Physica A 387 (2008) 4376–4386www.elsevier.com/locate/physa

Turbulent statistical characteristics associated to the north windphenomenon in southern Brazil with application

to turbulent diffusion

Maria Cristina Andres Arbage, Gervasio Annes Degrazia∗, Guilherme Sausen Welter,Debora Regina Roberti, Otavio Costa Acevedo, Osvaldo Luiz Leal de Moraes,

Simone Teleginski Ferraz, Andrea Ucker Timm, Virnei Silva Moreira

Universidade Federal de Santa Maria, Departamento de Fısica 97105-900, Santa Maria, RS, Brazil

Received 21 February 2008Available online 4 March 2008

Abstract

A parameterization for the transport processes in a shear driven planetary boundary layer (PBL) has been establishedemploying turbulent statistical quantities measured during the north wind phenomenon in southern Brazil. Therefore, observedone-dimensional turbulent energy spectra are compared with a spectral model based on the Kolmogorov arguments. The goodagreement obtained from this comparison leads to well defined formulations for the turbulent velocity variance, local decorrelationtime scale and eddy diffusivity. Furthermore, for vertical regions in which the wind shear forcing is relevant, the eddy diffusivityderived from the north wind data presents a similar profile to those obtained from the non-extensive statistical mechanics theory.Finally, a validation for the present parameterization has been accomplished, using a Lagrangian stochastic dispersion model. ThePrairie Grass data set, which presents high mean wind speed, is simulated. The analysis developed in this study shows that theturbulence parameterization constructed from wind data for north wind flow cases is able to describe the diffusion in a high windspeed, shear-dominated PBL.c© 2008 Elsevier B.V. All rights reserved.

Keywords: Shear driven turbulence; Turbulent parameterization; Lagrangian stochastic dispersion model

1. Introduction

Frequently, in a regional scale at the center of Rio Grande do Sul state (southern Brazil), a phenomenon knownas north wind flow (NWF) occurs. It is characterized by intense warm winds from the north, presenting low relativehumidity, associated to meteorological frontal systems. During this phenomenon, occurring most frequently in wintermonths (June, July and August), a system of low pressure is observed in the atmospheric circulation with low levels(1000 mb) to the South of the South America and high pressure in the extratropical Atlantic. The relative humidityis lower than the average with anomalies around −10% in the Rio Grande do Sul, and winds from the north more

∗ Corresponding author. Tel.: +55 5532208616; fax: +55 5532208616.E-mail address: degrazia@ccne.ufsm.br (G.A. Degrazia).

0378-4371/$ - see front matter c© 2008 Elsevier B.V. All rights reserved.doi:10.1016/j.physa.2008.02.068

M.C.A. Arbage et al. / Physica A 387 (2008) 4376–4386 4377

Fig. 1. Characteristic values of the mean wind speed in m s−1 for a North Wind case; (b) characteristic wind direction in degrees for a North Windcase.

Fig. 2. Turbulent heat flux in W m−2.

intense than the average in the same region. The wind anomaly shows that this north flow is accelerated by the actionof the local topography, which reinforces its magnitude, transporting drier air from northern regions. Figs. 1a, b show,respectively, the magnitude (ms−1) and direction (degrees) of the NWF for a typical day in Paraıso do Sul city (centerof Rio Grande do Sul). It can be seen that the magnitude of the wind at certain moment changes abruptly from lowto high values, characterizing the presence of this particular flow. Furthermore, observation of Fig. 1b confirms thedominant north direction associated with this flow. For the same case exhibited in Fig. 1a and 1b, Fig. 2 shows theturbulent sensible heat flux (W m−2). From this figure it can be seen that there is a negative sensible heat flux occurringnear noon. This anomalous behavior associated with the heat flux shows that dry and warm air has been transportedby this intense NWF and has created a situation, in which the near surface temperature is higher than those in thesoil. This sensible heat flux pattern identifies a stable boundary layer, which usually occurs at night. However, inthe presence of the NWF, an unexpected situation arises, during which a negative turbulent heat flux is observed atdaytime.

