Atmospheric Environment 34 (2000) 3575}3583 Turbulence parameterisation for PBL dispersion models in all stability conditions G.A. Degrazia!,*, D. Anfossi", J.C. Carvalho#, C. Mangia$, T. Tirabassi%, H.F. Campos Velho& !Departamento de Fn& sica, Universidade Federal de Santa Maria, Santa Maria, Brazil "CNR, Istituto di Cosmogeoxsica, Torino, Italy #Departamento de CieL ncias Atmosfe& ricas, Universidade de SaJ o Paulo, Instituto AstronoL mico e Geofn& sico, SaJ o Paulo, Brazil $CNR, ISIATA, Lecce, Italy %CNR, ISAO, Bologna, Italy &Instituto Nacional de Pesquisas Espaciais, LAC, SaJ o Jose& dos Campos, Brazil Received 2 September 1999; received in revised form 5 December 1999; accepted 13 January 2000 Abstract Accounting for the current knowledge of the planetary boundary layer (PBL) structure and characteristics, a new set of turbulence parameterisations to be used in atmospheric dispersion models has been derived. That is, expressions for the vertical pro"les of the Lagrangian length scale l and time scale ¹ and di!usion coe$cient K , i"u, v, w, are proposed. i i i The classical statistical di!usion theory, the observed spectral properties and observed characteristics of energy containing eddies are used to estimate these parameters. The results of this new method are shown to agree with previously determined parameterisations. In addition, these parameterisations give continuous values for the PBL at all elevations (z )z)h, z ) and all stability conditions from unstable to stable, where h and z are the turbulent heights in 0 i i stable or neutral and convective PBL, respectively, and ¸ is the Monin}Obukhov length. It is the aim of this work to present the general derivations of these expressions and to show how they compare to previous results. Finally, a validation of the present parameterisation applied in a Lagrangian particle model, will be shown. The Copenhagen data set is simulated. This data set is particularly suited for this validation, since most of the Copenhagen tracer experiments were performed in stability conditions that are the result of the relative combination of wind shear and buoyancy forces. As a consequence, a parameterisation scheme, able to deal contemporary with neutral and slightly convective condition, is to be preferred. ( 2000 Elsevier Science Ltd. All rights reserved. Keywords: Planetary boundary layer; Turbulence parameterisation; Statistical di!usion theory; Turbulent velocity spectra; Dispersion models 1. Introduction In the atmospheric dispersion models turbulence parameterisation is a key parameter. The reliability of each model strongly depends on the way turbulent parameters are calculated and related to the current understanding of the planetary boundary layer (PBL). Most of the turbulence parameterisation used in advanced dispersion * Corresponding author. models are based on PBL similarity theories (Hanna, 1982; Stull, 1988; Holtslag and Moeng, 1991; Kaimal and Finnigan, 1994; Sun, 1993; Rodean, 1994): the turbulence parameterisation is directly related to the basic physical parameters describing the turbulence state of the PBL. Throughout classical statistical di!usion theory (Batchelor, 1949), it is possible to relate turbulent parameters to spectral distribution of turbulent kinetic energy (TKE). Following this approach, Degrazia and Moraes (1992), Degrazia et al. (1997, 1998) developed a model for the turbulent spectra in both a purely shear and 1352-2310/00/$ - see front matter ( 2000 Elsevier Science Ltd. All rights reserved. PII: S 1 3 5 2 - 2 3 1 0 ( 0 0 ) 0 0 1 1 6 - 3 3576 G.A. Degrazia et al. / Atmospheric Environment 34 (2000) 3575}3583 buoyancy dominated PBL, and proposed a formulation for turbulent parameters to be used in air quality dispersion models in these extreme turbulent regimes. However, in many real cases, dispersion of atmospheric e%uents occurs in an intermediate PBL that is driven by both shear and convective forcing. The aim of this study is to use Taylor's di!usion theory together with a model for turbulent spectra in a shearbuoyancy driven PBL, in order to generate a turbulence parameterisation that gives continuous values for the PBL at all elevations and all stability conditions from unstable to stable. The very stable cases are excluded