An Algebraic Subgrid-Scale Model for Large-Eddy Simulations of the Atmospheric Boundary Layer

Transcription

1 An Algebraic Subgrid-Scale Model for Large-Eddy Simulations of the Atmospheric Boundary Layer Rica Mae Enriquez and Robert L. Street Bob and Norma Street Environmental Fluid Mechanics Laboratory, Stanford University & Technical Report Number EFML February 2017

2 Abstract Large-eddy simulation (LES) resolves large eddies in the flow while modelling e ects of smaller motions (turbulence) on those larger eddies. Although the turbulence model can significantly a ect the accuracy of an LES, turbulence models which are known to be flawed are widely used. Here, a linear algebraic subgrid-scale model, that actively couples momentum and heat transport, is presented as an alternative. This non-eddy-viscosity model accounts for additional transport processes, including production, dissipation, pressure redistribution, and buoyancy terms. With the inclusion of an actively coupled turbulent heat flux, the current algebraic model is applicable to a range of atmospheric stability conditions for the dry atmosphere. LES using a mesoscale non-hydrostatic code at various resolutions in a neutral boundary layer flow shows that the algebraic model is a more physically complete model that accounts for near-wall anisotropies and generates better logarithmic-layer velocity profiles than simpler models. Comparison of spectra for the resolved vertical velocity showed that the new model supports more energetic representation of the smaller resolved scales. LES of a moderately convective boundary layer demonstrated that the model predicted the evolution of resolved quantities at least as well as simulations with simpler models, while including additional physics. Simulations of stable boundary layers complete the demonstration of the linear algebraic model. Finally, subgrid-scale anisotropy is shown to vary significantly as a function of grid anisotropy, but to have a relatively small e ect on mean profiles. Keywords Anisotropy Atmospheric boundary-layer Large-eddy simulation Subgrid-scale modelling Turbulence IntroductionandObjectives Large-eddy simulation (LES) originates from Deardor s simulations of turbulent channel flow (Deardor, 1970) and planetary boundary layers (Deardor, 1972), in which he used Lilly s (1967) eddy-viscosity model for subgrid-scale (SGS) turbulence. With its agreeable results and ability to elucidate the characteristics of eddies, LES and SGS turbulence modelling became an active area of fluid dynamics research. LES applies a spatial filter on the Navier-Stokes equations to numerically resolve larger energy containing eddies, while necessitating parametrizations of smaller-scale motions. In succeeding years many variations on Deardor and Lilly s original themes have appeared. They include dynamic and dynamic-mixed models (Chow et al., 2005; Zang et al., 1993); nonlinear models (Kosović, 1997); truncated equation models (Ramachandran and Wyngaard, 2011); and algebraic models (Rasam et al., 2013), for example. Meneveau and Katz (2000), Sagaut (2006), and Pope (2000) cover most of the field. Ludwig et al. (2009) provide a discussion of the suitability and inadequacies of several models. Generally speaking, most eddy viscosity models employ a scalar eddy viscosity so that the turbulent stresses are linearly related to the strain rate, which typically is not correct. Both nonlinear and algebraic models remove this restriction. The mixed models approach the problem di erently, e.g., in the dynamic-mixed model of Chow et al. (2005) the flow is separated into subgrid scale (SGS) and subfilter scale (SFS) domains. The latter is solved by reconstructing the SFS motions, while the SGS motions are modeled using the dynamic eddy viscosity method. Chow and Street

3 (2009) demonstrated that other SGS models can be used profitably, e.g., a TKE 1.5 method; see comment below regarding Enriquez et al. (2010). Most recently, Ramachandran and Wyngaard (2011) presented an approach in which they integrated a truncated set of subfilter-scale (SFS, which is used in place of SGS) conservation equations with applications to the atmospheric boundary layer (ABL). They stated: While including the dominant SFS production terms improves the parametrization of the SFS stresses and fluxes, the benefits of solving for them prognostically are less clear. One way to avoid solving extra transport equations would be to set the time derivative terms in the SFS conservation equations equal to zero and then solve the resulting set of equations algebraically for the unknown SFS stresses and fluxes. Such an algorithm will entail inverting matrices and will likely give rise to a fresh set of problems, such as, singularity of the matrix being inverted. Nevertheless, a truncated version of the SFS conservation equations that can be solved algebraically could, in principle, retain important SFS physics without incurring very high computational costs. Contemporaneously (Enriquez et al., 2010, 2012), we were developing a method exactly along these lines, viz., a general linear algebraic subgrid-scale model. We report here on this work. After presentation of model equations, we apply them in a mesoscale code to prove the concept and show relevant applications. The concept of algebraic stress models is not new (Rodi, 1976). The Rodi (1976) approach has been built upon by others. Building on the work of their colleagues, Marstorp et al. (2009) applied concepts of algebraic Reynolds stress models (ARSM) or explicit ARSM (EARSM) to LES (as did Findikakis and Street, 1979), and Rasam et al. (2013) coupled the Marstorp et al. (2009) EARSM with a passive Reynolds-averaged Navier-Stokes (RANS) algebraic heat flux model to study LES of channel flow. The models include production, pressure redistribution, and dissipation terms, and Rasam et al. (2013) s model was capable of producing SGS scalar flux anisotropy and predicting scalar profiles reasonably well, as compared with filtered DNS data. In a historical perspective, after unsatisfactory attempts in simulating a stable layer over a convective layer with a constant coe cient eddy-viscosity model, Deardor (1973) sought a more sophisticated treatment of the subgrid Reynolds fluxes. For the dry atmosphere, Deardor (1973) solved ten SGS turbulence transport equations: six SGS stress equations, three SGS heat flux equations, and one SGS potential temperature variance equation. For the moist atmosphere, Deardor (1974) had an additional five transport equations: three SGS water vapour flux equations, one SGS water vapour and potential temperature covariance equation, and one SGS water vapour variance equation. Although the transport equations could simulate the convective boundary layer, at the time, their implementation was cost prohibitive, and Deardor returned to using eddy-viscosity models. However, Deardor (1974) strongly suggested that a truncated version of the transport equations be developed for LES. Our work builds upon Findikakis and Street (1979) and Rodi (1976), rather than on Marstorp et al. (2009) and Rasam et al. (2013), because we choose to solve a set of algebraic equations for the SGS terms and not to create an approximate EARSM. Thus, our model is an implicit, generalized, linear, algebraic,

