Employing heisenberg’s turbulent spectral transfer theory to
ARTICLE IN PRESS Atmospheric Environment 41 (2007) 7059–7068 www.elsevier.com/locate/atmosenv Employing Heisenberg’s turbulent spectral transfer theory to parameterize sub-ﬁlter scales in LES models Gervasio A. Degraziaa,Ã, Andre B. Nunesb, Prakki Satyamurtyb, ´ ´ Otavio C. Acevedo , Haroldo F. de Campos Velhoc, ´ a Umberto Rizzad, Jonas C. Carvalhoe a ´ Departamento de Fısica, Universidade Federal de Santa Maria, Santa Maria, RS, Brazil b ´ CPTEC, Instituto Nacional de Pesquisas Espaciais, Sa Jose dos Campos, SP, Brazil ˜o c ´ LAC, Instituto Nacional de Pesquisas Espaciais, Sa Jose dos Campos, SP, Brazil ˜o d ´ Istituto di Scienze dellAtmosfera e del Clima, CNR, Lecce, Italy e Faculdade de Meteorologia, Universidade Federal de Pelotas, Pelotas, RS, Brazil Received 11 October 2006; received in revised form 23 April 2007; accepted 2 May 2007Abstract A turbulent subﬁlter viscosity for large eddy simulation (LES) models is proposed, based on Heisenberg’s mechanism ofenergy transfer. Such viscosity is described in terms of a cutoff wave number, leading to relationships for the grid meshspacing, for a convective boundary layer (CBL). The limiting wave number represents a sharp ﬁlter separating large andsmall scales of a turbulent ﬂow and, henceforth, Heisenberg’s model agrees with the physical foundation of LES models.The comparison between Heisenberg’s turbulent viscosity and the classical ones, based on Smagorinsky’s parameteriza-tion, shows that both procedures lead to similar subgrid exchange coefﬁcients. With this result, the turbulence resolutionlength scale and the vertical mesh spacing are expressed only in terms of the longitudinal mesh spacing. Through theemployment of spectral observational data in the CBL, the mesh spacings, the ﬁlter width and the subﬁlter eddy viscosityare described in terms of the CBL height. The present development shows that Heisenberg’s theory naturally establishes aphysical criterium that connects the subgrid terms to the large-scale dimensions of the system. The proposed constrain istested employing a LES code and the results show that it leads to a good representation of the boundary layer variables,without an excessive reﬁnement of the grid mesh.r 2007 Elsevier Ltd. All rights reserved.Keywords: LES subﬁlter; Heisenberg model; Convective boundary layer1. Introduction (Deardorff, 1973; Moeng, 1984; Moeng and Wyn- gaard, 1988; Schmidt and Schumann, 1989; Schu- Large eddy simulation (LES) models represent a mann, 1991; Mason, 1994). In LES, only the energy-well-established technique to study the physical containing eddies of the turbulent motion are explicitlybehavior of the planetary boundary layer (PBL) resolved and the effect of the smaller, more isotropic eddies, needs to be parameterized. Modelling these ÃCorresponding author. residual turbulent motions, which are also referred as E-mail address: firstname.lastname@example.org (G.A. Degrazia). subgrid-scales or sub-ﬁlter motions (Pope, 2004) is in1352-2310/$ - see front matter r 2007 Elsevier Ltd. All rights reserved.doi:10.1016/j.atmosenv.2007.05.004
ARTICLE IN PRESS7060 G.A. Degrazia et al. / Atmospheric Environment 41 (2007) 7059–7068large part a phenomenological procedure based on horizontal subgrid characteristic length scales andheuristic arguments (Sullivan et al., 1994). the ﬁltered wavelengths of the inertial subrange In general, the governing equations in the LES from the observed turbulent spectrum in the CBL.models are the incompressible Navier–Stokes equa- As a phenomenological consequence, the magnitudetions described for a horizontally homogeneous of Dx, Dy and Dz are ﬁxed in terms of these residualboundary layer. The resolved turbulent ﬂow quan- turbulent wavelengths and described as a smalltities (wind components, pressure, temperature, etc.) fraction of the dominant convective length scale.are obtained by the application of a low-pass spatial In order to test the grid mesh spacing proposed,ﬁlter of characteristic width, the turbulence resolu- the LES model developed by Moeng (1984) istion length scale (Pope, 