## 1. Introduction

Prediction is a principal aspect of the atmospheric sciences and is traditionally interweaved with weather forecasting (Richardson 1922), but later is also associated with climate projections (Phillips 1956; Smagorinsky 1963). A key element in accurate projections is the understanding of the relevant physical processes. This two-part paper aims to advance the modeling of turbulence, which is one of the most consequential physical processes in geophysics. Specifically, the central theme is the modeling of turbulence within the context of large-eddy simulation (LES). The main focus of the investigation concerns the atmospheric boundary layer, since atmospheric turbulence is most prominent in this lowermost layer of the atmosphere.

The value and necessity of the LES modeling technique is a consequence of the large range, *O*(10^{8}), of sizes of motions in the atmosphere compared to the current computational capabilities, *O*(10^{3}). A viable modeling paradigm is to employ the Russian doll approach, where the entire range of scales is divided into subsets that are tractable by appropriate techniques, which often rely on information from a finer-scale model or parameterization. Within this context, LES is used in the development and evaluation of weather or climate model parameterizations by resolving quantities that are heavily dependent on small-scale statistics, such as entrainment rates (e.g., Tiedtke 1989; Neggers et al. 2009). In turn, LES relies on closures or subgrid-scale (SGS) turbulence models for the unresolved motions.

Here, we provide extensive evaluation and assessment of such an SGS model, the buoyancy-adjusted stretched-vortex model (BASVM), that was presented in Chung and Matheou (2014, hereafter Part I). BASVM employs flow physics ideas in the form of vortical structures, extensively utilizes the current knowledge of stratified turbulence, and takes into account the anisotropic character of the unresolved (subgrid) motions. Keeping up with the Russian doll approach, BASVM development is informed by a series of direct numerical simulations (DNS) of homogeneous stratified shear turbulence (Chung and Matheou 2012).

The overarching goal is to advance the state-of-the-art LES modeling of atmospheric boundary layers by considering contributions in two areas. First, we aim to capture diverse atmospheric conditions using an identical model setup, that is, without using any flow-adjustable parameters of the SGS model. Second, we aim to demonstrate grid convergence of low-order statistics, for example, horizontal means and second-order moments, and suggest an empirical grid convergence criterion suitable for LES.

The present investigation has strong links with the study of Matheou et al. (2011), where the impact of numerical discretization and grid spacing on model prediction is discussed, and Matheou et al. (2010), where grid convergence and convergence criteria in LES using the stretched-vortex model are presented.

The focus of the simulations is on the representation of turbulence in the LES, thus complex interactions and feedbacks with other processes are excluded. The present LES runs include the physics of density stratification, moist physics that include latent heat exchange due to condensation or evaporation, warm-rain microphysics for convective clouds, and simplified radiation for long-lived stratiform clouds.

The paper is organized as follows: The problem of grid convergence and a convergence criterion are discussed in section 2. The details of the LES implementation including specific details pertaining to the SGS models are presented in section 3 followed by the simulation results in section 4. A brief synthesis of the results is presented in section 5. Finally, the conclusions are presented in section 6.

## 2. Grid convergence for LES

A central question in LES, but also in numerical modeling in general, is how the predicted statistics depend on the grid resolution. Several previous investigations have explored the impact of grid resolution, numerical discretization, and SGS model (e.g., Mason and Thomson 1992; Vreman et al. 1996; Brown 1999; Stevens et al. 1999, 2002; Chow and Moin 2003; Pope 2004; Cullen and Brown 2009; Matheou et al. 2011; Sullivan and Patton 2011). The coveted result in these investigations is an LES framework where predictions (i.e., mean wind and thermodynamic–variable profiles and turbulent fluxes given initial and boundary conditions and large-scale forcing) are insensitive to the aforementioned simulation aspects. Simply, results should depend only on physical (e.g., atmospheric stability) and not numerical parameters (e.g., grid resolution). Before these aspects are explored, a convergence criterion is discussed, since, in practice, the quality of the LES prediction should be assessed without performing computationally expensive runs with varying grid resolution.

Grid convergence is a general term that must be qualified by the type of flow statistic that exhibits convergence. Moreover, convergence in one statistic does not imply convergence of another (Brown 1999). It is expected that convergence will be achieved at coarser resolutions for lower-order statistics (mean profiles) than for higher-order statistics (turbulent fluxes, triple correlations, etc.). The present discussion concerns only very high Reynolds number flows, such as those encountered in the atmosphere, consequently inferences from DNS results (e.g., Meyers et al. 2003) are not directly applicable.

