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VALIDATION OF A NEW TURBULENT PARAMETERIZATION FOR DISPERSION MODELS IN CONVECTIVE CONDITIONS G. A. DEGRAZIA Universidade Federal de Santa Maria, Santa Maria, Bolsista do CNPq, Brazil

U. RIZZA and C. MANGIA Institute ISIAtA of CNR, Lecce, Italy

T. TIRABASSI Institute FISBAT of CNR, Bologna, Italy (Received in final form 10 June, 1997)

Abstract. A new parameterization for turbulent dispersion in a convective boundary layer is proposed. The model is based on turbulent kinetic energy spectra and Taylor’s diffusion theory. The formulation, included in an advanced dispersion model, has been tested and compared with vertical and lateral dispersion schemes reported in the literature, using data from field experiments. The application of a statistical evaluation shows that the proposed parameterization has the best overall fit to the data. Key words: Air pollution modelling, Dispersion parameterization, Spectral theory, Eulerian dispersion models

1. Introduction The Eulerian dispersion model concept is important for estimating ground-level concentrations due to tall stack emission and it is usually suitable for regulatory use in air quality models. In these models the main scheme for closing the advectiondiffusion equation is to relate concentration turbulent fluxes to the gradient of the mean concentration by eddy diffusivities (K theory). Improved dispersion algorithms in updated diffusion models calculate the lateral dispersion parameter (y ) and the eddy diffusivities in terms of distinct scaling parameters for turbulence (Holtslag and Nieuwstadt, 1986). These scaling parameters, including friction velocity (u ), Obukhov length (L), convective velocity scale (w ) and convective boundary-layer height (zi ), are frequently used in expressions to calculate the lateral dispersion parameter and the eddy diffusivities (Degrazia, 1989, Holstlag and Moeng, 1991; Pleim and Chang, 1992). In the convective boundary layer (CBL) the scalar turbulent transport is also represented by a non local eddy diffusivity described in the form of a similarity profile, using the convective velocity scale and the CBL height (Holtslag and Moeng, 1991). Dispersion experiments show that in the planetary boundary layer (PBL) the lateral profile of a plume resembles a Gaussian distribution (Siversten, 1978; Gryning et al., 1978; Nieuswstadt and van Duuren, 1979). Therefore, knowing the lateral dispersion parameter and the crosswind-integrated concentration at the surface Boundary-Layer Meteorology 85: 243–254, 1997. c 1997 Kluwer Academic Publishers. Printed in the Netherlands.

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G. A. DEGRAZIA ET AL.

Cy (x; 0), we can calculate the surface concentration at any point. Thus, for convective conditions, Briggs (1985) and Hadfield (1994) suggest formulations for y in terms of w and zi .

Recently Rizza et al. (1996) have developed an operative model for evaluating ground-level concentration from elevated sources (virtual height dispersion model, VHDM). This model is based on a Gaussian approach for transport and vertical diffusion, and a scaling classification for turbulent regimes. In the present paper, through the VHDM model, a new turbulent parameterization for dispersion models is presented and validated. We use the convective similarity theory and Taylor’s statistical diffusion theory to derive, under unstable conditions, expressions for the lateral dispersion parameter and the non-local vertical eddy diffusivity. The new vertical turbulent parameterization is evaluated against the Wyngaard and Brost (1984) formulation and the horizontal parameterization against the Gryning et al. (1987) scheme, both using ground level concentration measurements obtained from atmospheric dispersion experiments carried out in the northern part of Copenhagen under moderately unstable conditions. 2. The Parameterization of Eddy Diffusivities and Dispersion Parameters from Taylor’s Theory The dispersion parameters are statistical quantities of much interest in diffusion modelling (Batchelor, 1949). They are defined in terms of the second moments of particle displacements. Our formulation starts with the equation for the generalized dispersion parameters 2 as given by Pasquill and Smith (1983),

2 Z 1 E = i i2 F (n) 0 i 2

2

"

sin2 (nt= i )

n2

#

dn;

(1)

