has been recently used for the one-dmensonal, steady case, by Franzese et al. (1999). These authors made the assumpton that the drft term can be expressed as a second order polynomal n the velocty a(u, x) = α(x)u 2 + β(x)u + γ(x), and the parameters are determned dervng from the relevant FPE, 54, expressons for the frst three moments of u. These expressons are of course functons of the parameters, whch are determned equatng the expressons to the measured (or estmated) moments. Ths assumpton has been shown to work approxmately n a lmted velocty range, so ts applcablty s problem dependent. As outlned before, the soluton for a s not unque, and t s unclear how to ntroduce crtera for selectng among the dfferent solutons. Equvalent solutons for Gaussan p E has been dscussed by some authors (see for nstance Borgas and Sawford (1994)). To date, however, the WMC represents the way to utlze at the best all the avalable nformatons about the Euleran velocty feld. 6.3 A soluton wth Gaussan p E A smple model for p E s the Gaussan dstrbuton for the velocty u = u+u, wth average u(x, t) and covarance matrx V j (x, t) = u u j : ( 1 p E (u) = (2π) 3/2 exp 1 ) det(v ) 1/2 2 u (V 1 ) j u j (57) A soluton for a reads: a = C ε 2 (V 1 ) k u k + Φ p E (58) [ (V 1 ) lj Φ ( Vl V l = 1 + u p E 2 x l t + u u l + x ) l t + u V l m + u ] u j + x m x j 11

13 + 1 2 (V 1 ) lj V l x k u ju k (59) If the correlaton terms can be neglected, then the three-dmensonal process reduces to three one-dmensonal ndependent processes. 6.4 A smple exercse: the 1D, homogeneous steady case wth zero mean velocty Let us smplfy the prevous case to one dmenson and put u =. Then p E reads: ) 1 p E (u) = exp ( u2 (6) (2π) 1/2 u21/2 2u 2 The drft term becomes: a = C ε 2u u = u (61) 2 T beng T the ntegral Lagrangan tme scale (see Eq. 43). The full model reads: dx = v dt (62) ( ) 1/2 dv = v T dt + (C ε) 1/2 dw = v T dt + 2u 2 dw (63) T wth ntal condtons x() = and v() v, wth v() =. Here v s the velocty of the sample partcle. The soluton for the model s as follows. The velocty varance: the tme correlaton: v 2 (t) = u 2 (u 2 v 2 ) exp( 2t/T ) (64) R(τ) = v(t + τ)v(t) = v 2 exp( τ/t ), τ > (65) and the Lagrangan structure functon of second order: (v(t) v ) 2 = u 2 (u 2 v 2) exp( 2t/T ) + v2 [1 2 exp( t/t )] (66) show that partcle velocty goes to equlbrum wth the flow velocty n half the tme scale, and that the correct behavour n the nertal subrange s recovered: (v(t) v ) 2 2u 2 t T, t < T (67) 12

14 The poston varance s: x(t) 2 = T 2 (v 2 u2 )[1 exp( t/t )] 2 + 2u 2 T t 2u 2 T 2 [1 exp( t/t )] (68) The ballstc t << T and asymptotc t >> T regmes can be found v 2 (t) v 2, x2 (t) v 2t2 (69) v 2 (t) u 2, x 2 (t) 2u 2 T t (7) and the transton occurs wth a t 3 correcton x 2 (t) v 2t2 (v 2 23 ) t 3 u2 T (71) whose sgn s postve or negatve dependng on the value of the ntal velocty wth respect to the Euleran equlbrum velocty. If the source s n equlbrum wth the flow, the coeffcent of t 3 s gven by C ε/ The mean shear example Let us consder the smple case of homogeneous steady turbulence (pdf(u ) ndependent on poston and tme) n a mean shear flow u 1 (x 3 ) = Γx 3. For ths case the exact soluton s known: Monn and Yaglom (1971, pag. 556). Then u = (Γx 3 +u 1, u 2, u 3 ) (Γz +u, v, w ) and compute the second order moments of the postons x x j. The ntal poston s x() = : x 2 = From Eq. 72 xz = dt 1 dt 2 [Γz(t 1 ) + u (t 1 )][Γz(t 2 ) + u (t 2 )] (72) dt 1 dt 2 [Γz(t 1 ) + u (t 1 )]w (t 2 ) (73) x 2 = dt 1 dt 2 u (t 1)u (t 2), = 2, 3 (74) Γ x 2 (t) = Γ2 3 (t s) 2 R 31 (s) ds + Γ ( 2t 3 3t 2 s + s 3) R 33 (s) ds + 13 (t 2 s 2 )R 13 (s) ds + 2 (t s)r 11 (s) ds (75)

