Nonlocal boundary layer: The pure buoyancy-driven and the buoyancy-shear-driven cases

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1 JOURNAL OF GEOPHYSICAL RESEARCH, VOL. 114,, doi:10.109/008jd01068, 009 Nonlocal boundary layer: The pure buoyancy-driven and the buoyancy-shear-driven cases N. M. Colonna, 1 E. Ferrero, 1 and U. Rizza Received 30 June 008; revised 7 October 008; accepted 5 November 008; published 4 March 009. [1] A Reynolds-averaged Navier-Stokes (RANS) model including nonlocality was developed and tested. The aim of this paper was to investigate different boundary layer conditions with the RANS model. In particular we focused our attention on boundary layers where convection plays a major role: the shear-free buoyancy-driven boundary layer and the buoyancy-shear-driven one. We presented a new model which solves dynamical equations for all the turbulence moments up to the third order. The model assumes the fourth-order moments to be Gaussian-distributed according to the quasi-normal approximation. A new approach to avoid the unphysical growth of the third-order moments, connected to the quasi-normal hypothesis, is suggested. The new physical ingredient of this method is in essence the parameterization of the relaxation and dissipation time scales related to the return to isotropy terms in the triple-pressure correlations and to the dissipation rate of the potential temperature variance, respectively. This new parameterization takes into account the dependence of the third-order moments on the integral length scale of the dissipation rate, a quantity related to the eddy size variation across the boundary layer. The time scales are reduced as the integral length increases, and these reductions provide the proper damping of the turbulent vertical transport. In order to test our model, we carried out large eddy simulations (LESs) of boundary layers with shear and buoyancy. We present the results of the comparison with LESs, aircraft measurements, and DNS, which show that the model performs rather well in all the stability conditions and that the proposed damping gives reliable third-order moments that satisfy the realizability constraints. Citation: Colonna, N. M., E. Ferrero, and U. Rizza (009), Nonlocal boundary layer: The pure buoyancy-driven and the buoyancyshear-driven cases, J. Geophys. Res., 114,, doi:10.109/008jd Introduction [] It is generally recognized that nonlocal transport of turbulence is critically important in the PBL s mixing processes, especially in those fluxes where the buoyancy plays a major role. Among the several procedures well known and strongly validated in the literature, the Reynolds- Averaged Navier-Stokes (RANS) models, firstly proposed by Reynolds, represent a useful tool to describe many turbulent phenomena. [3] RANS models should be divided in local and nonlocal models. To the first category belong those models that include up to the second-order moments (SOMs) equations and adopt a parameterization for the third-order moments (TOMs) [Mellor and Yamada, 198; Byggstoyl and Kollmann, 1986; Durbin, 1993; Cheng and Canuto, 1994; Canuto et 1 Dipartimento di Scienze e Tecnologie Avanzate, Universitá del Piemonte Orientale, Alessandria, Italy. Consiglio Nazionale delle Ricerche, Istituto di Scienze dell Atmosfera e del Clima, Lecce, Italy. Copyright 009 by the American Geophysical Union /09/008JD01068 al., 1999; Speziale et al., 1996; Jones and Musange, 1988; Trini-Castelli et al., 001]. The usual parameterizations employed in local models are not appropriate when applied to convective fluxes since they do not properly account for the complex mixing processes that affect the turbulent quantities [Moeng and Wyngaard, 1989; Holstag and Boville, 1993]. In fact, in a local model it is usually assumed a one to one correspondence between turbulent fluxes at a given height and other parameters of the flow, namely the local gradients of the mean fields, at the same height. Such an approximation is justified only when the turbulent mixing length is much less than the length scale of heterogeneity of the mean flow. However, observations and Large Eddy Simulation (LES) studies have demonstrated that turbulence in the convective boundary layer is associated with the nonlocal integral properties of the boundary layer [Moeng and Sullivan, 1994; Holstag and Moeng, 1991; Schmidt et al., 006]. This means that mixing is present also in those regions of the boundary layer where there is no local production of turbulence, thanks to nonlocal effects of the vertical transport. These effects are essentially due to the presence in the convective boundary layer of large-scale semiorganized structures, namely buoyancy-driven cells, 1of13

2 that embrace the entire boundary layer. A similar effect is present also in the sheared flows due to large turbulent structures like rolls or streaks [Moeng and Sullivan, 1994]. Recent studies [Ferrero and Racca, 004; Ferrero, 005] have demonstrated that nonlocality plays an important role in these kind of flows even though the structures entailed are smaller than those present in convective boundary layers. As a matter of fact, a model including the TOMs is able to better determine the boundary layer height in the neutral case [Ferrero and Racca, 004]. [4] Therefore, to properly represent nonlocality it requires that the model would be able to encompass all the effects from the bottom of the boundary layer to the height where the fluxes are computed. Several improvements of local models have been made by including nonlocal adjusting terms [Deardorff, 197; Holstag and Moeng, 1991; Wyngaard and Weil, 1991; Canuto et al., 005] but to overcome the shortcomings of local models, one should