Turbulence dissipation rate derivation for meandering occurrences in a stable planetary boundary layer

A new formulation for the turbulence dissipation rate ε occurring in meandering conditions has been presented. The derivation consists of a MacLaurin series expansion of a lateral dispersion parameter that represents cases in which turbulence and oscillatory movements associated to the meandering events coexist. The new formulation 5 presents the identical physical premises contained in the classical and largely used one, but the new formulation derived from meandering situations is expressed in terms of the loop parameter m that controls the absolute value of the negative lobe in the meandering autocorrelation function. Therefore, the m magnitude regulates the turbulence dissipation rate. This dissipation rate decreases for cases in which turbulence 10 and low frequency horizontal wind oscillations coexist and increases for a fully developed turbulence. Furthermore, a statistical comparison to observed concentration data shows that the alternative relation for the turbulent dissipation rate occurring in situations of meandering enhanced dispersion is suitable for applications in Lagrangian Stochastic dispersion models.


Introduction
Lagrangian turbulent velocity correlation coefficient is an important physical quantity in turbulent diffusion problems.Generally, the use of this autocorrelation coefficient in the Taylor statistical diffusion theory allows calculating the dispersion parameters associated to the turbulent diffusion modeling studies in the Planetary Boundary Layer (PBL) (Taylor, 1921).
For a stationary and isotropic turbulence, the two-time Lagrangian velocity correlation coefficient may be written as

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where τ is the time lag, u ′ i (t) is the Lagrangian turbulent velocity of a fluid particle at time t, i = 1, 2 and 3; u ′ i represents the turbulent wind components in x, y and z directions(u Based on the Taylor statistical diffusion theory, the following formula has been proposed by Frenkiel (1953) to represent the turbulent Lagrangian autocorrelation coefficients This functional form is composed of the product of the classical exponential function (representing the autocorrelation function for a fully developed turbulence) by the cosine function (describes the meandering phenomenon associated to the observed low frequency horizontal wind oscillations).This latter allows Eq. ( 2) to reproduce the negative lobes observed in the meandering autocorrelation functions (Anfossi et al., 2005).The Frenkiel function is a hybrid formula described in terms of T L i , the Lagrangian integral time scale for a fully developed turbulence, and m, the loop parameter, which controls the meandering oscillation frequency associated to the horizontal wind.Indeed, the m parameter controls the negative lobe absolute value in the autocorrelation function and hence establishes the meandering phenomenon magnitude (Anfossi et al., 2005).Recently, Oettl et al. (2005) and Goulart et al. (2007) proposed a theory on meandering atmospheric flow in low wind speed situations.This theory shows that, in simplified conditions, the forecast equation for mean wind in the PBL (Stull, 1988) provides a particular solution that is capable of describing the meandering behavior.Here, we succinctly present this derivation employing the three-dimensional Reynolds-average momentum conservation equation in an atmospheric boundary layer Holton (1992).Neglecting viscosity terms, this equation that describes the horizontal mean wind may Introduction

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Interactive Discussion EGU be written as where i , j =1, 2 and 3; u i represents the mean wind components in x, y and z directions (u 1 = u, u 2 = v and u 3 = w) averaged over a certain time interval, f c is the Coriolis parameter, ε i j 3 is the Levi-Civita tensor, ρ is the mean density,p i is the mean pressure and g is the gravity acceleration.
The complexity of Eq. ( 3) does not allow it to be analytically solved, except under some appropriate simplifying assumption yields from Eq. ( 3) an analytical solution.However, we assume that all the horizontal and vertical gradients of the horizontal wind velocity components and pressure may be considered as constant.Furthermore, the w magnitude, which characterizes subsidence in fair weather boundary layer conditions, may range from 0 to 0.1ms −1 (Stull, 1988).Thus, assuming these values in Eq. ( 3), the terms containing w and horizontal gradients of w could be neglected.With these simplifications and assuming the hydrostatic balance condition yields where Considering the a 1 , a 2 , b 1 , b 2 , c 1 and c 2 terms as constants, the system Eq.( 4) becomes a first-order linear differential equation system, where the horizontal wind components in Eq. (4a) may be written as a second-order linear ordinary differential where According to the values of roots r 1 and r 2 from the auxiliary equation, there are three cases in the solution of Eq. ( 5).We select the case that presents oscillatory behavior, which means the condition B 2 − 4C < 0, in the auxiliary (or characteristic) equation for the given differential Eq. ( 5), . Therefore, the following solution for u (t) and v (t) is obtained and If the horizontal and vertical gradients of the turbulent momentum fluxes may be disregarded, a scale analysis allows the derivation of the following simplified form for Eq.(6a) and Eq.(6b) (Oettl et al., 2005;Goulart et al., 2007) and