Before we consider a convergence criterion, we briefly discuss what convergence should *not* depend on. For instance, it is quite straightforward to say that the grid size should be 100 times larger than the smallest flow scale. However, such a criterion is unpractical because it violates the predetermined design of LES, that is, a simulation without the smallest flow scales. Convergence (e.g., of turbulent fluxes that determine the mean profiles) should depend on the characteristics of turbulence, and turbulence depends on the large-scale parameters rather than molecular dissipation [see Frisch (1995, p. 67) for further discussion and references].

*l*, then the LES must resolve this length scale, the meaning of which is made more concrete by the following estimate based on turbulent kinetic energy (TKE). Consider an estimate for the resolved TKE fraction using the Kolmogorov energy spectrum:where

*z*) becomes comparable to the grid size. Therefore, the ratio of kinetic energy is not expected to accurately capture the resolution adequacy of the LES in this case.

The dynamics of stable density stratification can also result in small length scales in the vertical direction, that is, smaller *l*. The resolution requirement becomes more stringent in this regime if one were to “resolve” the turbulent eddies in the sense used above. This is the approach taken for the nocturnal stratocumulus [Second Dynamics and Chemistry of the Marine Stratocumulus (DYCOMS-II) field study RF01] case, in which the sharp density interface characterized by small turbulent eddies is resolved (resolved TKE fraction of 90% when

The kinetic energy criterion has two important advantages compared to previous propositions: it is flow configuration and SGS model independent. For instance, the criterion of Sullivan and Patton (2011) includes a flow length scale and a model constant, while Stevens et al. (2002) refer to (dimensional) grid spacings for a specific flow. Note also that the kinetic energy criterion that is used in the present study is nondimensional.

In the variable grid resolution simulations of this study, we set the LES filter width, or cutoff scale *x*. In general, these two parameters can be varied independently, and their ratio has been shown to affect the numerical solution through the interaction between numerical errors and SGS model (Ghosal 1996; Chow and Moin 2003). In the grid resolution study of Part I, it is shown that results remain unchanged when *x* is varied (

## 3. Large-eddy simulation implementation

To a large part, the LES computer code used in the present simulations is new, yet, as is often the case, it has evolved from previous LES implementations. The current LES implementation is a descendent of the University of California, Los Angeles, Large-Eddy Simulation (UCLALES) code (Stevens et al. 1999, 2005a; Stevens and Seifert 2008) and, in particular, the variant used in Matheou et al. (2011). The most important changes pertaining to the flow physics are the addition of a high-order advection discretization and the stretched-vortex SGS model.

The LES code numerically integrates the filtered (density weighted) anelastic approximation of the Navier–Stokes equations (Batchelor 1953; Ogura and Phillips 1962). The conservation equations for mass, momentum, liquid water potential temperature, total water, and additional microphysical quantities are solved in a doubly periodic domain in the horizontal directions, utilizing the *f*-plane approximation, with impermeable top and bottom boundaries. The complete system of governing equations is documented in appendix A. A sponge region is utilized near the top of the domain to minimize undesirable gravity wave reflection. The flow fields are discretized on an Arakawa C grid (Arakawa and Lamb 1977). The fourth-order fully conservative scheme of Morinishi et al. (1998) is used to approximate spatial derivatives except those related with the SGS terms where second-order centered differences are used. An exact Poisson solver using discrete Fourier transforms in the horizontal directions is used to compute the pressure and satisfy the anelastic constraint (Schumann 1985). The semidiscrete system of equations is advanced in time using the third-order Runge–Kutta method of Spalart et al. (1991). In the cases where the process of precipitation is included, the double-moment bulk microphysical parameterization of Seifert and Beheng (2001) is used (Stevens and Seifert 2008). To preserve conservation of water, a second-order monotonized central (MC) flux-limited scheme (Van Leer 1977) that ensures monotonicity is used to advect rain mass and raindrop number. For all other prognostic variables advection is nondissipative; thus, any dissipation arises purely from the subgrid-scale closure (see also section 5b).

In the present simulations, two turbulence closures are used: the BASVM subgrid-scale model (Misra and Pullin 1997; Voelkl et al. 2000; Pullin 2000; Part I) and the constant coefficient Smagorinsky (1963)–Lilly (1962) model. The focus of this study is the results using BASVM, but two cases are repeated using the Smagorinsky model such that the effect of the SGS model can be discerned. The constant coefficient Smagorinsky model is chosen because it has been widely used in LES of the boundary layer and of its very simple formulation, such that the rather complex BASVM can be compared with a simpler SGS model. Recent modifications to Smagorinsky–TKE-type SGS models (specific references are cited in the simulations section for individual flow configurations) are expected to perform better. Moreover, results from model intercomparisons that are compared with BASVM in the following sections include LES implementations with the Smagorinsky SGS.