employ a nonlocal model by including the dynamical equations for the TOMs [Canuto, 199; Cheng and Canuto, 1994; Zilitinkevich et al., 1999; Canuto et al., 001; Cheng et al., 005; Ferrero and Racca, 004;Ferrero, 005; Gryanik et al., 005; Ferrero and Colonna, 006]. [5] In this paper we focus our attention on the turbulence closure problem for convective sheared and unsheared boundary layers. We present simulations performed with a third-order closure model whose TOMs equations were recently proposed and tested by us [Ferrero and Colonna, 006]. This model is able to account for both the turbulent production processes, namely shear and buoyancy, at the same time thus allowing to simulate more complex flows corresponding, for example, to the actual atmospheric boundary layer. [6] The first flow investigated is a buoyancy-driven shear-free boundary layer (hereafter case B) while the second one is a buoyancy-driven sheared flow (hereafter case SB). Most of the boundary layer flows studied in literature are, for sake of simplicity, pure convective and pure shear cases which represent very particular conditions. On the contrary, the case SB represents a realistic combination of shear and convection such as the one that characterizes the actual boundary layer, thus representing a case of particular interest. [7] In section the model equations system is presented. The problem of turbulent damping of the TOMs is discussed in section 3 and a new approach to solve the problem is suggested. large eddy simulations and data for comparison are presented in section 4. Finally, in section 5, the TOM model settings are discussed and the results are presented. For sake of the completeness the details related to the equations have been listed in Appendix A, while in Appendix B a preliminary test of the model, where the results are compared with a Direct Numerical Simulation (DNS) data set, is shown.. Model [8] We consider a horizontally homogeneous boundary layer. The model consists of a system of 34 dynamical equations derived from the Navier-Stokes equations following the procedure described in the work of Canuto [199]. The model numerically solves the equations for mean fields, SOMs and TOMs. A closure scheme is required for the fourth-order moments (FOMs) appearing in the TOMs equations. In this work we have adopted the quasi-normal approximation though it exhibits some well known shortcomings (divergences of the TOMs in neutral and convective boundary layers and spurious oscillations in the stable regions of the flow) [Canuto et al., 005; Cheng et al., 005; Moeng and Randall, 1984]. For this reason, a new approach, based on physical grounds, is employed to avoid the anomalous growth of the TOMs due to the use of the quasi-normal approximation. The model does not include rotation and viscosity effects since their contributions are negligible in the atmospheric boundary layer. Exception is made for the mean wind equations in which rotation effects due to the Coriolis force are included..1. Mean Fields and SOM [9] According to the horizontal homogeneity hypothesis the ensemble averaged equations of motion with the hydrostatic approximation can be written @z f V g V @z þ f U g U : ðþ where the upper cases indicate the mean values while the lower cases stand for their fluctuations. U and V are the components of the mean wind velocity, U g and V g are the geostrophic wind velocity components and f is the Coriolis parameter ( f =10 4 s 1 ). The dynamical equation governing the mean potential temperature, Q, where qw is the potential temperature flux. [10] The first moments entail second-order correlations, such as uw, vw and qw. The dynamical equations for the SOMs are given by the following i j ¼ u j u k þ u i u iu j u k þ l j u i q ij i P ij þ l i u j q ð3þ ð4þ ¼ u i u qu iu j P q i þ l i q ¼ u i q : ð6þ where l i = g i a, being g i = (0, 0, g) the gravitational acceleration and a the thermal expansion coefficient, ij and q are the Reynolds stress and the potential temperature variance dissipation rates respectively. The first can be of13

3 Table 1. Constants of the Model Case c 4 c 5 c 6 c 7 c 10 c 11 c * a 1 a c m a e a 0 a 1 a B SB related to the turbulent kinetic energy dissipation rate,, by the following expression: ij k ¼ 3 d ij ð7þ Hanjalic and Launder [197, 1976], analyzed in detail by Zeman and Lumley [1979]. The FOMs are taken to be Gaussian-distributed, according to the quasi-normal approximation and therefore each fourth-order term can be expressed as a products of SOMs. while the second is given by abcd ¼ ab cd þ ac bd þ ad bc ð13þ q ¼ ð8þ [11] Concerning the dissipation term q Zeman and Lumley [1976] proposed a dynamical equation, but here we simply adopt the parameterization suggested by André et al. [198]: q ¼ c t 1 q where c is a coefficient depending on the integral length scale, as proposed by Ferrero and Colonna [006] and discussed in detail in section 3. [1] P ij and P i q, representing the pressure correlation terms, are discussed in detail in Appendix A. For the dissipation rate of the turbulent kinetic energy,, we have adopted the following dynamic equation in the standard form, as suggested by Lumley and Khajeh-Nouri j ¼ a 1 t 1 ðp q þ P s Þ a t 1 ð10þ where t = q e, q being u i u i =E (E is the turbulent kinetic energy), is the characteristic turbulence life time. [13] In the LHS of equation (10) the second term represents the divergence of the turbulent flux of (see Appendix A for discussion) while the terms P q and P s, in the RHS of equation (10), represent the buoyancy and the shear production, respectively. These terms are defined as the trace of the following tensors. P q ij ¼ l iqu j þ l j qu i ; i P ij ¼ u i u k þ u j u k : k.. Third-Order