The aim of this study is to use Taylor’s statistical diffusion theory together with a model for the turbulent spectrain a shear driven planetary boundary layer (PBL), in order to evaluate the turbulent statistical parameters associated tothe NWF occurring in the central region of the Rio Grande do Sul state. Relevant micrometeorological quantities, suchas the friction velocity, turbulence dissipation rate and the spectral peak frequencies were measured and employed tocalculate the turbulent velocity variances and the decorrelation time scales for the north wind occurrences. Theseturbulent parameters derived from the north wind were introduced into a Lagrangian stochastic dispersion model to

4378 M.C.A. Arbage et al. / Physica A 387 (2008) 4376–4386

Table 1

Mean quantities (U, T ) and the turbulent statistical parameter (u∗0, w′θ ′, L) for the 7 NWF cases

U (m s−1) T (◦C) u∗0 (m s−1) w′θ ′ (W m−2) L (m)

7.24 24 0.61 −64.09 197.25

simulate the ground level concentrations measured in the Prairie-Grass classical experiment under windy conditions.The simulations accomplished in this study were generated by the Lagrangian Stochastic particle model LAMBDA.This model is based on the Langevin equation, whose parameters are derived by solving the Fokker–Planck equation.

2. Experimental micrometeorological data associated to the NWF events

In this section micrometeorological statistical parameters associated to the NWF phenomenon are obtained andanalyzed. Data were collected at a continuously measuring site, at Paraıso do Sul (S 29◦44

′

39.6′′

, W 53◦09′

59.8′′

),in southern Brazil. The micrometeorological tower is a part of CT-HIDRO project, a Brazilian-wide study with thepurpose of describing the surface conditions for distinct ecosystems in the country. The tower is located at a flat site,with a hill to the north. Turbulence is measured at the 10 m height by a Campbell 3-D sonic anemometer employinga sample rate of 16 Hz. Other measurements include the radiative energy budget components, soil temperature andmoisture content at five levels, as well as slow response wind, temperature, relative humidity and pressure.

The data used in this study consist of 7 NWF cases from May, July and August, 2004. Each case has a durationperiod of approximately 4 days. Evolutions of the micrometeorological parameters that need the estimation ofturbulence statistics were determined using 30 min data windows for the mean removal [1]. The windows are centeredat a given data point which advances with 3 min time steps for each series. Before the analyses the individual windvectors were rotated into the mean wind direction in such a way that v = w = 0. Then, the mean values of wind speedand temperature, surface layer turbulent statistics (u∗0, w′θ ′, L) and standard deviations of the turbulent velocities(σi (i = u, v, w)) were calculated, where u∗0 is the surface friction velocity, w′θ ′ is the surface sensible turbulent heatflux and L is the Obukhov length scale. A data quality analysis was applied to detect any spikes, kinks or missingportions in the data.

For characterizing the surface layer structure, turbulent fluxes of heat and momentum were estimated using theeddy correlation technique. The scaling of velocity (surface friction velocity) has been evaluated, at a height of 10 m,from the expression

u∗0 = (w′u′2+ w′v′

2)1/4 (1)

where w′u′ and w′v′ are the surface turbulent momentum fluxes in the direction of the u and v components of thewind vector velocity, respectively. Another important surface turbulent parameter is the Obukhov length scale, whichwas obtained for each test run using

L = −ρcpu3

∗0

κ(g/T0)w′θ ′(2)

where κ is the von-Karman constant, g/T0 is the buoyancy parameter (g is the acceleration due to gravity and T0 isthe mean surface temperature), cp is the specific heat at constant pressure and ρ is the air density).

Table 1 shows the mean wind speed U and temperature T and the turbulent statistical parameter (u∗0, w′θ ′, L)for the 7 NWF cases. These magnitudes represent average values, calculated considering all events of NWF withmean wind speed greater than 6.0 ms−1. The values of the distinct parameters in Table 1 characterize a shear drivendominated PBL (neutral stability).