No modifications are required in order to use BASVM in the LES of the atmospheric boundary layer. The generic buoyancy variable *b* that was used in Part I is identified with the normalized virtual potential temperature (*gθ*_{υ}/*θ*_{0}) here (see appendix A for further details).

*τ*

_{ij}is the SGS tensor. Substituting the expression of the BASVM stress tensor,where

*K*

_{s}is the SGS turbulent kinetic energy,

*e*

_{i}are the components of the unit vector aligned with the most extensional eigenvector of

*λ*

_{3}≥ 0 the corresponding eigenvalue. (In the anelastic approximation the trace

The Smagorinsky model is discretized using the “SMC” implementation of Matheou et al. (2011), where the approximations of the components of the rate-of-strain tensor are computed at the grid cell centers. The model constant is set to *C*_{S} = 0.23, and the turbulent Prandtl number is Pr_{t} = 0.33. A similar discretization is employed for BASVM, which results in multiple evaluations of the resolved rate-of-strain tensor and its eigenvectors at four locations per grid cell instead of the single evaluation used in a collocated variable arrangement.

For the Smagorinsky model, a damping function is used near the surface to reduce the magnitude of the turbulent diffusivity by modifying the characteristic resolved-scale length scale (Mason and Thomson 1992). For BASVM no modification of the cutoff scale

For both SGS models, the subgrid condensation scheme is “all or nothing” (i.e., no partially saturated air in the grid cells) (e.g., Cuijpers and Duynkerke 1993). The thermodynamic state at the cell center is used to classify each grid cell as saturated or not and determine the corresponding thermodynamic coefficients for all variables, including those residing at the cell’s vertices.

## 4. Simulations

The main purpose of the simulations is to validate the new SGS model, assess its performance for a series of configurations of practical interest, demonstrate that the current LES implementation can successfully predict diverse conditions without any parameter tuning, and show that the results are independent of grid resolution. Accordingly, the setup of the LES remains the same for all simulations. We use an advection scheme for momentum and scalars that has no dissipation, even for cases with sharp inversions, such as stratocumulus-topped boundary layers. This may not be an optimal setup for some cases, and we do not suggest that this setup is followed by other investigators for all cases. All dissipation in the LES originates from the SGS model, therefore we view the present simulations as a particularly difficult test of the SGS model (see also section 5b).

All simulations, but the free convection case, are based on model intercomparisons of the Global Water and Energy Experiment (GEWEX) Cloud System Studies (GCSS) project. The only case that partially deviates from the GCSS setup is the stratocumulus boundary layer where surface fluxes are calculated using the Monin–Obukhov theory rather than using the prescribed values. Table 1 summarizes the simulations.

Summary of the cases simulated. The references correspond to the case specification where details pertaining to LES setup and results from several models can be found. The grid spacing is uniform in all cases, Δ*x* = Δ*y* = Δ*z*, *L*_{x,y,z} is the domain length, and *N* is the number of grid points.

The simulation domains are larger than the corresponding intercomparison setups in order to achieve more representative statistics. Moreover, all grids used here are uniform with equal spacing in all directions, Δ*x* = Δ*y* = Δ*z*. Most previous LES studies have used anisotropic grids with Δ*z* < Δ*x* to better resolve thin inversion layers. However, the primary difficulty in strongly stably stratified flows is the anisotropy of motions; thus, countering flow anisotropy with grid anisotropy is not a viable solution for LES. As shown by Waite (2011), high grid anisotropy can suppress overturning motions and lead to spurious results. The SGS model cutoff scale is set equal to the grid length

In all cases, when turbulence statistics are reported (e.g., the turbulent kinetic energy, velocity variances 〈*ww*〉), they include subgrid-scale contribution. The angle brackets denote horizontal averages or (for time periods when the flow is stationary) a horizontal and time average. For clarity, the angle brackets are omitted for all mean fields, and the primes denoting fluctuations from the mean are omitted from the turbulent fluxes.

The simulations are presented in order of increasing complexity and difficulty, beginning with single-phase flows, followed by cloudy convective cases, and finally with a stratocumulus-topped boundary layer. For all cases, the current LES results are plotted against the observations and/or results from previous model runs, and in two cases, are compared with theoretical relations.

