ROTATION OF TRAJECTORIES IN LAGRANGIAN STOCHASTIC MODELS OF TURBULENT DISPERSION

ROTATION OF TRAJECTORIES IN LAGRANGIAN STOCHASTIC MODELS OF TURBULENT DISPERSION B.L. SAWFORD CSIRO Atmospheric Research, Aspendale, Vic 3195, Australia Received in final form 27 July 1999) Abstract. We present a new measure for the rotation of Lagrangian trajectories in turbulence that simplifies and generalises that suggested by Wilson and Flesch Boundary-Layer Meteorol. 84, 411 426). The new measure is the cross product of the velocity and acceleration and is directly related to the area, rather than the angle, swept out by the velocity vector. It makes it possible to derive a simple but exact kinematic expression for the mean rotation ds of the velocity vector and to partition this expression into terms ds that are closed in terms of Eulerian velocity moments up to second order and unclosed terms. The unclosed terms ds arise from the interaction of the fluctuating part of the velocity and the rate of change of the fluctuating velocity. We examine the mean rotation of a class of Lagrangian stochastic models that are quadratic in velocity for Gaussian inhomogeneous turbulence. For some of these models, including that of Thomson J. Fluid Mech. 180, 113 153), the unclosed part of the mean rotation ds vanishes identically, while for other models it is non-zero. Thus the mean rotation criterion clearly separates the class of models into two sets, but still does not provide a criterion for choosing a single model. We also show that models for which ds = 0 are independent of whether the model is derived on the assumption that total Lagrangian velocity is Markovian or whether the fluctuating part is Markovian. Keywords: Trajectories, Dispersion, Rotation, Lagrangian models. 1. Introduction In a first-order Lagrangian stochastic model for the trajectories of fluid particles in a turbulent flow, the change in velocity du over the time interval dt is specified by a stochastic differential equation of the form Thomson, 1987) and where du i = a i u, x,t)dt + C 0 ɛdξ i t) 1) dx i = u i dt, 2) a i P E = 1 2 C 0ɛ P E u i + ϕ i, 3) The CSIRO right to retain a non-exclusive royalty-free license in and to any copyright is acknowledged. Boundary-Layer Meteorology 93: 411 424, 1999. 1999 Kluwer Academic Publishers. Printed in the Netherlands.

412 B. L. SAWFORD ϕ i = P E u ip E, 4) u i x i and P E u, x, t) is the Eulerian probability density function for the velocity. C 0 is the universal constant associated with the Lagrangian velocity structure function, ɛx, t) is the mean rate of dissipation of turbulence kinetic energy and dξ is the vector incremental Wiener process Gardiner, 1983). The terms involving C 0 and ϕ in 1) and 3) are usually associated with fluctuating pressure-gradient and viscous forces, while ϕ represents the large scale forcing of the turbulence. In general 4) does not determine ϕ uniquely. Consider, for example, Gaussian inhomogeneous turbulence where P E is given by P E u) = λ1/2 2π) 3/2 exp 1 2 u i u i )λ ij u j u j )), 5) where λ ij is the inverse stress tensor and λ is its determinant. For later convenience we also introduce here the stress tensor σ ij. Sawford and Guest 1988) presented two different solutions to 3) and 4) due to Thomson 1987) and unpublished work by Borgas respectively) and showed that they differ non-trivially. Borgas et al. 1997) described a simple stochastic model for axisymmetric turbulence and derived formal analytical solutions for the velocity components from which they determined the Lagrangian velocity covariance tensor and the dispersion tensor. The trajectories spiral about the axis of symmetry with the result that dispersion in the plane perpendicular to that axis is reduced compared with that in the direction of the axis of symmetry. The significance of that work is that it demonstrates that statistics of the velocity field alone are insufficient to describe turbulent dispersion and that the non-uniqueness is related to physical properties unconnected with P E. In particular, for their model at least, some measure of the rotation of trajectories is necessary to define the stochastic model and to quantify the dispersion. Wilson and Flesch 1997) proposed the change of the orientation of the Lagrangian velocity fluctuation vector as a useful measure of the rotation of trajectories. For simplicity they presented their analysis for two-dimensions, suggesting that extension to three dimensions is straightforward. They showed that for the simple case of stationary, horizontally homogeneous turbulence in two dimensions the mean rotation of trajectories according to this measure is zero for Thomson s model and non-zero for Borgas s model. Furthermore, they showed that Thomson s model agrees well with wind tunnel measurements of the dispersion of heat from a line source within a model crop canopy Legg et al., 1986), whereas the Borgas model significantly underestimates the observed dispersion. Calculations for a different model Flesch and Wilson, 1992) in which ϕ is constrained to be parallel to the velocity vector so that the mean rotation of trajectories is zero) also agree well with the observations. The implication of these results is that models in which the mean rotation of the velocity fluctuation vector is zero perform better than do those

