1. Introduction
Since then, Eq. (1) has been tested and verified in independent field campaigns. For example, profiles of 
Selected field experiments with vertical velocity variance profile measurements listed in chronological order. The ARM(Clear) and ARM(ShCu) experiments refer to the clear-sky and shallow cumulus studies at the ARM SGP site by Berg et al. (2017) and Lareau et al. (2018), respectively. The “time” column indicates the period of time chosen for the analysis, with ARM(Clear) including general daytime hours. The stability parameters −zi/L were not explicitly given for the two ARM studies but were expected to include a wide range given the yearlong data analyzed in both studies.
Besides field experiments, L80’s profile has also been verified in laboratory experiments (Willis and Deardorff 1974; Kumar and Adrian 1986; Adrian et al. 1986; Hibberd and Sawford 1994; Fedorovich et al. 1996), and numerical studies using large-eddy simulation (LES) and direct numerical simulation of dry CBL cases (e.g., Deardorff 1974; Moeng 1984; Schmidt and Schumann 1989; Sullivan and Patton 2011; Garcia and Mellado 2014; Salesky et al. 2017), as well as shallow cumulus and stratocumulus cases (e.g., Deardorff 1980; Siebesma et al. 2003; Zhu and Albrecht 2003; Heinze et al. 2015). With abundant evidence from the field, laboratory, and numerical simulations, it is extremely likely that under rather general circumstances, 
The robustness of 
Judging from observations, an Eq. (1) type of scaling in the CBL requires the following external conditions: first, a relatively flat and horizontally homogeneous surface. Locations of all field experiments listed in Table 1 including ocean, farmland, and plain satisfy this criterion. Both topography and heterogeneous surface can influence the dynamics of the boundary layer, affecting vertical velocity variances in complex ways (Kaimal and Finnigan 1994, chapters 4 and 5). Second, quasi-steady-state conditions, as almost all field experiments selected data from the early noon to afternoon period when the CBL is well developed, such that second-order turbulence statistics reach steady state despite the slowly evolving mean flow. Berg et al. (2017) showed that 
In the absence of mesoscale disturbances and over horizontally homogeneous land surfaces, do internal CBL dynamics affect 
Among the field experiments listed in Table 1, the observed ζ varies from near-neutral to highly convective values 
Unlike the vertical, the horizontal component of TKE 
Motivated by the universal profile of 
2. Case description and numerical setup
This study is based on LESs of seven idealized CBL cases and a neutral boundary layer (NBL) case. A list of model parameters is presented in Table 2. The model setup follows Shin and Hong (2013) with prescribed surface heating and large-scale pressure gradient forcings. The “B” cases are driven by a constant heat flux of 0.20 K m s−1. Case B1 represents an almost free-convective boundary layer with a weak 1 m s−1 geostrophic wind. The B5 and B10 cases have 5 and 10 m s−1 geostrophic winds, respectively. Case B10 has a bulk stability parameter ζ around 20, almost the same as that of the “moderately convective” case of L12. The prescribed surface heat flux for the four “S” cases is 0.05 K m s−1. Case S10 with a geostrophic wind of 10 m s−1 has 
List of model parameters and some statistics averaged from the last 2 h of the simulations. The initial letter in the case name indicates the prescribed heat fluxes, which are 0.20, 0.05, and 0 K m s−1 for the B, S, and N cases, respectively. The numerical value at the end of the case name indicates the prescribed geostrophic wind speeds (m s−1). The SG10 is a baroclinic case, while the rest are barotropic cases.
All CBL cases are performed on a doubly periodic domain of 5.04 km × 5.04 km × 2 km, with Rayleigh damping applied to the top 500 m. Uniform 10-m horizontal and 4-m vertical grid spacings are adopted below 1.3 km, so that the CBL is well resolved with a negligible contribution from the subgrid scale (SGS), except for the first few grid points above the ground. Above 1.3 km, the vertical grid is stretched from 4 to 50 m over 24 grid points. The NBL case is performed with a finer 8-m isotropic grid spacing since the energy-containing turbulence length scale under neutral stratification is smaller than that under convective conditions. The domain size for the NBL is 9.216 km × 9.216 km × 1.5 km. The 1.5-order TKE closure of Moeng (1984) is adopted to parameterize SGS turbulence. The CBL reaches a statistical quasi-steady state after 6τ, where 
The Advanced Regional Prediction System (ARPS) is used for the simulations. ARPS was developed at the Center for Analysis and Prediction of Storms at the University of Oklahoma. It is a nonhydrostatic mesoscale and convective-scale finite-difference numerical weather prediction model that is also suitable for LES (Chow et al. 2005). More details on ARPS are documented in Xue et al. (2000, 2001).