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which is expressed in an analytical functional form as Employing Eq. ( 8) into Eq.( 1), the following turbulent velocity correlation coefficient is obtained considering the real part of the Eq. ( 9) yields which presents the same functional form as the autocorrelation function proposed by Frenkiel Eq. ( 2).Anfossi et al. (2005) employed Frenkiel classical mathematical expression to reproduce autocorrelation functions observed during meandering periods, which presented negative lobes.Therefore, based on observational evidences (Anfossi et al., 2005) and on the fact that the general form of the autocorrelation coefficient proposed by Frenkiel is derived from the first principles (forecast equation for mean wind), we assume in this study that the Frenkiel autocorrelation function captures well the physical properties of a fully developed turbulence as well as hybrid flow cases, in which turbulence and meandering occurrences coexist.Indeed, the Frenkiel formulation Eq. ( 2) presents an empirical flexibility that allows represent observations in the PBL (Manomaiphiboon and Russel (2003) and Anfossi et al., 2005).This imparts to Eq. ( 2) a heuristic validity.
The turbulence dissipation rate is a quantity frequently used in turbulent parameterizations applied to Lagrangian stochatic dispersion models.Indeed, formulations for the turbulence dissipation rate are extensively employed in turbulent diffusion models to simulate the transport of passive scalars and the dispersion of contaminants in a fully developed turbulence (Thomson, 1987;Luhar and Britter, 1989;Sawford, 1991;Wilson and Sawford, 1996;Yeung, 2002;Degrazia et al., 2005).In this study, employing the EGU physical contents as those classical derived by Hinze (1975) and Tennekes (1982), however the dissipation rate could be described not only for a fully developed turbulence, but also for situations in which the turbulence weakness allows the occurrence of the meandering phenomenon.With the above statements becomes evident the existence of a strong constraint between the turbulence dissipation rate and the Lagrangian stochastic dispersion models.Therefore to investigate the influence of this new dissipation rate in turbulent diffusion problems, it is introduced in a Lagrangian stochastic dispersion model in order to simulate diffusion experiments performed under low wind speed meandering conditions.

Theoretical development
The lateral dispersion parameter σ y is a statistical quantity fundamental for the dispersion modeling and for the derivation of turbulence dissipation rate functional form (Tennekes, 1982;Degrazia et al., 2005).From the Taylor statistical diffusion theory, this important parameter may be described as where t is the travel time of the fluid particle and σ v is the standard deviation of the turbulent lateral velocity.The substitution of the Frenkiel autocorrelation function into Eq.( 11) yields Printer-friendly Version Interactive Discussion