since the Monin}Obukhov scaling does not apply in these extreme conditions (Mahrt, 1999; Gryning, 1999). Expressions for the vertical pro"les of the Lagrangian length and decorrelation time scales and eddy di!usivity are proposed to be used in atmospheric dispersion models. Employing simulations with a stochastic Lagrangian model of turbulent di!usion, the new turbulent parameterisation is evaluated against ground-level concentration data from atmospheric dispersion experiments that were carried out in the Copenhagen area under neutral to unstable conditions (Gryning and Lyck, 1984, 1998) and is compared with Hanna (1982) parameterisation. 2. Analytical derivation of the Lagrangian length scale, decorrelation time scale and eddy di4usivity Part of this derivation was already proposed by Degrazia et al. (1998) and is resumed here for sake of clarity of presentation. According to Weil (1989), the dispersion rate of a particle ensemble in a turbulent #ow can be expressed, for large travel time (tA¹ ), as i A B P d x2 = i "p2 o (t) dt"p2¹ , i i i i dt 2 0 (1) where i"u, v, w, x2 is the variance of particle positions, i p2 corresponds to the Lagrangian variance of the ith i component of the turbulent wind "eld, o (t) is the nori malised Lagrangian autocorrelation function and ¹ is i the Lagrangian decorrelation time scale. Concerning the horizontal ¹ components, the larger scales due to the i meandering (scales greater than the spectral gap of Van der Hoven wind velocity spectrum (Stull, 1988)) are not included. Following Tennekes and Lumley (1972), the Lagrangian length scale l can be de"ned in terms of the Lagrani gian variance and decorrelation time scale, namely l "p ¹ . (2) i i i On the basis of Taylor's theory, Batchelor (1949) (see also Degrazia and Moraes, 1992) proposed the following relationships for the rate of dispersion for an ensemble of particles for any travel time t: A B P d x2 p2b = FE(n) sin(2pnt/b ) i " i i i i dn, (3) dt 2 2p n 0 where b is the ratio of the Lagrangian to the Eulerian i time scales, FE(n) is the Eulerian spectrum of energy i normalised by the velocity variance and n is the frequency. Wandel and Kofoed-Hansen (1962), Angell (1974), Pasquill (1974, p. 89) and Hanna (1981) have suggested the following expression for b : i ; b "d , (4) i p i where ; is the mean wind speed and d is a constant whose numerical value is given by a large number of both experimental and theoretical works. In this paper, we use the value d"0.55 estimated by Degrazia and Anfossi (1998). The "lter spectrum in Eq. (3) has its major passband around zero frequency. In this case, the eddy di!usivity for large times depends on the behaviour of the spectrum near the origin, so that the concept of "ltering applied to a one-dimensional turbulent velocity "eld allows to select the energy-containing eddies (Degrazia and Moraes, 1992; Degrazia et al., 1996, 1997). This means that the "lter chooses the characteristic frequency (nP0) describing these eddies. As a consequence, 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: A B d x2 p2b FE(0) i " i i i . dt 2 4 (5) Therefore, with this information we can construct and calculate various fundamental parameters (eddy diffusivities and Lagrangian length and time scales) associated to energy-containing eddies and that describe the turbulent transport process in the PBL. From Eqs. (1), (2) and (5), the Lagrangian length scale can be expressed as b p FE(0) l" i i i i 4 (6) that yields a Lagrangian decorrelation time scale given by b FE(0) l (7) ¹" i" i i . i p 4 i It is worthwhile to point out the bene"ts of using the parameterisation given by Eq. (5): Taylor's theory is valid for homogeneous turbulence only, whereas Eqs. (5)}(7), referring to the eddy di!usivity, Lagrangian length scale and Lagrangian decorrelation time scale, respectively, can be used in non-homogeneous turbulence as well, thus resulting in more general application. G.A. Degrazia et al. / Atmospheric Environment 34 (2000) 3575}3583 3. Turbulent velocity spectra in a shear/buoyancy driven PBL It is well known that turbulent dispersion in the PBL is generated by two di!erent processes: mechanical and thermal. The former is related to wind shear, and it is most e!ective close to the ground. The latter results