2004), smaller than the employed, where the subgrid parameterization isscales of the resolved turbulent motions. This low- based on Sullivan et al. (1994).pass spatial ﬁlter width has the same order ofmagnitude as the numerical grid dimensions and, 2. Subgrid scale model and spatial scales in a CBLbased on Kolmogorov spectral characteristics, canbe expressed in terms of the inertial subrange scales. In LES studies, one is trying to model eddies of According to Wyngaard (1982), ‘‘the parameter- inertial-range scales and to resolve explicitly theization of the residual stress term in the large eddy largest energy-containing eddies. For a well-devel-equation is dynamically essential; it causes the oped CBL, the typical Reynolds number is greatertransference of kinetic energy to smaller scales. than 107 and the inertial subrange is deﬁned by theThus, its parameterization is a key step in develop- interval Lblbm, where L is the integral scale, ling a large eddy model’’. Interpreting the turbulence characterizes the size of the inertial subrange eddies,ﬁeld as a superposition of myriads of energy modes, and m is Kolmogorov’s microscale. The integralthe parameterization of residual stress means to scale characterizes the energy-containing eddies,model the physical effects of a large number of which perform most of the turbulent transport,degrees of freedom (turbulent scales), which, by while Kolmogorov’s microscale deﬁnes the size ofvirtue of the ﬁltering application are not explicitly the viscous eddies that dissipate the turbulentresolved in a numerical simulation. energy. For a fully developed CBL, LE102 and The purpose of the present study is to show that mE10À3 m, and the eddy size range in the CBLHeisenberg’s classical theory for inertial transfer of covers roughly ﬁve decades (Sorbjan, 1989).turbulent energy (Heisenberg, 1948) can be used to Furthermore, since the Reynolds number is veryparameterize the residual stress tensor in LES high for the CBL, the effect of molecular diffusivitymodels applied to a convective boundary layer is irrelevant compared to the turbulent diffusivity,(CBL). The idea is motivated by the fact that in the and hence molecular viscosity will have negligibleinertial subrange the Heisenberg’s theory establishes direct inﬂuence on the resolved-scale motionsthat the transfer of energy from wave numbers (Mason, 1994). Therefore, the convergence of thesmaller than a given particular value to those larger statistics of the studied quantities with LES is notcan be represented as the effect of a turbulent expected to be affected by the dissipative scales.viscosity. This introduces a division between scales A frequently used parameterization for the residualat any arbitrary wave number in the inertial stress tensor tij in the large eddy models is expressedsubrange. This scale separation is indeed naturally by (Smagorinsky, 1963; Deardorff, 1973; Moeng,relevant in the LES frame, where a cutoff wave ´ 1984; Sullivan et al., 1994; Lesieur and Metais, 1996)number is arbitrarily chosen in the inertial range, qmi qmj introducing then an artiﬁcial sharp division, to which tij ¼ ÀnT þ , (1)Heisenberg’s approach seems well suited to be qxj qxiapplied. Therefore, in this paper, we employ where mi and mj are resolved velocity components, i,Heisenberg’s theory to obtain a kinematic turbu- j ¼ 1,2,3, corresponding to the x, y, z directions,lence viscosity (KTV), which represents the effects respectively, and nT is the turbulent eddy viscosityof the ﬁltered eddies of the inertial subrange. which is expressed as From a physical point of view, the expression forKTV provides a cutoff or limiting wave number, nT ¼ ck l 0 e1=2 , (2)separating the resolved scales from those ﬁltered. It where ck ¼ 0.1 is a constant, l0 is a mixing-lengthallows to establish a relationship between the scale related to the ﬁlter operation (Mason, 1994),