15 From Eq. 73 xz(t) = Γt (t s)r 33 (s) ds + whereas for y 2 and for z 2 the soluton s stll: x x j = 2 (t s)[r 13 (s) + R 31 (s)] ds (76) R j (s)(t s)ds (77) n partcular, for long tmes x 2 1 u 2 3 t3. For the correspondng stochastc model, n case of no cross correlatons u u j = the drft term reads (T defned as n Eq. 43): a 1 = v 1 u 1 (x 3 ) T 1 + du 1 dx 3 v 3 v 1 u 1 T 1 + Γv 3 (78) a = v T, = 2, 3 (79) and shows that the couplng of the mean flow gradent wth the fluctuatons n the drecton of the gradent s necessary to fulfll the WMC. The velocty gradent term ntroduces a tme scale T s = ( du 1 / dx 3 ) 1, on whch nhomogenety acts. Fg. 1 shows some smulatons of dsperson from a pont source, for a unform flow and for three dfferent gradents. The poston varance departs from the unform case as the tme elapsed s comparable wth T s. 6.6 Dscusson of the non Gaussan case For non Gaussan cases (whch are expected to be frequent n geophyscal flows) n general a soluton lke 58 and 59 s not avalable. Some hnt can be obtaned lookng at 54 n the one dmensonal, steady case. It reads: and thus ap E = C ε p E 2 u + Φ, Φ u = u p E x (8) u Φ = u p E x du = u u p E du (81) x In general a numercal ntegraton needs. For nstance, f the Maxmum Mssng Informaton pdf Jaynes (1957) s used: 14

16 1 1 ballstc (u 2 t 2 ) dffusve (2 <u 1 2 > T1 t) shear ( t 3 ) Γ T 1 =.1 Γ T 1 =.5 Γ T 1 =1. <x 1 2 >/(<u1 2 >T1 ) t/t 1 Fgure 1: Varance of the poston of flud partcles normalzed over the varance of turbulent fluctuatons and the correlaton tme squared, as a functon of tme normalzed over the correlaton tme. The homogeneous case (no-shear) s reported as reference. The three shear case (wth shear tme scales of 1, 2 and 1 the correlaton tme) depart from the no-shear case at decreasng tmes. 15

17 p E (u, x) = exp ( ) M λ k (x)u k k= (82) wth M even and the coeffcents λ k obtaned from the normalzaton condton and from the knowledge of the frst M moment. For M = 2 ths pdf reduces to a Gaussan dstrbuton. Usng 82 a polynomal expresson for the drft term n the homogeneous non Gaussan case, from 8, s obtaned: a = C ε 2 M kλ k u k 1 (83) k=1 7 A dfferent model formulaton In geophyscal applcatons, Lagrangan Stochastc Models are usually formulated n terms of fluctuaton about a mean flow. The mean flow can be the flow feld that results from a crculaton model or a measured feld (e.g., from drfters). Mean flow from models s ntended as an ensemble average when Reynolds-averaged equatons (.e., when stresses depend on the flow and not on the grd dmenson) are consdered. On the other hand, n cases of nstantaneous fltered flow feld (Large Eddy Smulatons), fluctuatons are a measure of the subgrd turbulence and depend on the grd sze (e.g., for the Smagornsky model, to consder a very smple example). An applcaton of the Thomson (1987) formalsm to LES s presented by Wel et al. (24). For sake of smplcty let us consder a statstcally statonary and homogeneous fluctuatng feld u (wth Gaussan dstrbuton) about a mean flow u functon of poston x and tme t. Thus: u(x, t) = u(x, t) + u, p E(u ) t = p E(u ) x = (84) The formulaton of the stochastc model for the velocty, Eq. 39, can be made n terms of the fluctuatng components: Consstently, Eq. 38 reads dv = a (x, v ) dt + (C ε) 1/2 dw (85) dx = [ u (x, t) + v ] dt (86) 16