ROTATION OF TRAJECTORIES IN LAGRANGIAN STOCHASTIC MODELS 413 for which it is non-zero. Wilson and Flesch 1997) suggested that the constraint of zero mean rotation be applied to choose between different models that satisfy 4). They recognised however, that this additional constraint does not define a unique solution to 4), nor could they provide an explicit specification for ϕ that gives zero mean rotation according to their definition. This appealingly simple picture is complicated somewhat by a series of comparisons between various models and experimental dispersion data summarised by Reynolds 1998). He confirmed the findings of Wilson and Flesch for canopy flows, but suggested that for a neutral boundary layer again under horizontally homogeneous conditions) some two-dimensional models with non-zero rotation perform better than, for example, Thomson s model. He argued that this discrepancy might be due to the presence of coherent structures in the neutral boundary layer, which are suppressed by the roughness elements in the canopy flow. These conjectures await confirmation but we are unable to address them here. Reynolds 1999) also indicates that for Thomson s model the mean rotation in three dimensions is non-zero according to the Wilson and Flesch 1997) measure. For more general flows in two dimensions, and in three dimensions, both Thomson s model and Borgas s model give a non-zero mean rotation according to the Wilson and Flesch 1997) measure. The main purpose of the present paper is to propose an alternative measure for the rotation of trajectories, which is more natural and simpler than that proposed by Wilson and Flesch, and which discriminates more clearly between different stochastic models. Using this new measure, we derive an exact kinematic result partitioning the mean rotation of the total velocity vector into mean-mean, mean-turbulence and turbulence-turbulence terms. For simple shear flows we use this exact result to show that the interaction of the mean flow and the turbulence produces a non-zero mean rotation of the total velocity vector. This result confirms, in a much simpler and more transparent way, that derived by Wilson and Flesch for a simple version of Thomson s model. Our main result is a simple demonstration that according to this new measure, the turbulence-turbulence part of the mean rotation which corresponds to the component considered by Wilson and Flesch) always has zero mean rotation for Thomson s model in Gaussian inhomogeneous turbulence. We apply our new rotation measure to the class of quadratic form models for Gaussian inhomogeneous turbulence and to some other specially constructed models. In doing so, we consider models based on alternative Markov assumptions. In particular, we consider both the class of models which assume the total velocity is Markovian and those models which assume that the fluctuating velocity is Markovian. We show that some models including Thomson s model) have the same form regardless of whether they are formulated in terms of the total velocity or in terms of the fluctuating velocity. This is in some sense a Galilean invariance property that seems a desirable feature.

414 B. L. SAWFORD 2. A New Measure for Rotation of Trajectories 2.1. PREAMBLE: THE MEANING OF EULERIAN AND LAGRANGIAN In this paper, and more generally throughout the literature, the terms Eulerian and Lagrangian tend to be used in at least two different ways. Therefore it is worthwhile for us to clarify those meanings here in order to prevent later confusion. The first sense in which these two terms are used is in the definition of the reference frame in which the flow is described. In an Eulerian sense, we describe the flow in terms of the changes at a fixed point in an inertial reference frame. The independent variables are the location of the point and the time. Conversely a Lagrangian description is in a reference frame moving with the fluid in particular following a fluid element). In this case the time is the only independent variable and the position of the fluid element is a dependent variable. The stochastic differential equation 1) is a Lagrangian description of the flow and the equivalent Fokker-Planck equation which underlies 3) and 4)) is an Eulerian description. Thus while we speak of Lagrangian stochastic models because they are usually implemented in the trajectory formulation of the stochastic differential equation, these models are not inherently Lagrangian because they could always be formulated and in principle implemented) in the corresponding Eulerian formalism of the Fokker Planck equation. The second sense in which we use these terms is in describing the type of statistics with which we are dealing. Eulerian statistics are generally taken to be those at a fixed point, whereas we generally think of Lagrangian statistics as averages over trajectories, which emanate from some fixed point. While the correspondence between these two senses of these terms is clear, there is potential for confusion because the correspondence is not perfect. In particular, it is possible to calculate both Eulerian and Lagrangian statistics from a description of the flow in both Eulerian and Lagrangian reference frames. Thus, for example, in the Lagrangian or trajectory description, if we average over trajectories with a fixed initial location we obtain Lagrangian statistics. Alternatively if we average over those trajectories which pass though the fixed point and time without concerning ourselves where they came from originally) we obtain Eulerian statistics. Similarly, both Eulerian and Lagrangian statistics can be calculated from the Eulerian Fokker- Planck description by choosing appropriate initial conditions. Because both descriptions can be used to calculate both sorts of statistics, we do not use a different notation for Eulerian and Lagrangian velocities. In most cases in the rest of this paper we will be working in a Lagrangian frame, but dealing with Eulerian statistics.