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Therefore, the relation Eq. ( 12) may be considered as a lateral dispersion parameter applied to different types of turbulence.For the special case m = 0, the Eq. ( 12) reproduces the classical expression for the lateral dispersion parameter valid to a fully developed turbulence, which is obtained by substituting the classical exponential autocorrelation function e −τ/−T L v into Eq.( 11) (Pope, 2000).Thus, the Eq. ( 12) represents a more general formulation since it describes hybrid cases, in which turbulence and oscillatory movements associated to the meandering occurrences coexist (weak turbulence conditions).
For t<T L v , a MacLaurin series expansion of the general dispersion parameter Eq. ( 12) may be obtained The comparison of Eq. ( 13) with the Taylor theorem Eq. ( 11) for small diffusion times shows that the negative term in the right side of Eq. ( 13) contributes for the decrease on the hybrid dispersion parameter.From the physical point of view, representing the Taylor model in the spectral form (Degrazia et al., 2005) may be seen that this negative contribution results in the suppression of a number of degrees of freedom of the turbulent field associated to the high-frequency harmonics.As a consequence, it is reasonable to relate the term to the inertial subrange high frequency eddies.This relationship was firstly obtained by Tennekes (1982) through the use of the Lagrangian structure function, the Lagrangian autocorrelation function and the inertial subrange Lagrangian turbulent spectrum.The Tenneke's development establishes the following formulation for the Lagrangian autocorrelation function in terms of inertial subrange quantities: 6 t 3 , which after comparison with the truncated hybrid dispersion parameter Eq. ( 13), leads to the following fundamental relation As seen from Eq. ( 15), this new formula for the turbulence dissipation rate preserves the fundamental physical premise that turbulence is dissipated at a rate proportional to the energy available and inversely proportional to the time scale of the energy containing eddies (Tennekes, 1982).However, as a new element in the relation Eq. ( 15), the parameter m appears, which establishes a new formulation for the turbulent dissipation rate applied to hybrid situations, including the presence of low-frequency horizontal wind oscillations (meandering).In comparison with the classical turbulent dissipation rate derived from the exponential autocorrelation function for a fully developed turbulence, the m magnitude in the denominator controls the turbulence dissipation rate that will be larger for the fully developed turbulence case (small m) and smaller for the meandering case (large m).Therefore, we expect that the new functional form, Eq. ( 15), when employed in Lagrangian stochastic particle models, reproduces the dispersion process occurring under different types of turbulent flows in a more robust manner.
Recently, Manomaiphiboon and Russel (2003), through consideration of mathematical and physical requirements, evaluated Eq. ( 2) for the special case m = 1.They concluded that for this particular m value, the Eq. ( 2) is appropriate to be used in turbulence studies because it complies well with most requirements.According to this procedure, we can see that the more general autocorrelation form, as given by Eq. ( 2), satisfies the condition so that the Frenkiel formula is valid as long as T L v is well defined.From the physical point of view, Eq. (2) satisfies the inertial subrange conditions as proposed by Kolmogorov (1941, K41 theory) and suggested by Tennekes (1979); Lagrangian velocity fluctuations should be in accordance with EGU spectrum.This is shown in Figs. 1, 2, and 3 where a comparison between the energy spectra derived from both models (classical exponential autocorrelation function and Eq. 2) is presented.From these figures, we observe that the Frenkiel's formulation for distinct m values (including m values associated to strong negative lobes) captures the n −2 falloff.Therefore, Eq. ( 2) agrees with the K41 theory.Furthermore, it can be also seen from Figs. 1, 2, and 3 that as m increases, passing from nonmeandering to meandering conditions, the turbulent energy associated to the inertial subrange frequencies (scales) decreases abruptly.Figure 4 exhibits an autocorrelation function characterized by a large negative lobe, which is associated to the meandering phenomenon.The minimum value of this negative lobe, and the general form in Fig. 4 agrees well with the meandering autocorrelation functions observed by Anfossi et al. (2005).Therefore, for large m values characterizing the dominant presence of the meandering phenomenon in comparison with the fully developed turbulence, the turbulent energy of the inertial subrange becomes negligible and the low frequency horizontal wind oscillations (meandering) represent the fundamental energy source for the velocity variance observed in low wind speed conditions.From the analysis above, we conclude that the Frenkiel hybrid formula Eq. ( 2), derived from a simplified solution of the forecast equation for mean wind in the PBL, represents a general expression that could be used to reproduce autocorrelation functions observed in the PBL.As a consequence, the turbulence dissipation rate derived from Frenkiel autocorrelation function and given by Eq. ( 15 The purpose of this section is to test our parameterization as given by Eq. ( 15) in a practical application and to show how it works.Therefore, a Lagrangian stochastic dispersion model using the turbulent dissipation rate containing the loop parameter m (Eq.15) has been employed to simulate the measured concentration data.
The concentration data employed in the comparison with the model were obtained from the low wind speed experiment performed in a stable boundary layer from the series of field observations conducted at the Idaho National Laboratory -INEL (Sagendorf and Dickson, 1974).