18 whch actually states that the moton of a passve tracer has to be consdered as the moton followng the mean flow plus a fluctuaton. (Because fluctuatons are correlated n tme as a result of Eq. 85, the model has to be consdered Markovan for the jont varable (x, v) where v u + v ). As the formulaton of the stochastc model requres the use of a Fokker- Plank equaton assocated to the LE, the correct formulaton requres the correct defnton of the process equatons. The velocty ncrement s: dv = du + dv (87) where the mean velocty varaton along a flud partcle trajectory s [ u du = t + ( ] u k + v k ) u dt (88) x k Therefore, the fnal form of the stochastc model s dx = [ u (x) + v ] dt (89) [ ] u dv = t + u u k + v u k + a x k x (x, v ) dt + (C ε) 1/2 dw (9) k n whch the frst two terms on the r.h.s. of 9 evdences the contrbuton from the spatal varaton of the mean flow. The assocated Fokker-Planck Eq.37, n the steady case gves the soluton for a ; usng 58 and 59 we get so that a = v + u T t + ( u k + v k ) u (91) x k a = v T (92) accordng to 9. The thrd term of the r.h.s. of Eq. 9 descrbes the couplng between the fluctuatng feld and the nhomogeneous mean feld. The nhomogenety ntroduces a tme scale T s = u / x k 1, so ths couplng may be neglected f T << T s and we are nterested n tmes smaller than T s. 7.1 Some comments The prevous approach s straghforward as the velocty feld can be dvded n a resolved, determnstc part and a unresolved part. Ths s the case for RANS or LES numercal ntegratons, as far as dsperson applcaton are 17

19 concerned.the queston concerns the determnaton of the drft and dffuson coeffcents. Wel et al. (24) use the Lagrangan tme gven by Eq. 21, but the velocty varance s that of the unresolved (sub-grd) moton. Attenton must be pad f the determnstc part of the flow feld s modelled by more than one stochastc process. An example s made by Pasquero et al. (21). See also Maryon (1998). The need of splttng the flow feld nto two parts wth dfferent correlaton tmes arses from geophyscal observatons. The descrpton by the sum of two stochastc processes can be adopted f they are ndependent each other, but the correct defnton of the drft and dffuson coeffcents needs attenton. 8 The stochastc model for a dffuson process: the case wth N = 3 The model wth N = 6 has a lmtng behavour consstent wth a dffuson process: for t > T the partcles forget the ntal condtons and ther varance grows as t. It s nterestng to nvestgate the possblty to develop a model (wth N = 3) for ths stage. LE reads: dx = v dt + β j dw j (93) wth reference to Eq. 26, wth ntal condton x () = x. We start wth the smple case of moton ndependent on each drecton: β j = β δ j and look for a proper defnton for β. (Note that [β] = LT 1/2 : the square root of a dffuson coeffcent.) The soluton for the frst and second moment s gven by Gardner (199, pag. 12) (note that v s a nonrandom functon of tme): x (t) = x + [x (t) x (t)] 2 = v dt (94) β (t ) 2 dt (95) If v = and β s constant along the trajectory: x (t) 2 = β 2t. Comparng wth the exact soluton for homogeneous steady flow by Taylor (1921) for t > T we dentfy v as the mean flow velocty u and β = 2u 2 T = 2D, constant n tme and space. Thus the model wth N = 3 holds for long tmes. 18

20 The correspondng FPE can be derved now. We have: and the FPE reads z = x (96) a = v u (97) β j = 2u 2 T δ j (98) p t = u p + 2 (D p) (99) x x x and ths s the advecton-dffuson equaton for the mean concentraton c: c t + u c 2 c = D x x 2 (1) as the dffuson coeffcent s ndependent on poston (consstent wth the prevous assumptons of steadyness and homogenety). In the most general case n whch we do not specfy D j = βj 2 /2, the FPE equaton reads: The same equaton holds for c: c t = x p t = v p x + 2 x x j (D j p) (11) [( v D j x j ) ] c + ( ) c D j x x j (12) Eq. 12 shall be compared wth the Reynolds averaged equaton for c: c t = x j ( u j c + u j c ) where the turbulent flux term c u j appears. Thus, the dffuson coeffcent arses from a flux-gradent relatonshp: (13) u j c = D j c x (14) and the drft velocty accounts for the mean flow velocty and the nhomogenetes of the dffuson coeffcent: v = u + D j x j (15) 19