### a. Free convection

Free convection over a uniformly heated surface, topped by a layer of uniformly stratified fluid, with zero-mean flow is a fundamental configuration of the atmospheric boundary layer and has been widely studied using LES (e.g., Moeng and Wyngaard 1988; Ebert et al. 1989; Schmidt and Schumann 1989; Nieuwstadt et al. 1993; Khanna and Brasseur 1998). In this first and simplest case, we consider only a dry atmosphere, where conditions are such that water always remains in the gas phase. Dry convection is often used to evaluate different aspects of the LES technique, including grid convergence of various flow statistics (Mason and Brown 1999; Brown et al. 2000; Sullivan and Patton 2011). Accordingly, the current simulations aim to assess BASVM on an unstably stratified flow and provide a reference for comparison with the more complex runs that follow.

In LES of buoyancy-driven flows using the standard stretched-vortex model (i.e., without the buoyancy adjustment), Chung and Pullin (2010) showed good agreement with direct numerical simulation results. BASVM for unstably stratified flows accounts for the additional SGS turbulence buoyancy production, whereas this term is neglected in the standard SGS model.

The setup of the simulations is the same as in Matheou et al. (2011). The domain size is (10.24)^{2} × 4 km^{3}. The initial potential temperature lapse rate is 2 K km^{−1}, with *θ*(*z* = 0) = 297 K. The initial total water mixing ratio lapse rate is −0.37 × 10^{−3} km^{−1} up to *z* = 1350 m and −0.94 × 10^{−3} km^{−1} higher up with *q*(*z* = 0) = 5 × 10^{−3}. The temperature and total water surface fluxes are 0.06 K m s^{−1} and 2.5 × 10^{−5} kg kg^{−1} m s^{−1}, respectively. Four simulations at resolutions Δ*x* = 10, 20, 40, and 80 m are performed.

Figure 1 shows a comparison of the current results with the measurements reported in Lenschow et al. (1980) and the convergence of LES profiles for potential temperature, vertical velocity variance, TKE, and the vertical flux of temperature. The agreement of the LES with observations is overall good, with all resolutions located within the scatter of the observations. Moreover, turbulence statistics agree well with previous LES (e.g., Schmidt and Schumann 1989; Sullivan and Patton 2011) (not shown in Fig. 1). Results using the Smagorinsky SGS model for an identical flow configuration are reported in Matheou et al. (2011). The most notable difference between the current and previous LES results is that the maxima of the vertical velocity variance 〈*ww*〉 and (consequently) TKE are somewhat larger in the current LES.

Overall, grid convergence is achieved, but, as expected, not vall quantities converge at the same rate. The mean potential temperature is virtually unchanged when the resolution varies by a factor of 8, whereas the peak of the vertical velocity variance drifts. For all cases the inversion height *z*_{i}, defined as the minimum of the buoyancy flux, is the same; thus, the normalization of height in Fig. 1 is inconsequential.

*C*is a constant (Lumley and Panofsky 1964, p. 109). The ratio of the

*θ*

_{υ}gradient to the rhs without the constant is plotted in Fig. 2 with respect to height normalized by the convection length

*z*

^{+}≡

*z*[〈

*wθ*

_{υ}〉

_{(z=0)}/(Pr/

*ν*)

^{3}]

^{1/4}≈ 8000 wall units, which implies very coarse resolution of the surface layer. The derivatives of temperature and temperature turbulent flux exhibit oscillations near the surface that often occur in LES; see results from various LES models and discussion in Brasseur and Wei (2010). The magnitude of oscillations is comparable to previous results with the stretched-vortex model using similar wall functions (Pantano et al. 2008). The temperature gradient profiles show good agreement with respect to grid resolution; however, for the region that is resolved, the constant

*C*somewhat varies with height within the surface layer.

The question of what is an adequate grid resolution for the flow is contingent on the quantity of interest. For instance, the boundary layer depths are the same for all resolutions (not shown in Fig. 1), whereas TKE converges for Δ*x* < 20 m. The boundary layer is poorly resolved in the coarse-resolution run with *z*_{i}/Δ*x* ≈ 20 at *t* = 8 h. The ratio of SGS to total TKE is shown in Fig. 3. Except the coarse-resolution run, the fraction of the SGS TKE is less than 10%. As expected, the peaks of the TKE ratio for the high-resolution cases show that the flow is less resolved near the surface and in the interfacial layer at the top of the boundary layer.