Moments [14] The dynamic equations for the third-order correlations can be derived from the SOMs equations as in the work of Canuto [199]. The TOMs imply the fourth-order correlations, u i u j u k u l, u i u j u k q, u i u j q and u i q 3 which need to be modeled. We have adopted the approximation of [15] With these and the horizontal homogeneity assumptions, the equations for the TOMs i t 1 3 u i u j u k ¼ u i u j u 3 þ u k u i u 3 þ u j u k u j u i u k u i u i u 3 þ u k u 3 þ u j u 3 þ ð1 c 11 Þ l i qu j u k þ l j qu k u i þ l k qu i u j 3t d ijq u k þ d ki q u j þ d jk q u i t 1 3 u i u j q ¼ u i u j u j u i u 3 þ ð1 c 11 u i u 3 j þ u j u 3 i u j þ u j u 3 þ qu 3 Þ l i q u j þ l j q u i þ c * t d ijq þ t 1 3 þ t 1 q u i ¼ qu i u 3 u 3 i u þ ð1 c 11 Þl i q u i u 3 q ; 3 þ c 10 t 1 3 q 3 ¼ u 3 3u 3 : c 3 ð17þ where c 10 is a constant of the model (see Table 1) while t q = t/(c ) is a time scale related to the turbulent damping [Ferrero and Colonna, 006] discussed in the next Section. The dynamical equations for the TOMs have been obtained by modeling the pressure triple correlation terms following André et al.[198] (see Appendix A for details). 3. Turbulence Damping: A New Parameterization of the Relaxation and Dissipation Time Scales [16] As pointed out in previous works, those of Cheng et al. [005], Canuto et al. [007], and Moeng and Randall 3of13

4 [1984], the use of the quasi-normal approximation for the fourth-order correlations can cause an unexpected and unphysical growth of the TOMs. This singular behavior is due to the fact that the quasi-normal approximation does not provide sufficient damping for the third-order correlations which, especially in convective boundary layer, may diverge. To avoid this from happening, several approaches have been suggested as the clipping approximation [André et al., 1976], the Eddy Damped quasi-normal Markovian model [Orszag, 1977; Lesieur, 1990] and ad hoc procedures like that proposed by Canuto et al. [001] for the unstable boundary layer. Thanks to these procedures the eddy sizes are chopped down and the transport is properly weakened. In spite of its shortcomings the quasi-normal approximation is still largely employed [Vedula et al., 005]. Cheng et al. [005] have suggested a new, nonlocal formulation for the FOMs based on a top-down approach, a procedure that, unlike the bottom-up one, starts from the higher-order moments to obtain information about the lower moments. Gryanik et al. [005] assumed that a measure of deviation of statistics from the Gaussian behavior is provided by the skewness of vertical velocity and potential temperature. [17] As stated in the previous section, in this work we have adopted the quasi-normal approximation with a new method providing the correct turbulence damping. This method has already been presented by Ferrero and Colonna [006]. It should be underlined that while in the present paper we have performed simulations of the planetary boundary layer in which the dynamic equations are numerically solved for mean fields, SOMs and TOMs, in a previous paper [Ferrero and Colonna, 006], we presented a series of validations of the TOMs. They were carried out by numerically solving the dynamical equations of the TOMs only, using as input of the model LES profiles of the mean components and the SOMs. We tested this new approach over a wide range of atmospheric flows [Ferrero and Colonna, 006]. On the contrary, the model presented here solves all the mean and turbulent quantities and LES profiles are used only for sake of comparison. [18] The LHS of equations (14) (17) show terms in which the relaxation time scale, t 3, and the dissipation time scale, t q, appear. Both these parameters depend on the characteristic turbulent life time, t, and on the two constants c 8 and c : t 3 ¼ t c 8 ; t q ¼ t c ð18þ ð19þ [19] The terms involving these two time scales follow from the return-to-isotropy terms of the triple-pressure correlations and from the dissipation rate of the potential temperature variance respectively. They control the turbulent damping of the TOMs [André et al., 198; Lesieur, 1990]. A proper choice of t 3 and t q may therefore avoid the unphysical growth of the TOMs. [0] The basic idea of the new approach proposed in this work is to consider c 8 and c varying across the whole planetary boundary layer depending on a physical parameter. This parameter is represented by the integral length scale for the dissipation rate [Ferrero and Colonna, 006]: 3 L ¼ q ð0þ This length scale, which varies with the height, represents the size of the eddies in the boundary layer. Since the size of the eddies is strictly related to the turbulent transport the variation across the boundary layer of L implies that the turbulence damping should be different at different heights. Therefore c 8 and c are taken to depend on the height and on L as follow [Ferrero and Colonna, 006]: c 8 ¼ c A 8 1 þ L c A 8 Z ; ð1þ i c ¼ c C 1 þ L c C Z : ðþ i where Z i is the boundary layer height while c A 8 = 8 and c C =.5 are the standard values suggested for this two constants by André et al. [198] and Canuto [199] respectively. As can be seen from equations (1) and () the new expressions reduce to these values if L Z i. Equations (1) and () properly modulate the value of c 8 and c at different levels without differing excessively from the literature values of the two constant. As a matter of fact, the linear dependence on L assures a slight modulation of the constants across the boundary layer. This first-order correction should properly damp the turbulent transport at different heights thus avoiding overestimation and underestimation of TOMs vertical profiles. It is worth noticing that Z i is a constant