from a buoyancy forcing mechanism and it is generally responsible for convective transport of momentum, heat or other scalars. These two forcing mechanisms produce a wide range of turbulent eddies and consequently a spectral distribution of turbulent kinetic energy (TKE) over a broad range of scales. The #ow patterns resulting from interactions among shear-buoyancy turbulent eddies are quite complex, and even if they fall between the extreme cases (purely mechanic or purely convective), they show structures which are not present in either the two limiting cases (Moeng and Sullivan, 1994). An hypothesis of linear combination of the two mechanism can be assumed only when there is statistical independence between their Fourier components. This happens when the energy containing wave-number ranges are well apart for the two spectra. However, it has been shown (H+jstrup, 1982; Berkowicz and Prahm, 1984; Moeng and Sullivan, 1994) that a simple TKE model, based on neglecting the interaction e!ects between shear and buoyancy, can be regarded as a good approximation for an intermediate PBL. Thus, assuming the hypothesis of superposition of the e!ects produced by two forcing mechanisms, we can write the dimensional Eulerian spectra as SE(n)"SE (n)#SE (n), i i" i4 (8) where the "rst term on the r.h.s. represents the buoyancy production, the second one is the mechanical component, the subscript b is for buoyancy, s for shear and i"u, v, w. Following Degrazia et al. (1998) the dimensional convective turbulence spectra in the PBL can be written as nSE (n) 1.06c f (t z/z )2@3 i" " i e i , w2 [( fH )#]5@3M1#1.5f/[( fH )#]N5@3 H .i .i (9a) where w is the convective velocity scale, f"nz/; is the H reduced frequency, ; is the mean wind speed in the convective PBL, z is the height above the surface, ( fH )# is .i the reduced frequency of the convective spectral peak, t "e z /w3 is the adimensional dissipation rate funce " i H tion, e is the buoyant rate of TKE dissipation and z is " i the convective boundary layer (CBL) height. The mechanical component of the dimensional spectrum is the same as the neutral limit of Degrazia and Moraes (1992): nSE (n) 1.5c f/2@3 i4 " i e . u2 [( fH )n`s]5@3M1#1.5f5@3/[( fH )n`s]5@3N H .i .i (9b) 3577 where u is the friction velocity, the dissipation rate H / "e kz/u3 is adimensionalized with surface layer scale 4 H ing parameters, e is the mechanical rate of TKE dissipa4 tion, i is the von Karman constant and ( fH )n`s is the .i reduced frequency of the neutral or stable spectral peak. For both Eqs. (9a) and (9b), c "a a (2pi)~2@3 and i i u a "0.5$0.05 and a "1, 4, 4 for u, v and w compou i 3 3 nents, respectively (Champagne et al., 1977; Sorbjan, 1989). By analytically integrating the Eulerian spectra given by Eqs. (9a) and (9b) over the whole frequency domain, one can obtain the buoyant and mechanical wind velocity variances P p2 " i" = 1.06c t2@3w2 (z/z )2@3 i e H i SE (n) dn" i" [( fH )#]2@3 0 .i (10a) and P = 2.32c /2@3u2 i e H (10b) SE (n) dn" i4 [( fH )#]2@3 0 .i and, moreover, the total wind velocity variance (sum of the two parts): p2 " i4 p2"a2w2 #b2u2 , (11) i i H i H where a and b are the buoyancy}force and shear}stress i i terms, respectively, to be determined. In all these derivations it has been assumed that the Lagrangian and Eulerian variances of the turbulent wind "eld are equal. This assumption is commonly made and is based upon the fact that the turbulent kinetic energy is the same for both approaches (Hanna, 1982, p. 177). By means of the above hypothesis concerning the statistical independence of convective and mechanical spectra, we can handle each spectrum individually, neglecting the interactions among shear}buoyancy turbulent eddies. As a consequence, each turbulent contribution to the normalised Eulerian spectra can be scaled with the respective variance. Thus, we can write SE SE (12) FE(n)" i" # i4 . i p2 p2 i4 i" Considering Eqs. (9a), (9b), (10a), (10b) and (12), the value of the normalised Eulerian energy spectrum at the origin can be given by z 0.64z FE(0)" # . i ;( fH )# ;( fH )n`s .i .i (13) 4. The turbulence parameterisation It is the aim of this section to derive a turbulent parameterisation giving continuous values for the PBL at all elevations (z )z)h, z ) and all stability condi0 i tions from unstable to stable. From Eqs. (4), (6) and (13) 3578 G.A. Degrazia et al. / Atmospheric Environment 34 (2000) 3575}3583 a Lagrangian length scale for all stability conditions and that gives continuous values for all the PBL elevations can be expressed as GC D H !