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  • View in gallery

    Two-dimensional energy spectra of (a) vertical velocity w and (b) horizontal velocity u at (top to bottom) z/zi = 0.8 [E(κ) × 1000], z/zi = 0.5 [E(κ)], and z/zi = 0.1 [E(κ) × 0.001]: case BT (thick solid black), BF (thick dotted black), SW (thick dotted–dashed gray), and SS (thin solid black). Slope of the straight lines is κ−5/3. Wavenumbers corresponding to 2ΔLES and 6ΔLES wavelengths are marked by arrows.

  • View in gallery

    A schematic diagram illustrating the three length scales in section 2d: D and ΔLES are the horizontal domain size and the grid size of the benchmark LES, respectively, and Δ ranges between D and ΔLES. The k index is used for Δ in the LES domain of size D × D, while the j index is used for ΔLES in each subdomain of size Δ × Δ (cf. section 2d). Figure is modified from Dorrestijn et al. (2012).

  • View in gallery

    Ratio of resolved (red) and subgrid-scale (blue) TKE to total TKE as functions of dimensionless grid size: for case BT, (a) in the mixed layer (0.10 ≤ z/zi ≤ 0.80) and (b) in the entrainment zone (0.85 ≤ z/zi ≤ 1.00) (cf. appendix). The red and blue solid lines are derived from the reference data of this study. The black lines represent the grid-size dependency functions of HMC11: mean (solid) and the first and last vigintiles (dotted).

  • View in gallery

    (a) Two-dimensional energy spectra at z/zi = 0.5, normalized by total TKE. (b) Ratio of resolved (red) and subgrid-scale (blue) TKE to total TKE as functions of dimensionless grid size (Δ/zi): for cases BT (thick solid), BF (thick dotted), SW (thick dotted–dashed), and SS (thin solid).

  • View in gallery

    Ratio of resolved (red) and subgrid-scale (blue) parts to the corresponding total as functions of dimensionless grid size (Δ/zi): vertical transports of (a) potential temperature, (b) u wind, and (c) bottom-up diffusion scalar, for the BT (thick solid), BF (thick dotted), SW (thick dotted–dashed), and SS (thin solid) cases in the mixed layer. In (b), the case BT is excluded (cf. section 3).

  • View in gallery

    Vertical profiles of the length scale L, where the subgrid-scale TKE, and subgrid-scale vertical fluxes of θ, u, and cbu equal the corresponding resolved parts (i.e., the subgrid-scale parts are 50% of the corresponding total). For 〈wu′〉, the case BT is excluded (cf. section 3).

  • View in gallery

    (a) Coverage fraction of coherent structures, and contribution of nonlocal vertical transport to corresponding total vertical transport for (b) heat and (c) bottom-up scalar: the BT (thick solid), BF (thick dotted), SW (thick dotted–dashed), and SS (thin solid) cases. Two conditional sampling methods are used: M1 (black) and M2 (gray) (cf. Table 2). In (b), data for 0.75 < z/zi < 0.90 are omitted owing to their sharp fluctuations.

  • View in gallery

    Ratio of resolved (red) and subgrid-scale (blue) transports to total (resolved plus SGS) transport, for (a),(c) nonlocal and (b),(d) local components: (top) potential temperature and (bottom) bottom-up diffusion scalar. For the BT (thick solid), BF (thick dotted), SW (thick dotted–dashed), and SS (thin solid) cases in the mixed layer. M1 is used for conditional sampling. The gray lines in each figure show the ratio of resolved and subgrid-scale transports before their separation into the nonlocal and local components (i.e., identical to Fig. 5a for θ, and Fig. 5c for cbu).

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Analysis of Resolved and Parameterized Vertical Transports in Convective Boundary Layers at Gray-Zone Resolutions

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  • 1 Department of Atmospheric Sciences, Yonsei University, Seoul, South Korea
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Abstract

The gray zone of a physical process in numerical models is defined as the range of model resolution in which the process is partly resolved by model dynamics and partly parameterized. In this study, the authors examine the grid-size dependencies of resolved and parameterized vertical transports in convective boundary layers (CBLs) for horizontal grid scales including the gray zone. To assess how stability alters the dependencies on grid size, four CBLs with different surface heating and geostrophic winds are considered. For this purpose, reference data for grid-scale (GS) and subgrid-scale (SGS) fields are constructed for 50–4000-m mesh sizes by filtering 25-m large-eddy simulation (LES) data.

As relative importance of shear increases, the ratio of resolved turbulent kinetic energy increases for a given grid spacing. Vertical transports of potential temperature, momentum, and a bottom-up diffusion passive scalar behave in a similar fashion. The effects of stability are related to the horizontal scale of coherent large-eddy structures that change in the different stability. The subgrid-scale vertical transport of heat and the bottom-up scalar are divided into a nonlocal mixing owing to the coherent structures and remaining local mixing. The separate treatment of the nonlocal and local transports shows that the grid-size dependency of the SGS nonlocal flux and its sensitivity to stability predominantly determine the dependency of total (nonlocal plus local) SGS transport.