Model Description a. Horizontal Components
Recently, Carvalho and Vilhena (2005) suggested an approach to obtain the solution of the Langevin equation for low-wind dispersion.The solution consists of the linearization of the Langevin equation as stochastic differential equation: which has the well-known solution determined by the integrating factor e t t 0 In order to embody the low wind speed condition in the Langevin equation, it is assumed that the function f (t) is a complex function.Therefore, the exponentials appear- In Eq. ( 20), the term e −pt cos(qt) is analogous to the autocorrelation function suggested by Frenkiel (1953) and written in a different way by Murgatroyd (1969), where p and q are given by p = 1 (m 2 +1)T and q = m (m 2 +1)T and T is the time scale for a fully developed turbulence.
Using the Lagrangian particle models, the turbulence is considered as Gaussian in the horizontal directions (i = 1, 2) and, therefore, Eq. ( 20) may be written as: . Introduction

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The Picard Method is applied to Eq. ( 21), assuming that the initial guess is determined from a Gaussian distribution.The generic iterative step is written as: Concerning the m values, it is important to note two distinct cases about Eq. ( 22).For m > 1, the meandering phenomenon has an effect upon the dispersion process.On the other hand, for m = 0, the meandering phenomenon is vanished and Eq. ( 22) is written in terms of the exponential autocorrelation function (e −t / τ l ), which Lagrangian particle models are usually employed in windy conditions.Therefore, the approach Eq. ( 22) is capable of simulating the contaminant dispersion in the PBL in both cases, that is, when the contaminant plume evolution is governed by eddies associated to a fully developed turbulence and exhibits a 'fanning' kind of behavior (typical of windy conditions) and when the plume evolution is governed by low frequency horizontal wind oscillations (meandering behavior associated to low wind conditions).
For applications, the values obtained for the parameters m and T are calculated by the empirical formulation employed by Carvalho and Vilhena (2005) ) and T * = 200m + 500.b.Vertical Component For the vertical component, the Langevin equation is solved by the approach as suggested by Carvalho et al. (2005).While for the horizontal directions the PDF is considered as Gaussian, for the vertical direction the PDF is assumed to be non-Gaussian (to deal with non-uniform turbulent conditions and/or convection).In both cases, a Gram-Charlier PDF, which is given by the series of Hermite polynomials, may be adopted (Anfossi et al., 1997;Ferrero and Anfossi, 1998).
The Gram-Charlier PDF truncated to the fourth order is given by the following ex-

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pression (Kendall and Stuart, 1977): where r i = u i σ i , σ i is the turbulent velocity standard deviation, H 3 and H 4 are the Hermite polynomials and C 3 and C 4 their coefficients.In the case of Gaussian turbulence, Eq. ( 23) becomes a normal distribution, considering C 3 and C 4 equal to zero.The third order Gram-Charlier PDF is obtained with C 4 = 0. Applying the Eq. ( 23) in the Fokker-Planck equation (Rodean, 1996), the Langevin equation may be written as: , j may assume 1,2,3 and j = i and f i , g i and h i are expressions written as: Multiplying the integrating factor [exp( ] by all terms in Eq. ( 24), the following integral equation is obtained: Tables Figures Printer-friendly Version Interactive Discussion

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The Picard Method is now applied to Eq. ( 28), assuming that the initial value for the turbulent velocity is a random value supplied by a Gaussian distribution: The generic iterative step is written as: .( 29)