ROTATION OF TRAJECTORIES IN LAGRANGIAN STOCHASTIC MODELS 415 Figure 1. Schematic illustrating the relationship of the cross product u du to the area of the triangle swept out as u moves to u +du over time dt. Note that the length of the line AA isgivenbydu sindα) =u +du) sindθ), so that as dt 0, u du = u du sindα)=u 2 dθ. 2.2. DEFINITION OF ROTATION We consider a fluid trajectory passing through the point x at time t with velocity ux, t) and acceleration du/dt, where the substantive derivative d dt = + u i 6) x i is the rate of change following the fluid. Thus the acceleration is a Lagrangian quantity in the first sense discussed above, since it is defined in a reference frame moving with the mean flow. However, it has a well-defined value at every point in the fluid at every time, which is determined by its value along the trajectory passing through a point at a particular time. Thus we can define both Lagrangian and Eulerian statistics of the acceleration i.e., in the second sense of those words discussed above). Throughout this paper we will be concerned with Eulerian statistics. We follow Wilson and Flesch 1997) and define rotation as a change in the direction of the velocity vector at x, t) over the time interval dt. Clearly if the acceleration is in the same direction as the velocity there is no rotation and we can express this mathematically by saying that the cross product of the velocity and acceleration vectors is zero. More generally, we use this cross product as a measure of the rotation, ds i = ɛ ij k u j du k, 7) where ds = u 2 i dθ is the vector area swept out by the velocity vector over time dt,dθ is the vector angle between u +du and u see Figure 1) and ɛ ij k is the alternating tensor. The components of ds are given by the projections onto the three planes defined by the basis vectors and can be expressed somewhat awkwardly in terms of projection operators Temple, 1960) ds i = Q i) jk u kq i) jl u ldθ i, 8)

416 B. L. SAWFORD where Q i) jk projects a vector onto the plane perpendicular to the i)-direction and there is no summation over the superscripted index in braces. Explicitly, the three components are ds 1 = u 2 2 + u2 3 )dθ 1, 9) ds 2 = u 2 3 + u2 1 )dθ 2, 10) ds 3 = u 2 1 + u2 2 )dθ 3, 11) For the two-dimensional case only one of these components is involved, say ds 1 = u 2 2 +u2 3 )dθ 1, and Wilson and Flesch 1997) expressed the equivalent result in terms of the corresponding angle dθ 1 =ds 1 /u 2. However, the present formulation in terms of the area is simpler and more natural and we will show below that it is more useful. Since the alternating tensor is antisymmetric it follows from 7) that the rotation is zero if the tensor u i du j is symmetric, and that only the antisymmetric part of u i du j contributes to the rotation. 2.3. EXACT KINEMATIC EXPRESSIONS FOR THE MEAN ROTATION We can derive a formal expression for the Eulerian mean rate of rotation by combining 6) and 7) to give ds i =ɛ ij k u j uk ) u k + u l dt. 12) where the angle brackets formally denote an Eulerian average for example over an ensemble of unbiased trajectories passing through the point x at time t.) In practice, the averages might be measured as spatial averages over homogeneous regions or time averages for stationary flows. Writing the total velocity u as a sum of mean and fluctuating components u and u respectively, we have ds i = ɛ ij k u j u k + u j u k u l + + u j u l u k + u l x u j u k l x l u u k j + u u k j u l + u j u l u ) k dt. 13) In 13) the first two terms on the right correspond to the rate of change of the mean flow acting on the mean velocity and the second two terms correspond to interactions between the mean flow and the turbulence. The last three terms represent the mean rotation due to the rate of change of the fluctuating velocity acting on the fluctuating velocity. For convenience we partition the total mean rotation into the sum of the mean-mean plus mean-turbulence terms which we denote by ds