= x; y; z i = u; v; w where FiE (n) is the value of the Eulerian spectrum of energy normalized by the Eulerian velocity variance, i is defined as the ratio of the Lagrangian to the Eulerian integral time scales, i is the Eulerian standard deviation of the turbulent wind speed, n is the frequency and t is the travel time. An expression for the time-dependent exchange coefficients K can be expressed with

by (Batchelor, 1949; Degrazia and Moraes, 1992),

i2 i Z 1 E sin(2nt= i ) Fi (n) dn: 2 0 n According to Wandel and Kofoed-Hansen (1962) i can be written as, K =

d 1 2 dt 2

i =

U2 16 i2

!1=2

=

(2)

;

(3)

VALIDATION OF A NEW TURBULENT PARAMETERIZATION

245

where i is a function of the mean wind speed and the turbulence intensity and varies from 10, for stable conditions, to 2 for convective conditions (Panofsky and Dutton, 1984). For large diffusion travel times (t ! 1), the filter function in the integral of Equation (2) selects FiE (n) at the origin of the frequency space, such that the rate of dispersion becomes independent of the travel time from the source and can be expressed as a function of local properties of turbulence as follows,

K =

d 1 2 dt 2

=

i2 i FiE (0) Z 1 sin(2nt= i ) dn 2 n 0

(4)

where FiE (0) is the value of the normalized Eulerian energy spectrum at n = 0. By using the residue theorem (Boas, 1983, p. 605) it can be shown that the integral in Equation (4) is equal to /2, for t > 0. Therefore, the eddy diffusivity for large travel time assumes the simple form given by,

2 F E (0) K = i i i : 4

(5)

The Eulerian velocity spectra under unstable conditions can be expressed as a function of convective scales (Olesen et al. 1984; Degrazia et al. 1996) as follows,

nSiE (n) w2

=

2=3 z 2=3 f 1:5 f 5=3 " 1+ 5=3 q (fm )i qi qi zi i (fm )i 0:98c

(6)

where c is equal to 1 (2k ) 2=3 , k is the von Karman constant, 1 is determined from observation to be about 0.5 for the u spectrum (Champagne et al., 1977), implying that c = 0.3 for this spectrum, and c = 0.4 for v and w spectra, U is the mean wind speed, f is the reduced frequency (nz=U ), z is the height above the surface, (fm )i is the frequency of the spectral peak in neutral stratification, qi = (fm )i (fm )i 1 is a stability function where (fm )i is the frequency of the spectral peak regardless of the stratification, " = "zi =w3 is the non-dimensional molecular dissipation rate function, and " is the ensemble-average rate of dissipation of turbulent kinetic energy. By analytically integrating Equation (6) over the whole frequency domain, one can obtain the variance that is used to normalize the spectrum,

2=3 z 2=3 " i = w2 2=3 qi zi (fm )i 2

0:98c

(7)

so that the value of the normalized Eulerian spectrum can be given by

FiE (n) =

z 1:5 nz 5=3 1+ : (fm )i Uqi (fm )i Uqi 1

(8)

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A general formulation for the dispersion parameters can now finally be obtained from Equations (1), (3), (7) and (8) and be expressed as,

2 zi2

1:5(z=zi )2 16qi2 (fm )2i

=

0

Z1

2

sin

4

p(0:98c)1=2 f(f )2 q2 g1=3 Xn0 ! mi i " 1:5(z=zi )2=3

0

n02(1 d+nn0)5=3

(9)

where X = xw =zi U can be thought of as a nondimensional time since it is the ratio of travel time x=U to the convective time scale zi =w , x is the dimensional distance downwind and

n0 =

1:5z

n:

fm )i Uqi

(

Thus, we first consider the lateral dispersion parameter from an elevated continuous source in an unstable PBL, where elevated here means that at this height we can idealize the turbulent structure as homogeneous with the length scale of the energy containing eddies being proportional to the CBL height zi . Therefore, using the peak lateral wavelength (m )v = 1:5zi (Kaimal et al., 1976), c = 0.4 and (fm )v = 0.16 (Olesen et al., 1984), the lateral dispersion parameter can be obtained by Equation (9) with, (f ) qv = m v (fm )v

z 0:16(m )v

=

=

4:16

z ; zi

(10)

and be expressed as

y2 zi2

=

Z 1 sin2 f2:26 1=3 Xn0 g " dn0 0 (1 + n0 )5=3 n02

0:21

(11)

where the dissipation function " is given by (Højstrup, 1982) 1=3

"

"

=

1

#1=2 z 2 z 2=3 + 0:75 : zi L

(12)

In convective conditions, the turbulent eddy diffusivities can be derived from Equations (5) and (8), and inserting of i yields

K =

p z i

16(fm )i qi

:

(13)

For horizontal homogeneity the CBL evolution is driven mainly by the vertical transport of heat. Therefore, the analysis will focus on the vertical eddy diffusivity.