Corresponding author address: Song-You Hong, Dept. of Atmospheric Sciences, College of Science, Yonsei University, 50 Yonsei-ro, Seodaemun-gu, Seoul 120-749, South Korea. E-mail: songyouhong@gmail.com

Abstract

The gray zone of a physical process in numerical models is defined as the range of model resolution in which the process is partly resolved by model dynamics and partly parameterized. In this study, the authors examine the grid-size dependencies of resolved and parameterized vertical transports in convective boundary layers (CBLs) for horizontal grid scales including the gray zone. To assess how stability alters the dependencies on grid size, four CBLs with different surface heating and geostrophic winds are considered. For this purpose, reference data for grid-scale (GS) and subgrid-scale (SGS) fields are constructed for 50–4000-m mesh sizes by filtering 25-m large-eddy simulation (LES) data.

As relative importance of shear increases, the ratio of resolved turbulent kinetic energy increases for a given grid spacing. Vertical transports of potential temperature, momentum, and a bottom-up diffusion passive scalar behave in a similar fashion. The effects of stability are related to the horizontal scale of coherent large-eddy structures that change in the different stability. The subgrid-scale vertical transport of heat and the bottom-up scalar are divided into a nonlocal mixing owing to the coherent structures and remaining local mixing. The separate treatment of the nonlocal and local transports shows that the grid-size dependency of the SGS nonlocal flux and its sensitivity to stability predominantly determine the dependency of total (nonlocal plus local) SGS transport.

Corresponding author address: Song-You Hong, Dept. of Atmospheric Sciences, College of Science, Yonsei University, 50 Yonsei-ro, Seodaemun-gu, Seoul 120-749, South Korea. E-mail: songyouhong@gmail.com

1. Introduction

Turbulence in planetary boundary layers (PBLs) consists of different-sized eddies, which range in scale from boundary layer depth zi to the Kolmogorov scale. Turbulent flows are responsible for vertical transport in PBLs, and the transport can be decomposed into two categories according to the scales of eddies: nonlocal transport and local transport (Siebesma and Cuijpers 1995; Siebesma et al. 2007). Coherent structures (or organized structures) of the turbulence in convective boundary layers (CBLs) (e.g., thermals, rolls, and cells) correspond with spectral energy peaks and largely contribute to the vertical turbulent transport in a nonlocal way (e.g., LeMone 1976; Young 1988; Siebesma et al. 2007; Couvreux et al. 2010). Depending on how the turbulent flows are generated, the coherent structures that appear in CBLs show different features (e.g., LeMone 1973; Moeng and Sullivan 1994, hereafter MS94; Pino et al. 2003). Buoyancy-driven updrafts (thermals) are dominant in strongly convective PBLs. Horizontal roll vortices are organized when shear forcing is sufficiently high and surface heating is weak: i.e., 0.35 < u*/w* < 0.65 or 1.5 < −zi/LMO < 9.3, where u* is the friction velocity, w* is the convective velocity, zi is the PBL height, and LMO is the Monin–Obukhov length (cf. MS94). On the other hand, small eddies in the remaining turbulent field locally mix CBL properties.

There are two broad classes of methods for modeling PBLs over domains ranging from a few kilometers to a global area, depending on the size of the energy- and flux-containing large eddies l and the typical grid or spatial filter scale Δ (Wyngaard 2004). In numerical weather prediction (NWP) models or general circulation models (GCMs), l/Δ ≪ 1. In these models only mean flows are explicitly calculated, and one-dimensional PBL schemes parameterize all turbulence, including the coherent structures (i.e., large eddies). The nonlocal mixing due to the organized structures is expressed using a mass-flux term (e.g., Pleim 2007; Siebesma et al. 2007) or a gradient-adjustment gamma (γ) term (e.g., Troen and Mahrt 1986; Holtslag and Boville 1993; Hong et al. 2006), while an eddy diffusivity term is used for the local mixing. The other approach is a three-dimensional large-eddy simulation (LES). The LES technique is justified when Δ is in the inertial subrange (i.e., l/Δ ≫ 1). The organized structures and the relevant nonlocal vertical transport are explicitly calculated, and only small-scale eddies are parameterized. Wyngaard (2004) noted that neither of the two methods were designed to operate at a resolution where l and Δ are comparable (i.e., l/Δ ~ 1), which he termed “terra incognita.” At this resolution, the large eddies are partly grid scale (GS) and partly subgrid scale (SGS): the resolution falls into the gray zone (Hong and Dudhia 2012).

For the development of a SGS parameterization for the gray-zone resolutions, an investigation of how the characteristics of the SGS turbulences change according to the grid size must take precedence. Honnert et al. (2011, hereafter HMC11) constructed reference data, describing how resolved/subgrid-scale vertical fluxes and variances depend on mesh size in Wyngaard's terra incognita for dry and cloudy CBLs. A unified scale-dependency function is suggested for each variable at each layer (e.g., surface layer, mixed layer, entrainment zone) for different types of convective PBLs. For example, a single function explains the grid-size dependency of TKE in the mixed layer for CBLs that differ in stability (i.e., differ in u*/w* or −zi/LMO). The first objective of this study is to further classify the function by the stability.

Compared to their grid-size dependency functions, HMC11 evaluated the two types of existing PBL modeling methods: one-dimensional PBL parameterization and LES. They indicated the importance of accurately representing SGS thermals by taking grid size into account when developing a SGS turbulence model for the gray-zone resolutions. The second objective of this study is to separately analyze the grid-scale dependencies of nonlocal and local transports. The separate treatment is meaningful, considering that nonlocal and local parts are separately parameterized in NWP models and GCMs.