ROTATION OF TRAJECTORIES IN LAGRANGIAN STOCHASTIC MODELS 417 and the turbulence-turbulence terms which we denote by ds. We see that ds is closed in terms of the mean flow and the stress tensor and their derivatives. Note that if the rotation is expressed in terms of the angle dθ, partitioning in this way is more complicated because the total mean rotation involves the magnitude of the projected velocity Q i) jk u kq i) jl u l, whereas dθ as defined by Wilson and Flesch involves the magnitude of the projected velocity fluctuation vector Q i) jk u k Qi) jl u l. In general the mean-mean and mean-turbulence terms are non-zero. For example, consider a simple shear flow a fully developed neutral boundary layer, say) in which U i = δ i1 δ j2 Sx 2 ) 14) and σ 11 x 2 ) σ 12 x 2 ) 0 u i u j = σ 12 x 2 ) σ 22 x 2 ) 0. 15) 0 0 σ 33 x 2 ) Note that in meteorological terminology, x 2 is the vertical coordinate and σ 22 is the variance of the vertical velocity fluctuations, more familiarly denoted by σ 2 w. In this case the mean-mean terms vanish and the mean-turbulence terms reduce to ɛ ij k u l u j u k + u j u k u l = ɛ ij 1 u 2 u j U 1 x 2 + ɛ i1k U 1 u k u 2 x 2. 16) That is, in this case ds 1 /dt = ds 2 /dt =0and ds 3 dt = σ 22 Sx 2 ) + U 1 σ 22 x 2. 17) Thus in a neutral surface layer where σ 22 constant) or in a homogeneous shear flow, the shear causes trajectories to rotate clockwise about the 3-axis. This exact result confirms the result inferred by Wilson and Flesch in much more complex form) from their analysis of Thomson s model. If the turbulence is inhomogeneous the gradient of the stress tensor also contributes to the mean rotation. In principle it is also possible to determine the unclosed term ds i =ɛ ij k u j du k =ɛ u k ij k u j + u j u k u l + u j x u l l u ) k dt 18) directly by Eulerian measurement or by DNS calculations. In general there is no reason to suppose that this term must vanish.

418 B. L. SAWFORD 2.4. ROTATION IN THOMSON S MODEL Thomson s 1987) model for Gaussian inhomogeneous turbulence takes the form a i = 1 σ ij + u i + u j u { i 1 2 2 C 0ɛλ ik u i x k 1 2 λ km σim + u j σ im )} u k u k ) + 1 2 λ σ ij jm u k u k )u m u m ), 19) x k where we recall that σ ij = u i u j is the stress tensor and λ ij is its inverse. Substituting into 1) and 7) and averaging over the velocity and the Wiener process, we obtain ds i = ɛ ij k u j u k { 1 2 C 0ɛλ kn σ nj 1 2 λ nmσ nj + u j u l u k + u j x u l u k + u j u k u l l σkm )}) dt. 20) + u l σ km By comparing 20) with 13) we see that Thomson s model reproduces the correct mean-mean and mean-turbulence components of the mean rotation and that it represents the turbulence-turbulence component ds i = ɛ ij k u j du k by the closure ds i dt = ɛ ij k { 1 2 C 0ɛδ jk 1 2 σjk + u l σ jk )}, 21) where we have used the result δ ij = σ ik λ jk. Thus for Thomson s model the turbulence turbulence component of the mean rotation vanishes because the term in brackets is symmetrical. This neat result is not obtained if the angle dθ is used as the measure of rotation as proposed by Wilson and Flesch, 1997) since then the factor Q i) jk u k Qi) jl u l ) 1 is included in the average over the velocity distribution and the symmetry is destroyed. It can be shown that in general dθ /dt is non-zero for Thomson s model. 2.5. ROTATION IN BORGAS S MODEL Borgas s model for Gaussian inhomogeneous turbulence Sawford and Guest, 1988) takes the form a i = σ ij + 1 λ 2 λ 1 σ ij + u i + u j u i { 1 2 C u m 0ɛλ ik σ ij λ km 1 2 λ km u k u k ) + 1 2 λ km σim + u j σ )} im σ ij u k u k )u m u m ). 22)