### b. Stable boundary layer

Most of the buoyancy adjustment development of the stretched-vortex model pertains to the stably stratified case and the second flow considered here aims to assess the stably stratified branch of BASVM. We follow the setup of Kosovic and Curry (2000) and Beare et al. (2006) that corresponds to a weakly stable arctic boundary layer, where the term weakly stable refers to conditions that can sustain turbulence without the creation of intermittent laminar–turbulent layers. The flow becomes nearly stationary after 8 h with a boundary layer depth that is about double the Obukhov length *L* based on the surface fluxes. As shown in Fig. 4, the combination of surface shear and cooling and planetary rotation results in a fairly complex mean flow.

The accurate simulation of this flow has been challenging, as demonstrated by the differences between various LES models and the lack of grid convergence in the intercomparison study (Beare et al. 2006), with some more recent results showing improved grid convergence characteristics (Stoll and Porté-Agel 2008; Huang and Bou-Zeid 2013). Previous studies have used grid resolutions in the range of 1–12 m, with Δ*x* ≈ 4 m the most common grid size. Here, we carry out four simulations at Δ*x* = 2, 4, 8, and 16 m with the stretched-vortex model and a second set of runs with the Smagorinsky model for comparison. The current domain size is about twice as large in the horizontal directions than in the intercomparison study, but about the same size as in the later studies.

Figure 4 shows a comparison of the BASVM profiles with the LES results reported in Beare et al. (2006). Only 2-m results from Beare et al. (2006) are shown. The current results are well within the spread of the intercomparison and agree with the model mean.

The BASVM mean profiles of wind and potential temperature are in good agreement for Δ*x* ≤ 8 m. The grid with Δ*x* = 16 m predicts a somewhat higher boundary layer depth, but overall follows closely the profiles of the finer grids. The discrepancy is within 2Δ*x* and can be considered small compared to the resolution. Moreover, the coarsest resolution can marginally resolve the finer features of the flow, such as the jet of the υ velocity at the boundary layer top; therefore, differences are expected. The momentum and buoyancy fluxes also show good convergence properties. Several models (Beare et al. 2006) predict the linear variation of the buoyancy flux; however, the present simulations show good collapse of the profiles with respect to resolution.

The 〈*wθ*〉 and 〈*uw*〉 fluxes show a spurious overshoot at the first level above the surface. This results from an overestimate of the SGS part of the fluxes. Similar to the oscillations of *dθ*_{υ}/*dz* in Fig. 2, oscillatory behavior near the surface is sometimes observed in LES solutions [see Brasseur and Wei (2010) and references therein]. While improving the oscillations should be possible, because only one vertical level is affected and there is no negative effect on the overall quality of the solution, it was not deemed critical for this paper.

As an additional validation check, in Fig. 5 the MO functions for wind shear and potential temperature are compared with their commonly used empirical forms *ϕ*_{m}(*z*/*L*) = 1 + 4.8*z*/*L* and *ϕ*_{h}(*z*/*L*) = 1 + 7.8*z*/*L*, respectively. The deviations from Monin–Obukhov scaling near the surface are small, and less than the majority of the results for neutral flows reported in Brasseur and Wei (2010). Similar to the free convection temperature gradient profiles near the surface (Fig. 2), the differentiation results in less smooth curves than the corresponding variable profile (Fig. 4). Moreover, resolution in the surface layer is coarse with the first grid point of the highest-resolution run at *z*^{+} ≡ *zu*_{*}/*ν* ≈ 2000 (note that the diffusive length scale is defined differently in shear and pure convective flows). Grid convergence is maintained as the profiles approach the surface, but the convergence of potential temperature is non-monotone with the 16-m run being in better agreement with the highest-resolution run compared to the 8-m run.

The subgrid to total TKE ratio of Fig. 6 shows that the stable boundary layer case is less resolved than the free convection case of the previous section since for the highest-resolution run about 6% of the TKE remains unresolved. As can be seen from the profiles of Fig. 6, for Δ*x* = 16 and 8 m near the top of the boundary layer (*z* > 180 m) the SGS model predicts null fluxes because of the strong stability.

Although previous studies have reported LES predictions for the first GEWEX Atmospheric Boundary Layer Study (GABLS) case using various models, we carry out a set of runs using the constant coefficient Smagorinsky model in order to provide a more representative comparison of the effect of the SGS model with all other LES configuration choices being identical. The mean profiles are shown in Fig. 7. An LES run at Δ*x* = 16 m with the Smagorinsky model was not feasible because the model predicts no fluctuations. In contrast, simulations with BASVM do not laminarize. A run with Δ*x* = 32 m was carried out using BASVM, and even though the quality of the prediction is poor, the flow shows turbulent fluctuations that are comparable to the finer resolutions. This is an advantage of BASVM that consistently predicts a turbulent state regardless of the grid resolution in this case.