3. Observed turbulent velocity spectra for the NWF cases

In this section, turbulent spectral curves for the 7 NWF cases, whose mean values of the distinct variables aredescribed in Table 1, are presented within the framework of surface layer similarity. The employed data set to generatethe spectral curves was comprised of 48 neutral runs. Each run consisted 215

= 32,768 points (≈28 min). For spectrum

M.C.A. Arbage et al. / Physica A 387 (2008) 4376–4386 4379

Fig. 3. Spectrum of the longitudinal turbulent velocity of the NWF normalized by the mechanical rate of TKE dissipation and for the frictionvelocity. The dashed line represents the spectral model as given by Eq. (4) and the crosses represent the observed spectral data.

analysis, averaging times of 60–90 min are usually required [2], which often conflicted with non-stationarity caused bydiurnal variations. The spectra obtained in this study with the above record lengths have clear positions for the energypeak and roll-off on the low-frequency side of the spectrum. These are clear indications of an adequate averaging timeand a reasonably good approach to stationarity.

The spectra were computed using the Fast-Fourier technique and following the steps suggested in Stull [3]. Thespectral estimates are block-averaged over 22 non-overlapping frequency bands [4]. The u (longitudinal), v (lateral)and w (vertical) spectra were normalized by u2

∗0φ2/3ε (φε is the dimensionless dissipation rate of turbulent kinetic

energy). This procedure to normalize the spectral curves has been used in the literature [2,5]. The dimensionlessdissipation rate is defined as

φε =κzε

u3∗0

(3)

where ε is the mean turbulent kinetic energy dissipation per unit time per unit mass of fluid, with the order ofmagnitude of ε determined only by those quantities that characterize the energy-containing eddies.

Following this framework for setting spectral models and the observed turbulent energy spectra in the surfacelayer [6], when neutral spectra are plotted against the dimensionless frequency f = nz/U (n being the cyclicfrequency, U be mean wind speed and z the observation height) on a log–log scale, they are characterized by a low-frequency slope of +1, and by a high-frequency slope of −2/3 ( f −2/3 Kolmogorov’s law for the inertial subrange).

Figs. 3–5 present, respectively, the normalized turbulent spectra for the u, v, w velocity components. In thesefigures, crosses represent the observed spectra while the continuous lines are obtained from the following expressionderived by Degrazia et al. [7] and that represents a model to describe measured turbulent spectra in a shear driven PBL

nSi

u2∗

=1.5ciφ

2/3ε f(

1 +1.5 f 5/3

[( fm )i ]5/3

) [( fm)i

]5/3(4)

where i = u, v, w, u2∗ = u2

∗0 (1 − z/h)1.7 is the local friction velocity for a neutral PBL [7], ( fm)i is the non-dimensional frequency of the neutral spectral peak and ci = αiαu (2πκ)−2/3, αu = 0.5 ± 0.05, and αi = 1, 4/3, 4/3for u, v and w components, respectively [8,9].

It can be seen that all the spectra at the high-frequency follow approximately Kolmogorov’s −2/3 slope.Furthermore, the model as given by (4) fits well the observed spectra in the frequency range containing the mostenergy.

The peak frequencies ( fm)u , ( fm)v and ( fm)w are important for studies of scalar turbulent transport in the PBLand merit close analysis. For the neutral case, these peak frequencies describing the spatial and temporal characteristic

4380 M.C.A. Arbage et al. / Physica A 387 (2008) 4376–4386

Fig. 4. Spectrum of the lateral turbulent velocity of the NWF normalized by the mechanical rate of TKE dissipation and for the friction velocity.The dashed line represents the spectral model as given by Eq. (4) and the crosses represent the observed spectral data.

Fig. 5. Spectrum of the vertical turbulent velocity of the NWF normalized by the mechanical rate of TKE dissipation and for the friction velocity.The dashed line represents the spectral model as given by Eq. (4) and the crosses represent the observed spectral data.

of the energy-containing eddies, are expressed by the following formulation [3,7,10–12]

( fm)i = ( fm)0i

(1 + 0.03ai

fcz

(u∗)0

)(5)

where ( fm)0i is the frequency of the spectral peak in the surface, fc = 10−4 s−1 is the Coriolis parameter, withau = 500, av = 1094, and aw = 3889. For the NWF events, measured in this study, the peak frequencies inthe surface occurred in the range 0.015 < ( fm)0u < 0.060; 0.085 < ( fm)0v < 0.30; 0.19 < ( fm)0w < 0.70with mean values of ( fm)0u = 0.040, ( fm)0v = 0.10 and ( fm)0w = 0.33. These estimated values of the spectralpeak frequencies are in fair agreement with the values obtained from the experimental data measured in Kansas andMinnesota experiments [6].