4 subgrid scale model; iglass hereafter. The Smagorinsky eddy-viscosity model (Smagorinsky, 1963), most commonly applied to LES for its simplicity, excludes backscatter, has a highly dissipative nature, and mistakenly aligns the strainrate and stress tensors. The iglass SGS turbulence model proposed here addresses these issues. In addition, the iglass SGS model can be used in conjunction with reconstruction to create a mixed model as demonstrated in Enriquez et al. (2010). There, we combined the RSFS model of Chow et al. (2005) with the iglass model (in the context of the Carati et al., 2001, formulation) and assessed the combined models performance in the neutral boundary layer, but do not present those results here. Wyngaard (2004) revisited the potential of abbreviated transport equations when analysing Horizontal Array Turbulence Study (HATS) data. He found that production terms in the scalar flux transport equation were important in accurately predicting the turbulent scalar flux. He created a simple rate-equation model, which included these production terms, a time-change term, and a simple pressure-destruction term. Hatlee and Wyngaard (2007) further examined HATS data to determine if a fuller rate-equation model would improve prediction of the turbulent scalar flux and stress. This fuller rate-equation model incorporates advection, turbulent di usion and transport, and buoyant production terms in addition to the simple rate-equation. They concluded that a simple rate-equation model with advection was adequate for parametrizing scalar flux, but to accurately predict turbulent stress, they also needed to 1) include rapid-mean-shear and buoyancy terms in modelling the pressure-destruction term, and 2) add the buoyant production term. Implementation of the extended simple rate-equation models into LES of the ABL by Ramachandran and Wyngaard (2011) neglects the inclusion of rapid-mean-shear and buoyancy terms in modelling the pressuredestruction term, but includes advection and buoyancy production terms. Their comparisons of turbulent statistics of LES and HATS demonstrate that additional terms in SGS models allowed for anisotropic production, but underpredicted turbulent quantities. They hypothesize that these underestimates are due to omission of the rapid-mean shear term. In this paper, we begin with development of the iglass model. We then present an analysis of the SGS stress model in the neutral boundary layer. Next, we show the usability of the model in LES of convective and stable boundary layers. For the neutral and stable cases, the anisotropy of the SGS stresses as a function of grid anisotropy and buoyancy is examined TheSubgrid-Scale(SGS)TurbulenceModel The dry-air dynamics (under the Boussinesq approximation with buoyancy e ects and Coriolis forces) and thermodynamics equation set is the basis for the LES; see Enriquez (2013) for details. The SGS terms for the Reynolds stress R ij and heat flux H i are given by: R ij = u i u j u i u j, (1) 135 and H i = u i u i. (2)

5 The evolution equations for R ij and H i can be created with a methodology similar to those of Lilly (1967) and Wyngaard (2004), where the overbar,, denotes filtered variables. We neglect Coriolis terms for the turbulent scales. Our SGS stress and heat flux equations to model + u k = R i R kj + 1 * k i i + j j j + + Hi j3 + H j k ij u i u j u k u k u i u j u j u i u k u i u j u k + 2u i u j u k uj p u j p ik + u i p u i p jk, k o 140 i i i + u k = R ik H k + 1 * k i - ( + @u i k - o u i u k u i u k u k u i u i u k + 2u i u k ( + k p k o ik. (4)! The set of equations is written in Einstein notation, where u i represents velocity, o reference density, p pressure, kinematic viscosity, gravitational acceleration, o reference potential temperature, potential temperature, thermal di usivity, and ik is the Kronecker delta. Wyngaard (2004) corrected the SGS stress by also including the slow production term and creating an implied tensor eddy-di usion model. This production term appeared to be significant when his simple rate-equation model was analysed with Horizontal Array Turbulence Study (HATS) data. Hatlee and Wyngaard (2007) expanded this rate-equation model to include buoyancy generation and the rapid production term, which showed substantial improvement in calculating the mean rate of energy transfer over the simple rate-equation. Accordingly, using this information and guided by the experiences of others cited above, we chose to retain production, pressure redistribution, dissipation, and buoyancy generation, while transport and di usion, and pressure transport terms are neglected; thus: 0 = R k 2 i R kj + 1 * k j + k - o i + j j + - Hi j3 + H j i3, (5)