In section 2, reference data for four CBL cases with various surface heating and geostrophic wind conditions are constructed using an LES model for quantifying the resolved and subgrid-scale energy and transports in the gray zone. In section 3, we discuss the grid-size dependencies of GS and SGS vertical transports and the effects of stability on the dependencies. In section 4, conditional sampling is used to select the coherent structures from the LES and to then divide the total transport into nonlocal and local components. We provide the grid-size dependencies of SGS nonlocal and local transports separately. Summary and concluding remarks follow in the final section.

2. Construction of the reference data

The reference turbulence fields are produced for horizontal grid sizes ranging from 25 to 4000 m [according to Dorrestijn et al. (2012)] for four convective boundary layer cases. For this purpose, four CBLs are simulated by an LES model with 25-m grid spacing as the benchmark simulations. The reference data for 50–4000-m grid sizes are then constructed by spatially filtering the 25-m benchmark LES output. The data for each grid size (Δ) describe how the resolved and parameterized turbulence should be represented at each grid size. This section provides brief descriptions of the LES model [the Weather Research and Forecasting Model (WRF)], the experimental setup and characteristics of the benchmark simulations, and the reference data obtained from the simulations.

a. Model description

WRF is used as an LES model. WRF calculates the fully compressible and nonhydrostatic governing equations that are formulated using a terrain-following mass vertical coordinate (Skamarock et al. 2008). The model can be operated at any spatial resolution since it has no hydrostatic assumption constraints. Given a proper subgrid-scale parameterization designed for LES and a resolution in the inertial subrange of target phenomena, the model can be used as an LES (Antonelli and Rotunno 2007; Moeng et al. 2007; Zhu 2008; Rotunno et al. 2009; Catalano and Moeng 2010). A three-dimensional SGS parameterization using a turbulent kinetic energy (TKE) equation (Deardorff 1980) is selected for our study. The diffusion coefficients chosen are proportional to the grid sizes with some modifications for considering the effects of grid anisotropies (Catalano and Moeng 2010; Scotti et al. 1993).

The third-order Runge–Kutta time-integration scheme is selected for the temporal discretization of the governing equations for low-frequency modes. High-frequency modes are calculated with a shorter time step; namely, a time-split integration scheme is adopted. For the spatial discretization, WRF uses an Arakawa C grid staggered in the horizontal direction, and the fifth-order (third order) accurate finite-differencing advection scheme is applied for horizontal (vertical) advection. The fifth-order horizontal advection scheme inherently induces sixth-order numerical diffusion in the horizontal direction (the third-order vertical advection scheme inevitably includes fourth-order numerical diffusion in the vertical direction) with a viscosity proportional to the Courant number (Skamarock et al. 2008).

b. Design of benchmark simulations

The model domain is 8 km × 8 km × 3.5 km along the x, y, and z directions, respectively. The number of grid points is 320 × 320 × 120, with a horizontal grid spacing of 25 m (i.e., ΔLES = 25 m). In the vertical direction, the lowest model level height is approximately 12 m, and the deepest layer thickness is approximately 35 m. Periodic boundary conditions are used in the x and y directions, and Rayleigh damping is adopted in the upper 1000 m. The PBL is driven by a constant surface heat flux (subscript 0 refers to the surface) and geostrophic wind in the x direction Ug. The Coriolis parameter is set as f = 10−4 s−1. Small random perturbations of [−0.05, 0.05 K] are imposed on the initial temperature fields at the lowest four levels. The initial sounding of the potential temperature is
e1

Four benchmark simulations are conducted for different values of and Ug (Table 1). The BT case is purely buoyancy driven (B) and organized thermals (T) appear: [, Ug] = (0.20 K m s−1, 0.0 m s−1). The BF case is buoyancy driven (B), as well as wind forced (F); therefore, it is identical to BT, except that Ug = 10.0 m s−1. The weaker-shear (SW) and stronger-shear (SS) experiments are conducted using [, Ug] = (0.05 K m s−1, 10.0 m s−1) and (0.05 K m s−1, 15.0 m s−1), respectively. The boundary layer statistics are listed in Table 1. From the statistics, it is expected that the dominant organized structures in the BT simulation are thermal circulations and the convective rolls appear in the SW and SS simulations (cf. MS94). The estimated large-eddy turnover time (τ* ~ zi/w*) is approximately 600 s for the BT and BF cases and 800 s for the SW and SS cases. The benchmark LES fields are analyzed for 2t0 seconds between t0 and 3t0: t0 = 6τ*—the time required for the dynamic flow field to reach a statistically quasi-steady state (MS94).

Table 1.

Summary of benchmark simulations with boundary layer statistics at 2t0 (t0 = 6τ*, where τ* is the large-eddy turnover time). The coherent structures appearing in the BF are between the BT and SW cases (cf. section 3).

Table 1.

In addition to the dynamic fields, a bottom-up diffusion passive scalar cbu is included in the benchmark simulations. The scalar is used for conditional sampling (section 4a) as well as for the comparison of the grid-size dependencies among the different variables (section 3). The filtered conservation equation of the scalar cbu is
e2
where an overbar with superscript r indicates resolved (or filtered) variables, and fi is the SGS scalar flux in the i direction. For the bottom-up diffusion scalar, a constant surface flux of = 0.001 kg kg−1 m s−1 is imposed. Initially, cbu = 1 kg kg−1 for the whole vertical domain.