The simplest measure of turbulent velocity fluctuation levels are the variances or mean-square fluctuations σ 2u ,

σ 2v and σ 2

w. These parameters represent quantities that allow the turbulence intensity estimation. Therefore, they areimportant in determining the longitudinal, lateral and vertical dispersion of contaminants. From the experimentalspectral data, the following values for the turbulent velocity variances were calculated: σu = 2.67u∗0, σv = 1.95u∗0and σw = 1.39u∗0. Results for these quantities published in Panofsky and Dutton [13], and applied in situations ofstability in which mechanical purely turbulence prevails, suggest the following values: σu = 2.40u∗0, σv = 1.90u∗0and σw = 1.30u∗0. This comparison shows that the turbulent velocity variances, measured during the NWFphenomenon, are in good agreement with the values found in the literature when mechanical turbulence is dominant.

M.C.A. Arbage et al. / Physica A 387 (2008) 4376–4386 4381

It is important to note that to calculate the above velocity variances from the observed turbulent spectra, values of thedimensionless dissipation rate, φε, are necessary. Thusly, the dissipation rate of turbulent kinetic energy for each testrun has been obtained from the inertial subrange of the one-dimensional spectra of vertical velocity fluctuations usingKolmogorov’s law [14].

At neutral stability it is expected that φε approaches unity due to the balance between shear production and viscousdissipation of turbulence in the absence of any buoyant production and transport. Our result φε(z/L → 0) = 1.1 is ingood agreement with Kansas results [15] as well with the theoretical predictions [9].

4. Determination of the Lagrangian decorrelation time scale for the NWF phenomenon

Based on the model for the spectra of turbulent kinetic energy and Taylor statistical diffusion theory [16] derived aformulation for the Lagrangian decorrelation time scales and wind velocity variances for fully developed turbulence.

For a non-homogeneous turbulence these decorrelation time scales can be expressed as:

TL i =βi F E

i (n → 0)

4(6)

where βi = 0.55 Uσi

[17–20] is defined as the ratio of the Lagrangian to the Eulerian decorrelation time scale and

F Ei (n → 0) =

Si (n→0)

σ 2i

represents the spectra in which the high frequencies were filtered.

On the other hand, analytical integration of Eq. (4) over the whole frequency domain leads to the following turbulentvelocity variance

σ 2i =

2.32ciφ2/3ε u2

∗

( fm)2/3i

(7)

that is employed to normalize the spectrum. The normalized spectrum can now be written as follows:

F Ei (n → 0) =

Si (n → 0)

σ 2i

=0.64z

( fm)iU. (8)

In neutral conditions, the local decorrelation time scales can be derived from Eqs. (6) and (8) as

TL i = 0.088z

σi ( fm)i. (9)

Therefore, Eqs. (7) and (9) can be used in a Lagrangian stochastic dispersion model to reproduce observedconcentrations of contaminants released in a turbulent field generated by NWF.

As a test for the present model we calculate from Eq. (7) the turbulent velocity variances for the NWF. Therefore,we substitute in Eq. (7) ( fm)0u = 0.04, ( fm)0v = 0.10, ( fm)0w = 0.33 and φε(z/L → 0) = 1.1, to obtain,respectively σu = 2.46u∗, σv = 2.0u∗ and σw = 1.38u∗. Comparing the values of the turbulent velocity standarddeviations obtained from Eq. (7) employing the observed turbulent parameters for the NWF cases with those extractedfrom the experimental spectral curves (Section 3), it can be seen that the measured turbulent velocity standarddeviations at the surface can be well reproduced by the Eq. (7) derived from the spectral model given by Eq. (4).Therefore, based on this analysis, the typical velocities, length scales and time scales associated with the turbulentfield generated by the NWF can be obtained using the formulations given by Eqs. (4), (7) and (9). Furthermore, fromEqs. (5), (7) and (9), using ( fm)0w = 0.33, u2

∗ = u2∗0 (1 − z/h)1.7 and φε(z/L → 0) = 1.1, we can derive a vertical

eddy diffusivity for a shear-dominated PBL. This eddy diffusivity describes the magnitude of the turbulent transportand can be written as

Kz

u∗0h∼=

0.4z/h(1 − z/h)0.85[1 + 15 fcz

u∗0

]4/3 (10)

where h is the depth of the neutral PBL which is given by [13,21]

h =0.2u∗0

| fc|. (11)

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Fig. 6. Vertical eddy diffusivity profile calculated from Eq. (12).