ARTICLE IN PRESS G.A. Degrazia et al. / Atmospheric Environment 41 (2007) 7059–7068 7061and e is the turbulent kinetic energy (TKE) of rated Fourier elements of the spectrum (statisticalsubﬁlter scales (SFSs). For neutral (zero surface independence between turbulent scales), is a prop-turbulent heat ﬂux) or unstable (positive turbulent erty of the inertial subrange turbulent eddies, whosesurface heat ﬂux) conditions, l0 is equal to the low- scales range from an arbitrary but ﬁxed wavepass LES ﬁlter width D. number to inﬁnity (since dissipative scales and Following Weil et al. (2004), the low-pass LES molecular viscosity are irrelevant, as mentionedﬁlter width is given by before). Therefore, the magnitude of KTV will #1=3 depend on the inertial subrange eddies that cause 3 2 the viscosity, and hence must depend on eddies withD¼ DxDyDz , (3) 2 wave numbers in the range (kc,N), where kc is a cutoff or limiting wave number for the inertialwhere Dx, Dy and Dz are the computational mesh subrange, which can be determinated from thesizes in the three coordinate directions x, y, z and experimental TKE spectra.the constant (3/2)2 accounts for the dealiasing. An eddy viscosity is the product of a character-Therefore, the ﬁlter width D is of the order of the istic turbulent length scale and a velocity, and thusnumerical grid dimension and the choice of their dimensional analysis yields (Heisenberg, 1948—seevalue may be guided by the typical scales of the also Hinze, 1975)CBL ﬂow. In fact, this choice will depend on the sﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ Z 1available computer power and the range of scales EðkÞthat needs to be actually resolved for the pertinent nT ¼ CH dk, (5) kc k3description of the investigated problem. For a LESresolving the energy-containing eddies in a CBL the where k is the wave number, CH is Heisenberg’sﬁlter width D can be chosen in the inertial subrange, dimensionless spectral transfer constant and E(k) iscloser to the integral scale, and far from Kolmogor- the three-dimensional (3-D) turbulence energyov’s dissipative scale. spectrum in the inertial subrange, with the following Furthermore, the turbulence dissipation rate e is form (Kolmogorov, 1941):parameterized as (Moeng, 1984) EðkÞ ¼ aK 2=3 kÀ5=3 , (6) 3=2 e where aK is the Kolmogorov constant. Assuming ¼ c , (4) l that the small-scale turbulence (inertial subrange)where ce is a constant (0.93). should act on the large-scale turbulence like an additional eddy viscosity we substitute Eq. (6) into3. Heisenberg model for the KTV Eq. (5), where CHE0.47 and aKE1.52 (Muschinski and Roth, 1993; Corrsin, 1963), to obtain In his classical work, based on intuitive argu- À4=3 nT ¼ 0:441=3 kc . (7)ments, Heisenberg (1948) assumed that the processof the energy transfer from the small to the large It is important to note that Eq. (7) was ﬁrstlywave numbers in a Kolmogorov turbulent spectrum derived by Muschinski and Roth (1993).is similar to the conversion of mechanical energy Following the philosophy of Kraichnan’s eddyinto thermal energy through the agency of mole- ´ viscosity in spectral space, Lesieur and Metaiscular viscosity. In other words, the physical picture (1996) present an eddy viscosity expressed bythat forms the basis of Heisenberg’s theory is that, 2 À3=2 Eðkc Þ 1=2 in the energy cascade process within the kinetic n T ¼ aK , (8) 3 kturbulent spectrum, the mechanism of inertialexchange of energy from large to small eddies is where E(kc) is the kinetic energy spectrum at thecontrolled by a KTV. Thus, the effect of the inertia cutoff kc.term can be regarded as equivalent to a virtual Now, substituting Eq. (6) with k ¼ kc in Eq. (8)turbulent friction, nT, produced by the small-scale yieldsturbulence (inertial subrange eddies), acting on the À4=3 nT ¼ 0:441=3 kc (9)large-scale turbulence (energy-containing eddies). nTrepresents the KTV, which as a consequence of the suggesting that Heisenberg’s eddy viscosity islack of correlation between large and small sepa- ´ identical to the one proposed by Lesieur and Metais