scale parameter whose value is not calculated run time by the model but merely derived as a bulk characteristic of the specific case considered. Although, in an actual boundary layer, Z i can vary, the simulation performed by the RANS model refers to a stationary condition and therefore we can assume that the boundary layer height is constant, although it could have been easily determined dynamically. Furthermore, the important parameter is the ratio between L and Z i which variation could be due to one of the two considered quantities or to both of them. If the model would be applied to a non stationary case the possible Z i variations should be taken into account. [1] An important point related to the use of the quasinormal approximation is represented by the problem of the model realizability. Gryanik et al. [005] demonstrated that any model using quasi-normal approximation can be realizable only if the skewness S satisfies the relation S. We verified that our RANS model, with the new damping method, is consistent with these quasi-normal approximation constraints. The vertical velocity skewness, S w, satisfies the realizability condition, while for the potential temperature skewness, S q, the condition is not verified (according to Gryanik et al. [005]) only near the surface and near the capping inversion. 4of13

5 Table. External LES (CFR) Parameters Case N (L x L y L z ) (km) U g (ms 1 ) Q (ms 1 K) * B 56 (5, 5, ) SB 56 (5, 5, ) [] As said before, the effectiveness of the method proposed here was demonstrated by Ferrero and Colonna [006] with the direct comparison of tests performed with constant values of c 8 and c versus a test with values variable following equations (1) (). However, in order to further verify the model, full boundary layer simulations, under various conditions, are indispensable. 4. Data for Comparison [3] Since previous studies have highlighted that turbulent quantities obtained with LESs may show trends in contrast with experimental data [Moeng and Rotunno, 1990], it becomes necessary to compare the results of our RANS model not only with LES data but also with aircraft measurements. Furthermore, the Direct Numerical Simulation (DNS) results of a channel flow [Moser et al., 1999], were used for preliminary comparison of the RANS model performance (see Appendix B). In order to obtain all the turbulent quantities for the comparison of the model results we have appositely ran two LESs (hereafter CFR). The LES data considered include both the resolved and the subgridscale (SGS) statistics, except for the TOMs. In fact, the subgrid-scale (SGS) model is able to predict only the turbulence kinetic energy and hence it cannot contribute to the TOMs. Further, the high resolution of the LES reduces the SGS contribution to the predicted flow. For sake of comparison we have also used, when available, the LES data of Moeng and Sullivan [1994] (hereafter referred to as MS). [4] Our LES model is based on the incompressible Boussinesq form of the Navier-Stokes equations and considers a horizontally homogeneous boundary layer. The boundary layer variables (mean wind components, pressure, potential temperature etc..) are spatially filtered to define resolved component and SGS component. More details of the LES model used in the present work can be found in the works of Moeng [1984], Moeng and Sullivan [1994], and Rizza et al. [006]. [5] In this work, in order to obtain a complete data set for the comparison with the output of our third-order model, we reproduced two types of atmospheric boundary layers: a buoyancy-driven boundary layer with weak shear effects (case B) and a flow with both shear and convection (case SB) respectively. As a matter of fact, the turbulent kinetic energy is produced by two distinct mechanisms, namely shear and buoyancy. The turbulent boundary layer can be driven either by both the two (SB), which is the most frequent and typical atmospheric situation, or by one of them as the buoyancy (B). Even if this last is for the atmosphere an idealized case, it can occur when the solar radiation is intense and the wind weak (low wind). [6] The dimension of the domain was 5 5km in the horizontal directions and km in the vertical direction. Simulation started from barotropic conditions. Parameters like the extension of domain, grid size, geostrophic winds and surface heat flux are listed in Table. In order to obtain a quasi-stationary boundary layer we advanced in time the LES code for more than ten large eddy turnover times, t * = Z i /w * (Tables 3 and 4). [7] As experimental data we have used the aircraft profiles taken during the ARTIST (Arctic Radiation and Turbulence Interaction Study) campaign [Hartmann et al., 1999]. These measurements reproduce the polar convective boundary layer during a cold outbreak over the ocean at moderate wind (8 1 ms 1 ). The stability parameter Z i /L (L the Monin Obukhov length) ranged from 10 to 0, and the corresponding surface parameter u * /w * ranged from 0.7 to This values indicate that convective instability played the dominant role in organizing the turbulence [Etling and Brown, 1993; Moeng and Sullivan, 1994; Lupkes and Schlunzen, 1996]. The meteorological conditions of the field experiment correspond to a shear-buoyancy case even though the boundary conditions cannot be exactly the same as the LESs and the RANS simulation. However the experimental data are considered for comparison using non dimensional coordinates. Moreover, due to the light wind, the measured data can be used as a benchmark also for the purely buoyancy case, which is an idealized case. 5. Numerical Solutions and Results [8] As said before, in order to test the new ensemble average model we have investigated