Therefore, the substitution of Eq. (11) into Eq. (10) results in

Kz

u∗0h∼=

0.4z/h(1 − z/h)0.85

[1 + 3.0z/h]4/3 . (12)

Fig. 6 shows the vertical eddy diffusivity profile calculated from Eq. (12). At this point, it is important to note thatEq. (12) provides a Kz profile in lower and central regions of the neutral PBL with the same structure as proposedby Campos Velho et al. [22] which is based on non-extensive statistical mechanics [23]. Futhermore, the neutraleddy diffusivity provided by Eq. (12) presents a profile and magnitudes that agree with the formulation proposed byGarrat [21].

5. Test of the proposed parameterization with the Prairie Grass tracer experiment

In this section, we test the values of φε and ( fm)0i measured from the NWF events in an application to aircontaminant dispersion. Therefore, a Lagrangian stochastic dispersion model employing the measured values of( fm)0u = 0.04, ( fm)0v = 0.10, ( fm)0w = 0.33 and φε = 1.1 in Eqs. (7) and (9) has been utilized to simulatethe measured concentration data. Lagrangian stochastic particle models are powerful computational tools for theinvestigation of the atmospheric dispersion process [24]. In these models, the fluid particle displacements are producedby stochastic velocities and the movement evolution of a particle can be considered a Markov process [25] in whichpast and future are statistically independent when the present is known. This method is based on Langevin equation,which is derived from the hypothesis that the velocity is given by the combination between a deterministic term anda stochastic term [26]. Each fluid particle moves taking into account the transport due to the mean wind velocity andthe turbulent fluctuations of the wind velocity components. From the spatial distribution of the particles it is possibleto determine the contaminant concentrations. The implementation of the Lagrangian stochastic dispersion modelpermits the taking into account of complex situations such as the presence of different topographies [27], low windvelocities [28] and spatial and temporal variations of the meteorological fields. These models can simulate sourcesof different forms and dimensions, as well as continuous, variable in time and instantaneous sources. For a betterunderstanding of these models we suggest the important review paper by Ref. [29]. This work presents historical andbasic description about Lagrangian stochastic dispersion models in the PBL.

The actual atmospheric conditions associated with the NWF cases are characterized by intense mean wind speedsand, consequently, for these situations, wind shear forces generate a dominant mechanical turbulence. Therefore, oneof the main peculiarities of the present turbulent parameterization (obtained from the NWF cases) is to be able todeal with turbulent dispersion in neutral situations. From these arguments we decided to simulate the classical PrairieGrass tracer dispersion experiments [30] that occurred under high mean wind speed conditions.

The Prairie Grass experiment was performed in O’Neill, Nebraska, in 1956 and it is described in detail by Ref. [30].The contaminant (SO2) was emitted without buoyancy at a height of 0.5 m and it was measured by samplers at a height

M.C.A. Arbage et al. / Physica A 387 (2008) 4376–4386 4383

Table 2Meteorological parameters and ground-level cross-wind-integrated concentration (Cy ) measured during the Prairie Grass experiment

Run h (m) u∗0 (m s−1) U10m (m s−1) Q (gs−1) 50 m (gm−2) 100 m (gm−2) 200 m (gm−2) 400 m (gm−2) 800 m (gm−2)