ARTICLE IN PRESS7062 G.A. Degrazia et al. / Atmospheric Environment 41 (2007) 7059–7068(1996, Eqs. (3.5) and (4.1)). A remarkable point here imposed by the observed horizontal homogeneity.is the consistency of the theoretical model (8) with This homogeneous behavior for a CBL sets anthe experimental value for the Heisenberg’s con- approximately constant value for lI for the u and vstant. components, valid for all heights in a CBL. At this point, spectral turbulent viscosities, such Horizontal homogeneity, therefore, naturally im-as those given by Eqs. (5) and (8) can be compared poses Dx as the characteristic length scale of thewith those given by Eq. (2) (Smagorinsky, 1963; ﬂow and makes kc ¼ p/Dx a well-deﬁned cutoffMoeng, 1984; Sullivan et al., 1994; Weil et al., 2004). wave number. This controlling length scale Dx canFor establishing this comparison we rewrite be derived from observational data for the CBL andEq. (4) as related to a characteristic scale of the physical 1 system, such as the mixed layer depth. This is done 1=3e1=2 ¼ 1=3 l 0 (10) in the next session. C 1=3 Now, inserting relation (12) into Eq. (7), Heisen-and substitute it into Eq. (2) berg’s eddy viscosity can be written as nT 4=3 nT ¼ 0:102l 0 . (11) ¼ 0:096ðDxÞ4=3 . (13)1=3 1=3 Next, one assumes that the cutoff wave number in For convective conditions, where l0 ¼ D, theFourier space comparison of Eqs. (11)–(13) leads to pkc ¼ . (12) Dx D ¼ l 0 ﬃ 0:96Dx % Dx. (14) In expression (12), Dx is the longitudinal mesh Taking into account relation (14), we can choosespacing, and lies within the kÀ5/3 Kolmogorov kc ¼ p/D, then one would ﬁnd ðnT ÞHeis =ðnT ÞSmag ¼spectrum (Eq. (6), inertial subrange). Furthermore, 0:96 % 1. It means that the models are actuallyall degrees of freedom (turbulent energy modes) equivalent, probably within the error bars of thewith k4kc. have been ﬁltered and, in this case, the experimental determination of the phenomenological ´ﬁlter is sharp in Fourier space (Lesieur and Metais, constants appearing in each model (CH and Ck).1996; Armenio et al., 1999). However, for deriving (14) an a priori assumption is The argument for the choice of Dx in Eq. (12) made, which consists in assuming the equivalencecomes from the physical properties associated to the between Smagorinsky and Heisenberg models. This isCBL. Observations show that a CBL presents a expected since both models arise from the samehomogeneous character in the horizontal directions. simple dimensional analysis arguments.Analyzing the turbulent velocity spectra, the vertical Considering that observational studies show thecomponent shows the largest height dependence existence of horizontal homogeneity of the turbu-among the three velocity components (Kaimal et al., lent properties of the CBL, one can assume that1976; Caughey, 1982). In fact, throughout the CBL Dx ¼ Dy. This means that the spectral peak for the udepth, both u (longitudinal velocity) and v (lateral and v velocity components occurs in the samevelocity) spectra display a well-established z-less frequency, and that this frequency is almost con-limiting wave number lI for the inertial subrange. stant with height (Kaimal et al., 1976; Caughey,This inhomogeneity of the turbulence in the vertical 1982). Employing Dx ¼ Dy in Eq. (3) and substitut-direction in a CBL makes the inertial subrange ing in Eq. (14), the following relation between Dzlimiting wavelength associated to the vertical and Dx results:turbulent velocity vary with height. As a conse-quence, for the turbulent vertical velocity, an 0:4Dx for D ¼ 0:96Dx; observational value for lI valid throughout the Dz ﬃ (15a, b) 0:44Dx for D ¼ Dx:depth of the CBL cannot be clearly established.Therefore, a relationship such as kc ¼ p/Dz is ill Expressions (14) and (15) directly relate D and Dzdeﬁned since lI for the vertical velocity is strongly to the longitudinal mesh spacing Dx. This resultheight dependent in the lower layers of the CBL. warrants that only one spatial scale is necessary toThis discussion shows that the choice of the determine all the mesh sizes. From Eq. (14): DEDx,horizontal length scale Dx as a characteristic scale we can choose kc ¼ p/D. Therefore, this choice willof the ﬂow is not an arbitrary assumption, but it is reduce the inaccuracies for the velocity ﬁeld in the