6 5 155 and 0 = k ( + ) i H k + 1 * k @u i k - o i3. (6) These SGS stress equations can be solved algebraically (i.e., by matrix inversion and so implicitly) if pressure redistribution and dissipation terms are modelled. Moelcular di usion of heat is neglected and, the last term above (SGS heat flux buoyant production) is neglected also, as Ramachandran and Wyngaard (2011) showed it to be small compared to other terms. It is worth noting that contraction of Eqs. 5 and 6 yields the SGS TKE and temperature variance equations, in which the pressure-strain terms do not contribute to the SGS kinetic energy. Consequently, the TKE represents a balance between production, dissipation and buoyancy terms and backscatter is possible; however, in the neutral boundary layer (NBL), this model does not allow backscatter when used as a pure SGS model. When the SGS model is combined with reconstruction, backscatter would be allowed in the NBL; see Enriquez et al. (2010). Pressure redistribution terms are the sum of rapid pressure-strain and slow pressure-strain terms. Separation of the pressure redistribution term into rapid and slow components is commonly done in RANS modelling (Pope, 2000). To extend the model to buoyant flows, an additional buoyant force term is included (Gibson and Launder, 1978). We justify using a RANS model for the LES SGS pressure-strain tensor because it is the high frequency (i.e., the SGS) fluctuating components that are modelled, and Findikakis and Street (1979), Marstorp et al. (2009), and Perot and Gadebusch (2009) argue that the same modeling approach can be used in both cases. The SGS stress total pressure-strain term, ij, is qualitatively broken into a slow pressure-strain term, which involves interactions of fluctuating quantities, a fast pressure-strain term, which involves interactions with the mean rate of strain, a buoyant force term, and a wall e ect term of the slow and rapid pressure-strain. Since the SGS model includes production, pressure redistribution, dissipation, and buoyancy generation terms, it is applicable to a range of atmospheric stability conditions for the dry atmosphere. The SGS stresses are solved as a system of linear equations and are fully coupled to the set of equations that model the active SGS heat flux. Thus, 0 = R k R k + ij 2 3 ij + o Hi j3 + H j i3 (7) 186 and 0 = k H k + i, (8)

7 6 c 1 c 2 c 3 c 4 c 5 c 6 c 7 c 8 c c 1 c Table 1 iglass model coe cients where D ij = P ij =!!! 2 ij = c 1 R ij e 3 e 2 ij c 2 P ij 3 P 2 ij c 3 es ij c 4 D ij 3 P ij {z } {z } Slow Pressure-Strain! 2 c H i j3 + H j i3 o 3 H 3 i3 ij {z } Buoyant Force Rapid Pressure-Strain!! 2 + c 5 R ij e 3 e ij + c 6 P ij c 7 D ij + c 8 es ij f (z), (9) {z } Wall-Pressure E ects R k + R jk, S ij i R k + R k j, P = R i!, f (z) = 0.27 z z. The SGS heat flux pressure redistribution term, i, is modelled i i = c 1 H i + c 2 H k. (10) {z e {z k } Slow Pressure-Strain Rapid Pressure-Strain The wall function, f (z), relates local vertical grid scale, z, with distance from the surface, z. Modelcoe cients shown in Table 1 are distilled from Craft and Launder (2001); Hanjalić (2002); Launder et al. (1975); and Shabbir and Shih (1992). For dissipation,, based on the analysis of Moeng and Wyngaard (1988), we use the standard parametrization (see Moeng, 1984): 195 where = C e1.5, (11) B B = 8 > if N 2 apple 0, <>: q e 0.76 if N 2 (12) > 0, N and N is the Brunt-Väisälä buoyancy frequency, = ( x z ) 1/3 is the grid scale, and the x-directional, and y-directional scales are x and. In our earliest simulations (for the neutral boundary layer), the SGS turbulent kinetic energy (TKE or e), was obtained from the TKE transport equation contained in the Advanced Regional Prediction System (ARPS, see Sect. 3), which is based on Deardor (1980) and Moeng (1984) formulations. To match this model, we use C = For other simulations, rather than use the ARPS TKE equation,

8 we use the more e cient SGS TKE model from Yoshizawa (1986) and C = 1.2. We modified the TKE model to allow for buoyancy e ects, cf., the ARPS TKE implementation. The modification is analogous to adjustments that allow the Smagorinsky model to account for static stability changes, so e = 8 > <>: S 2 N 2 /Pr t if N 2 < 0, 9> = S 2 if N 2 0, > ; (13) 207 where Pr t is the turbulent Prandtl number (= 1/3 here) and S 2 = 1 2 S ijs ij ImplementationoftheiGLASSModel The Advanced Regional Prediction System (ARPS), a modular code appropriate for LES resolutions (Chow et al., 2005; Xue et al., 2000, 2001) was used. ARPS is a three-dimensional, compressible, non-hydrostatic, and parallelized code. Our simulations are on idealized periodic domains. Initialization and forcing are described for each application. For the neutral boundary layer, we used a modified version of ARPS in which density and potential temperature are constant (Chow et al., 2005). For the convective and stable boundary layers, the original compressible code was used. For resolved variables, ARPS accounts for acoustically active terms related to compressibility by using a mode-splitting scheme with smaller time steps for the acoustically active terms. For its SGS turbulence models, ARPS addresses compressibility by using the product of the local density and either the SGS stress or heat flux in computing the turbulent mixing term for resolved variables; divergence terms are small and are neglected in turbulence models. Our turbulence model was developed within an incompressible fluid framework (cf., Chow et al., 2005); however, we are able to account for local density di erences using the same approach as ARPS, and we make the same approximation to neglect divergence e ects. To solve for SGS stresses and heat flux terms we adopted ludcmp and lubksb subroutines from Numerical Recipes in Fortran edited by Press et al. (1992). At each grid point, a set of six linear equations are solved for the SGS stresses and a subsequent three linear equations are solved for the SGS heat fluxes. This is consistent with the solution sequence in ARPS for resolved variables; however, one could solve all nine equations simultaneously. Our tests showed that this does not change results, but increases cost due to the need to invert larger matrices. If a matrix is singular, we set SGS stresses and heat fluxes to zero. Setting SGS stresses and heat fluxes to zero when a matrix is singular can be considered a numerical regularization, cf., Gatski and Speziale (1993). Our regularization has a parallel in the clipping procedure for dynamic SGS models where the dynamic eddy-viscosity is prevented from becoming more negative locally than molecular viscosity. In the absence of clipping, a negative total viscosity leads to numerical instabilities (Chow et al., 2005; Zang et al., 1993). As a test, we ran the convective boundary layer case with the 0.3 K m s 1 surface heat flux discussed below for 500 simulation seconds. The planar- and time- averaged regularization probabilities were no more than 0.1% in any horizontal plane. Thus, the iglass model is infrequently bypassed because of matrix singularities, and use of LU decomposition with the chosen algorithms is stable for the flows tested.