5 780 0.40 7.0 78 3.30 1.80 0.81 0.29 0.0923.40 1.82 0.84 0.39 0.19

9 550 0.48 8.4 92 3.70 2.20 1.00 0.41 0.133.33 1.79 0.88 0.40 0.18

19 650 0.41 7.2 102 4.50 2.20 0.86 0.27 0.0584.24 2.19 1.07 0.50 0.28

20 710 0.63 11.3 102 3.40 1.80 0.85 0.34 0.132.40 1.28 0.72 0.34 0.17

26 900 0.45 7.8 98 3.90 2.20 1.04 0.39 0.1273.82 2.04 1.09 0.51 0.23

27 1280 0.44 7.6 99 4.30 2.30 1.16 0.46 0.1763.85 2.13 1.00 0.50 0.25

30 1560 0.48 8.5 98 4.20 2.30 1.11 0.40 0.103.57 1.93 0.90 0.41 0.24

43 600 0.40 6.1 99 5.00 2.40 1.09 0.37 0.124.81 2.30 1.22 0.62 0.29

44 1450 0.42 7.2 101 4.50 2.30 1.09 0.43 0.144.17 2.24 1.10 0.51 0.27

49 550 0.47 8.0 102 4.30 2.40 1.16 0.45 0.153.68 1.86 1.03 0.48 0.25

50 750 0.46 8.0 103 4.20 2.30 0.91 0.39 0.113.68 1.90 0.96 0.51 0.27

51 1880 0.47 8.0 102 4.70 2.40 1.00 0.38 0.0843.50 2.01 0.96 0.47 0.27

61 450 0.53 9.3 102 3.50 2.10 1.14 0.53 0.203.10 1.67 0.84 0.45 0.21

Observed concentrations are in the first line and predicted concentrations in the second line.

of 1.5 m at five downwind distances (50, 100, 200, 400, 800 m). The Prairie Grass site was flat with a roughnesslength of 0.6 cm. From the Prairie Grass runs we select thirteen cases in which the mean wind speed was greater than6.0 m s−1 with values of (u∗)0 > 0.4 m s−1. Table 2 provides the values of the micrometeorological parameters forthe selected Prairie Grass runs. The values of U and u∗0 expressed in Table 2, represent characteristic values of aneutral PBL [21].

The Prairie Grass tracer dispersion was simulated by the Lagrangian particle model LAMBDA [31,32]. For adetailed presentation and discussion of LAMBDA we refer to the quoted papers. The actual version of the LAMBDAdispersion model is based on the generalized Langevin equation [33], whose coefficients are obtained by solvingthe Fokker–Planck equation, and satisfies the well-mixed condition. It can use, as input, higher-order moments ofthe atmospheric probability density function (PDF) of wind velocity. In the present application, LAMBDA used aGaussian PDF on the horizontal plane and a Gram-Chalier PDF, truncated to the fourth order, in the vertical.

All the available data (see Table 2) were used to create an input file for the simulations. The profiles of windstandard deviations (σi ) and the local decorrelation time scales (TLi ), which were used in the LAMBDA dispersionmodel, were calculated according to Eqs. (7) and (9), with ( fm)i given by Eq. (5) and considering a neutral PBLu2

∗ = u2∗0 (1 − z/h)1.7 [7,34].

LAMBDA simulation results are presented in Table 2. The model performances are shown in Table 3 andFig. 7. Table 3 exhibits the result of the statistical analysis made with observed and predicted values of ground-level cross-wind integrated concentration (Cy). Additionally, Table 3 presents a comparison between the turbulentparameterization derived from the NWF cases (present parameterization) with the one from Hanna [35] for the neutralcondition. This last parameterization was chosen for the comparison because it is well known and widely used in theliterature. The statistical indices in Table 3 are suggested by Hanna [36]:

N M SE = (Co − C p)2/(Co · C p) (Normalized Mean Square Error)

F B = (Co − C p)/(0.5(Co + C p)) (Fractional Bias)

4384 M.C.A. Arbage et al. / Physica A 387 (2008) 4376–4386

Fig. 7. Scatter diagram between observed and predicted ground-level cross-wind integrated concentrations (Cy ) for the Prairie Grass experiment.

Table 3Statistical evaluation indices considering Hanna’s model for the Prairie Grass experiment

NMSE COR FA2 FB FS

NWF 0.04 0.99 0.92 0.08 0.14Hanna (1982) 0.40 0.95 0.78 0.41 0.39

F S = 2(σo − σp

)/(σo + σp

)(Fractional Standard Deviation)

R = (Co − Co)(C p − C p)/σoσp (Correlation Coefficient)

F A2 = 0.5 ≤ Co/C p ≤ 2 (Factor 2)

where C is the analyzed quantity (concentration) and the subscripts “o” and “p” represent the observed and thepredicted values, respectively. The overbars in the statistical indices indicate averages. The statistical index FBindicates if the predicted quantity underestimates or overestimates the observed one. The statistical index NMSErepresents the quadratic error of the predicted quantity in relation to the observed one. The statistical index FS indicatesthe measure of the comparison between predicted and observed plume spreading. The statistical index F A2 providesthe fraction of data for which 0.5 ≤ Co/C p ≤ 2. The nearer to zero are the NMSE, FB and FS and the nearer tounity are the R and FA2, the better are the results. Fig. 7 shows the scatter diagram between observed and predictedground-level cross-wind integrated concentrations for the Prairie Grass concentration data set.