ARTICLE IN PRESS G.A. Degrazia et al. / Atmospheric Environment 41 (2007) 7059–7068 7063horizontal direction (since D is slightly smaller than curves are analytical functions approximating theDx), and it will preserve the relation DzE0.4Dx. data. These curves should be, therefore, viewed as reasonable approximations to the observed spectra.4. An expression for Dx and nT in terms of the CBL On the other hand, considering the spectral plotsheight provided by Kaimal et al. (1976) for the u and v velocity components the lower frequency limit (nzi/ Since the parameterization for the residual stress U) of the inertial subrange could be as large as 30,tensor must be based on inertial subrange properties which would result in lIﬃ0.03zi. If one uses this(small eddies) and that modeling the effects of the result in the Dx expression, Dx could be 0.015ziresidual turbulent motions on the resolved motions rather than 0.05zi. Although there is considerableis a somewhat empirical method (Sullivan et al., uncertainty in the lI estimate, which translates into1994), we can now utilize spectral observed data for uncertainty in the required grid size, our relationsa CBL to estimate Dx and consequently D and Dz in for Dx, D and Dz obtained from Heisenberg eddyterms of a limiting or cutoff wavelength for the viscosity and constructed from the experimentalinertial subrange. Accepting the observational data of the observed turbulent spectra in a CBL, canevidence that lI is approximately constant for the be now compared with those found in the literature.u and v components, as discussed in the previous With this purpose, we choose zi ¼ 1000 m to obtainsection, kc can be written as kc ¼ 2p/lI and there- Dx ¼ Dyﬃ50, Dﬃ48 and Dzﬃ20 m. These meshfore a comparison of this relation with kc ¼ p/Dx spacing sizes are in agreement with those selected byleads to Moeng and Sullivan (1994) and Weil et al. (2004), which were employed in a LES model to numeri- lI cally simulate the CBL. Indeed, the grid sizesDx ¼ Dy ¼ , (16) 2 selected by these authors support the conclusionwhich establishes a direct relationship between mesh established by Kaimal et al. (1976), meaning that forspacing and inertial subrange scales. According to the horizontal velocity components in the mixedatmospheric measurements (Kaimal et al., 1976; layer lIﬃ0.1zi. This observed limiting wavelengthKaimal and Finnigan, 1994; Caughey and Palmer, has also been assumed in another recent study1979; Caughey, 1982), the spectra of the velocity (Elperin et al., 2006).components in the bulk of the CBL can be Finally, we can introduce l0 ¼ D ¼ 0.048zi andgeneralized within the framework of mixed layer Dx ¼ 0.05zi, respectively, in Eqs. (11) and (13) tosimilarity. Therefore, experimental observations in obtain an unique expression for the subﬁlter eddythe CBL allow the determination of lI. Following diffusivity. In terms of the CBL height zi, thisKaimal et al. (1976), the onset of the inertial turbulent viscosity can be written assubrange for the u and v spectra in the CBL, occurs nT 4=3at limiting wavelength lIﬃ0.1zi, where zi is deﬁned ¼ 0:0018zi . (17) 1=3as the height of the lowest inversion base in a CBL.At this point, the substitution of lI ¼ 0.1zi in Eq. The fact that zi is present in the expressions for(16) yields Dx ¼ Dyﬃ0.05zi and, as a consequence Dx, D, Dz and nT is a consequence imposed by theof Eqs. (14) and (15), Dﬃ0.048zi and Dzﬃ0.02zi. observations accomplished in the CBL. In spite ofTherefore, Eq. (16) along with the relationship the inertial subrange convective length scale beinglI ¼ 0.1zi, imparts a physical spatial constraint, small, it can still be described in terms of thewhich helps the correct choice of the dimension of controlling convective scale zi, and this can bethe numerical grid in LES models. understood as a direct consequence of the fact that There is an approximate nature in the arguments all turbulent energy modes must scale as a functionof Kaimal et al. (1976). Although they conclude that of the energy-containing eddies.lIﬃ0.1zi in the mixed layer (0.1pz/zip1.0), they The present analysis considers only a highly CBL,state that the limiting wavelength (upper value) of where Àzi/L is quite large. Indeed, in a CBL withthe inertial subrange is lIp0.1zi. Furthermore, there signiﬁcant wind shear effects (Àzi/Lo10), theare at least two approximations in these results: (1) horizontal spectra near the surface (z/zio0.1), willn/U is used as an approximation to the wavelength, not look as similar to those above that height as dowhere n is the frequency and U is the horizontal spectra in the highly CBL. For example, themean wind speed; (2) Kaimal et al. (1976) spectral horizontal velocity spectra computed from LES by