9 LESoftheNeutralBoundaryLayer First, we simulate the rotation-influenced neutral boundary layer (NBL) used by others (Andrén et al., 1994; Chow et al., 2005; Kosović, 1997; Lu and Porté-Agel, 2010; Ludwig et al., 2009; Porté-Agel et al., 2000; Sullivan et al., 1994). The setup of our simulations is similar to runs by Chow et al. (2005) and Ludwig et al. (2009). The major di erence is that we used larger domains in order to generate turbulent statistics for our coarse resolution runs. Table 2 summarizes parameters used for these NBL runs. Runs were initialized with an Ekman-like spiral for mean velocities and a small random perturbation was added to the x-component of the velocity, u. The flow was driven by a constant pressure gradient that matches a geostrophic wind of (U, V ) =(10,0ms 1 )atalatitudeof45n. The Coriolis parameter, is f, is on the order of 10 4 s 1. Surface fluxes were computed with an instantaneous logarithmic drag law, in which a drag coe cient analogous to a roughness length of 0.1 m was used. Three di erent domains are used in order to detect e ects of resolution and aspect ratio on LES results. Table 3 lists run parameters altered for each run. We ran a suite of simulations with a horizontal domain size of 2.6 km 2.6 km for the higher horizontal resolution cases of x = 8 m, 16 m, and 32 m. We also ran simulations using the dynamic Wong and Lilly (1994), and Smagorinsky (1963) SGS turbulence models with this domain size. A domain that is twice the size, 5.2 km 5.2 km, is used to accommodate larger horizontal grid cells and to look at simulations with aspect ratios (ARs) = 3.2, 6.4, and 9.6. With this larger domain we assessed iglass with x =32m,64m,and96m,andaminimum vertical resolution of z,min = 10m.Additionally,weexaminedtheiGLASSmodel performance with x =32mand z,min = 5 m to obtain AR = 6.4 on the larger domain. We ran some special cases with a domain size of 1.3 km 1.3 km in order to see if grid anisotropy a ects SGS and resolved stress anisotropies. An isotropic grid with x = 8 m is used with iglass8b. Runs iglass16c - iglass16e use x = 16 m and varying z,min from 6.7 m to 10 m. In the vertical, a stretched grid is used, with z,min spacing near the bottom and a maximum grid spacing near the top of the domain. The stretching is defined by a hyperbolic tangent function (see CAPS, 1995) with an average spacing, z,a for each simulation equal to the domain height, H = 1500 m divided by nz - 3. Specific vertical spacing parameters for each simulation can be found in Table 3. iglass model coe cients stated in Table 1 are used for all resolutions and ARs. The Smagorinsky model coe cient, C s is set to 0.18 (Chow et al., 2005; Sullivan et al., 1994). The dynamic Wong-Lilly (DWL) run uses the Brown et al. (2001) near-wall canopy model with a scaling factor of 0.5 and a cuto height of 4 x. Mean and turbulent statistics were gathered after statistically steady-states were reached. We use non-stationarity measures C u and C,definedbyAndrénetal. (1994) or Chow et al. (2005) to locate these times (Enriquez, 2013). As expected, inertial oscillations were present. While C u and C did not reach unity, we use solutions as close to a statistically steady-state as possible in order to lower e ects of unsteadiness on first-order quantities (cf., Andrén et al., 1994). For example, our simulation run times were about three times as long as those of Andrén et al. (1994) for the 32-m resolution runs. As resolution increases, these inertial oscillations are damped much quicker and averaging periods become shorter.