247


This diffusivity can be derived from Equation (13) by assuming that (fm )w (Sorbjan, 1986) and

qw

= =

z fm )w (m )w z 1:68 1 zi (

exp

4

z zi

0:0003 exp

=

8z 1 ; zi

0:33

(14)

where (m )w is the value of the vertical wavelength at the spectral peak, which was obtained from the empirical relation proposed by Caughey and Palmer (1979). Finally, the vertical eddy diffusivity for convective conditions can be obtained from Equations (7), (13) and (14),

Kzz w zi

=

1=3

0:15 "

1

exp

z 4 zi

8z 4=3 0:0003 exp : zi

(15)

In analogy with the previous case the vertical turbulent eddy diffusivity can be derived from Equations (13) and (14), using w given by the empirical relation proposed by Sorbjan (1989, p. 113),

z 2=3 z 2=3 z 2=3 z 2=3 2=3 = cwb 1 + cwt R 1 +D : zi zi zi zi Using D = 0, cwt = 0:5, cwb = 1 and R = 0:2 yields, z z 1=3 w = 1:08 1 w : zi zi w2 w2

(16)

(17)

Finally, the vertical eddy diffusivity can be written as,

Kzz w zi

=

z 1=3 z 1=3 0:22 1 8z zi z zi 1 exp 4 z 0:0003 exp z : i i

(18)

The turbulent vertical exchange coefficients in Equations (15) and (18) represent a non-local closure explicitly describing the fact that the energy-containing eddies are scaled by the convective velocity scale and by the height of the convective PBL. 3. The Dispersion Model The virtual height dispersion model (VHDM) is an operative short range model for evaluating ground level concentration from industrial sites. It applies the following

248


Gaussian formulation (for the calculation of the integrated crosswind concentration), in which the real source is replaced by a virtual source height function of the vertical profiles of wind and eddy diffusivity (Lupini and Tirabassi, 1981; Tirabassi and Rizza, 1994),

Cy (x; 0) =

Q pxu K

s s

exp

s s 4xKs =us

(19)

where Q is the source strength, us and Ks are the wind speed and the eddy diffusivity at the source height respectively, s and s are two virtual source heights defined as,

Z Hs u(z ) K 1=2 s s = dz; Kzz (z ) us z0

Z Hs u(z ) s = dz z0 us

(20)

where z0 is the roughness length and Hs is the effective release height. The model accepts both experimental and theoretical profiles for the eddy diffusivity Kzz (z ) and wind velocity u(z ), provided the integrals in Equation (20) exist. The lateral dispersion parameter is simulated by a purely Gaussian term so that the three dimensional model is given by,

C (x; y; 0) =

p Cy exp 2y

!

y2 : 2y2

(21)

4. Field Experiment The turbulent parameterization performances have been tested through VHDM using the well known dataset for Copenhagen (Gryning, 1981), according to the protocol agreed at the Manno workshop (Cuvelier, 1994). The field campaign took place in the suburbs of Copenhagen, known as Gladsaxe, in 1978 under moderately convective conditions. A SF6 tracer was released without buoyancy from a tower at a height of 115 m and collected at ground-level positions in up to three crosswind arcs of tracer sampling units. The sampling units were positioned at 2, 4, and 6 km from the release point. The site was mainly residential with a roughness length of 0.6 m. In order to evaluate the model performances in convective conditions, we selected from the original data set four experiments for which zi =L < 10, following the scaling approach proposed by Holtslag and Nieuwstadt (1986). Table I contains the meteorological data relative to the experiments utilized in our study.

249


Table I Copenhagen data set. Input meteorological parameters for selected experiments used by models Exp. No.

u (m s 1 )

1 3 7 8

0.36 0.38 0.64 0.69

L (m) 37 71 104 56

w (m s 1 )

zi (m)

1.7 1.1 2.1 2.1

1980 1120 1850 810

zi L

1

53.5 15.8 17.8 14.5

Figure 1. Scatter diagram between observed and predicted cross-wind integrated concentrations normalized to the emission rate Q. Model I is given by using Equation (15 ), Model II by using Equation (18) and Model III using Equation (22). Data between dotted lines correspond to the ratio Cy (pred)=Cy (obs) [0:5; 2].