the development of an unsheared (case B) and a sheared (case SB) buoyancydriven boundary layer, both heated from below at a constant and uniform rate switched on at the initial time. The physical parameters and the initial and boundary conditions were chosen to simulate the conditions of our large eddy simulations. Equations (1) (), (3), (4) (6) and (14) (17) were numerically solved using a standard finite differences numerical scheme, with forward derivatives in space and time. The horizontal homogeneity hypothesis allows all the spatial derivatives but the vertical to be set to zero. Further all the terms involving the mean vertical velocity are neglected. We have adopted a staggered vertical grid: the SOMs and the dissipation,, were evaluated on the outer grid, while the mean fields and the TOMs were computed on the inner grid. Therefore boundary conditions for the SOMs and the dissipation rate are required only. They were set as a function of the surface layer parameters and of the stability (see Table 5): Q * is the surface layer heat flux, w * = [gaq * Z i ] 1/3 the convective velocity scale, u * the friction velocity and q * = Q * /w * the potential temperature scale. The values assigned to these parameters are listed in Tables 3 and 4 respectively. As discussed in detail in next subsection, for the case B only five boundary conditions are reported since in this simulation it is not needed to solve all the Table 3. Numerical External Parameters of the Simulations Case dt (s) dz (m) Z i (m) Q (ms 1 K) * U g (ms 1 ) V g (ms 1 ) B SB of13

6 Table 4. Numerical Internal Parameters of the Simulations Case u * (ms 1 ) w * (ms 1 ) t * (s) q * (K) B SB SOMs equations. It can be noted that, in case SB, the coefficient of u is about twice that of v as suggested by Dobrinski et al. [004]. The coefficients for the momentum fluxes are chosen so that the surface value of vw is one tenth of uw according to experimental and simulation data [Moeng and Sullivan, 1994; Hartmann et al., 1999]. At the upper boundary all turbulent quantities are taken to be zero. [9] As said before, the simulations results have been compared with the LES (CFR) (the profile of mean components, SOMs and TOMs were available) and, for the available quantities, with the LES (MS) by Moeng and Sullivan [1994]. All the simulations were carried out for about 8 large eddy turnover time (see Table 4 for numerical values), in order to match the potential temperature profile of the LES. [30] The model resolution was 6 m for the B simulation and 13 m in the SB simulation consequently, the number of grid points in the BL was about 40 and 77 respectively (see Table 3), allowing to well resolving the boundary layer Unsheared Buoyancy-Driven Boundary Layer: Case B [31] Firstly we simulated a buoyancy-driven boundary layer without shear. To simulate this case of pure convection, we took U =0,V = 0; this simplification implies that the model does not solve the full system of 34 dynamic equations but only some of these. In particular, besides the equations for the mean potential temperature, there are four SOMs to be considered, the equations for the dissipation rate and five TOMs. Q; u ; w ; qw; q ;;u w; w 3 ; qw ; q w; q 3 : ð3þ where q = w +u according to Cheng et al. [005]. The values of the model parameters and constants are listed in Tables 1, 3 and 4. [3] In Figures 1 and, the results of a simulation performed with a second-order model based on the closure scheme for the TOMs proposed by Daly and Harlow [1970] are included for sake of comparison (hereafter DH model). [33] The model initial condition corresponds to a potential temperature profile increasing with height which represents a stable condition of the flow. At the top of the boundary layer, a strong thermal inversion characterizes the profile. Then the heating of the boundary layer from below at a constant rate, Q * (see Table 3), generates a region of instability near the surface in which the profile of potential temperature decreases with height. Above this layer the buoyancy forces provide and maintain a wellmixed layer in which the potential temperature is quite uniform. Figure 1 shows the final profile of potential temperature; as can be seen, the agreement of the model output with the LES is good, the superadiabatic condition near the surface is reached and a strong mixing layer is obtained; in the upper boundary layer, at the inversion layer, the model output accurately reproduces the LES data. [34] In Figures and 3, the SOMs and TOMs profiles are compared with the two LESs: CFR (solid line) and, when available, MS (thin dotted line), with the aircraft measurements (crosses) and with the second-order DH model (dots). Each turbulent quantity is properly normalized with appropriate combinations of the convective velocity (w * ) and the potential temperature (q * ) scales (see Table 4 for their numerical values). [35] The q profile (Figure a) shows efficient mixing in the middle part of the boundary layer and the model reproduces well this feature although underestimates the LES (CFR) profile. The agreement with the data LES (MS) is satisfactory. [36] The heat flux vertical profile, reported in Figure b, decreases linearly with height and becomes negative at the inversion layer, which is associated to the entrainment of the overlying air. The agreement with both the LESs and the aircraft data is good. From Figures 1 and, it can be argued that the TOM model is generally superior in reproducing the LES data than the lower order closure (DH model). [37] The TOMs are reported in Figure 3. As said in the previous sections, the triple correlations are essential for a correct description of the nonlocal properties of the flow. The