10 9 Domain height H =1500m Small domain: width length 2.6 km 2.6 km Large domain: width length 5.2 km 5.2 km Special domain: width length 1.3 km 1.3 km Geostrophic wind (U, V ) =(10,0)ms 1 Coriolis parameter f (45 N) 10 4 s 1 Lateral boundaries Periodic Bottom boundary Rigid, rough; Log BC Roughness length 0.1 m Table 2 NBL LES general parameters. Run name (nx, ny, nz) x (m) z,min, AR t,l (s) t,s (s) z,a (m) iglass8 (323, 323, 163) 8 2.5, iglass16 (163, 163, 83) 16 5, iglass16b (163, 163, 83) 16 16, DWL16 (163, 163, 83) 16 5, iglass32 (83, 83, 43) 32 10, DWL32 (83, 83, 43) 32 10, Smag32 (83, 83, 43) 32 10, iglass32b (163, 163, 43) 32 10, iglass32c (163, 163, 83) 32 5, iglass64 (83, 83, 43) 64 10, iglass96 (43, 43, 43) 96 10, iglass8b (163, 163, 53) 8 8, iglass16c (83, 83, 43) 16 10, iglass16d (83, 83, 53) 16 8, iglass16e (83, 83, 63) , Table 3 NBL LES run parameters include nx, ny, nz, thenumberofgridnodesineach direction, x,thehorizontalresolution, z,min, theminimumverticalresolution,ar, theaspect ratio, t,l,thelargetimestep,and t, s, thesmalltimestep.thetopsectionlistsrunsdone on the smaller 2.6 km 2.6 km domain. The middle section identifies runs done on the larger 5.2 km 5.2 km domain. The bottom section records special tests done on a 1.3 km 1.3 km domain Velocity Profiles Blackadar and Tennekes (1968) showed, via scaling analysis, that the near-wall region ( 10% of the boundary layer depth) of the turbulent Ekman layer should follow a logarithmic law. Typical eddy-viscosity models overpredict shear in the model, with velocity too low at the wall and too high further away from it (Andrén et al., 1994). Figure 1 shows the mean velocity, U M, profiles normalized by the friction velocity, u, using Smagorinsky, DWL, and iglass SGS models. In Fig. 1a, the Smagorinsky model overestimates mean velocities greatly in the upper portion of the logarithmic zone. DWL and iglass models adhere to the theoretical logarithmic law closely in this zone, so individual lines are di cult to di erentiate. Normalized velocities of coarser iglass runs on the larger domain also appear to adhere to the logarithmic law (Fig. 1b). However, at a constant x = 32m, an increase in AR, from 3.2 to 6.4 (comparison of iglass32b and iglass64), leads to an underestimation of normalized velocities at points nearest to the surface. If z,min remains the same, and x increases, discrepancy from the logarithmic law is minimal (comparisons of iglass32b to iglass64 to iglass96).

11 10 (a) (b) U M u * -1 U M u * iglass8 iglass16 iglass32 DWL16 DWL32 Smag32 Exact 10 iglass32b, AR = 32:10 = 3.2 iglass32c, AR = 32:5 = 6.4 iglass64, AR = 64:10 = 6.4 iglass96, AR = 96:10 = 9.6 Exact zh zh -1 Fig. 1 Mean wind speed, U M,profileswiththeexactlogarithmicvelocitylawfor(a)Smagorinsky (Smag), DWL, and iglass models and (b) iglass models with di erent aspect ratios (ARs). The friction velocity, u,andtheboundarylayerheight,h,normalizeu M and z Increasing AR can reduce required computational time, but can alter solutions. As AR increases, grid cells become more pancake-like and resolved eddies become distorted. Consequently, LES results will depend more on the SGS model closer to the surface (cf., Brasseur and Wei, 2010). In order to examine e ects of AR, we use a larger 5.2 km 5.2 km domain (see Enriquez, 2013). Profiles of iglass32 and iglass32b match well, and inform us that the larger domain size does not a ect results. As AR increases, disagreement with the logarithmic law slightly increases, but at AR = 9.6 with x =96m, iglass outperforms the Smagorinsky model at AR = 3.2 and x =32min predicting a logarithmic behaviour for the NBL Vertical Velocity Snapshots, Correlograms, and Energy Spectra Spatial patterns of vertical motions vary with altitude and SGS model (Ludwig et al., 2009). They reported unrealistic elongated stripes in Smagorinsky plots, but smaller contiguous areas with more complete SGS models and higher resolution. Figure 2 depicts snapshots of vertical velocity, w, patterns of simulations with x =32m,16m,and8matelevations z 10 m and 150 m. The iglass model incorporates more physics, so at all resolutions, it allows representation of granular behaviour and smaller resolved scales near the surface, i.e., having a more active spectra at the smaller scales of the resolved motions. Figure 7 of Ludwig et al. (2009) suggested that more sophisticated turbulence models resolved smaller formations. Kirkil et al. (2012) reported that the size of Smagorinsky instantaneous stream-wise velocities were much larger than laboratory tests and that more sophisticated turbulence models produced structures that were more realistic in size. While SGS model contributions are concentrated in unresolved regions, such as near the surface, SGS models may influence flow field far from the wall (Juneja and Brasseur, 1999; Ludwig et al., 2009). The Ludwig figures at z 150 m show that di erences in patterns are still present at higher altitudes.

12 Vertical Velocity (cm s ) at z 10 m km iglass8 2.6 km iglass km iglass32 11 Vertical Velocity (cm s ) at z 150 m Fig. 2 Vertical velocity, w, at z 10 m (left panel) and z 150 m (right panel) with black contour lines of w = 0 cm s 1 for iglass32, iglass16, and iglass8 at t = 300,000 s.

13 12 Fig. 3 Normalized correlograms of vertical velocity at z 10 m, 50 m and 150 m for iglass32, iglass16, iglass8, DWL16, and DWL32. Isopleths are shown for correlations of 0.4 (largest curves), 0.6, and 0.8 (smallest curves). The isopleth diagram planar scale is non-dimensionalized by grid size,,asindicatedonthescalingbar;thus,the0.4isoplethof DWL32 at z 10 m is physically about twice the size of the same isopleth for DWL While these instantaneous w plots provide a good representation of di erent patterns that these models may yield, the normalized correlogram of w (Fig. 3) can provide a more quantitative view of structures. Correlograms show isopleths of w correlation coe cients between a point and surrounding points, as derived from all points at each elevation. Each correlogram is an average from nine di erent snapshots, and is normalized by. Data every 1250 s, from t =135, ,000 s, is used for iglass8 results. iglass16 and DWL16 results are sampled every 2500 s, from t = 150, ,000 s. Data every 5000 s, from t = 200, ,000 s, is used for iglass32 and DWL32 results. Isopleths, moving from the outside to the inside, are associated with correlation coe cients of 0.4, 0.6, and 0.8. In general, w correlation structures get larger with increased z. Athigheral- titudes, because allowable scales in the flow grow with distance from the ground, there are relatively fewer small scales and more larger flow-features. This a ects the pattern of the correlogram, making it larger relative to the grid scale. We normalize correlograms by in order to understand how iglass and DWL use information from neighbouring points. At the highest elevation, DWL correlograms do DWL16 iglass32 Fig. 4 Normalized correlograms of vertical velocity at z 10 m for iglass32 and DWL16. Isopleths are shown for correlations of 0.4 (largest curves), 0.6, and 0.8 (smallest curves).