LAMBDA model, employing the present parameterization, simulates quite well the experimental concentrationdata for the neutral Prairie Grass tracer experiments and, indeed, presents results comparable or even better than thoseobtained by the Hanna’s turbulent parameterization model. The statistical analysis reveals that all indices are withinacceptable ranges, with NMSE, FB and FS values are relatively near to zero and R and FA2 are relatively near to 1.

6. Conclusions

In this study, we employed Taylor’s statistical diffusion theory, a model that describes observed turbulent spectraand a Lagrangian stochastic particle model, to investigate turbulent statistical characteristics and diffusive propertiesassociated with the north wind phenomenon. This NWF consists of a warm air flow that occurs in a regional scalein the winter period in southern Brazil. It is characterized by high mean wind magnitudes and generates a shear-dominated atmospheric boundary layer. Wind turbulent data for seven NWF cases were utilized to generate one-dimensional turbulent energy spectra, experimental values of the surface spectral peak frequencies, and velocityvariances. The values of these observed variances and spectral peak frequencies exhibit a good agreement with the

M.C.A. Arbage et al. / Physica A 387 (2008) 4376–4386 4385

results of literature. Thusly, the comparison indicates that the transport process in the PBL generated by the NWF isdominated by mechanical turbulence.

The peak frequencies and velocity variances describe the temporal and spatial scales of the energy-containingeddies and therefore they are important for studies of turbulent transport in the PBL. Thusly, to obtain a turbulenceparameterization, the NWF energy spectra were compared with fitting curves obtained from a spectral model based onKolmogorov hypothesis for a fully developed turbulence (Eq. (4)). The comparison shows that, in the turbulent motionfrequency range, the observed energy spectra are well represented by the curves generated by the spectral model. Asa consequence of this result, formulations for the velocity variances (Eq. (7)), and local decorrelation time scales(Eq. (9)), expressed in terms of the observed statistical variables associated with the NWF cases (φε, ( fm)0i ), becomewell defined and can be used as physical turbulent parameters in dispersion models. Therefore, a vertical turbulentdiffusivity has been derived (Eq. (12)) and compared with an eddy diffusivity derived from the non-extensive statisticalmechanical concepts that describes turbulent transport generated by mechanical forcing. This comparison shows thatboth values of eddy diffusivity exhibit a similar profile in vertical regions in which the wind shear force is dominant.

The present turbulent parameterization representing the dispersion condition in a shear dominated PBL occurringduring the NWF cases, was evaluated and validated through a comparison with experimental concentrationdata. Particularly, the results obtained by the Langrangian stochastic particle model LAMBDA, incorporating theparameterization derived from the turbulent statistical characteristics associated to the NWF, agree very well with theexperimental concentration data, indicating that the model represents the diffusion process correctly in intense windspeed neutral conditions. It is also possible to verify that the simulated results employing the parameterization derivedfrom the NWF data are better than those obtained by the Hanna’s parameterization model.

Considering the statistical evaluation and the arguments discussed above the parameterization for the NWFphenomenon allied to the Lagrangian dispersion model LAMBDA is found to be suitable to simulate the diffusion ofpassive scalars in a intense wind speed neutral PBL.

Acknowledgements

The authors acknowledge the financial support provided by CNPq (Conselho Nacional de DesenvolvimentoCientıfico e tecnologico), CAPES (Coordenacao de Aperfeicoamento de Pessoal de Nıvel Superior) and FAPERGS(Fundacao de Amparo a Pesquisa do Estado do Rio Grande do Sul). The Paraıso do Sul dataset is part of CT-HIDROproject, coordinated by Dr. Carlos Nobre.

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