3. Results and discussion
The objective of this study is to understand the universal profile of the buoyancy-normalized vertical velocity variance over a broad range of CBL stability. As such, the sole premise of this work is that profiles of 
Traditionally, Eq. (1) has been regarded as a mixed-layer scaling (Stull 1988, chapter 9.6.3). The mixed layer is located between the unstable surface layer (occupying roughly the bottom 10% of the CBL) and the stably stratified entrainment zone (usually in the top 15%–20% of the CBL), and is characterized by intense turbulent mixing. Nevertheless, in many field observations, Eq. (1) often fits 
Next, we define the parameter range of ζ over which the universal scaling applies. As discussed in section 1, observational evidence suggests that L80 scaling applies for 
Furthermore, we clarify the extent of “universality” of 
Normalized vertical profiles of the horizontally and time-averaged resolved turbulent variances of (a) vertical velocity and (b) horizontal velocity for the seven CBL cases arranged in decreasing order of ζ. The thick gray line in (a) represents the L80 profile [see Eq. (1)]. Angle brackets indicate horizontal averaging. Note the different x axes in (a) and (b).
Citation: Journal of the Atmospheric Sciences 76, 5; 10.1175/JAS-D-18-0325.1
a. Mean profiles
Horizontally and time-averaged resolved variance profiles of 
The bulk stability parameter ζ of the seven CBL cases ranges from 2.4 for the weakly unstable case S15 to 1.5 × 103 for the nearly free-convective case B1. General agreement in 
The vertical velocity spectra illustrate the partition of variances among scales. Figure 2 presents the two-dimensional time-averaged spectra of w obtained through a horizontal Fourier transformation following Sullivan and Patton (2011). The wavenumber-weighted spectra kE are presented to reveal the energy-containing scales (Kaimal and Finnigan 1994, chapter 2.3). Spectra are plotted at two selected elevations, one slightly above the surface layer at 0.15zi and the other well inside the mixed layer at 0.5zi. The normalized spectra exhibit −5/3 inertial subrange scaling at both elevations. The spectra of all seven cases agree well with each other except at the high-wavenumber end where the spectra drop off sharply. Such differences approaching the grid-cutoff resolution could be an artifact of the finite-difference numerical scheme. The B1 spectrum is somewhat different than the other six cases with a higher spectral density at wavenumbers on the order of the CBL depth (i.e., 
Normalized time-averaged wavenumber-weighted vertical velocity spectra as a function of normalized wavenumber 
Citation: Journal of the Atmospheric Sciences 76, 5; 10.1175/JAS-D-18-0325.1
Figures 1a and 2 together reveal that not only the vertical velocity variances collapse to a near-universal profile, so do their spectral densities, although some moderate differences are found at the high and low ends of ζ (i.e., cases B1 and S15). Next, responses of turbulent velocity variances to shear and buoyancy forcings are quantified. In our idealized case setup (see section 2), shear and buoyancy are determined by the prescribed geostrophic winds and surface heat fluxes. The normalized profiles of wind shear 
Vertical profiles of the horizontally and time-averaged shear 
Citation: Journal of the Atmospheric Sciences 76, 5; 10.1175/JAS-D-18-0325.1
Next, we move on to investigate the response of turbulent velocity variances to the bulk stability parameter ζ. Since ζ is a measure of the interplay between buoyancy and shear forcings in the CBL, the effects of buoyancy and shear are first isolated and then examined separately. First, the influence of buoyancy is isolated by taking the difference of the turbulent velocity variances from three pairs of cases with the same geostrophic winds but different surface heating. Figure 4 presents the differences of the vertical and horizontal velocity variances between (B5, S5), (B10, S10), and (S10, N10). The former member of each pair is driven with a larger prescribed surface heat flux than the latter (see Table 1). With fixed geostrophic winds, turning up the surface heat flux increases both the vertical and horizontal turbulent velocity variances throughout the depth of the CBL. For all three cases, 
Vertical profiles of the horizontally and time-averaged resolved (a) horizontal and (b) vertical turbulent velocity variance differences between the (B5, S5), (B10, S10), and (S10, N10) pairs. Members of each pair share the same geostrophic wind forcings but have different surface heat fluxes. Note that the variances in this figure are not normalized.
Citation: Journal of the Atmospheric Sciences 76, 5; 10.1175/JAS-D-18-0325.1
Next, the effect of wind shear is examined by comparing cases with the same surface heating but different geostrophic winds. Our case setup offers quite a few possible combinations, and we present five such pairs3 in Fig. 5. In each pair, the former member is forced with a stronger geostrophic wind. The S10 and SG10 pair is differentiated by the presence of geostrophic wind shear. Compared to Fig. 4, responses of turbulent velocity variances to shear are rather different from those to buoyancy. With stronger shear, 
As in Fig. 4, but for the (B5, B1), (B10, B5), (S10, S5), (S15, S10), and (S10, SG10) pairs. Members of each pair share the same surface heat flux but have different geostrophic wind forcings. The black dashed–dotted line in (a) marks the location of zero.