¸ z 0.64 1@2 1 i # , (14) z !¸ ( fH )# ( fH )n`s i .i .i where !¸/z is an average stability parameter for the i convective PBL. The term in square brackets has been introduced in order to give a continuous transition from the neutral to the convective conditions. From Eqs. (10a), (10b) and (14) the eddy di!usivity K and Lagrangian i decorrelation time scale assume the following expressions, respectively: l "0.14z i G w ((!¸/z )z /¸)1@2(t z/z )1@3 i i e i H [( fH )#]4@3 .i /1@3u e H # (15) [( fH )n`s]4@3 . K "p l "0.14Jc z i ii i H and G z 0.14((!¸/z )z /!¸)1@2 l i i ¹" i" i p [( fH )#]2@3w (t#z/z )1@3 Jc i . i H e i i 0.059 # . (16) [( fH )n`s]2@3(/n`s)1@3u .i e H To construct turbulence parameterisations from Eqs. (14)}(16) for PBL dispersion models it is necessary to have expressions for w , u , t#, /n`s, ( fH )# and ( fH )n`s. H H e e .i .i For a convective PBL (t#)2@3+0.75 (Caughey and e Palmer, 1979; H+jstrup, 1982; Wilson, 1997), w " H (u ) (!z /i¸)1@3 and, recalling that ( f H )#"z/(j ) and H0 i .i .i that (j ) is the peak wavelength of the turbulent velocity .i spectra, ( f H)# expressions for i"u, v, w can be derived. .i According to Kaimal et al. (1976), Caughey (1982) and Degrazia and Anfossi (1998), H (j ) "(j ) "1.5z .u .v i and C (17) A B A BD z z (j ) "1.8z 1!exp !4 !0.0003exp 8 .w i z z i i so that A B A BD B A z f z ( fH )n`s"( f )n 1#a # #3.7 , .i . *4 iG K (21) where ( f )n is the frequency of the spectral peak in the . i4 surface for neutral conditions, G is the geostrophic wind speed and f "10~4 s~1 is the Coriolis parameter. Ac# cording to Olesen et al. (1984) and Sorbjan (1989), ( f )n "0.045, ( f )n "0.16 and ( f )n " 0.33. Fur. u4 . v4 . w4 thermore, a "500 (Hanna, 1968, 1981); as a consew quence of the Blackadar (1962) mixing length hypothesis (i.e., the asymptotic length scale l +G/f is limited = # by a constant value, equal for all the components) we found a "3889 and a "1094. Then, by writing u v f z/G"[ f z/(u ) ][(u ) /G] where (u ) /G is the neu# # H0 H0 H0 tral geostrophic drag coe$cient, Eq. (21) results A B A B f z (u ) z H 0 #3.7 . ( f H )n`s"( f )n 1#a # (22) .i . i4 i (u ) G K H0 By considering (u ) /G"0.03 (Tennekes, 1982; Hanna, H0 1982), Eq. (22) can be written as f z z ( f H )n`s"( f )n 1#0.03a # #3.7 . (23) .i . i4 i (u ) K H0 It is to point out that the meaning of continuous parameterisation for all stabilities in the present paper is the following. At the same time stable and unstable conditions cannot coexist. Unstable and neutral e!ects may be jointly take into account due to the contemporary presence of mechanical and convective turbulence. Stable and neutral e!ects may also jointly be accounted for because of the competition between wind-shear generated turbulence and stabilising e!ects of strati"cation. Therefore, in unstable-neutral conditions the last term of Eq. (21) is set equal to zero, whereas in the neutral-stable conditions the term 1/( f H )# is set to zero. .i (18) ( f H)#"z/(B z ) (19) .i i i with B " B " 1.5 and u v !4z 8z B "1.8 1!exp !0.0003 exp . (20) w z z i i For a neutral or stable PBL, /n`s can be written e (Sorbjan, 1989) as /n`s"/n(1#3.7z/K), where e /n"1.25 and K"¸(1!z/h)(1.5a1 ~a2 ) (Degrazia and e Moraes, 1992) is the local Monin}Obukhov length. For a shear-dominated stable boundary layer, a "1.5 and 1 a "1.0 (Nieuwstadt, 1984). Furthermore, for a neutral 2 C or stable PBL, u2 "(u2 ) (1!z/h)a1 in which a "1.7 1 H H0 for the neutral case (Wyngaard et al., 1974). Then, following Delage (1974), Stull (1988) and Sorbjan (1989), it is obtained 5. Comparison with previous parameterisations For horizontal homogeneity the PBL dynamics is driven mainly by the vertical turbulent transport. Therefore, the present analysis will focus on the vertical exchange turbulent parameters. In convective conditions (¸(0), the vertical Lagrangian length scale can be derived from Eqs. (14), (19) and (20) as A B C A BD A B z 1@2 4z l# "0.25z 0.01 i 1!exp ! 8 i !