ARTICLE IN PRESS7064 G.A. Degrazia et al. / Atmospheric Environment 41 (2007) 7059–7068Khanna and Brasseur (1998), for Àzi/L ¼ 8, when S2 have very similar proﬁles for the average zonalcompared with those from Kaimal et al. (1976), wind speed, as expected. Simulations S3 and S4present a different pattern of spectra curves. Close present a zonal wind vertical gradient inside theto the ground, there is a disparity in length scale mixed boundary layer greater than in the previousbetween the u and w spectra (Khanna and Brasseur, simulations, and a lower gradient in the entrainment1998). In fact, in the surface CBL, the horizontal, region. This gradient is steeper in S4 than in S3. Theunstable spectra appear to have a somewhat average zonal wind speed in the mixed layer isdifferent structure with a tendency toward having greater when the computational grid is coarser.two peaks (Panofsky and Dutton, 1984). Hojstrup However, these differences are not due to changes in(1981), describes these horizontal spectra by sums of the geometry (boundary layer height, for example)low-frequency (convective part) and high-frequency or the dynamic conditions in the system, since the(mechanical part) portions. The methodology pre- energy provided to the system is the same. There-sented in this study does not apply to a moderately fore, the differences shown in Fig. 1 are due to aunstable CBL in which shear is an important wrong representation of the CBL induced by a badturbulence production mechanism. choice for the mesh discretization. A comparison among the potential temperature5. Numerical experiments proﬁle is shown in Fig. 2. Again, it is noted that for coarser the grid resolution, the simulated CBL In order to provide numerical experimentation temperature decreases. There is a similarity betweenfor the approach developed in this paper, the LES the proﬁles S1 and S2, and between S3 and S4. It ismodel developed by Moeng (1984) is used here, with also noted that for a lower r (ﬁner grid) there is athe subgrid turbulence parameterization based on steeper gradient in the entrainment region, i.e., theSullivan et al. (1994). boundary layer height is better identiﬁed for the Some numerical experiments were carried out, ﬁnest grid resolution.where different grid spacing were adopted (Table 1). The SFS TKE fraction varies little with heightThe grid spacing is indicated by the ratio rDx/(zi)0, (Fig. 3), being approximately 0.11 over the bulk ofwhere (zi)0 is the initial height of the CBL, and Nj is the simulated CBL for cases S1 and S2 (thosethe number of grid points for each direction (j ¼ x, presenting smaller r ratios). This behavior is similary, z). The same domain (5 km Â 5 km Â 2 km) was to those simulated and discussed by Weil et al.considered for all simulations. As indicated, a value (2004). On the other hand, in the case of simulationsr ¼ 0.04 was employed in the simulation S1, whiler ¼ 0.05 was used in the simulation S2. Moeng andSullivan (1994, 2002) considered r ¼ 0.05, but in theliterature other values for r have been used, such asrE0.156 (Moeng, 1984) or rE0.1 (Hadﬁeld et al.,1991; Brown, 1996; De Roode et al., 2004).Antonelli et al. (2003) employed r ¼ 0.04, andSullivan et al. (1996) used several values rangingfrom r ¼ 0.01 up to 0.06. The simulations were performed to represent2.5 h of fully developed turbulence in the CBL. The initial boundary layer height is the same forall simulations. Fig. 1 shows that simulations S1 andTable 1Description of the simulationsSimulations r ¼ Dx/(zi) (Nx, Ny, Nz) (zi)0 (m)S1 0.040 (128,128,128) 1000S2 0.050 (96,96,96) 1000S3 0.078 (64,64,64) 1000S4 0.156 (32,32,32) 1000 Fig. 1. Average zonal wind speed for the simulations S1–S4.