The differences with respect to resolution for the Smagorinsky SGS model are larger than the runs with BASVM. The runs with Δ*x* = 2 and 4 m predict the same boundary layer depth, with fair overall agreement. Only the 2-m Smagorinsky run captures the small jet of the *υ* velocity component at *z* ≈ 200 m.

Before discussing more complex flows, we briefly comment on the features of the instantaneous flow for a grid-converged LES. Figure 8 shows the instantaneous potential temperature at *t* = 9 h for two grid resolutions Δ*x* = 2 and 8 m using BASVM. Although the flow is different, the main features remain unchanged with the finer resolution capturing more detail. For example, because shear is predominantly along the *x* direction (Fig. 4), *θ* exhibits ramp-and-cliff structures on the *x*–*z* plane.

### c. Shallow cumulus

Following the validation of BASVM for convective and stable boundary layers, the next three cases consider moist convection. First, we model a cumulus-topped boundary layer using the setup based on conditions from the Barbados Oceanographic and Meteorological Experiment (BOMEX) campaign (Holland and Rasmusson 1973). BOMEX is likely the most prevalent shallow cumulus case and one of the early LES studies of cloudy boundary layers (Cuijpers and Duynkerke 1993). Here, we follow the configuration of Siebesma et al. (2003), with a larger domain of (20.48)^{2} × 3 km^{3}, and carry out three simulations with Δ*x* = 20, 40, and 80 m. Typical resolutions used in previous studies have Δ*x* ≈ 40 m (e.g., Brown 1999; Neggers et al. 2003; Siebesma et al. 2003; Riechelmann et al. 2012).

Figure 9 shows the current BASVM LES profiles plotted against the model intercomparison results of Siebesma et al. (2003). Similar to the stable case, the results are well within the spread of the intercomparison and the two highest resolutions agree with the model mean, except the liquid water profile. Previous LES studies have shown good agreement between different model implementations (i.e., narrow spread of model intercomparison results for potential temperature and total water). Thus, this is not considered a particularly challenging case for LES, and the agreement of the mean profiles of potential temperature and total water is somewhat expected, with the exception of the Δ*x* = 80-m grid, which is typically considered a coarse resolution. The grid convergence of liquid water is more significant than the other mean profiles because it is related to the variance of water—a higher-order moment.

The turbulent fluxes and TKE exhibit differences with respect to grid resolution that are larger than the stable and free convection cases. The ratio of SGS to total TKE (Fig. 10) shows that the shallow cumulus case is less resolved than the previous flows with only the highest-resolution grid (Δ*x* = 20 m) resolving more than 90% of the total TKE.

### d. Precipitating shallow cumulus

While the conditions of the preceding BOMEX case result in fair-weather, nonprecipitating, cumulus clouds, precipitation in trade wind cumulus clouds is ubiquitous and exhibits subtle dependence on the meteorological environment with deeper clouds (*z*_{cloud} > 3 km) readily precipitating (Nuijens et al. 2009). The conditions of the simulations are based on a period from the Rain in Shallow Cumulus over Ocean (RICO) campaign (Rauber et al. 2007) and follow the setup of the model intercomparison case (vanZanten et al. 2011). The main characteristic of the case is the interaction of precipitation with convection, which results in difficulties in the accurate prediction of the boundary layer. Precipitation rates vary considerably between the models of the intercomparison study and show strong dependence on the numerical methods used (Matheou et al. 2011).

The present runs can be considered an extension of Matheou et al. (2011) as the setup is identical with a domain size of (20.48)^{2} × 4 km^{3} and grid resolutions of Δ*x* = 20, 40, and 80 m. In addition to the use of BASVM, a higher-order advection is used presently; therefore, a second set of runs with the Smagorinsky model is also carried out. The focus of the results is on the evolution of precipitation during the 24-h run; thus, only time traces are reported here. The convergence characteristics of the profiles are similar to the BOMEX case.

Figure 11 shows the radiance distribution at the top of the atmosphere illustrating the realistic structure of cloud field [cf. images in Snodgrass et al. (2009)], the formation of cold pools due to precipitation, and a few small cloud anvils forming at the boundary layer top. The radiance field was calculated with the Monte Carlo code for physically correct tracing of photons in cloudy atmospheres (MYSTIC), a three-dimensional radiative transfer model (Mayer 2009; Buras and Mayer 2011). The solar zenith angle is 45°, the solar azimuthal angle is 25° (i.e., the sun is southwest), and the sensor is due north, looking toward the south (top of figure) at a 45° viewing angle from nadir direction. The calculation is for a red wavelength of 671 nm and assumes a Gamma size distribution for the cloud droplets with effective radius *r*_{eff} = 10 *μ*m and variance of 0.1 to convert liquid water content into cell opacity. The dark areas in Fig. 11 correspond to the cloud shadows on the ocean surface, which is assumed uniform and modeled as a rough Fresnel interface with the Cox–Munk distribution of slopes for a wind speed of 10 m s^{−1}.