ROTATION OF TRAJECTORIES IN LAGRANGIAN STOCHASTIC MODELS 419 It differs from Thomson s model in the coefficient of the quadratic term in the velocity, in the coefficient of the mean shear in the term linear in the velocity and in the constant term. For this model the mean rotation can be partitioned into the correct mean-mean and mean-turbulence components and a turbulence-turbulence component given by the term ds i dt { u m u k = ɛ ij k σ kl λ nm σ nj σ lj = ɛ ij k { σ kl u j σ lj u k } }, 23) which is non-zero because the term in brackets is antisymmetric. Thus the new criterion for rotation distinguishes clearly between Thomson s model, for which the turbulent-turbulent component of the mean rotation is zero and Borgas s model for which it is non-zero. The original criterion based on the angle of rotation does not distinguish so neatly between the two models, since in general dθ /dt is non-zero but different) for both models. For the simple shear flow characterised by 14) and 15) Borgas s model gives ds 1 = ds 2 =0and ds 3 dt = 2σ 22 Sx 2 ), 24) which is precisely enough to reverse the sign of the first term when added to 17). That is, for homogeneous turbulence in a shear flow, the closure term 24) for Borgas s model exactly reverses the rotation of the closed terms 17). This seems unlikely to be correct. 2.6. ROTATION IN OTHER QUADRATIC MODELS For Gaussian inhomogeneous turbulence there is a class of six solutions to 3) and 4) in which a i is a quadratic function of the velocity as in 19) and 22)). These six models arise from combinations of two forms for the coefficient in the term that is linear in velocity and three forms for the coefficient in the term that is quadratic in velocity. The constant term is not independent, but is linked to the quadratic term. The two versions of the linear terms and two of the versions of the quadratic terms are those in the second and third lines respectively of 19) and 22). For the third version of the quadratic term the indices k and m in 19) are interchanged and since these are contracted with a pair of velocity components the quadratic term is unaltered. Rodean 1996) gives the details. Thus there are only four distinct quadratic models and the remaining two can be obtained by exchanging the linear terms in 19) and 22). Since it is the linear term which gives rise to the non-zero rotation in Borgas s model, it follows that of these two extra models one, which has

420 B. L. SAWFORD the same linear term as Thomson s model, will have a zero turbulence turbulence component to the mean rotation. It takes the form a i = σ ij + 1 λ 2 λ 1 σ ij + u i + u j u i { 1 2 C 0ɛλ ik u i 1 x k 2 λ σim km + u j σ )} im u k u k ) + 1 2 λ σ ij km u k u k )u m u m ). 25) The other, which has the same linear term as Borgas s model, will have the same non-zero turbulence turbulence component to the mean rotation as Borgas s model. This model is a i = 1 σ ij + u i 2 1 2 λ km σim + u j u i + u j σ im { 1 2 C 0ɛλ ik σ ij λ km u m )} u k u k ) + 1 2 λ σ ij jm u k u k )u m u m ). 26) x k Thus, imposing the additional constraint that the turbulence-turbulence component of the mean rotation be zero or some other value) is not sufficient to uniquely define a stochastic model even within the restricted class of quadratic models. Reynolds 1998) has considered the infinite set of quadratic models constructed from linear combinations of the two sets of linear and quadratic coefficients. That is he considered models of the form a i = 1 σ ij 2 σij + λ 1 λ σ ij + 1 2 C 2 { 1 2 C 0ɛλ ik u i 1 x k 2 λ km u i +C 1 σ ij λ km u m + 1 2 1 C 2 )λ jm σ ij x k x k σim ) + u i ) u k u k ) + C 2 λ km σ ij + u j σ im + u j u i )} u k u k ) ) u k u k )u m u m ). 27) Thomson s model is retrieved with C 1 = C 2 = 0, Borgas s model with C 1 = C 2 =1, model 25) with C 1 =0andC 2 = 1 and model 26) with C 1 =1andC 2 =0.The mean rotation in this general case can be partitioned into the correct form for ds and a turbulence-turbulence component given by ds i dt = C 1 ɛ ij k { σ kl u j σ lj u k }. 28)