c. Spectral analysis of benchmark simulations

Two-dimensional energy spectra of vertical and horizontal velocities are calculated from the four benchmark simulations (Fig. 1). The wavenumber κ is defined as κ = (κx2 + κy2)1/2, where κx (=2π/λx) and κy (=2π/λy) are wavenumbers in the x and y directions, respectively (λx,y are wavelengths in the x and y directions, respectively). For the vertical velocity spectra (Fig. 1a), the spectral peaks are near κzi between 3 and 4 (wavelength between 1500 and 2000 m) (e.g., at z/zi = 0.5 and z/zi = 0.8) and define the gray-zone resolution at those levels. The grid size has already been reached by fine-mesh mesoscale modeling studies (e.g., Bryan et al. 2003; Fiori et al. 2010; LeMone et al. 2010). Our benchmark simulations capture the κ−5/3 slope at the middle of PBL as well as a broadening of the peak and the peak's shift to smaller scales near the top of the surface layer (i.e., at z/zi = 0.1) (cf. Sullivan and Patton 2011). However, energy damps more rapidly than the κ−5/3 slope when the horizontal wavelength is shorter than approximately 6ΔLES owing to the implicit dissipation inherent in odd-order advection schemes (Skamarock 2004). Mechanical forcing determines w variance at low wavenumbers (e.g., BF ≈ SW), while buoyancy is important near the spectral peaks in all levels (e.g., BT ≈ BF and SW ≈ SS). Because of the implicit dissipation proportional to the Courant number, the slopes of spectral tails depend on the horizontal winds.

Fig. 1.
Fig. 1.

Two-dimensional energy spectra of (a) vertical velocity w and (b) horizontal velocity u at (top to bottom) z/zi = 0.8 [E(κ) × 1000], z/zi = 0.5 [E(κ)], and z/zi = 0.1 [E(κ) × 0.001]: case BT (thick solid black), BF (thick dotted black), SW (thick dotted–dashed gray), and SS (thin solid black). Slope of the straight lines is κ−5/3. Wavenumbers corresponding to 2ΔLES and 6ΔLES wavelengths are marked by arrows.

Citation: Journal of the Atmospheric Sciences 70, 10; 10.1175/JAS-D-12-0290.1

The spectral energy peak of the horizontal velocity (Fig. 1b) increases as z decreases from 0.5zi to 0.1zi (cf. Sullivan and Patton 2011). The relative impacts of the mechanical and buoyancy forcing are different from those for the vertical velocity. At low wavenumbers, the amplitude of the energy is largest in the BF and SS cases, followed by the SW and BT cases. At wavenumbers higher than energy-peak wavenumbers, the power increases with wind shear near the surface, while surface heating becomes increasingly important as z increases (e.g., analogy between the BF and BT cases, and SW and SS cases, at higher levels).

d. Construction of the reference data for grid scales between 50 and 4000 m

Here, the method of Dorrestijn et al. (2012) is described. Dorrestijn et al. (2012) defined three different length scales (Fig. 2). The first scale, D, refers to the horizontal domain size of the benchmark LES; D = 8 km in this study. At the resolution comparable to D, the use of the 1D PBL parameterizations is reasonable (i.e., D > l). The second scale, ΔLES, is the horizontal LES grid size used in the benchmark simulations: ΔLES = 25 m. The third length scale, Δ, ranges between D and ΔLES and therefore covers the gray zone. The LES domain (D × D) consists of K grids (or subdomains) of grid size Δ × Δ: K = (D/Δ)2. On the other hand, J LES grids of size ΔLES × ΔLES compose each subdomain Δ × Δ: J = (Δ/ΔLES)2. The small letter k is used to refer to the kth subdomain in the LES domain, and j refers to the jth LES grid in the kth subdomain.

Fig. 2.
Fig. 2.

A schematic diagram illustrating the three length scales in section 2d: D and ΔLES are the horizontal domain size and the grid size of the benchmark LES, respectively, and Δ ranges between D and ΔLES. The k index is used for Δ in the LES domain of size D × D, while the j index is used for ΔLES in each subdomain of size Δ × Δ (cf. section 2d). Figure is modified from Dorrestijn et al. (2012).

Citation: Journal of the Atmospheric Sciences 70, 10; 10.1175/JAS-D-12-0290.1

The reference (REF) data for grid-scale (subdomain scale) variables at coarse resolutions (Δ) are constructed by applying a box average to the benchmark LES results. For a variable ,
e3
Here, the overbar with the superscript Δk refers to the average over the kth subdomain of size Δ × Δ.
Vertical transport of the variable over the whole domain (i.e., 〈wϕ′〉) consists of resolved and SGS parts for each resolution Δ [designated by superscripts R(Δ) and S(Δ), respectively]:
e4
e5
e6
The angle brackets refer to an 8 km × 8 km domain average, and the prime designates perturbations from the domain average. The double prime refers to perturbations of LES grid values from the corresponding subdomain average : . In Eq. (6) is the SGS vertical transport for Δk, while it is resolved vertical transport for ΔLES. Also in Eq. (6) is the SGS vertical transport for ΔLES and is calculated by the LES SGS model (cf. section 2a).

Figure 3 shows the comparison between the BT case and HMC11 with respect to the ratio of GS/SGS TKE to total TKE. The free-convection case has u*/w* = 0.097 and −zi/LMO = 430.31 (cf. Table 1), which is among the three dry CBL cases of HMC11 (u*/w* = 0.02–0.16 and −zi/LMO = 94–4715). Our grid-size-dependent functions partitioning the resolved/subgrid-scale TKE are in accordance with the HMC11 functions in the mixed layer (Fig. 3a), but not in the entrainment zone (Fig. 3b). For example, the ratio of grid-scale TKE in the entrainment zone equals the ratio of the subgrid-scale counterpart at Δ/zi ~ 0.35 in our experiment; however, it is at Δ/zi ~ 0.45 in HMC11. This inconsistency can be attributed to differences in the SGS parameterization, the spatial discretization method, the grid system (i.e., staggered grids), and numerical diffusion, as well as vertical resolution in the entrainment zone. In regard to this, Cheng et al. (2010) mentioned that the combination of the numerics and the grid mesh implicitly defines a filter, which is spectrally imprecise.