The variability, especially in precipitation, is shown in the time traces for both BASVM (Fig. 12) and Smagorinsky (Fig. 13) SGS models. To reduce the large small-time variability of the statistics of Figs. 12 and 13, a 1-h rolling average is employed.

The time traces show that BASVM becomes more energetic at coarser resolutions, predicting higher values of TKE. The same trend is observed in the TKE profiles for the BOMEX case (Fig. 9). The increase of TKE for Δ*x* = 80 m interacts with the precipitation dynamics, creating large swings in the cloud and rain liquid water path (LWP) and surface precipitation rates. The amount of cloud liquid remains relatively constant with resolution; thus, most of the variation in TKE and precipitation is attributed to the flow organization (fewer and larger clouds versus more and smaller) rather than changes in the total cloud amount.

In this case, the corresponding results from the model intercomparison are not shown because the domain size is likely to play a role in the precipitation statistics. The expected surface precipitation rate *P*_{srf} = 35 W m^{−2} from the analysis of the observations (Nuijens et al. 2009) is used as a validation metric. The highest-resolution run produces the expected surface precipitation rate during the last 4 h of the simulation.

The variability of the time traces makes the assessment of grid convergence less straightforward than the other cases considered in this study. BASVM traces are in better agreement than the corresponding runs with the Smagorinsky model (Fig. 13), especially for *t* < 16 h. Precipitation is weaker (less rain water path and surface precipitation rate) in the Smagorinsky model, with an increasing tendency with finer grid resolutions, whereas precipitation statistics do not show monotone variations with BASVM—grid convergence is not always monotone. For the highest-resolution run, Δ*x* = 20 m, the precipitation statistics are in good agreement between the two SGS models.

### e. Stratocumulus

The last case we consider is that of a stratocumulus-topped boundary layer. Simulations of the stratocumulus-topped boundary layer have been the subject of increased interest because of the importance of this cloud regime to Earth's energy budget and challenges in its accurate modeling using LES. The set of physical processes encountered is complex and includes strong temperature inversions, latent heat exchange, radiation, and microphysical and aerosol interactions (e.g., Brost et al. 1982; Lenschow et al. 1988; Stevens et al. 2003a, 2005b; Zheng et al. 2011; Wood 2012).

The relatively simple case of a nocturnal boundary layer corresponding to the first research flight (RF01) of the DYCOMS-II field study (Stevens et al. 2003a) is simulated. The setup follows Stevens et al. (2005a) with the exception of the surface fluxes. The surface fluxes are computed using the Monin–Obukhov similarity theory (MOST) with Charnock’s roughness length (Charnock 1955) using SST = 292.5 K (Stevens et al. 2005a). The same roughness length is used for momentum and heat fluxes. MOST inherently adjusts to the variable grid spacing, in contrast to using drag coefficients. The domain-mean surface fluxes calculated from MOST differ from the ones specified in the model intercomparison study and they are decreasing with time (see appendix B for more details). As in the intercomparison study, a simplified radiation parameterization is used, while droplet sedimentation and drizzle are neglected.

Previous LES studies of nonprecipitating stratocumulus (e.g., Moeng 1986; Moeng et al. 1992, 1996; Stevens et al. 1996; Wang et al. 2003; Yamaguchi and Feingold 2012) captured the complex interactions between turbulence and radiation, but also showed that simulations presented more challenges than dry or cumulus convection cases. The main difficulty arises from the presence of a sharp interface between cloud and clear air at the top of the boundary layer. Several physical processes (i.e., strong stratification, latent heat exchange, and radiative cooling) occur within a thin layer (~10 m) at the boundary layer–free troposphere interface and strongly affect the entrainment of dry air into the boundary layer. These challenges often manifest in the spread of predictions between different models but also for different configurations of the same model (Stevens et al. 2005a; Savic-Jovcic and Stevens 2008).

Because of the fine vertical structure near the cloud top, previous studies have used anisotropic grids with typical aspect ratios from 2.5 to 10 and vertical resolutions from 1 to 25 m near the inversion. The grid sensitivity runs of the DYCOMS-II RF01 case of Yamaguchi and Randall (2008) show good agreement for 5-m vertical grid spacing and horizontal spacing of 30 and 50 m. In the present LES, the grid aspect ratio is unity, and three grids are used with Δ*x* = 2.5, 5, and 10 m.