14 E w u * -2 z iglass8 iglass16 iglass32 DWL16 DWL32 Smag κz Fig. 5 Averaged one-dimensional spectra of vertical velocity obtained from iglass8, iglass16, iglass32, DWL16, DWL32, and Smag32 at z 150 m not change much with, suggesting that correlations are still dependent on grid resolution. In addition, normalized correlograms from the DWL model are larger than those for iglass. At x = 32 m, iglass correlograms are 5 smaller than DWL correlograms; at x = 16m, thedi erence drops to 4. The test filtering used in the DWL is greater than and a function of grid size, probably leading to the larger-scale correlated structures. The iglass model is a more localized model, as evidenced from smaller correlograms, consistent with the results of Kosović (1997). Likewise, Ludwig et al. (2009) showed that better models produce smaller and more isotropic correlograms. While adding additional physics to the turbulence model, iglass at a given resolution is able to resolve structures that are about the same size as structures resolved by DWL at twice the resolution. Examination of the correlograms of these simulations (Fig. 4), taken at t = 200,000satz 10 m under neutral conditions, shows that iglass32 contours are of essentially the same diameter as those of the DWL16 simulation. The cost of producing these smaller structures at 16 m resolution using DWL, is 14X (1400%) more than using iglass at 32 m. Using iglass is a much more economical choice, while also providing improved simulation results. In order to further illustrate di erences between SGS models, we examine energy spectra of w at z 150 m for runs done on the smaller domain. Spectra are calculated from one-dimensional Fourier transforms of w that are then averaged in the horizontal direction and in time. At each resolution, we use the same time averaging, as described for the correlograms. Spectra are normalized by u 2 and

15 the height at which spectra were sampled. Then, the wavenumber,, is normalized with this height yielding a nondimensional plot. Figure 5 confirms that as x decreases, smaller scales are represented, with the tails of the spectra ending at 2 x. The changes in the spectra can be seen in previous instantaneous vertical velocity plots. Di erences between iglass, DWL, and Smagorinsky spectra are most noticeable at the smaller resolved scales. While Bryan et al. (2003) pointed out that di erences in the spectrum region less than 6 x can be influenced by computational mixing and should not be used for comparison, we can still see that at a given resolution, the iglass models have a more energetic spectrum than DWL and Smagorinsky models do, i.e, extending the representation of the smaller resolved scales Anisotropy in the Neutral Boundary Layer Coherent structures and anisotropic turbulence are an integral part of the atmospheric boundary layer (Dubos et al., 2008). While Sullivan et al. (2003) show the anisotropy of turbulence in the HATS near-ground field data, Biferale et al. (2004) cite the quantification of anisotropic e ects in small-scale turbulence as a theoretical and practical challenge and a first-order question for near-wall LES. A method to quantitatively measure anisotropy of SGS turbulence and the connections amongst SGS components is by examining two invariants of the anisotropy tensor, b ij (Lumley, 1979). These invariants take into account all six of the SGS stresses, not just the normal stresses. As defined in Pope (2000), the anisotropy tensor, b ij, and its invariants, and are: b ij = R ij R kk 1 3 ij (14) 2 = 1 3 ( ) (15) 3 = ( ). (16) and 2 are the first and second eigenvalues of the anisotropy tensor. All realizable sets of and lie within the Lumley triangle (Pope, 2000) and characterize the state of anisotropy (Figs. 6-7). The location of and can illustrate if turbulence is stretched in one dominant direction, stretched in two dominant directions, or is isotropic. Simonsen and Krogstad (2005) describe the shape of the stress tensor in the context of the triangle, and we adopt their view rather than attempting description of eddy shapes. Points that lie on the left side of the Lumley triangle define stress tensors with the shape of pancakes (two dominant directions), while those on the right side define rods (one dominant direction). Based on Simonsen and Krogstad (2005), rod-like tensors have their greatest principal stress aligned with the streamwise direction in space, while the smallest principal stress will be in the stream-normal (i.e., typically vertical) direction for pancakes. Coloured shapes on the Lumley triangle in Fig. 6 are at six vertical points nearest the surface (Table 3). We see that iglass8, iglass16, and iglass32 data fall within the same areas (Fig. 6a). Turbulence steadily moves from a region of two-component anisotropy to isotropy (toward the origin in the figure) with distance away from the surface. Turbulence anisotropies for AR = 6.4 (iglass32c