Fig. 3.
Fig. 3.

Ratio of resolved (red) and subgrid-scale (blue) TKE to total TKE as functions of dimensionless grid size: for case BT, (a) in the mixed layer (0.10 ≤ z/zi ≤ 0.80) and (b) in the entrainment zone (0.85 ≤ z/zi ≤ 1.00) (cf. appendix). The red and blue solid lines are derived from the reference data of this study. The black lines represent the grid-size dependency functions of HMC11: mean (solid) and the first and last vigintiles (dotted).

Citation: Journal of the Atmospheric Sciences 70, 10; 10.1175/JAS-D-12-0290.1

3. Analysis of resolved and subgrid-scale vertical transports

HMC11 addressed dependency on a normalized grid size (i.e., Δ/zi) for GS/SGS TKE, heat and moisture fluxes, and potential temperature and mixing ratio variances in CBL cases using the mesoscale nonhydrostatic (Meso-NH) LES model. Therefore, we restrict ourselves to discussing the effects of stability on grid-size dependency. Hereafter, an overbar or angle brackets without any superscripts/subscripts refer to an 8 km × 8 km domain average. Symbols with superscripts or subscripts will be defined later as necessary.

Figure 4a shows the energy spectrum E(κ) normalized by total TKE for the four stability cases. The SGS TKE at Δ is physically equal to the integral of the energy spectrum from κ → ∞ to κ = 2π/Δ: that is, . The comparison between BT and BF shows that the contribution of the large eddies (i.e., eddies corresponding to κ < κmax, where κmax is the horizontal wavenumber that corresponds to the energy peak) increases from BT to BF, while the contribution of the small eddies (i.e., κ > κmax) decreases. From BF to SW, the contribution of the large eddies increases and κmax itself moves to a larger scale. This trend is consistent with the effects of the stability on the grid-size dependency (Fig. 4b). That is, at an identical Δ, the ratio of the resolved part is smallest in the BT case, while it is largest in the SW and SS cases.

Fig. 4.
Fig. 4.

(a) Two-dimensional energy spectra at z/zi = 0.5, normalized by total TKE. (b) Ratio of resolved (red) and subgrid-scale (blue) TKE to total TKE as functions of dimensionless grid size (Δ/zi): for cases BT (thick solid), BF (thick dotted), SW (thick dotted–dashed), and SS (thin solid).

Citation: Journal of the Atmospheric Sciences 70, 10; 10.1175/JAS-D-12-0290.1

Figure 5 shows grid-size dependencies of grid-scale and subgrid-scale vertical transports for potential temperature, u wind, and bottom-up diffusion scalar in the middle of PBL. Note that in the no-mean-wind case (i.e., the case BT), there are no meaningful correlations between w′ and u′. In other words, the spatial distributions of wu′ are random over the domain, and the domain-averaged correlation [i.e., 〈wu′〉] approaches zero. Therefore, the discussions on the momentum flux are excluded for the case BT. In general, the vertical transports are resolved for more than 90% at Δ ~ 0.05–0.1zi, and more than 90% of them are SGS at Δ = 2zi. As wind shear increases [i.e., u*/w* increases (or −zi/LMO decreases)], the resolved vertical fluxes normalized by the corresponding total values gradually increase for the identical Δ/zi, except that the differences between SW and SS are negligible.

Fig. 5.
Fig. 5.

Ratio of resolved (red) and subgrid-scale (blue) parts to the corresponding total as functions of dimensionless grid size (Δ/zi): vertical transports of (a) potential temperature, (b) u wind, and (c) bottom-up diffusion scalar, for the BT (thick solid), BF (thick dotted), SW (thick dotted–dashed), and SS (thin solid) cases in the mixed layer. In (b), the case BT is excluded (cf. section 3).

Citation: Journal of the Atmospheric Sciences 70, 10; 10.1175/JAS-D-12-0290.1

The effects of stability can be interpreted in terms of organized structures that appear in each simulation. Organized structures account for more than half of the total vertical fluxes in CBLs (LeMone 1976; Young 1988; Couvreux et al. 2010). In the absence of mean shear (e.g., the BT case), thermal circulations are typical coherent structures. On the other hand, convective rolls dominate the SW and SS simulations. The BF case shows large-eddy structures between the BT and SW cases; in other words, eddies near the surface show neither clear polygonal spoke patterns nor a preferred horizontal elongation. For reference, Dosio et al. (2003) classified a case with u*/w* = 0.27 and −zi/LMO = 18 (~BF; Table 1) into a shear–buoyancy-driven CBL group, while this case is classified as a free-convection case in MS94. The horizontal length scale of the thermal circulations is roughly 1.5zi, which is smaller than the scale of the roll vortices (i.e., 3zi) (Stull 1988; MS94). Therefore, given the same grid spacing, the contributions of the resolved parts are larger in the SW and SS cases than those in the BT and BF cases.