21 A smlar dentfcaton s made by Thomson (1995). The same couple of equatons 93 and 12 are dscussed by Mazzno et al. (25), ther Eqs. 11 and Reducton of Eq. 49 to Eq. 11 Remember that Eq. 49 s satsfed bt n gven by Eq. 5: n t = (v n) (a n) + C ε 2 n (16) x v 2 v v and that the average concentraton s gven by Eq. 51. We can ntegrate Eq. 16 over the velocty n order to get an equaton for the average concentraton. For nstance: d 3 v n t = c (17) t Assumng that n = at the boundary of the ntegraton volume n d 3 v, namely for nfnte modulus of the velocty, and that n/ ν = on the same boundary, beng ν the outward normal to the boundary, we can use Gauss and Green theorems to reduce Eq. 16 to c t = d 3 v (v n) (18) x Let us wrte the partcle velocty v = u + v. In the ntegraton, dv = dv. Thus c t = u c d 3 v ( v x x n ) (19) The ntegral n the last term s dentfed as the turbulent flux. 9 Relatve dsperson modellng: the Thomson (199) model Relatve dsperson s a process that depends on the combnaton of the Euleran and Lagrangan propertes of the turbulent flow. (If partcle separaton falls n the nertal subrange, the Euleran spatal structure affects the dsperson.) On the bass of the general results n Thomson (1987), Thomson (199) extended the method for the selecton of sngle partcle Lagrangan Stochastc Models to models for the evoluton of partcle par statstcs. Now, the 2

22 state of a partcle par s represented by the jont vector of poston and velocty (x, u) (x (1), x (2), u (1), u (2) ), where the upper ndex denotes the partcle, whose evoluton s gven by the set of Langevn equatons (as n the absolute dsperson case): dx = u dt (11) du = a (x, u, t) dt + b (x, t) dw (t) (111) where now, j = 1,..., 6. The coeffcents a and b are determned through the WMC and the consstency wth the nertal subrange scalng, respectvely. Thus b = C ε,, j = 1,..., 6 (112) wll be used accordng to the usual scalng of Lagrangan structure functon. In the WMC, the consdered pdf s the one-tme, two-pont jont pdf of x (p) and u (p), p = 1, 2, accountng for the spatal structure of the turbulent flow consdered. Remember that nertal subrange theory gve a prescrpton not only for the second order structure functon v 2 = C K (ε r) 2/3 (113) for the modulus of the Euleran velocty dfference v = v(r + r) v(r) at separaton r = r x (1) x (2), but also for the thrd order moment v 3 = 4 ε r (114) The Lagrangan-to-Euleran scale rato In order to hghlght the effect of turbulence features on the model formulaton (see Maurz et al. (24)), characterstc scales for partcle par moton: a velocty scale σ, a Euleran space scale λ and a Lagrangan tme scale τ must be dentfed and used respectvely to make velocty u, poston x and tme t non-dmensonal. They also form a non-dmensonal parameter β = στ λ (115) whch can be recognsed as (a verson of) the well known Lagrangan-to- Euleran scale rato. In non-dmensonal form, the Langevn equaton now reads dx = βu dt, du = a dt + 2 dw (t) (116) 21

23 where, wth a change of notaton wth respect to 11 and 111, all the quanttes nvolved are nondmensonal. The assocated Fokker-Planck equaton s p L t + βu p L + a p L = 2 p L (117) x u u u where p L s the pdf of the Lagrangan process descrbed by 116 for some ntal condtons. Usng the WMC, a can be wrtten as where Φ p E u a = ln p E u = p E t + Φ p E (118) βu p E x (119) and p E s the Euleran one-tme, two-pont jont pdf of x and u. Ths result shows that, gven a Euleran pdf, once the non-unqueness problem s solved by selectng a sutable soluton to 118, any soluton of 117 wll depend on one parameter only, namely on the Lagrangan-to-Euleran scale rato. It can also be observed that ths dependence s completely accounted for by the non-homogenety term, whch s an ntrnsc property of the partcle par dsperson process n spatally structured velocty felds. In smple words, f β 1 the correlaton along the trajectory of each partcle and the one between the two partcle dsappear almost at the same phase of the dsperson process. If β >> 1 the spatal correlaton dsappears n a phase n whch the tme correlaton s expected to be stll not neglgble, so we expect a transton towards a process wth ndependent partcles. Note that we can estmate (but other choces are possble) the tme and lenght scales from nertal subrange second order structure functons, 113 for the Euleran velocty dfference and 41 for the Lagrangan one: the length scale λ s defned n the Euleran frame, so that n the nertal subrange (namely, for η r λ where η s the Kolmogorov mcroscale) the structure functon for each component may be wrtten as v 2 = 2σ 2 ( r λ )2/3 (12) where σ = v 2 /3. The Lagrangan tme scale τ can be defned n a smlar way. For τ η t τ, t results u 2 = 2σ 2 t τ (121) 22