results obtained with the TOM model generally agree satisfactorily with both the LES and the aircraft data. Concerning u w (Figure 3a), the profile predicted by the model is more similar to the aircraft data, especially in the lower part of the boundary layer, than to the LES data. The profile rapidly decreases in the middle of the boundary layer underestimating both the LES profiles. This may be due to the fact that the third-order model simulated a shearfree flow, while the LES, although very weakly, accounts for shear effects. In Figure 3b the w 3 profile is shown. The general trend of the LESs is well reproduced by the model even if the maximum of the curve occurs at higher level than in the LES data. [38] In Figures 3d, 3e, and 3f, the TOMs involving the potential temperature field are shown. The q w profile predicted by the TOM model differs from the LES (CFR) near the surface, as also found in the works of André etal. [198] and Cheng and Canuto [1994], but agrees with the aircraft data. Concerning qw and q 3, it can be noted that the Table 5. Boundary Conditions Case u v w uv uw vw qw qu qv q e B 13.8u * w * Q * q * 5.1w 3 * /Z i SB 4.9u *.85u * 0.06w * 0.08u * 0.4w * 0.04w * Q * 1.7Q * 0.1Q * 4.5q * 1.3w 3 * /Z i 6of13

7 Figure 1. Case B: vertical profile of mean potential temperature. Circles represent the third-order model output; solid line, our LES (CFR); and dotted line, the model output of Daly and Harlow [1970] (DH). results compare well with both the LES (CFR) and the experimental measurements. The model output for q 3 reproduces well the aircraft data near the surface while the LES (CFR) shows a different behavior. However, it is not unusual that the TOM model (and measurements) features cannot be reproduced by the LES near the surface [Moeng and Rotunno, 1990] since the LES is not completely reliable in the surface layer because of the subgrid limitations. 5.. Sheared Buoyancy-Driven Boundary Layer: Case SB [39] The second kind of boundary layer we have investigated represents a more realistic flow characterized by a mixture of shear and convection. A flow with similar characteristics was considered for model comparison in the work of Gryanik and Hartmann [00] but it was analyzed according to a mass-flux approach. In a previous paper [Ferrero and Colonna, 006], we tested our TOMs in this kind of flow, as above described. Numerical parameters of the present simulation can be found in Tables 3 and 4 while the model constants are listed in Table 1. The model results were compared with our LES (CFR) and LES (MS, case SB1) [Moeng and Sullivan, 1994] and the aircraft measurements utilized for the previous simulation although the flow they represent is characterized by weak shear effects. The initial condition corresponds, as in the previous simulation, to a stable profile for the mean potential temperature with a capping inversion at the top of the boundary layer. Furthermore, for this simulation the mean velocity components were also initialized with values equal to the geostrophic wind (see Table 3 for numerical values). As in the case B, a simulation using the DH second-order model was also carried out. The results are shown in Figures 4 and 5 together with the TOM model results. [40] Figure 4 displays the mean potential temperature and the mean velocity components final profiles. The agreement with the LES (CFR) is satisfactory across the whole domain and all the features are well reproduced by the model although small discrepancies appear in the layer very close to the surface, probably due to the boundary conditions. It can be noted that the three profiles of mean quantities show a pronounced mixing layer comparable to that of a pure convective flow. This indicates that even a relatively small buoyancy force is strong enough to create a considerable mixing process in the boundary layer. [41] The SOMs are shown in Figure 5. The agreement of q (Figure 5a) with both the LESs can be considered satisfactory. Concerning the heat flux vertical profile (Figure 5b), the third-order model output shows a different slope respect to the LES (CFR) which may depend on the surface heat flux used in the simulation, which is slightly different than that of the LES simulation. In Figures 5c and 5d, the momentum fluxes uw and vw are reported. The typical negative decreasing trend expected for these fluxes is well reproduced by the model and the agreement with the LES (CFR) data is good. With respect to the aircraft data, the uw profile yields larger (negative) values, while vw fits the experimental data. [4] Mean values and all the SOMs are better predicted by the TOM model than by the second-order DH model, except for the case of qw where the latter reproduces better the aircraft data but it shows an overestimation of the LES (CFR) data. Figure. Case B: SOMs normalized vertical profiles of (a) q and (b) qw. Circles represent the thirdorder model output; solid line, LES (CFR); thin dotted line, LES (MS); crosses, aircraft measurements; and dotted line, the DH model output. 7of13