The effects of the stability on the bottom-up scalar flux are similar to those on the heat flux (cf. Figs. 5a and 5c), although the bottom-up scalar transport is resolved more than the heat transport for a fixed Δ/zi for all four cases. Previous studies of Jonker et al. (1999) and de Roode et al. (2004) demonstrated that particular scalars could have larger structures than momentum fields and gradually become dominated by mesoscale fluctuations over time. They linked this enlargement to the geometry of the flux profiles (i.e., the ratio of the entrainment flux to the surface flux, r). Length scale of a variable is smallest when r = −0.25 (mesoscale fluctuations are negligible) and becomes larger as r approaches ±∞ (de Roode et al. 2004). In our study, r ranges from −0.2 to −0.4 for θ, and it is 0 for cbu. This explains the small differences between the grid-size dependencies of the potential temperature and the bottom-up scalar.

HMC11 defined a dimensionless length scale L as the normalized grid size where the grid-scale and subgrid-scale portions are equal. For example, L for TKE [i.e., L(TKE)] in the mixed layer is roughly 0.3, where the red and blue lines in Fig. 3a cross. Figure 6 shows the vertical profiles of L for TKE and vertical transports of θ, u, and cbu. We use L as a measure for the scale of the gray zone. As in HMC11, through analyzing the height dependency of L for a variable, the height dependency of the gray-zone grid size for the variable can be investigated. The height variations of L for TKE and 〈wθ′〉 are relatively small within 0.1 ≤ z/zi ≤ 0.8 and 0.2 ≤ z/zi ≤ 0.6, respectively. The small height variations suggest that simple functions that are dependent on grid size (Δ/zi) can explain how much portion of TKE and heat flux should be parameterized at gray zones, roughly throughout the mixed layer (cf. appendix). On the other hand, L(TKE) and L(〈wθ′〉) reach their maxima at z/zi ~ 0.9, and decrease above. The presence of the inversion would restrict the size of eddies that are generated in proximity to the inversion, whereas the cospectum between w and θ shows that the large-scale eddies, which are originated from the surface, are mainly responsible for the negative entrainment flux. The larger L at z/zi ~ 0.9 is consistent with the increase of the horizontal scale of the coherent structures according to height: plumes in the surface layer, thermals in the mixed layer, and mergence of the thermals as they rise (Stull 1988). L(TKE) and L(〈wθ′〉) become larger as u*/w* increases (cf. Figs. 4b and 5a) except near the surface, where small-scale along-wind streaks are dominant in the SW and SS.

Fig. 6.
Fig. 6.

Vertical profiles of the length scale L, where the subgrid-scale TKE, and subgrid-scale vertical fluxes of θ, u, and cbu equal the corresponding resolved parts (i.e., the subgrid-scale parts are 50% of the corresponding total). For 〈wu′〉, the case BT is excluded (cf. section 3).

Citation: Journal of the Atmospheric Sciences 70, 10; 10.1175/JAS-D-12-0290.1

For 〈wu′〉, L increases from z = 0 to z = 0.8zi in the three shear-present cases. In each of the BF, SW, and SS cases, the contribution of small-scale eddies to the covariance between w and u becomes smaller as the height increases, whereas large-scale eddies contribute more to the covariance (not shown). This confirms the change in the dominant size of eddies found in the three shear-present cases: from shear-driven small eddies near the surface to larger eddies at higher levels. This interpretation of L(〈wu′〉) is somewhat contradictory to the relatively small height variation in L(〈wθ′〉). A closer examination of the cospectra shows that as z increases, the correlation between w and θ becomes smaller at whole wavenumbers, unlike the scale-dependent reduction in the wu cospectra (not shown). For the passive scalar, it is necessary to consider height dependency in all four stability cases to accurately estimate subgrid-scale scalar fluxes, as L(〈wcbu′〉) increases according to height. From de Roode et al. (2004) (cf. their Fig. 7), it can be inferred that the scales of scalar structures increase with height up to z = 0.6–0.9zi.

4. Decomposition of vertical transport: Separate analysis of nonlocal and local vertical transports

The total turbulent flux of any arbitrary variable can be decomposed into three terms (Siebesma and Cuijpers 1995; Siebesma et al. 2007):
e7a
e7b
e7c
The subscript cs (e) denotes an average over the coherent-structure (remaining environment) area. An overbar with the superscript cs (e) refers to the average of fluctuations with respect to the coherent-structure (remaining environment) averaged values. Variable a is the fractional area covered by the coherent structures. The first term in Eq. (7a) describes the nonlocal (NL) vertical transport by strong updrafts of coherent structures (e.g., thermals, rolls, and cells) [i.e., Eq. (7b)]. The second and third terms in Eq. (7a) are the correlated fluctuations within the coherent structures and the environment, respectively, and constitute the local (L) vertical flux [i.e., Eq. (7c)].

As indicated previously, the gray-zone resolution of CBLs depends on the stability, which is relevant to large-eddy structures. Therefore, in this section, we describe conditional sampling, which is conducted in order to select the organized structures from the four benchmark LES fields; characteristics of the nonlocal vertical transport are briefly discussed. The sampling is repeated for the 50–4000-m Δ reference fields and then separate analysis on grid-size dependencies of the sampled SGS nonlocal and local transports follows.

a. Conditional sampling: Vertical transport by coherent structures

To detect the coherent structures and separate the corresponding nonlocal vertical transport from the total transport, two conditional sampling methods are applied to the benchmark LES (cf. Table 2). The first method (M1) is based on Couvreux et al. (2010), and an LES point (x, y, z) is grouped into the coherent-structure (CS) category if
e8
where is the scalar concentration perturbation from an domain average 〈cbu〉 at level z and is the standard deviation of the scalar concentration. In contrast to our scalar fields [cf. our equation for the scalar, Eq. (2)], Couvreux et al. (2010) used a tracer emitted with a constant surface flux and that undergoes radioactive decay with a time constant. An additional tracer with the radioactive decay is not used in this study because the calculated nonlocal transport is not significantly sensitive to the time constant for the radioactive decay [cf. Fig. 7 in Couvreux et al. (2010)], while the additional tracer increases computing time for the benchmark LES runs.
Table 2.