16 15 (a) (b) Components Components η η iglass8 iglass16 iglass32 HATS Data iglass8b iglass16b 0.1 Isotropic Isotropic 0.1 ξ (c) ( Components η η Isotropic 0.1 ξ AR = 1.0 AR = 1.6 AR = 2.0 AR = 2.4 AR = 3.2 Fig. 6 Lumley triangle of iglass SGS stress anisotropy tensors. Colours symbolize SGS anisotropy at six points nearest to the surface. Red shapes are closest while violet shapes are furthest from the surface. Data shown are from runs with (a) aspect ratio (AR) = 3.2 (iglass8, iglass16, and iglass32), (b) AR = 1.0 (iglass8b and iglass16b) with a HATS datum at z = 6 m (Chen et al., 2009), and (c) AR = 1.0, 1.6, 2.0, 2.4, and 3.2 for iglass runs with a horizontal resolution of 16 m and iglass64) and AR = 9.6 (iglass96) show the same movement and also pancake-shaped stress tensors. Figure 7 shows that SGS HATS data describe SGS turbulence as rod-like, while the iglass SGS turbulent stress is more of a pancake shape in this near-wall region. Chen et al. (2009) report that the SGS turbulence of the Smagorinsky model, split model (Sullivan et al., 1994), and nonlinear model (Kosović, 1997) also have two dominant components. Other LES turbulence models not listed here also provide anisotropy, e.g., the subfilter-scale stress reconstruction model of Chow et al. (2005) provide stress anisotropy because the calculation reconstructs subfilter-scale terms using resolved velocities, which contain most of the anisotropy. Interestingly, the level of grid anisotropy in all of these simulations (i.e., the AR) is on the order of 3.0. Pancake SGS turbulence, seen near the surface in

17 Components η 0.1 Resolved SGS HATS Data iglass Smagorinsky Kosović 0.1 Isotropic 0.1 Fig. 7 Lumley triangle of stress anisotropy tensors. Squares represent resolved anisotropy while circles represent SGS stress anisotropy. HATS data at z = 6 m (Chen et al., 2009) are shown as black shapes, along with SGS anisotropy data. ξ simulations, may be due to the fact that resolved variables drive SGS models, which then reflect the pancake-shape of LES grids. HATS measurements are not so restricted. As seen in Fig. 6a, as we move further from the surface, points migrate toward the isotropic state. While this behaviour is in large part a reflection of the physics, in iglass simulations, the grid is stretched in the vertical and so, becomes more isotropic with height, and turbulence is more resolved. If we compare resolved turbulence structure from HATS and iglass, we see that they agree well, but resolved eddies are still pancake-shaped (Fig. 7). Kaltenbach (1997) notes that dividing an isotropic turbulence field in resolved and subgrid scales using an anisotropic filter can misrepresent the anisotropy of the resolved stresses as well as SGS stresses. We ran two isotropic grids to show the anisotropy iglass produces with an isotropic grid. The first one has x =8mand z,min =8m(iGLASS8b),andthe second one has x =16mand z,min =16m(iGLASS16b).iGLASS16bhasthe coarsest z,min of all runs and may help elucidate whether or not stress anisotropies are influenced by grid anisotropy or turbulence model. Figure 6b shows that with an isotropic grid the SGS turbulent stress is more of a rod-like shape, which is to be expected in a region near the surface. Similar to the results with anisotropic grids, the level of anisotropy decreases as one moves further away from the surface. Even with a vertical resolution of 16 m near the surface, the correct anisotropy shape can be recovered with an isotropic grid, thus, supporting the idea that grid AR plays an important role. While producing LES runs with AR = 1.0 is cost prohibitive, because of the need for fine vertical resolution, it becomes clear that one should consider the e ect of ARs when interpreting LES data. Furthermore, we see that for a given x = 16 m, turbulence moves from rod-like to pancake-like with increasing AR (Fig. 6c). Since AR = 1.6 still provides rod-like turbulence, results suggest that there is no single crossover point. Sullivan et al. (2003) plotted turbulence anisotropy for various HATS cases (using an isotropic filter), and showed, on average, that the SGS stress was rod-like. However, one case

18 17 Domain height H =2km Domain width length 10.2 km 10.2 km Large time step t,l =0.5s Small time step t,s =0.05s Horizontal resolution x =40m Average vertical resolution z,a =20m Minimal vertical resolution z,min =10m Reference potential temperature o =300K Geostrophic wind (U, V ) =(20,0)ms 1 Coriolis parameter f (40 N) s 1 Lateral boundaries Periodic Bottom boundary Rigid, rough; M-O BC Roughness length 0.01 m Table 4 General CBL LES characteristics with strongly stable conditions, displayed SGS eddies that were pancake shaped because vertical motions were prevented by stratification Summary for NBL The iglass prediction of a logarithmic velocity profile was better than that of the Smagorinsky model and comparable to that of the DWL model. While we have not formulated an LES setup that allows for total grid independence, iglass displayed near surface logarithmic behaviour well with a wide range of resolutions and aspect ratios that are commonly used in practice. Additionally, adjustments to model coe cients are not necessary to achieve these results. After examining vertical velocity snapshots, normalized correlograms, and energy spectra, we see that, at the same resolution, iglass allowed more energetic support of the smaller resolved eddies than did DWL and Smagorinsky models. Use of Lumley triangles showed the dependence of SGS anisotropy on grid aspect ratio. iglass provided near-wall stress anisotropies that eddy-viscosity models do not inherently have. On an anisotropic LES grid, iglass results were consistent with other SGS models that provide anisotropy. iglass LES runs with an isotropic grid showed that aspect ratios can have a great influence on the shape of the SGS stress anisotropy tensors, and led to better agreement with field data. iglass was consistent with other sophisticated SGS models, in regards to predicting the logarithmic velocity behaviour and providing surface stress anisotropies, but its advantage is that it meets both these criteria in a weather-forecastingmodel environment. Additionally, iglass may predict the logarithmic behaviour better than SGS models that do provide these stress anisotropies. For example, the Kosović (1997) model provided similar levels of anisotropy and predicted a logarithmic velocity profile well in a pseudo-spectral code, but simulations of the NBL in the Weather Research and Forecasting Model (WRF) led to large overshoots of the normalized velocity gradient, M, 1.5 and greater (Kirkil et al., 2012; Mirocha et al., 2010).