Dispersion simulation
The results of the proposed model are compared with concentration data collected under stable conditions in low wind speeds over flat terrain at the Idaho National Engineering Laboratory (INEL).The results have been published in a U.S. National Oceanic and Atmospheric Administration (NOAA) report (Sagendorf and Dickson, 1974).
For simulations, the turbulent flow is considered as inhomogeneous only in the vertical direction and the transport is performed by the longitudinal component of the mean wind velocity.The horizontal domain was determined according to sampler distances and the vertical domain was set as equal to the observed PBL height.The time step was maintained constant and it was obtained according to the value of the Lagrangian decorrelation time scale (∆t = τ L /c), where τ L must be the smallest value between τ L u , τ L v , τ L w and c is an empirical coefficient set as equal to 10.For each simulation, the number of particles released was 10 6 and the concentration was obtained by counting the number of particles in volumes with dimensions 5 m×5 m and 0.5 m in height.
Values of σ i and τ L i were parameterized according to scheme developed by Degrazia et al. (1996).For the horizontal components Eq. ( 22) the parameterization of C 0 ε has been calculated from the Eq. ( 15) employing m values given by the empirical formulation utilized by Carvalho and Vilhena (2005).For the vertical component, the Eq. ( 15) is employed with m = 0.The third moment of the vertical velocity component is assigned according to Chiba (1978) and the fourth moment is calculated based on method suggested by Anfossi et al. (1997).The integration method used to solve the integrals EGU appearing in Eq. ( 22) and Eq. ( 29) was the Romberg technique.Due to the wind direction variability, a full 360 • sampling grid was implemented.Arcs were laid out with radii of 100, 200 and 400 m from the emission point.Samplers were placed at intervals of 6 • on each arc for a total of 180 sampling positions.The receptor height was 0.76m.The tracer SF 6 was released at a height of 1.5 m.The 1 h average concentrations were determined by means of an electron capture gas chromatography.Wind speeds measured at levels 2, 4, 8, 16, 32 and 61 m were used to calculate the coefficient for the exponential wind vertical profile.According to Brusasca et al. (1992) and Sharan and Yadav (1998), the roughness length used was z 0 =0.005 m.The Monin-Obukhov length L and the friction velocity u * were not available for the INEL experiment but may be roughly approximated.Then, L may be written from an empirical formulation suggested by Zannetti (1990) and the stable PBL height h was determined according to expression suggested by Zilitinkevich (1972).
The model performance is shown in Table 1 and Fig. 5. Table 1 shows the result of the statistical analysis performed with observed and predicted ground-level centerline concentration values according to Hanna's (1989) statistical indices.Figure 5 shows the scatter diagram between observed and predicted concentration.Observing the results obtained, we may promptly conclude that the model simulates quite well the experimental data in low wind stable conditions.The statistical analysis reveals that all indices are within acceptable ranges, with NMSE, FB and FS values relatively near to zero and R and FA2 relatively near to 1.

Conclusions
In this study, a new formulation for the turbulence dissipation rate ε has been derived and tested.Unlike the classical form, which is derived from the exponential autocorrelation function, this new expression Eq. ( 15) is obtained from an autocorrelation function that describes negative lobes in the observed autocorrelation functions associated to the meandering phenomenon, and that has been derived from the forecast equation

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for mean wind in the PBL, when a number of simplifications are assumed.Therefore, the new formulation for the turbulence dissipation rate is expressed in terms of the loop parameter m and may be employed to parameterize the turbulence dissipation rate associated to the low frequency horizontal wind oscillations (meandering phenomenon) occurring in low wind stable conditions.The new parameterization may be applied in situations in which turbulence and meandering phenomenon coexist.Consequently, the turbulence dissipation rate as given by Eq. ( 15) constitutes a general functional form that may be used in distinct turbulent flow patterns.
The presence of m in Eq. ( 15) controls the turbulence dissipation rate.It will be larger for a well-developed turbulence (neglected m value) and smaller for the physical state described by the turbulence and meandering phenomenon coexistence (hybrid situation).
The present parameterization for the turbulence dissipation rate was evaluated and validated through the comparison with experimental data.Particularly, the results obtained by the Lagrangian stochastic dispersion model, employing the new formula for the turbulence dissipation rate, show that the model correctly represents the diffusion process in low wind speed stable conditions.
Considering the statistical evaluation and the scatter diagram given in Fig. 5, the new parameterization for the turbulence dissipation rate seemed to be suitable to simulate the meandering enhanced diffusion of passive scalars occurring in situations in which turbulence and meandering phenomenon coexist.Therefore, the main result of the present study is Eq. ( 15), which represents an alternative expression for the turbulence dissipation rate for hybrid flow cases and that has been derived from an autocorrelation function that reproduces experimental meandering data.
) represents a general functional form that converges to the classical relation ε = 2 a fully developed turbulence) when meandering effects are excluded (i.e. by setting m ∼ =0 in Eq. 2the derived parameterization with the INEL tracer experiment.
where c is a constant and F L v (n) is the Lagrangian lateral turbulent Introduction