2

5. Model Evaluation 5.1. VERTICAL DISPERSION The vertical eddy diffusivity formulations proposed have been compared with the Wyngaard and Brost (1984) eddy diffusivity model (based on their large eddy simulation data) utilizing the integrated crosswind concentration measured at Copenhagen. For convective conditions the above mentioned model gives,

Kzz = kw z (1 z=zi ):

(22)

250


Table II Statistical evaluation of model results for cross-wind integrated concentration. nmse is the normal mean squared error, cor is the correlation factor, fa2 is the fraction of predictions within a factor of two of observations, fb is the fractional bias and fs is the fractional standard deviation

Model I Model II Model III

nmse

cor

fa2

fb

fs

0.12 0.11 0.12

0.75 0.75 0.75

0.91 0.91 0.91

0.01 0.07 0.03

0.03 0.007 0.008

Figure 2. Plot of the nondimensional vertical eddy diffusivity profile according to Equation (15) (continuous line), Equation (18) (dotted line) and Equation (22) (dashed line) for each experiment considered.


251

Figure 3. Scatter plot between the observed and predicted ground level concentration using for lateral dispersion Equation (11) (a) and Equation (23) (b). Data between dotted lines correspond to ratio Cd (pred)=Cd (obs) [0:5; 2].

2

Table III Copenhagen convective experiments. Statistical evaluation of model results for ground level concentration along the arcs. nmse is the normal mean squared error, cor is the correlation factor, fa2 is the fraction of predictions within a factor of two of observations, fb is the fractional bias and fs is the fractional standard deviation

y -Degrazia y -Gryning

nmse

cor

fa2

fb

fs

0.39 0.74

0.92 0.84

0.63 0.55

0.14 0.23

0.21 0.24

Figure 1 shows the scatter diagram between the observed and predicted cross-wind integrated concentrations (Cy ) using three vertical eddy formulations. Model I is given by using Equation (15), Model II by using Equation (18) and Model III by Equation (22). The results given by simulations are quite satisfactory for all the models. This is also confirmed by the statistical indices contained in Table II. The performance measures obtained by using statistical evaluation procedure software described by Hanna (1989) are defined in the following way, nmse (normalized mean square) = (Co cor (correlation) = (Co

Co )(Cp

Cp )2 =Co Cp

Cp)o p,

fad2 = fraction of Co values within a factor two of corresponding Cp values, fb (fractional bias) = (Co

Cp)=(0:5(Co + Cp));

252


Table IV Copenhagen convective experiments. Statistical evaluation of model results for maximum arc-wise concentrations. Nmse is the normal mean squared error, cor is the correlation factor, fa2 is the fraction of predictions within a factor of two of observations, fb is the fractional bias and fs is the fractional standard deviation

y -Degrazia y -Gryning

nmse

cor

fa2

fb

fs

0.13 0.97

0.92 0.86

0.82 0.55

0.12 0.63

0.21 0.84

fs (fractional standard deviations) = 2(o

p )=(o + p );

where subscripts o and p refer to observed and predicted quantities, and an overbar indicates an average. It seems that the three different formulations do not influence VHDM results. This may be explained by the fact that the three vertical eddy diffusivity profiles do not differ substantially in the layer Hs =zi 0:15 that we consider for the integration of Equation (20). This can be seen in Figure 2, which illustrates a direct comparison of the vertical eddy profiles for each experiment. 5.2. LATERAL DISPERSION The lateral dispersion parameter (Equation (11)) has been compared against the one proposed by Gryning et al. (1987) to reproduce the ground level concentration and relative arcwise maxima. We compare the results given by Model I using, for y , first Equation (11) and then the model proposed by Gryning. The mentioned model gives,

y = v tfy (Tlv t)1=2 ;

(23)

where v is the standard deviation of the lateral wind fluctuations, fy is a function of the dimensionless travel time t=Tlv and Tlv is the Lagrangian time scale. The empirical formulation for fy for downwind distance less than 10,000 metres is given by Draxler (1976),

fy =

1 (1 + (t=2Tlv )0:5 )

where in accordance with Draxler the time scale can be taken as,

Tlv = 200 sec Tlv = 600 sec

(for ground level source, Hs =zi < 0:1), (for elevated source, Hs =zi > 0:1).