24 consstent also wth 43 when τ s concdent wth the ntegral tme scale. It should be observed that scales for the nertal subrange, at varance wth ther ntegral verson, can be defned ndependently of nhomogenety or unsteadness, provded that the scales of such varatons are suffcently large to allow an nertal subrange to be dentfed. As far as the velocty s concerned, σ can be recognsed as the approprate scale of turbulent fluctuatons n both descrptons. Wth these choces: β = C3/2 K, (122) 2C showng how to defne β from nertal subrange (unversal) propertes. 9.2 The spatal decorrelaton lmt In the lmt β, correspondng to a vanshng Euleran correlaton scale, the non-dmensonalsaton defned n the prevous secton fals to apply. However, n ths lmt, the process s descrbed by the Ornsten-Uhlenbeck (OU) process. It s worth notng that the OU process has sometmes been used to descrbe Lagrangan velocty n turbulent flows, for nstance by Gfford (1982), who poneered the stochastc approach to atmospherc dsperson. The Novkov (1963) model and the NGLS model (Thomson (199), pag.124) are smple applcatons of ths concept. Adoptng the choces made n the prevous Secton, but usng a spatal scale defned by τσ rather than the vanshng λ (thus, a Lagrangan spatal scale nstead of an Euleran one), the OU process equvalent to 116 s descrbed by the non-dmensonal set of lnear LE dx = u dt, du = u dt + 2 dw (123) where = 1,..., 6. The equatons for the relatve quanttes ( u, x ) can be obtaned from the dfference between quanttes relatve to the frst ( = 1, 2, 3) and second ( = 4, 5, 6) partcles. The resultng set of equatons reads d x = u dt, d u = u dt + 2 dw (124) where = 1,..., can be solved analytcally specfyng the correlaton functon and the varances Gardner (199). Some basc results are summarsed here. The second order moment of velocty dfference turns out to be an exponental functon dependent on the tme nterval only u u = u 2 exp ( t) (125) 23

25 By ntegratng 125, the dsplacement varance for a sngle component s ( x x ) 2 = ( u 2 2)(1 exp ( t)) 2 + 4t 4(1 exp ( t)) (126) For short tmes (but expandng 126 to the thrd power of t), t turns out that ( ) 4 ( x x ) 2 u 2 t u2 t 3 (127) From 127 t can be observed that, when the ntal relatve velocty u s dstrbuted n equlbrum wth Euleran statstcs (.e., u 2 = 2), a t 2 regme takes place wth a negatve t 3 correcton. On the other hand, f u 2 = the ballstc regme dsplays a t3 growth wth a coeffcent 4/3,.e., 2C ε/3 for the dmensonal verson: see Novkov (1963), Monn and Yaglom (1975), Borgas and Sawford (1991). Note that the denomnator 3 arses because we are dalng wth one velocty or poston component: n the separaton r s consdered, the coeffcent turns out to multpled by The model formulaton n the case of Gaussan p E In order to proceed wth the analyss of the model, we select as a possble soluton to 118, the expresson gven by Thomson (199) (hs Eq. 18) for Φ, consstent wth a Gaussan pdf n a flow wth zero mean velocty feld: Φ = 1 V l + 1 p E 2 x l 2 (V 1 V l ) lj t u j (V 1 V l ) lj u j u k (128) x k The spatal structure s accounted for usng the Durbn (198) formula for longtudnal velocty correlaton, whch s compatble wth the 2/3 scalng law n the nertal subrange. Although the Gaussan pdf s known not to satsfy completely the nertal subrange requrements (t prescrbes a Gaussan dstrbuton for Euleran velocty dfferences, whle nertal subrange requres a non-zero skewness), t has been used n basc studes Borgas and Sawford (1994) and applcatons Reynolds (1999). Note that the stochastc model s formulated for the varable (x, u) rather than for the varable ( x/ 2, u/ 2) as n Thomson s orgnal formulaton. In the present case, assumng homogeneous and sotropc turbulence, the covarance matrx V (x) of the Euleran pdf s expressed by ( V = I R (2,1) (x) R (1,2) ) (x) I (129) 24