¸ z i 8z !0.0003 exp , z i (24) G.A. Degrazia et al. / Atmospheric Environment 34 (2000) 3575}3583 in which a typical value !¸/z "0.01 was used. In very i unstable conditions (50(!z /¸(300; Weil, 1988), i this value yields a Lagrangian length scale similar to the mixing length proposed by Sun and Chang (1986) and Sun (1993), whereas in weak to moderate convection the magnitude of l# is smaller. w When the shear production is the only source of the TKE (neutral boundary layer or shear-dominated stable boundary layer, ¸'0), the vertical Lagrangian length scale can be derived from Eqs. (14) and (21) by assuming a "500 and ( f H )n "0.33, as w . w4 1 n`s 1 13.63 1 # # . (25) " l K 0.27z 5.43]10~4Gf~1 w # In neutral conditions, the vertical Lagrangian length scale is proportional to z for small heights, while for zPR l is proportional to Gf~1 (Blackadar, 1962). w # For a shear-dominated stable boundary layer the numerical coe$cient in the r.h.s. of Eq. (25), 13.63, indicates that the main e!ect of thermal strati"cation is to decrease the size of the turbulence length scale and, consequently, to increase the molecular dissipation. Therefore, the small eddy size indicates that the structure of turbulence does not respond directly to the ground conditions, bringing to the #ow a z-less characteristic (Grant, 1992). It is worth noting that the value of the numerical coe$cient of Eq. (25), 13.63, is in excellent agreement with the value 14.0 proposed by Delage (1974). Furthermore, an expression like (25) was also proposed by Stull (1988, p. 208). In unstable conditions (¸(0), the vertical eddy di!usivity can be derived from Eqs. (10a), (14) and (15): AB A B C A BD A B z 1@2 4z K# "0.16w z 0.01 i 1!exp ! w H i !¸ z i 8z 4@3 !0.0003 exp . (26) z i The eddy di!usivity resulting from Eq. (26) yields a wellbehaved exchange coe$cient like that one given by Holtslag and Moeng (1991); this last is a non-local eddy di!usivity derived from countergradient terms by using large eddy simulation data (Degrazia et al., 1997). When the shear production term is the only input to the reservoir of turbulent energy (neutral boundary layer or shear stable boundary layer, ¸'0), the vertical eddy di!usivity can be derived from Eqs. (15), (10b) and (23), as Kn`s" 8 0.4(1#3.7z/K)1@3u z H . z 4@3 1#15f z/(u ) #3.7 # H0 K C D (27) In the neutral limit (¸PR), from Eq. (27) we obtain: 0.4(1!z/h)0.85(u ) z H0 . Kn " 8 [1#15f z/(u ) ]4@3 # H0 (28) 3579 In the highly stable limit (z/¸PR), eddy vertical motion is strongly limited by the positive strati"cation and Eq. (27) results 0.4(1!z/h)3@4(u ) z H0 K4 " w 1#3.7z/¸(1!z/h)5@4 (29) and so we expect eddy sizes to be limited entirely by the stability and not by the distance from the surface. This is referred to as local-height-independent (z-less) scaling and will occur for ¸)z)h. In this case, Eq. (29) yields K4 "0.11(u ) ¸ and eddies will scale with ¸ rather than w H0 with z as in the neutral case. At larger values of z/¸ and z/h local similarity concepts are needed and Eq. (29) can be written as A B z 2 K4 "0.11(u ) ¸ 1! , w H0 h (30) where now z is large enough for surface #uxes to be locally irrelevant. In convective conditions (¸(0), according to Eqs. (16), (18) and (19), the following expression for the vertical decorrelation time scale can be obtained: A B C A BD A B z z 4z 1@2 ¹# "0.39 i 0.01 i 1!exp ! w w !¸ z H i 8z 2@3 . (31) !0.0003 exp z i For very unstable conditions (z /¸+100), in the central i region of the CBL, Eq. (31) can be approximated by z ¹# +0.33 i . (32) w w H It is worth noting that Hanna (1981) found, from Eulerian and Lagrangian observations, that for elevated heights in the CBL, the vertical Lagrangian decorrelation time scale was given by z (33) ¹# +0.17 i . w p w Following Hicks (1985) and Weil (1988), for elevated heights in the CBL (z'0.1z ), the turbulence structure i can be idealised as vertically homogeneous with p +0.59w . As a consequence, the last equation can be 8 H written as z ¹# :0.29 i (34) 8 w H showing that the present derivation (Eq. (32)) is in good agreement with Hanna's result. When the buoyancy term is zero or negative, the shear production term is the only input to the reservoir of turbulent energy. In this case, the vertical decorrelation 3580 G.A. Degrazia et