19 18 dθ = K m 1 dz z (m) z (m) 20 m s -1 Ɵ (K) Fig. 8 Initial profiles of potential temperature, wind velocity, U,forthesimulatedCBLcases. U g (m s -1 ),andthex-componentofthegeostrophic LESoftheConvectiveBoundaryLayer The dry convective boundary layer (CBL) includes a surface layer (the lowest 10%), a mixed layer, and an interfacial layer. The interfacial layer, also referred to as the entrainment zone, is a stable region, in which vertical mixing is inhibited. Here, air aloft mixes and becomes a part of the boundary layer. In this section, we assess the iglass model performance with the addition of the SGS buoyancy term. Previous studies (Huang et al., 2008; Nieuwstadt et al., 1993) with di erent eddy-viscosity models have shown consistent CBL behaviour, and thus, LES of the CBL is expected to be una ected by di erent SGS models except at the surface and interfacial layer. Here, we examine if iglass produces results similar to the TKE-1.5 model and can therefore be used to simulate the evolution of the CBL. In the CBL, atmospheric turbulence is strongly sensitive to buoyant forces and, in many cases, the buoyant surface heat flux is the primary driver in the development of turbulence features. However, CBLs exist in which the shear production of TKE is relatively strong and competes with the buoyancy production of TKE in driving the CBL. Fedorovich et al. (2004) and Conzemius and Fedorovich (2006) examined this competition. We use one of their setups so that direct comparisons can be made; the simulations were initialized with shear concentrated at the surface. A surface sheared CBL with a constant geostrophic wind of 20 m s 1 at a latitude of 40 Nissimulated.Atthislatitude,f s 1. A background free-atmosphere potential temperature stratification of K m 1 is applied. All simulations are done on a 10.2 km 10.2 km 2kmdomainwith x =40m and an average vertical resolution z,a = 20 m. Surface fluxes are computed at each point on the surface and at each time step with a stability-dependent drag coe cient using Monin-Obukhov similarity theory (Xue et al., 2001) and a roughness length of 0.01 m. Initial conditions are shown in Fig. 8, and general LES parameters are listed in Table 4. We use two versions of iglass, one that actively couples SGS heat flux with SGS stress and a passive version. The active version implements Eq. 7 & 8. These equations include buoyancy terms in the SGS stress equations, and allow the SGS heat flux to directly influence the SGS stress. The passive version does not include this buoyancy term and only allows the SGS stress to directly influence the SGS heat flux; thus, the SGS heat flux becomes a passive scalar. Additionally, we apply

20 19 Run name Q o (K m s 1 ) Coupling Type Entrainment Rates (m s 1 ) A Active A Active A Active P Passive P Passive P Passive TKE Passive TKE Passive TKE Passive Table 5 CBL LES run parameters. Q o is the surface heat flux and the coupling type refers to the turbulent heat flux model coupling three di erent surface heat fluxes: Q o =0.03,0.1,and0.3Kms 1.E ects of shear can be magnified with a weaker Q o. We compare iglass runs with standard TKE- 1.5 runs (Deardor, 1980; Moeng, 1984). The TKE-1.5 model solves a transport equation for the TKE, which is used as a velocity scale for an eddy-viscosity SGS stress model. The TKE-1.5 model is widely used in LES of the ABL (Moeng, 1984; Rizza et al., 2013; Sullivan and Patton, 2011). Table 5 lists nine CBL LES runs discussed herein. They are labelled with an A (iglass with active coupling), a P (iglass with passive SGS heat flux equation), or TKE (TKE-1.5), followed by the strength of the heat flux. However, we provide only representative samples here; the full set can be seen in Enriquez (2013). As noted in Sect. 2, we implemented a TKE parametrization based on a model from Yoshizawa (1986); that parametrization, Eq. 13, allows local buoyancy to contribute to the generation of TKE and helps us avoid stability issues near the inversion (cf., Ramachandran and Wyngaard, 2011, who used damping of the buoyancy term near the inversion). Later we will see that the buoyancy term a ects some mean profiles most significantly in this region Results Simulations are carried out over the time interval before the upper damping layer clearly impacts the CBL, which varies with the strength of Q o.forthesimulations with Q o =0.3Kms 1, data are sampled at t = 3000 s. Mean and turbulent results at t = 10,000 s are used for runs with Q o =0.1Kms 1,andatt = 20,000 s for runs with Q o =0.03Kms 1. Mean results are from planar averaged values at a single instant. We report on mean statistics that can demonstrate whether iglass can simulate a CBL, compared to the commonly used TKE-1.5 SGS model. The general shape of velocity profiles for each run remains the same, but there are di erences between iglass runs and TKE runs. The addition of the active buoyancy coupling does not appear to significantly change mean velocity profiles. For each case, iglass velocities are less than TKE velocities near the surface, most likely because iglass simulations are less resolved near the surface. Away from the surface, they agree well, except for the most convective case with Q o =0.3Kms 1, where at z 600 m, iglass velocities are about 10% greater than predicted TKE velocities. Data from Conzemius and Fedorovich (2006, see their Fig. 10a) for our least convective Q o =0.03Kms 1 run are shown here in Fig. 9a to illustrate the potential range of model results. Their simulation used