Overview of the conditional sampling methods used in this study.

Table 2.
The algorithm described by Siebesma et al. (2007) uses the p percentile of the w distribution at z [i.e., wp%(z)] to define strong updrafts at the height as follows:
e9
Over a horizontal surface at z, p% of the grid points has vertical velocities that exceed wp%(z). Here, p = 10 is used (i.e., the second method, M2; Table 2).

Figure 7a shows the coverage fraction occupied by the organized updrafts [i.e., a in Eq. (7)]. When M1 is applied, the coverage is approximately 0.15 near the surface and decreases below 0.07 at 0.9zi. The height dependency is similar to what Couvreux et al. (2010) reported, but the value of a is smaller than that of Couvreux et al. (2010). When the w10% method is used, a = 0.1 by definition. Young (1988) reviewed previous studies that had documented fractional area coverage of thermals and summarized that it is less than 0.5 throughout most of the CBL, but it depends on the thresholds forced on the indicator variables (e.g., , w).

Fig. 7.
Fig. 7.

(a) Coverage fraction of coherent structures, and contribution of nonlocal vertical transport to corresponding total vertical transport for (b) heat and (c) bottom-up scalar: the BT (thick solid), BF (thick dotted), SW (thick dotted–dashed), and SS (thin solid) cases. Two conditional sampling methods are used: M1 (black) and M2 (gray) (cf. Table 2). In (b), data for 0.75 < z/zi < 0.90 are omitted owing to their sharp fluctuations.

Citation: Journal of the Atmospheric Sciences 70, 10; 10.1175/JAS-D-12-0290.1

The contribution of the sampled nonlocal heat and bottom-up scalar transports to the corresponding total is shown in Figs. 7b and 7c, respectively. The coherent structures can account for more than half of the total heat transport in the lower part of the subcloud layer [Young (1988) and Couvreux et al. (2010) for the thermals; LeMone (1976) for the rolls]. The estimated nonlocal contribution increases above 0.1zi (decreases below 0.1zi) according to increases in u*/w* with the sampling method M1. When M2 is used, the nonlocal component (about 60%) is kept for almost all of the variables/stability, with the exception of near-surface levels.

The conditional sampling methods that only consider updrafts mirror a simplification in many PBL parameterizations and have limitations. This simplification is unreasonable for variables that are largely affected by the organized entrainment events. However, developing a new conditional sampling method considering downdrafts is out of the scope of this study. Therefore, we restrict ourselves to investigating only transports for heat and bottom-up diffusion scalar.

b. Decomposition of SGS vertical transport into nonlocal and local transports

To investigate the grid-size dependencies of the SGS nonlocal and local transports separately, the conditional sampling method is used for each subdomains for 50–4000-m Δ. Similar to Eq. (7), the SGS transport at each Δk [i.e., in Eq. (6)] is divided into the SGS nonlocal transport and SGS local transport as follows:
e10a
e10b
e10c
Here, it is assumed that all coherent structures are resolved at ΔLES; wholly contributes to the SGS local transport as in Eq. (10c). The domain-averaged SGS nonlocal and SGS local transports for Δ are
e11a
e11b
The corresponding domain-averaged resolved nonlocal and local transports can be derived using an assumption: the sum of GS nonlocal and SGS nonlocal transports over the same horizontal domain at a fixed level and at a particular time is kept, regardless of the grid size. That is,
e12a
e12b
The terms and can be obtained as described in section 4a, and is calculated by the LES SGS model. Equation (11) provides and . Therefore, and can be derived.

Now the total vertical flux for each Δ is partitioned into nonlocal and local mixing, and each part, in turn, is divided into grid-scale and subgrid-scale components [Eqs. (11) and (12)]. This allows for the investigation of the grid-size dependencies of nonlocal and local vertical transports separately from the total transport.

The grid-size dependencies of the resolved and subgrid-scale nonlocal and local vertical transports appear in Fig. 8. Here, the results from M1 are shown. The contribution of the SGS nonlocal flux decreases as the normalized grid size decreases (Fig. 8a for θ and Fig. 8c for cbu). Including the shear makes the contributions of the resolved nonlocal fluxes larger. One thing to note is that the grid-size dependency for the SGS nonlocal transport and effects of stability on that are very similar to those for the total (nonlocal plus local) SGS transport (i.e., similarity between the blue and gray lines in Figs. 8a and 8c). One exception is that L for the nonlocal transport (the grid size at which red and blue lines cross) is slightly larger than L for the total transport (the grid size at which gray lines cross). On the other hand, the SGS local transport shows quite different behaviors from the total SGS transport. The effects of stability are relatively small compared to the total SGS transport, except for the free-convection case. This implies that the grid-size dependency of the SGS vertical transport shown in section 3 largely depends on the nonlocal transport. This is consistent with what HMC11 investigated. They indicated the importance of accurately representing SGS thermals by taking grid size into account when developing an SGS turbulence model for the gray-zone resolutions. In the mixed layer (i.e., 0.2 ≤ z/zi ≤ 0.6), the grid-size dependencies of the nonlocal and local SGS transports for the BT case are quantified as
e13a
e13b
These functions are obtained by fitting to the reference data.
Fig. 8.
Fig. 8.