Figure 3 shows the scatter plot between the observed and predicted concentrations


253

for both simulations. Tables III and IV contain the statistical indices relative to the simulations of concentrations for each receptor along the arcs, and for the relative maxima respectively. 6. Conclusions For a CBL the validation of a new turbulent parameterization based on spectral properties and the Taylor statistical diffusion theory has been presented in this paper. The parameterization consists of non-local vertical eddy diffusivities (Equations (15 and 18)) and a lateral dispersion parameter (Equation (11)). These expressions are well-behaved and are in the form of a similarity profile, using the convective velocity scale w and the inversion height zi . All expressions utilize an empirical relation between stability and wavelength peak for the turbulent kinetic energy spectra obtained from experimental data. Therefore, the vertical eddy diffusivities and the lateral dispersion parameter are expressed in terms of the energy-containing eddies, which are responsible for turbulent transport in the CBL. As a test of the new approach we included this parameterization in an advanced dispersion model and, utilising data from field experiments, we compared the vertical eddy diffusivity formulation with the model suggested by Wyngaard and Brost (1984), and the lateral dispersion parameter with the Gryning et al. (1987) scheme. On analysing the results and relative statistics, we can see that the dispersion model reproduces adequately the experimental measurements with the parameterizations utilised, although better results for maxima concentrations are obtained with the parameterization proposed here. Therefore, the presented expressions (Equation (15)) and (Equation (18)) for the vertical eddy diffusivities, and (Equation (11)) for the lateral dispersion parameter may be suitable for applications in short- and long-range air quality dispersion models. References Batchelor, G. K.: 1949, ‘Diffusion in a Field of Homogeneous Turbulence, Eulerian Analysis’, Aust. J. Sci. Res. 2, 437–450. Boas M. L.: 1983, Mathematical Methods in the Physical Sciences, John Wiley & Sons, New York, 793 pp. Briggs, G. A.: 1985, ‘Analytical Parameterizations of Diffusion: The Convective Boundary Layer’, J. Atmos. Sci. 35, 1427–1440. Champagne, F. H., Friehe, C. A., Larve, J. C., and Wyngaard, J. C.: 1977, ‘Flux Measurements, Flux Estimation Techniques, and Fine Scale Turbulence Measurements in the Unstable Surface Layer Over Land’, J. Atmos. Sci. 34, 515–520. Caughey, S. J. and Palmer, S. G.: 1979, ‘Some Aspects of Turbulence Structure through the Depth of the Convective Boundary Layer’, Quart. J. Roy. Meteorol. Soc. 105, 811–827. Cuvelier, C. (Ed.): 1994, ‘Workshop on Intercomparison of Advanced Practical Short-Range Atmospheric Dispersion Models. Manno 1993’, Joint Research Centre (European Commission Institute for Safety Technology), EUR 15603 EN.