al. / Atmospheric Environment 34 (2000) 3575}3583 time scale can be derived from Eqs. (16) and (23) as 0.19z ¹n`s" . w u [1#15f z/(u ) #3.7z/K]2@3(1#3.7z/K)1@3 H # H0 (35) In the neutral limit (¸PR), we obtain from the previous equation 0.19z ¹n " w (u ) [1#15f z/(u ) ]2@3(1!z/h)0.85 H0 # H0 that can be also written as (36) 0.27z/p 8 ¹n " . (37) w [1#15f z/(u ) ]2@3 # H0 We notice that Eq. (37) has the same form as that one given by Hanna (1982). In the highly stable limit (z/¸PR), eddy vertical motion is strongly limited by the positive strati"cation and Eq. (35) results: 0.19z . (38) ¹4 " w (u ) [1#3.7z/K](1!z/h)3@4 H0 Eddies will scale with ¸ rather than with z as in the neutral case and will occur for ¸)z@h. In this case, Eq. (38) yields ¹4 "0.051¸/(u ) . w H0 At larger values of z/¸ and z/h, local similarity holds and the decorrelation time scale can be written as ¹4 "0.051 w ¸(1!z/h)1@2 . (u ) H0 (39) 6. Test of the proposal parameterisation with the Copenhagen tracer experiment In Section 4, we have derived the expressions (Eqs. (14)}(16) which allow the parameterisation of l , K and i i ¹ for the PBL at all elevations (z )z)h, z ) and all i 0 i stability conditions from unstable to stable. In Section 5, we have shown how this new parameterisation correctly recovers important results previously obtained by other researchers. It is the aim of this section to test our parameterisation in a practical application and to show how it works. The actual atmospheric conditions only very rarely are such as to be correctly represented by purely convective or stable or neutral situations. In the majority of cases the stability conditions and, consequently, the dispersive conditions, are a combination of wind shear and buoyancy forces. One of the main peculiarities of the present parameterisation is to be able to deal with such situations. As a consequence, to test our new parameterisation, we decided to simulate the Copenhagen tracer dispersion experiment (Gryning and Lyck, 1984, 1998). Copenhagen data set is particularly suited for this validation since most of these tracer experiments were performed in stability conditions that are the result of the relative combination of wind shear and buoyancy forces. Besides ground-level concentrations (g.l.c.s) at 3 arcs in the range 2000}6000 m from the source, located at 115 m above the ground, Copenhagen data set includes wind speed at 10 and 115 m, p and p at 115 m, ¸, (u ) and v w H0 z . All these data are on an hourly basis. i The tracer dispersion was simulated by the Lagrangian particle model LAMBDA (Ferrero et al., 1995; Ferrero and Anfossi, 1998a, b). For a detailed presentation and discussion of LAMBDA we refer to the quoted papers. Here we shall merely repeat the key points. The actual version of LAMBDA is based on the generalised Langevin equation (Thomson, 1987), whose coe$cients are obtained by solving the Fokker}Planck equation, and satis"es the well-mixed condition. It can use as input higher-order moments of the atmospheric probability density function (PDF) of wind velocity. In the present application, LAMBDA used a Gaussian PDF on the horizontal plane and a Gram}Charlier PDF, truncated to the fourth order, in the vertical. As far as the meteorological and turbulence input to LAMBDA is concerned, we proceeded as follows. Wind speed at 10 and 115 m were used to calculate the coe$cient for the exponential wind pro"le. p , p and ¹ v w i vertical pro"les were prescribed both with the present parameterisation and with the Hanna's (1982) one (neutral conditions). This last parameterisation was chosen for the comparison because it is well known and widely used in the specialised literature. The third- and fourth-order momentum of the vertical velocity (w3) and (w4) were prescribed according to Weil (1990) and Ferrero and Anfossi (1998a), respectively. p and p pro"les, either obtained with Hanna' scheme v w or with the present scheme, were multiplied by a proper coe$cient (di!erent for the two schemes) in order to exactly match their observed value at 115 m. LAMBDA simulation results are presented in Tables 1}3. These results are based on a model evaluation performed on all the pairs of observed and predicted values, grouping all together the various tracer