ROTATION OF TRAJECTORIES IN LAGRANGIAN STOCHASTIC MODELS 421 2.7. THE MARKOV ASSUMPTION APPLIED TO THE FLUCTUATING VELOCITY We have described these models in terms of a stochastic differential equation SDE) for the total velocity u and this is arguably most appropriate since the underlying assumption is that the total velocity is a continuous Markov process. However, we could also perhaps less logically) construct a SDE for the fluctuating velocity and treat the rate of change of the mean velocity separately through the deterministic equation d u i dt = u i + u j u i. 29) If we model the fluctuating velocity as a continuous Markov process, then we replace 1) 4) by and where and du i = a i u, x,t)dt + C 0 ɛdξ i t) 30) dx i = u i + u i )dt, 31) a i P E = 1 2 C 0ɛ P E u ϕ i, 32) i φ i u i = P E u i + u i )P E x i. 33) Note that the total velocity appears in the spatial derivative on the right-handside of 33) and we have expanded this term to emphasise the role of the mean velocity. These equations again give rise to four distinct quadratic models that correspond to 19), 22), 25) and 26) with all the terms involving derivatives of the mean velocity removed. For example, corresponding to 19) we have a i = 1 σ ij 2 { 1 2 C 0ɛλ ik 1 2 λ km u k u k ) + 1 2 λ jm σim + u j σ im )} σ ij u k u k )u m u m ). 34) x k We can then retrieve a model for the total velocity by adding back in the rate of change of the mean velocity from 29). Because the mean velocity derivative terms in 19) correspond precisely to those in 29), Thomson s model has the satisfying

422 B. L. SAWFORD property that exactly the same SDE is obtained for the total velocity whether we treat the total velocity or the fluctuating velocity as a Markov process. However, for Borgas s model 22), the form of the mean gradient in the term that is linear in the velocity is complicated by the factor σ ij λ km. Thus a model different from 22) is obtained if the rate of change of the mean velocity 29) is added to the SDE for the fluctuating velocity which corresponds to 22). Since this model is quadratic, it must be one of the four distinct models described above and in fact is the model 25). We can apply this analysis to all four distinct quadratic models. We find that both Thomson s model and the model 25) are independent of whether they are formulated directly in terms of the total velocity or in terms of a model for the fluctuating velocity to which is added the deterministic equation 29) for the mean velocity. On the other hand, both Borgas s model and the model 26) revert to the model 25) and Thomson s model respectively when they are formulated indirectly via the fluctuating velocity. Thus, as with the mean rotation criterion, this consistency property separates the four quadratic models into two pairs, but does not distinguish between Thomson s model and the model 25). 2.8. THE MODEL OF FLESCH AND WILSON Flesch and Wilson 1992) proposed a Markov model for the fluctuating velocity in two dimensions in which ϕ is parallel to u. Monti and Leuzzi 1996) extended this model to three dimensions, as a model for the total velocity in which ϕ is parallel to u. By definition, for these models ϕ cannot contribute to the mean rotation of the fluctuating velocity or the total velocity respectively. Furthermore, by forcing the mean rotation of the total velocity to vanish, for consistency with the exact result 13) the Monti and Leuzzi model must represent the turbulence-turbulence component by a term equal in magnitude, but opposite in sign, to the sum of the mean-mean and mean-turbulence components. Although this cannot be ruled out it seems unlikely. Clearly these models, formulated in terms of the fluctuating and total velocity respectively, are not equivalent. The constraint implied by these models is very severe. It means that the instantaneous rotation of the fluctuating velocity or the total velocity respectively is solely due to the terms involving C 0 and ɛ in 1) and 3), which are associated with fluctuating pressure-gradient and viscous forces. Nevertheless, Wilson and Flesch show that for a canopy flow the Flesch and Wilson model performs as well as Thomson s model. 3. Conclusions We have shown that defining the rotation of the velocity vector as the cross product of the velocity and the acceleration i.e., as the area, rather than the angle, swept out by the velocity vector) greatly simplifies the concept of the mean rotation of