ARTICLE IN PRESS G.A. Degrazia et al. / Atmospheric Environment 41 (2007) 7059–7068 7065 Fig. 2. Same as in Fig. 1, but for potential temperature. Fig. 4. SFS sensible heat ﬂux (S) and resolved scale sensible heat ﬂux (LE) as a function of height for the different simulations. with less TKE and sensible heat ﬂux is relevant, considering that in LES methodology one hopes that inertial subrange eddies contribute much less than the energy containing eddies. Therefore, from Figs. 1 to 4, it is possible to identify two distinct simulation groups: ﬁner resolu- tion (S1 and S2) and coarser resolution (S3 and S4). Additionally, it is veriﬁed that for an enhanced resolution (S1), the CBL properties are not much better represented than in the S2 simulation, i.e., it is not expected a better representation with a ﬁner resolution than that used in S2. Therefore, the criterion developed in this study (based on a physical constrain for the grid spacing) guides towards an optimized mesh size, which depends only on the CBL height. Recent testing of SFS models using observational data shows that the coefﬁcients Ck and Cs (Smagor-Fig. 3. SFS fraction of total TKE as a function of height for the inski coefﬁcient) in LES SFS models depend on thedifferent simulations. s2 and s2 are, respectively, the subﬁlter s LE ratio of lw/lI (Sullivan et al., 2003), where lw is theand resolved scale velocity variances. peak wavelength of the w-spectrum. For small lw/ lI, Ck and Cs approach zero and thus are notS3 and S4, that employed a larger r ratio, the SFS constant, as assumed in models with a sharp ﬁlterTKE fraction is larger, and, for S4, varies continually cutoff. Additionally, all of the SFS ﬂuxes vary withwith height. The SFS sensible heat ﬂuxes (Fig. 4) are lw/lI e.g., the SFS variances of the u and wlarger near the surface for simulations S3 an S4 than components are anisotropic for lw/lI near 1.0 andfor S1 and S2. In all cases, the linear sensible heat only tend toward the isotropic limit for lw/lI410 orﬂux proﬁle relationship is well simulated. The fact 20. However, ﬁeld observation in the bulk of a CBLthat, in simulations S1 and S2, the SFSs contribute (Caughey, 1979; Caughey, 1982) show that there is a
ARTICLE IN PRESS7066 G.A. Degrazia et al. / Atmospheric Environment 41 (2007) 7059–7068 models of a convective boundary layer (CBL). The theoretical framework is classical Heisenberg’s turbulent spectral transfer theory, which provides a kinematic turbulence viscosity (KTV), in terms of a cutoff or limiting wave number kc (Eq. (7)). In Heisenberg’s model, the KTV is invoked to explain the mechanism of inertial transfer of energy from large to small eddies. From a physical point of view, the KTV is assumed to represent the friction produced by the smaller eddies and acting on the larger eddies. The presence of this cutoff or limiting wave number in the subgrid turbulent viscosity introduced by Heisenberg’s model establishes aFig. 5. CPU-time for computational mesh with different number sharp ﬁlter in the turbulent energy modesof points in the grid. and, consequently, this theory is in good conformity with the main idea contained in LES models, in which energy-containing eddies (large scales)strong tendency for the peak wavelengths of all are explicitly resolved, whereas inertial subrangevelocity components to be the same and roughly eddies (small scales) are parameterized. Expressingequal to E1.5zi. Therefore, the substitution of the key physical quantity kc in terms of thelwE1.5zi and lIE0.1zi in the quotient lw/lI yields longitudinal mesh spacing Dx (Eq. (12)), wea value of approximately 15 