experiments performed. The statistical parameters considered are: mean value (MEAN), standard deviation (S.D.), correlation coe$cient (R), percentage of model predictions falling within a factor of two (FA2), fractional bias (FB) and normalised mean square error of the distribution (NMSE). Table 1 shows the results relative to the concentration maxima on the arcs normalised by the emission rate (ARCMAX/Q). We can see that the results of the present parameterisation are signi"cantly better than those obtained with Hanna scheme, except for the correlation coe$cient. Table 2 refers to the cross-wind integrated G.A. Degrazia et al. / Atmospheric Environment 34 (2000) 3575}3583 3581 Table 1 Statistical indexes for ARCMAX/Q Run Mean S.D. R FA2 FB NMSE Observed Hanna (1982) Present parameterisation 632.7 946.9 622.9 460.3 548.1 420.1 * 0.92 0.90 * 74 96 * 0.40 !0.02 * 0.24 0.10 Run Mean S.D. R FA2 FB NMSE Observed Hanna (1982) Present parameterisation 448.7 424.9 507.1 239.3 175.7 289.9 * 0.78 0.60 * 91 100 * !0.05 !0.15 * 0.12 0.08 Run Mean S.D. R FA2 FB NMSE Observed Hanna (1982) Present parameterisation 382.0 174.3 400.1 159.8 84.8 161.7 * 0.90 0.70 * 65 100 * !0.62 0.04 * 0.71 0.10 Table 2 As in Table 1 but for C /Q y Table 3 As in Table 1 but for p y concentrations at the various arcs normalised by the emission rate (C /Q). In this case the quality of the results y obtained with the two parameterisations is comparable. Finally, Table 3 illustrates the results relative to the cross-wind concentration standard deviations (p ). ExY cluding the correlation coe$cient, it can be seen that the proposed parameterisation outperforms the Hanna scheme. In conclusion it can be said that the general agreement between observed and predicted tracer g.l.c.s is good and that the present turbulence parameterisation generally performs better than the classical one. 7. Conclusions In this paper we presented a turbulence parameterisation for dispersion models in all stability conditions, excluding the very stable conditions. The present turbulence parameterisation is based on Taylor's statistical di!usion theory, in which the shear buoyancy PBL spectra are modelled by means of a linear combination of the convective and mechanical turbulent energy. Such an approach is strictly valid when the mechanical and convective wavelength peaks are well apart, so that the concept of localness can be applied. This means that in the turbulent spectrum, the energy #ux predominantly involves scales of comparable size. The parameterisation gives continuous values for the PBL at all elevations and all stability conditions from unstable to stable conditions. Furthermore, the derived turbulent parameters are well-behaved and presented in form of similarity pro"les, using the velocity (u , w ) and characteristic length H H scales (z , h, ¸, K). The parameterisation adequately rei produces important results previously obtained by other approaches. The formulation employs the empirical relationship between stability and wavelength peak for the turbulent kinetic energy spectra. Therefore, the turbulence parameterisation is expressed in terms of the energy-containing eddies, which are responsible for the main turbulent transport process in the PBL. The nature of the subject is not suited for a direct check between experiment and model. However, by using a Lagrangian particle model and a data set of di!usion experiments performed in a PBL generated both by thermal and 3582 G.A. Degrazia et al. / Atmospheric Environment 34 (2000) 3575}3583 mechanical forcing, the turbulent velocity variances and the Lagrangian decorrelation time scales derived in the present work were compared to the ones obtained by Hanna (1982) scheme. This last scheme was chosen since it is frequently used by the scienti"c community. Results obtained are quite encouraging. Therefore, the new turbulent parameters may be suitable for applications in regulatory air pollution modelling. Acknowledgements This work was partially supported by CNPq, FAPERGS and CAPES. References Angell, J.K., 1974. Lagrangian}Eulerian time-scale relationship estimated from constant volume balloon #ights past a tall tower, turbulent di!usion in environmental pollution. 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