26 where I s the dentty matrx and R (p 1,p 2 ) j (x) = u (p 1) u (p 2) j (13) where p 1, p 2 = 1, 2 (p 1 p 2 ) are the partcles ndces. The quantty u (p 1) u (p 2) j u (x (p1) )u j (x (p2) ) s the two-pont covarance matrx, whch s expressed n terms of longtudnal and transverse functons F and G (see, for nstance, Batchelor (1953) as and R j = F ( r) x x j + G( r)δ j (131) F = 1 f 2 r r G = f + r 2 f r (132) (133) It goes wthout sayng that R (p 1,p 2 ) j = R (p 2,p 1 ) j = R (p 1,p 2 ) j. As n Durbn (198), F and G are computed from the parallel velocty correlaton ( ) r 2 1/3 f( r) = 1 r 2 (134) + 1 whch s Kolmogorov (1941) complant for r 1. 1 The dffuson coeffcent n complex flows Although ts meanng s not clear, t s usual to treat many problems usng a dffuson coeffcent n complex flows (for nstance, n condtons of nhomogenety or unsteadness of the velocty feld). Is t possble to have some hnt about the structure of ths coeffcent? 1.1 The dffusve lmt of the Thomson (1987) soluton Thomson (1987) gves an estmate of the dffuson coeffcent of a process wth N = 6 n the lmt of the correlaton tme gong to zero: D j = (u u )G j d 3 u (135) where G k s a soluton of an equaton nvolvng the Euleran pdf (Thomson (1987), pag. 541). 25

27 There are analytcal soluton n few cases: f a s a lnear functon of u. In the one dmensonal case, n partcular: wth q = D = u 2q 2 C εp E du (136) (u u)p E du (137) from whch t s evdent that the dffuson coeffcent depends on the space and tme structure of the Euleran pdf of the flow. 1.2 The multple scale analyss for preasymptotc transport Further hnt n the problem can be obtaned lookng at a dfferent approach. Accordng to Mazzno et al. (25), f the velocty feld s decomposed n a large scale and a small scale parts v = U +ɛu, so that the dffuson equaton for the scalar c and the correspondng stochastc equaton c t + v c 2 c = D (138) x x x dx = v dt + 2D dw (139) may be wrtten for the large scale averaged scalar feld c L where c L t (E) U c L + x dx = U (E) dt + U (E) (E) = 2 Dj c L (14) x x j 2D (E) j dw j (141) = U + D j x j (142) D (E) j = D j + D j (143) 2 and D j s dependent on the large scale velocty feld and s defned by a proper equaton (a perturbatve soluton s descrbed n the cted paper). 26

28 Fgure 2: Effectve eddy dffusvty, after Mazzno et al. (25). Sold lne: approxmate soluton; dashed lne: exact soluton. The dsperson s essentally ndependent on the small-scale dffuson coeffcent D. A graph of D xx (E) for the partcular case of a small scale cellular velocty feld supermposed to a large scale shear n the x drecton U = (U sn(ky), ) (144) wth characterstc lenght scale L = 2π/k and ampltude U, s reported n Fg. 2. References Batchelor, G. K., 1953: The theory of homogeneous turbulence, Cambrdge Unversty Press. Borgas, M. S. and B. L. Sawford, 1991: The small-scale structure of acceleraton correlatons and ts role n the statstcal theory of turbulent dsperson. J. Flud Mech., 228,