for this ratio, meaning compare the Heisenberg’s turbulent viscositythat SFS motions are dominated by small-scale (Eq. (13)) with the classical one based on Smagor-turbulent eddies (Sullivan et al., 2003). As a insky’s model (Eq. (11)) (Moeng, 1984; Sullivanconsequence, in mid-CBL regions where the turbu- et al., 1994). The comparison showed that bothlence is well resolved (over the bulk of the simulated approaches (Eqs. (11) and (13)) provide a similarCBL, 0.1pz/zip1, the SFS TKE fraction is 0.11 in value for this subﬁlter eddy viscosity, and it leads tothe simulations S1 and S2), the local isotropy relationships for the ﬁlter width D (Eq. (14)) andassumption is consistent with most LES models. for the vertical mesh spacing Dz (Eq. (15)) only in Another important question to mention is the terms of Dx.CPU-time spent in the simulations. All simulations Finally, setting a relationship between Dx and thewere run in a sequential computer. The simulation approximately constant limiting wavelength for thetime could be reduced in a parallel machine, but the inertial subrange of a CBL (observational data,goal here is just a comparison among the simula- Eq. (16)) we describe Dx, D, and Dz as a minutetions. The comparison among the CPU-times is fraction of the CBL heigth zi. Therefore, withshown in Fig. 5. Dx, D, and Dz expressed in terms of zi an unique The simulation time is given in hours versus the formula for the turbulent viscosity, obtained fromnumber of the points in the computational grid. Eqs. (11) and (13), can be found. This KTVClearly, increasing the grid points more CPU-time is associated to the inertial subrange eddies can berequired (Fig. 5). There is a big difference among employed to parameterize the residual stress tensorthe CPU-time for the simulations, where S1 in LES models.simulation spent approximately 123 h and S2 The semiempirical analysis developed in thissimulation spent 37 h. Therefore, it is extremely work, which leads to the relationships between Dx,relevant to have a good representation for the D, Dz and nT in terms of zi shows that Heisenberg’sturbulence phenomena employing as little grids theory allied to observational data (heuristic argu-point as possible. ments) provides a physical basis to the choice of numerical values on the different formulas that6. Conclusion constitute the parameterization of the subﬁlter scales in LES models. In this study, we have developed relationships for From LES simulations, it was veriﬁed that if thethe subﬁlter scales in large eddy simulation (LES) physical constrain presented here is followed (S2
ARTICLE IN PRESS G.A. Degrazia et al. / Atmospheric Environment 41 (2007) 7059–7068 7067simulation), the result is similar to that obtained boundary layer. Part I: A small-scale circulation with zerowith a ﬁner grid (S1 simulation), but with signiﬁcant wind. Boundary-layer Meteorology 57, 79–114.reduction of the CPU-time. However, for S3 and S4 Heisenberg, W., 1948. Zur Statistischen Theorie der Turbulenz. Zeitschrift fur Physik 124, 628–657.simulations, Figs. 1 and 2 show different results Hinze, J.O., 1975. Turbulence. McGraw-Hill, New York.from those obtained with S1 simulation, indicating Hojstrup, J., 1981. A simple model for the adjustment of velocitythat both simulations are not a good representation spectra in unstable conditions downstream of an abruptfor modeling a CBL. In addition, in simulation S2, change in roughness and heat ﬂux. Boundary-layer Meteor-with the parameters suggested by the present study, ology 21, 341–356. Kaimal, J.C., Finnigan, J.J., 1994. 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