Citation: Journal of the Atmospheric Sciences 76, 5; 10.1175/JAS-D-18-0325.1
To summarize the findings from Figs. 4 and 5, 
Here we clarify an important point. Given the characteristic shape of the CBL mean shear profiles in Fig. 3, it is tempting to think that shear does not affect the vertical velocity variance profiles simply because shear production is small in the mixed layer. This would be a valid argument if the nature of turbulence in the CBL were local. In other words, forcings (in this case shear) at a certain elevation were only able to affect the turbulence intensity in the vicinity of that particular elevation. However, the nonlocal nature of CBL turbulence undermines the localness explanation. In the CBL, organized convective eddies span the depth of the boundary layer providing vigorous turbulent mixing (Hunt et al. 1988; Lothon et al. 2006). Such nonlocal characteristics are evidenced in the incremental variance profiles in Fig. 5; in the mixed layer (0.2zi–0.8zi) where shear is close to zero, 
b. Budget analysis
Based on the budget equations, we reinterpret the results of Figs. 4 and 5. When surface heat flux is increased, according to Eq. (2), the additional buoyancy production goes straight to increasing 
To evaluate the above interpretation, the return-to-isotropy term in Eq. (2) for all cases is presented in Fig. 6, 
Vertical profiles of the horizontally and time-averaged normalized vertical component of the return-to-isotropy terms.
Citation: Journal of the Atmospheric Sciences 76, 5; 10.1175/JAS-D-18-0325.1
Responses of the rest of the budget terms to changes in heating and shear are presented through three selected cases (B5, S5, S10) overlaid in Fig. 7. All terms are normalized by the surface buoyancy forcing 
Vertical profiles of the horizontally and time-averaged normalized forcing terms for (a),(b) 
Citation: Journal of the Atmospheric Sciences 76, 5; 10.1175/JAS-D-18-0325.1
In comparison, changes in surface heating between S5 and B5 have almost no effect on the normalized budget terms for both 
By analyzing the responses of TKE forcing terms to shear and buoyancy, it is found that the return-to-isotropy terms, which facilitate the conversion between 
c. A perspective from rotational and divergent winds
Normalized vertical profiles of the horizontally and time-averaged resolved turbulent variances of (a) divergent and (b) rotational flow for the seven CBL cases. Note the different x axes in (a) and (b).
Citation: Journal of the Atmospheric Sciences 76, 5; 10.1175/JAS-D-18-0325.1
The general patterns of the R&D variances presented in Fig. 8 support the hypothesis proposed at the end of section 3b that changes in shear only induce perturbations in the nondivergent (i.e., rotational) part of the horizontal turbulent velocity. As such, the return-to-isotropy term remains unchanged, and as a result, 
1) Characteristics of the rotational and divergent winds in the CBL
Before further investigation, it is worthwhile to briefly document the characteristics of 
Visualization of (a) w, (b) υ, (c) υd, and (d) υr at 0.15zi for case B10 at 4 h. All velocities components are normalized by w*. The color scale in (a) is different for contrast and clarity.
Citation: Journal of the Atmospheric Sciences 76, 5; 10.1175/JAS-D-18-0325.1
The spatial scales of the R&D winds are characterized in Fig. 10 through the energy spectra. At 0.15zi, the divergent 
As in Fig. 2, but for the horizontal turbulent velocity spectra 
Citation: Journal of the Atmospheric Sciences 76, 5; 10.1175/JAS-D-18-0325.1
To facilitate the upcoming discussion on momentum fluxes, the vertical energy spectra 
Height–wavenumber contours of the energy spectra of case B10 are presented in Fig. 11. As in Fig. 10, the boundary layer–scale energy peaks of 
Time-averaged height–wavenumber contours of the normalized (a) horizontal energy spectra, (b) vertical energy spectra, (c) divergent energy spectra, and (d) rotational energy spectra for case B10. All spectra are normalized by 
Citation: Journal of the Atmospheric Sciences 76, 5; 10.1175/JAS-D-18-0325.1
The horizontal energy spectra of case N10 is also examined to allow a general qualitative comparison of the R&D energy partition in the NBL and the CBL. Comparing Fig. 12 to Fig. 10, the most distinct difference is that 
As in Fig. 2, but for case N10. The energy spectra are normalized by the friction velocity squared 
Citation: Journal of the Atmospheric Sciences 76, 5; 10.1175/JAS-D-18-0325.1
The streamwise momentum fluxes associated with the R&D winds are presented in Fig. 13. The spanwise momentum fluxes are close to zero and are not presented. Momentum fluxes for B1 and B5 are not presented since they are very small, and show large wiggles when normalized with 
Vertical profiles of the horizontally and time-averaged resolved streamwise momentum fluxes of the (a) divergent, (b) rotational, and (c) total flow for five CBL and one NBL cases. The momentum fluxes are normalized by the friction velocity squared 