29 Borgas, M. S. and B. L. Sawford, 1994: A famly of stochastc models for two-partcle dsperson n sotropc homogeneous statonary turbulence. J. Flud Mech., 279, Durbn, P. A., 198: A stochastc model for two-partcle dsperson and concentraton fluctuatons n homogeneous turbulence. J. Flud Mech., 1, Franzese, P., A. K. Luhar, and M. S. Borgas, 1999: An effcent Lagrangan stochastc model of vertcal dsperson n the convectve boundary layer. Atmos. Envron., 33, Gardner, C. W., 199: Handbook of Stochastc Methods for Physcs, Chemstry and the Natural Scences, 2nd ed., Sprnger-Verlag. Gfford, F. A., 1982: Horzontal dffuson n the atmosphere: a Lagrangandynamcal theory. Atmos. Envron., 15, Gradshteyn, I. S. and I. M. Ryzhk, 198: Table of ntegrals, seres, and products, 198th ed., Academc Press. Jaynes, E. T., 1957: Informaton theory and statstcal mechancs. Physcs Revew, 16, Kaplan, H. and N. Dnar, 1993: A three-dmensonal model for calculatng the concentraton dstrbuton n nhomogeneous turbulence. Boundary- Layer Meteorol., 62, Kolmogorov, A. N., 1941: The local structure of turbulence n ncompressble vscous flud for very large reynolds numbers. Dokl. Akad. Nauk SSSR, 3, 31. Maryon, R. H., 1998: Determnng cross-wnd varance for low frequency wnd meander. Atmos. Envron., 32, Maurz, A. and S. Lorenzan, 21: Lagrangan tme scales n nhomogeneous non-gaussan turbulence. Flow, Turbulence and Combuston, 67, Maurz, A., G. Pagnn, and F. Tamper, 24: Influence of Euleran and Lagrangan scales on the relatve dsperson propertes n Lagrangan Stochastc Models of turbulence. Phys. Rev. E,

30 Maurz, A. and F. Tamper, 1999: Velocty probablty densty functons n Lagrangan dsperson models for nhomogeneous turbulence. Atmos. Envron., 33, Mazzno, A., S. Musaccho, and A. Vulpan, 25: Multple-scale analyss and renormalzaton for preasymptotc scalar transport. Phys. Rev. E, 71, Monn, A. S. and A. M. Yaglom, 1971: Statstcal flud mechancs, vol. I, MIT Press, Cambrdge, 769 pp. Monn, A. S. and A. M. Yaglom, 1975: Statstcal flud mechancs, vol. II, MIT Press, Cambrdge, 874 pp. Novkov, E. A., 1963: Random force method n turbulence theory. Sov. Phys. JETP, 17, Novkov, E. A., 1986: The Lagrangan-Eueleran probablty relatons and the random force method for nonhomogeneous turbulence. Phys. of Fluds, 29, Pasquero, C., A. Provenzale, and A. Babano, 21: Parameterzaton of dsperson n two-dmensonal turbulence. J. Flud Mech., 439, Reynolds, A. M., 1999: The relatve dsperson of partcle pars n statonary homogeneous turbulence. J. Appl. Meteorol., 38, Rsken, H., 1989: The Fokker-Planck Equaton. Methods of Soluton and Applcatons, 2nd ed., Sprnger-Verlag. Tamper, F., A. Maurz, and S. Albergh, 21: Lagrangan models of turbulent dsperson n the atmospherc boundary layer, Ingegnera del vento n Itala 2, Att del VI Convegno Nazonale ANIV, Genova gugno 2, G. G. Solar, L.C.Pagnn, ed., SGEdtoral, Padova. Taylor, G. I., 1921: Dffuson by contnuos movements. Proc. London Math. Soc., 2, Tennekes, H., 1982: Smlarty relatons, scalng laws and spectral dynamcs, Atmospherc turbulence and ar polluton modelng, F. T. M. Neuwstadt and H. van Dop, eds., Redel, pp Thomson, D. J., 1987: Crtera for the selecton of stochastc models of partcle trajectores n turbulent flows. J. Flud Mech., 18,

31 Thomson, D. J., 199: A stochastc model for the moton of partcle pars n sotropc hgh Reynolds-number turbulence, and ts applcaton to the problem of concentraton varance. J. Flud Mech., 21, Thomson, D. J., 1995: Dscusson. Atmos. Envron., 29, van Kampen, N. G., 1981: Stochastc Processes n Physcs and Chemstry, North-Holland, Amsterdam. Wel, J. C., P. P. Sullvan, and C.-H. Moeng, 24: The use of large-eddy smulatons n Lagragan partcle dsperson models. J. Atmos. Sc., 61,