8 Figure 3. Case B: TOMs normalized vertical profiles of (a) u w,(b)w 3,(c)q w,(d)q w,(e)qw, and (f) q 3. Circles represent the third-order model output; solid line, LES (CFR); dotted line, LES (MS); crosses, aircraft measurements. [43] This is a very important point, in fact, the results demonstrate that including TOMs in the RANS model allows not only to reproduce these moments but also improves the simulated SOMs respect to the lower order closure. [44] The TOMs obtained in the case SB are reported in Figure 6. With respect to the LES, the profiles of u w, v w and w 3 (Figures 6a, 6b, and 6c) simulated using the TOM model show an underestimation in the lower and in the mid part of the boundary layer and overestimation in the upper part. Concerning w 3 it can be noted that the model flattens the characteristic trend of this turbulent quantity. The comparison with the aircraft data is not particularly significant for this profile being the aircraft measurements mainly affected by the buoyancy than by the shear, which instead influences the model result. [45] The overestimation and underestimation found out for the TOMs previously shown, do not occur for the moments involving the potential temperature field, as can be observed in (Figures 6d, 6e, and 6f). The vertical flux of the potential temperature variance (q w) shows a satisfactory agreement with the LES and the experimental data. In the surface layer, as also in simulation B, the model differs from the LES reproducing better the experimental data. Concerning qw, the model agrees with the LES while it underestimates the aircraft measurements in which the buoyant effect prevails. Finally, the q 3 profile matches well both the LES and the aircraft data. Also for this profile, in 8of13

9 Figure 4. Case SB: vertical profiles of (a) mean potential temperature and (b) mean horizontal wind velocity components. Circles represent the third-order model output; solid line, LES (CFR); and dotted line, the DH model output. the surface layer, the model reproduces the trend of the aircraft measurements better than the LES. [46] The two time scales defined by the equations (1) and (), more accurately, their ratios to t, are, by physical arguments, reduced as the integral length increases. The results demonstrate that these reductions effectively damp the growth of the TOMs, as desired. Since the weakest part of the Mellor-Yamada type of second-order closure (SOC) model is the constant time scale ratios, there is a lot of room of improvement for the standard SOC models. As a matter of fact, recently Canuto et al. [008] parameterized the time scale ratio t pq /t (where t pq is the time scale of the return to isotropy term of the pressure-potential temperature correlation) and this single new ingredient made their earlier model [Cheng et al., 00] match data up to Richardson number Ri = 100 thus no Ri(critical) exists. 6. Conclusions [47] In this paper we have presented a Reynolds Stress model in which turbulence is treated nonlocally and the results of two complete simulations of turbulent boundary layers buoyancy and shear-buoyancy-driven respectively. [48] The model numerically solves a system of dynamical equations up to the TOMs. To close the system the quasinormal approximation is adopted for the FOMs. The known shortcoming of the quasi-normal approximation, namely the insufficient damping of the third-order correlations in presence of convection, is overcome by adopting a new approach to evaluate the relaxation and dissipation time scales, which are related to the TOMs turbulence damping. The new parameterizations are now functions of the integral length scale [Ferrero and Colonna, 006] and hence dependent on height rather than being constant. These dependencies provide a proper modulation of turbulent transport. The model is able to account for both shear and convection at the same time thus allowing to simulate complex flows such as, for example, the actual atmospheric boundary layer. [49] Nonlocal treatment of turbulent quantities is an essential requirement for a successful description of those flows involving convection. The intrinsic nature of these flows is in fact nonlocal being characterized by large energetic eddies which transport energy through the mixing layer. The nonlocality of turbulence is also important in the case of pure shear flows as shown by Ferrero and Racca [004] and Ferrero [005]. Many authors [Canuto, 199; Zilitinkevich et al., 1999; Gryanik and Hartmann, 00] also stressed that the triple correlations should be correctly treated to provide realistic Reynolds stress. [50] The results are compared with a LES performed by us (CFR), the LES of Moeng and Sullivan [199] (MS) and aircraft measurements [Hartmann et al., 1999] and preliminarily with a DNS [Moser et al., 1999]. Many of the turbulent quantities profiles presented in this work, in fact, show differences between LES and aircraft measurements and the third-order model often reproduces the experimental data rather than the LES, especially near the surface layer. Use of LES data and experimental measurements is particularly significant to test the validity of the model, as pointed out in previous work [Moeng and Rotunno, 1990]. Furthermore the comparison with two different LES data set have shown that there may be differences also between the large eddy simulations themselves, especially near the surface layer and at the bottom of the boundary layer, near the inversion layer. [51] Most of the profiles presented in the paper are in a satisfactory agreement with both LES and experimental data, thus demonstrating that our model is able to perform simulations of boundary layers where the turbulence is produced by mechanical and thermal processes separately or by a mixture of them as it occurs in the usual atmospheric flows. The success in correctly reproducing the TOMs confirms that the damping proposed in this work is correct, further it satisfies the realizability constraints. The reduction of scale ratios may be also useful for developing new second-order models. [5] Finally, the comparison with a second-order model [Daly and Harlow, 1970] demonstrates that to correctly reproduce the turbulence features in the atmospheric boundary layer, the nonlocal transport must be included, and this is accomplished here by accounting for the TOMs. Appendix A: Modeled Terms in the SOMS and TOMS Equations [53] In equation (4) the term including the pressure velocity correlation k 3 pu k) has been omitted [see Cheng et al., 005], its contribution being negligible. The pressure correlation terms P ij and P q i can be modeled 9of13