Ratio of resolved (red) and subgrid-scale (blue) transports to total (resolved plus SGS) transport, for (a),(c) nonlocal and (b),(d) local components: (top) potential temperature and (bottom) bottom-up diffusion scalar. For the BT (thick solid), BF (thick dotted), SW (thick dotted–dashed), and SS (thin solid) cases in the mixed layer. M1 is used for conditional sampling. The gray lines in each figure show the ratio of resolved and subgrid-scale transports before their separation into the nonlocal and local components (i.e., identical to Fig. 5a for θ, and Fig. 5c for cbu).

Citation: Journal of the Atmospheric Sciences 70, 10; 10.1175/JAS-D-12-0290.1

Although two conditional sampling methods produce different a and different contribution of nonlocal transport (cf. Fig. 7), they arrived at the same conclusions.

5. Summary and concluding remarks

In this study, the resolved and parameterized vertical transports in convective boundary layers were analyzed for grid sizes from 25 m (within the inertial subrange) to 4000 m (within mesoscale range). The main focus is the effects of the relative contributions of buoyancy and mechanical forcing on the grid-size dependency in terms of forcing-dependent large eddies (i.e., coherent structures). For this purpose, the benchmark LES runs were conducted with 25-m grid spacing for four CBL cases that differ in the imposed surface heating and geostrophic winds. The reference turbulence fields for grid spacings of 50–4000 m were deduced by averaging the 25-m LES output.

Results showed that the relative contributions of the resolved TKE and vertical fluxes to the corresponding total TKE and fluxes increase as wind shear becomes important for a given grid size. This is attributed to the larger horizontal scale of the rolls in shear–buoyancy-driven CBLs. A passive scalar with bottom-up diffusion behaves in a similar fashion, although the gray zone appears in a coarser resolution.

Since traditional one-dimensional PBL parameterizations treat the nonlocal and local vertical flux terms separately, it is meaningful to investigate the effects of the grid size and stability on each of the nonlocal and local transports. Using conditional sampling methods, the total vertical transport for the heat and bottom-up diffusion scalar is divided into the nonlocal component by thermals/rolls and the local component by a remaining turbulent field. It should be noted that the grid-size dependency of the total (nonlocal plus local) SGS transport and effects of stability on that largely depend on the nonlocal transport.

The above analysis suggests the following implications for a future SGS turbulence model at gray-zone resolutions for CBLs: 1) Taking stability into account is necessary for developing an SGS turbulence model at gray-zone grids, as has been done in one-dimensional PBL parameterizations. 2) As HMC11 discussed, adding resolution dependency to SGS nonlocal mixing is promising for adequately estimating SGS turbulences. 3) Different variables show different dominant scales, and therefore different grid-size dependencies of grid-scale and subgrid-scale partitions.

In this study, our main subject is the “domain averaged” grid-size dependency of the SGS vertical transports. Our results based on the domain-averaged quantities are enough to provide important clues to parameterization developers as above. However, there would be many feats to achieve before the development of a new parameterization. For example, the SGS transports over a domain vary from grid to grid at gray-zone resolutions. The large spatial variations of the SGS fluxes in the gray-zone simulations (simulations with Δ in the gray zone) require the representation of the SGS transport within each grid cell. Therefore, in addition to the domain-averaged grid-size dependency of the SGS transport, the way to map the spatial distributions of the SGS transport on the distributions of the resolved parameters is one of the key issues for the development of the SGS parameterization.

Acknowledgments

We acknowledge Jimy Dudhia and Chin-Hoh Moeng at NCAR and Yign Noh and Jinkyu Hong at Yonsei University for their comments during the early stages of our study. The authors would like to express their gratitude to Jongil Han at NCEP and Fang-Yi Cheng at National Central University (Taiwan) for their comments in the review of a thesis from which this work has been derived. The comments by the two anonymous reviewers improved the paper. This work was supported by the Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education, Science and Technology (MEST) (2013-004008) and also funded by the Korea Meteorological Administration (KMA) Research and Development Program under Grant CATER 2012-3084.

APPENDIX

Grid-Size Dependency Functions for Turbulent Kinetic Energy and Vertical Heat Transport

The grid-size dependency functions are fitted to the reference data from the WRF LES. The original forms of the functions suggested by HMC11 are maintained, and only the constants are tuned to minimize the root-mean-square error between the reference data and simple functions. For example, HMC11 suggested a grid-size dependency function for TKE in the entrainment zone in the form of
ea1
where P is the grid-size dependency function, x is the normalized grid size (i.e., Δ/zi), and a, b, c, and d are constants: a = 1.0, b 0.190, c 0.15, and d 0.333 in HMC11. For the BT case in this study, a = 1.0, b 0.145, c 0.172, and d 0.170.

Below are grid-size dependency functions for TKE and vertical heat transport, calculated according to the method above.

For turbulent kinetic energy in the free-convection case,
eA2a
eA2b
Note that the function for the ML is identical to that from the HMC11 (section 2d).
For vertical heat transport in the free-convection case,
eA3a
eA3b

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