ROTATION OF TRAJECTORIES IN LAGRANGIAN STOCHASTIC MODELS 423 particle trajectories in turbulence. Using this new measure of rotation, we have been able to derive an exact kinematic expression for the Eulerian mean rotation of the velocity vector. We partitioned this exact expression into terms closed in terms of Eulerian velocity moments up to second order and terms that are unclosed at that order. We showed explicitly for a simple shear flow that the closed part of the mean rotation is non-zero. The unclosed terms correspond to the mean rotation ds of the fluctuating velocity vector by the rate of change of the fluctuating velocity, a quantity which has been proposed as a discriminator between alternative Lagrangian stochastic models. We examined the mean rotation of a class of Lagrangian stochastic models that are quadratic in velocity for Gaussian inhomogeneous turbulence. The mean rotation predicted by all these models can be partitioned into the exact closed part plus a closure for the unclosed terms. Some of these models including Thomson s model) predict that the turbulence turbulence part of the mean rotation vanishes. Others, including the model derived by Borgas, predict a non-zero turbulence-turbulence component to the mean rotation. These findings simplify and generalise the results of Wilson and Flesch. These stochastic models are usually derived by assuming that the total Lagrangian velocity is Markovian. We also considered the class of models based on the assumption that the fluctuating part of the Lagrangian velocity is Markovian and found the interesting result that those models for which ds vanishes are independent of whether the fluctuating or total velocity is assumed to be Markovian. This is certainly a desirable property and invites the conclusion that these models are better based than other models. Despite the simplicity and generality of the results we have obtained using this new measure for the mean rotation of trajectories, prescribing a value zero or otherwise) for it still does not define a unique model. However, there is growing evidence from numerical comparisons against dispersion data and from the theoretical analysis undertaken here that models for which ds vanishes are the better models. Of these, until further progress is made, we recommend Thomson s model since it is the simplest. Acknowledgements It is a pleasure to acknowledge many stimulating and useful discussions of this work with Michael Borgas. References Borgas, M. S., Flesch, T. K., and Sawford, B. L.: 1997, Turbulent Dispersion with Broken Reflectional Symmetry, J. Fluid Mech. 332, 141 156.

424 B. L. SAWFORD Flesch, T. K. and Wilson, J. D.: 1992, A Two-Dimensional Trajectory-Simulation Model for Non- Gaussian, Inhomogeneous Turbulence within Plant Canopies, Boundary-Layer Meteorol. 61, 349 374. Gardiner, C. W.: 1983, Handbook of Stochastic Processes for Physics, Chemistry and the Natural Sciences, Springer-Verlag, Berlin, 442 pp. Legg, B. J., Raupach, M. R., and Coppin, P. A.: 1986, Experiments on Scalar Dispersion within a Model Plant Canopy, Part III: An Elevated Line Source, Boundary-Layer Meteorol. 35, 277 302. Monti, P. and Leuzzi, G.: 1996, A Closure to Derive a Three-Dimensional Well-Mixed Trajectory Model for Non-Gaussian, Inhomogeneous Turbulence, Boundary-Layer Meteorol. 80, 311 331. Reynolds, A. M.: 1998, On Trajectory Curvature as a Selection Criterion for Valid Lagrangian Stochastic Dispersion Models, Boundary-Layer Meteorol. 88, 77 86. Reynolds, A. M.: 1999, On the Non-Uniqueness of Lagrangian Stochastic Models, Fluid Dyn. Res. 25, 217 229. Rodean, H. C.: 1996, Stochastic Lagrangian Models of Turbulent Diffusion, Meteorological Monographs, Vol. 26, No. 48, American Meteorological Society, Boston, 84 pp. Sawford, B. L. and Guest, F. M.: 1988, Uniqueness and Universality in Lagrangian Stochastic Models of Turbulent Dispersion, in 8th Symposium on Turbulence and Diffusion, American Meteorological Society, Boston, MA, pp. 96 99. Temple, G.: 1960, Cartesian Tensors: An Introduction, Methuen, London, 92 pp. Thomson, D. J.: 1987: Criteria for the Selection of Stochastic Models of Particle Trajectories in Turbulent Flows, J. Fluid Mech. 180, 113 153. Wilson, J. D. and Flesch, T. K.: 1997, Trajectory Curvature as a Selection Criterion for Valid Lagrangian Stochastic Dispersion Models, Boundary-Layer Meteorol. 84, 411 426.