10 Figure 5. Case SB: SOMs normalized vertical profiles of (a) q,(b)qw, (c)uw, and (d) vw. Circles represent the third-order model output; solid line, LES (CFR); thin dotted line, LES (MS); crosses, aircraft measurements; and dotted line, the DH model output. (neglecting rotation) considering three distinct contributions accounting for the three sources from which pressure fluctuations arise: with b ij ¼ u i u j 1 3 q d ij ; ða6þ P ij ¼ P ðþ 1 ij þ P ðþ ij þ P ðþ 3 ij ; ða1þ P q i ¼ P q i1 þ Pq i þ Pq i3 ; ðaþ the turbulence self-interaction represented by Rotta return to isotropy term (P (1) ij and P q i1 ), the contribution due to the turbulence-buoyancy interaction (P () ij and P i q) and those (3) of the turbulence-mean flow interaction (P ij and P q i3 ), respectively. The explicit form of the pressure-velocitygradient P ij is prescribed following Canuto [199]. S ij i i ; R ij i : ða7þ ða8þ [54] For the pressure gradient-potential temperature correlation P i q we have [Canuto, 199] P ð1þ ij ¼ c 4 t 1 b ij ; ða3þ P ðþ ij ¼ c 5 l i qu j þ l j qu i 3 d ijl k qu k ; ða4þ P q i1 ¼ c 6t 1 u i q; P q i ¼ c 7l i q ; ða9þ ða10þ P ð3þ ij ¼ a 1 S ik b kj þ S kj b ik 3 d ijs kl b lk þ a R ik b kj þ R jk b ik þ 3 a 0q S ij : ða5þ P q i3 ¼ 4 5 a k d ik qu j 1 4 d kjqu i þ d ij qu k : ða11þ In equations (A3) and (A9) the turbulence life time is modified as in the work of Cheng et al. [005]. 10 of 13

11 Figure 6. Case SB: TOMs normalized vertical profiles of (a) u w,(b)v w,(c)w 3,(d)q w,(e)qw, and (f) q 3. Circles represent the third-order model output; solid line, LES (CFR); dotted line, LES (MS); crosses, aircraft measurements. [55] In the LHS of equation (10) the second term represents the derivative of the turbulent flux of, here evaluated as u j ¼ a n ða1þ where a is a constant (Table 1) and v t is the eddy viscosity, defined according to the Prandtl-Kolmogorov hypothesis as n t ¼ c m E ða13þ where c m is an empirical constant. The down gradient approximation for the flux may be a limitation for the nonlocal transport, although the dissipation is due to the smaller eddies and hence it can be considered a local phenomenon. [56] In the above equations c i (i =4,5,6,7),c *, a m, a 1, a, a 0 are constants of the model whose values are listed in Table 1. [57] As a matter of fact, the pressure models used in this paper have zero trace and therefore are pressure-strain models. The pressure transport is therefore being neglected (it has a nonzero trace). Pressure transport can be parameterized together with the TOMs since they are the same magnitude and act in similar ways physically. The pressure 11 of 13

12 Figure B1. Normalized vertical profiles of (a) mean horizontal wind velocity component U, (b) u,(c)v, (d) w, and (e) uw. Crosses represent the third-order model output; dotted line represents the DNS data. triple correlation terms present in the dynamical equations of the TOMs are modeled following André et al.[198]: P ijk ¼ t 1 3 u iu j u k þ c 11 u i u j ql k þ perm: ; ða14þ P q ij ¼ t 1 3 u iu j q þ c 11 l j u i q þ l i u j q 3 d ijl k u k q c * t 1 d ij q q; P qq i ða15þ ¼ t 1 3 u iq þ c 11 l i q 3 : ða16þ where t 3 = t/(c 8 ) and c 11, c * are constants whose values are reported in Table 1. c 8 is a coefficient, related to the turbulent damping, depending on the integral length scale discussed in section 3 [Ferrero and Colonna, 006]. It can be noted that the last term in the parenthesis, in the second of the above equations, can be set to zero in order to ensure the realizability conditions satisfaction, as suggested by Cheng et al. [005]. Appendix B: Preliminary Test With DNS Data [58] A preliminary test was performed by comparing the results of our model with a DNS data set [Moser et al., 1999]. This refers to a fully developed turbulent channel flow characterized by a Reynolds number Re = 590 (Re t = u * d/v). Because this DNS refers to a pure neutral flow, we used our model in its version for shear-driven boundary layer [Ferrero and Racca, 004] and the boundary con- 1 of 13

13 ditions were set as a function of the friction velocity. The wall-normal profiles of mean velocity and some SOMs are shown in Figure B1. It can be observed that the agreement between RANS model and DNS can be considered satisfactory except for a thin layer near the wall. This is not surprising because the spatial resolutions of the two model are very different. However the aim of our model is to correctly reproduce the main turbulent features in the whole boundary layer and this objective is achieved as is shown in this paper. Moreover, it should be stressed that the Reynolds number of the DNS is lower than that of a typical atmospheric flow (even if it is enough high to guarantee a fully developed turbulence), which can be obtained only using large eddy simulations. The lower Reynolds number can explain the small discrepancy found between DNS and RANS results, in fact, as shown in the Moser et al. 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Mironov (1999), Thirdorder transport and nonlocal turbulence closures for convective boundary layers, J. Atmos. Sci., 56, N. M. Colonna and E. Ferrero, Dipartimento di Scienze e Tecnologie Avanzate, Universitá del Piemonte Orientale, via Bellini 5, Alessandria I , Italy. (colonna@unipmn.it; enrico.ferrero@mfn.unipmn.it) U. Rizza, Consiglio Nazionale delle Ricerche, Istituto di Scienze dell Atmosfera e del Clima, Str. Prov. Lecce-Monteroni Km 1,00, Lecce I-73100, Italy. (u.rizza@isac.cnr.it) 13 of 13