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Degrazia, G. A: 1989, ‘Anwendung von Ahnlichkeitsverfahren auf die turbulent Diffusion in der Konvekyiven und stabilen Grenzschicht’, Institut f. Meteorologie und Klimaforschung, Univ. Karlsruhe, Report 12. Degrazia, G. A. and Moraes, O. L. L.: 1992, ‘A Model for Eddy Diffusivity in a Stable Boundary Layer’, Boundary-Layer Meteorol. 58, 205–214. Degrazia, G. A., Moraes, O. L. L., and Oliveira, A. P.: 1996, ‘An Analytical Method to Evaluate Mixing Length Scales for the Planetary Boundary Layer’, J. Appl. Meteorol. 35, 974–977. Draxler, R.: 1976, ‘Determination of Atmospheric Diffusion Parameters’, Atmos. Environ. 10, 95– 105. Gryning, S. E.: 1981, ‘Elevated Source SF6 -Tracer Dispersion Experiments in the Copenhagen Area’, Risø National Laboratory, Roskilde Denmark, Report R-446, 187 pp. Gryning, S. E., Lyck, E., and Hedegaard, K.: 1978, ‘Short-Range Diffusion Experiments in Unstable Conditions over Inhomogeneous Terrain’, Tellus 30, 392–403. Gryning, S. E., Holtstlag, A. A. M., Irwin, J. S., and Sivertsen, B.: 1987, ‘Applied Dispersion Modelling Based on Meteorological Scaling Parameters’, Atmos. Environ. 21, 79–89. Hadfield, M. G.: 1994, ‘Passive Scalar Diffusion from Surface Sources in the Convective Boundary Layer’, Boundary-Layer Meteorol. 69, 417–448. Hanna, S. R.: 1989, ‘Confidence Limits for Air Quality Models, as Etimated by Bootstrap and Jackknife Resampling Methods’, Atmos. Environ. 23, 1385–1395. Højstrup, J.: 1982, ‘Velocity Spectra in the Unstable Boundary Layer’, J. Atmos. Sci. 39, 2239–2248. Holtslag, A. A. M. and Moeng, C. H.: 1991, ‘Eddy Diffusivity and Countergradient Transport in the Convective Atmospheric Boundary Layer’, J. Atmos. Sci. 48, 1690–1698. Holtstlag, A. A. M. and Nieuwstadt, F. T. M.: 1986, ‘Scaling the Atmospheric Boundary Layer’, Boundary-Layer Meteorol. 36, 201–209. Lupini, R. and Tirabassi, T.: 1981, ‘A Simple Analytical Approximation of Ground Level Concentration for Elevated Line Source’, J. Appl. Meteorol. 20, 565–570. Kaimal, J. C., Wyngaard, J. C., Haugen, D. A., Cote, O. R., Izumi, Y., Caughey, S. J., and Reading, C. J.: 1976, ‘Turbulence Structure in the Convective Boundary Layer’, J. Atmos. Sci. 33, 2152–2169. Nieuwstadt, F. T. M. and van Duuren, H.: 1979, ‘Dispersion Experiments with SF6 from the 213 m Height Meteorological Mast at Cabauw in the Netherlands’, in Proc. of the 4th Symposium on Turbulence Diffusion and Air Pollution, Reno, Nevada, 15–18 January, pp. 34–40. Olesen, H. R, Larsen, S. E., and Hoistrup, J.: 1984, ‘Modelling Velocity Spectra in the Lower Part of the Planetary Boundary Layer’, Boundary-Layer Meteorol. 29, 285–312. Panofsky, H. A. and Dutton, J. A.: 1984, Atmospheric Turbulence, John Wiley & Sons, New York, 397 pp. Pasquill, F. and Smith, F. B.: 1983, Atmospheric Diffusion, John Wiley & Sons, New York, 473 pp. Pleim, J. E. and Chang J. S.: 1992, ‘A Non-Local Closure Model for Vertical Mixing in the Convective Boundary Layer’, Atmos. Environ. 26, 965–981. Rizza, U., Mangia, C., and Tirabassi, T.: 1996, ‘Validation of an Operational Model with Copenhagen and Kincaid Dataset’, in J. G. Kretzschmar and G. Cosemans (eds.), in Proc. 4th Workshop on Harmonisation within Atmospheric Dispersion Modelling for Regulatory Purposes, 1, pp. 41–48. Sorbjan, Z.: 1986, ‘Local Similarity of Spectral and Cospectral Characteristics in the StableContinuous Boundary Layer’, Boundary-Layer Meteorol. 35, 257–275. Sorbjan, Z.: 1988, Structure of the Atmospheric Boundary Layer, Prentice Hall, New York, 300 pp. Siversten, B.: 1978, ‘Dispersion Parameters Determined from Measurements of Wind Fluctuation ( ), Temperature and Wind Profiles’, in Proc. 9th International Technical Meeting on Air Pollution Modeling and its Application, Toronto (CANADA). Tirabassi, T. and Rizza, U.: 1994, ‘Applied Dispersion Modelling for Ground Level Concentration from Elevated Sources’, Atmos. Environ. 28, 611–615. Taylor, G. I.: 1921, ‘Diffusion by Continuous Movements’, Proc. London Math. Soc. Ser. 2, 20, 196–212. Wandel, C. F. and Kofoed-Hansen, O.: 1962, ‘On the Eulerian-Lagrangian Transform in the Statistical Theory of Turbulence’, J. Geophys. Res. 67, 3089–3093. Wyngaard, J. C. and Brost R. A.: 1984, ‘Top-down and Bottom-up Diffusion of a Scalar in the Convective Boundary Layer’, J. Atmos. Sci. 41, 102–112.