Citation: Journal of the Atmospheric Sciences 76, 5; 10.1175/JAS-D-18-0325.1
The divergent momentum fluxes 
The second notable feature of Fig. 13a is that 
We conclude this subsection by briefly summarizing the characteristics of the R&D winds in the CBL. The divergent wind contains the large coherent CBL structures that are essentially buoyancy driven. Without heating, 
2) Budget analysis of the rotational and divergent TKE
The 
Vertical profiles of the horizontally and time-averaged normalized forcing terms for (a) 
Citation: Journal of the Atmospheric Sciences 76, 5; 10.1175/JAS-D-18-0325.1
The budget of 
Budget analyses of Figs. 7 and 14 together suggest the following schematic of energy transfer in the CBL as illustrated in Fig. 15. Thermal energy enters the system by generating 
Schematic of energy conversion in the convective boundary layer. Processes I through IV represent buoyancy production, shear production, return to isotropy, and divergent and rotational energy conversion, respectively. Dashed arrows represent direct and indirect routes through which shear production might affect 
Citation: Journal of the Atmospheric Sciences 76, 5; 10.1175/JAS-D-18-0325.1
Going back to the question posed at beginning of this subsection, we examine the possible routes through which shear could have affected 
The indirect route involves shear production of 
4. Summary
The classic L80 profile of 
Large-eddy simulations of seven CBL and one NBL cases are performed with different thermal and mechanical forcings. By constructing the horizontal 
To understand the null response of the return-to-isotropy terms to shear, it is hypothesized that the incremental horizontal turbulent velocity as a result of the changes in shear is divergence free. Separation of the horizontal turbulent winds into rotational ur and divergent ud components and subsequent analysis confirms this hypothesis. It is found that similar to 
Further analysis of the spectral characteristics of the divergent and rotational winds relates to the former with boundary layer–scale organized convection and the latter with inertial subrange eddies. This suggests a connection of the reason why changes in shear do not induce horizontally divergent velocity perturbations to the dynamics and interactions of eddy structures. With the physical representation of ur and ud in mind, examination of the rotational 
- Direct shear production of
- Indirect effects of shear production on
With both limited direct and indirect influences of shear production on 
We thank Hao Fu for suggesting the R&D decomposition and his MATLAB code, and Prof. Ming Xue for helpful discussions. We thank Dr. Don Lenschow and two other reviewers for their constructive comments. This work was supported by the National Key R&D Program of China (Grant 2018YFC1506802), and the National Natural Science Foundation of China (Grant 41875011). Simulations are performed at the High Performance Computing Center of Nanjing University.
APPENDIX
Variance Budgets
Vertical budget profiles of 
Vertical profiles of the horizontally and time-averaged normalized forcing terms for (a) 
Citation: Journal of the Atmospheric Sciences 76, 5; 10.1175/JAS-D-18-0325.1
In section 3b, the residual method is used to compute the dissipation rate, which effectively absorbs the residual term into ε. Ratios of the resulting 
Vertical profiles of the horizontally and time-averaged ratio of 
Citation: Journal of the Atmospheric Sciences 76, 5; 10.1175/JAS-D-18-0325.1
Vertical profiles of the normalized conversion term for all CBL cases are presented Fig. A3. Case S15 aside, the normalized conversion terms exhibit near universality among the CBL cases above the surface layer. The positive signs of the conversion terms suggest that TKE flows from 
Vertical profiles of the horizontally and time-averaged normalized conversion term in Eq. (8). The x axis is cut off at −1 for clarity. The surface value of S15 extends to −3.5. The dashed–dotted black line marks the location of zero.
Citation: Journal of the Atmospheric Sciences 76, 5; 10.1175/JAS-D-18-0325.1
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Lareau et al. (2018) noted the exception that shallow cumulus of medium cloud cover (i.e., cloud fraction between 0.3 and 0.5) has a somewhat larger 
The bulk stability parameter is sometimes also expressed as 
Other combinations can be inferred from these five pairs, for example, the (S15, S5) pair is obtained by adding (S10, S5) and (S15, S10).
The return-to-isotropy term is sometimes also referred to as the pressure redistribution term (Stull 1988).
Strictly, there should be a third harmonic component (uh, υh) from the Helmholtz–Hodge decomposition. However, the harmonic component is identically zero for periodic domains (Cao et al. 2014).