Figure 14 shows a snapshot of the cloud field for the highest-resolution run, Δ*x* = 2.5 m, at the end of the run, *t* = 4 h. The cloud cover is virtually 100% with the cloud top being replete with cellular patterns of various sizes.

The differences of the time traces and profiles of Figs. 15 and 16 with respect to resolution show that the stratocumulus-topped boundary layer is a challenging case. Also shown in Figs. 15 and 16 are the model intercomparison results (Stevens et al. 2005a) and observations from the DYCOMS II campaign. The current results are in good agreement with the observations, especially in the amount of cloud liquid (top-right panel of Fig. 16). The largest differences between the current results and the intercomparison are in the cloud base and top (top-right panel of Fig. 15), likely because the fluxes computed by MOST deviate from those used in the intercomparison. The observations show a variable cloud thickness that ranges between the present results and those of the intercomparison mean. Cloud base and height do not change with respect to grid resolution and appear to slowly increase while keeping a constant cloud depth. Although the SGS to total TKE ratio (Fig. 17) in the mixed layer indicates that the flow in this layer is well resolved, the dynamics in the thin interfacial layer below the inversion seem to have a significant effect on the amount of liquid, vertical velocity variance, and TKE with all other profiles showing good agreement (Fig. 16). The variation of the amount of liquid and TKE is more evident in the time traces of Fig. 15, where after the initial transient, the boundary layer appears to attain a state of almost constant LWP and TKE, with Δ*x* = 10 m drifting to lower values and Δ*x* = 2.5 m drifting to higher values.

One of the principle quantities of interest in the stratocumulus-topped boundary layer is the entrainment rate. To compare the current LES results with the observations, we use the kinematic estimate (e.g., Stevens 2002) of the entrainment rate: *E*(*t*) = *dh*/*dt* + *Dh*(*t*), where *D* = 3.75 × 10^{−6} s^{−1} is the (prescribed constant) large-scale divergence and *h* is the height of the boundary layer, which we identify with the cloud-top height. The quantity *h*(*t*) is calculated using a linear fit of the LES data between hours 2 and 4 (Fig. 15). For all resolutions, the entrainment rate increases linearly with time. The two extreme values, at the beginning and end, of the time interval are Δ*x* = 2.5 m and *E* = (7.0, 7.1) × 10^{−3}; Δ*x* = 5 m and *E* = (6.8, 6.9) × 10^{−3}; and Δ*x* = 10 m and *E* = (6.7, 6.8) × 10^{−3} m s^{−1}. The corresponding value of the DYCOMS-II RF01 observations is 3.8 × 10^{−3} m s^{−1} (Stevens et al. 2003b), whereas most models in the intercomparison study predict values from 4 to 6 × 10^{−3} m s^{−1}.

## 5. Discussion

### a. Grid convergence

Using the results of the current simulations a grid convergence criterion that is based on the fraction of the resolved TKE is put forward. The purpose of the criterion is to help determine the accuracy of the LES results without performing multiple runs at different resolutions. Two overall trends can be discerned: first, lower-order flow statistics converge at coarser resolutions than higher-order statistics, and second, the convergence of mean fields is attainted when about 90% of the TKE is resolved, with the exception of the stratocumulus case. Although, the ratio of resolved TKE varies with respect to height, we refer to a single quantity that characterizes the entire simulation. We view this as a mean value given that no extensive layers have smaller ratios.

Note that unlike previous convergence criteria, the TKE criterion is not dependent on the flow configuration. For instance, although in the different configurations considered in this study the grid spacing varies significantly, the ratio of SGS to total TKE at which convergence is achieved does not change considerably. Moreover, the TKE criterion does not depend on the SGS model and can be used to determine the convergence of any LES model. However, it is important to note that we do not expect that all LES models will converge when 90% of the TKE is resolved.

In previous investigations of neutrally stratified turbulent shear flows with the stretched-vortex model (Matheou et al. 2010), the mean flow statistics converged when 80% of the TKE is resolved. For the current simulations, a resolution that captures only 80% of the TKE would be too coarse. It is likely that LES of atmospheric flows requires more resolution than free shear flows because of the interactions between additional physical processes, such as buoyancy forcing, latent heat exchange, and radiation. A prerequisite for the proposed convergence criterion is that all dynamically significant length scales must be resolved. In the case of the stratocumulus simulations this condition is not satisfied because of the shallow dynamics at the cloud top. Another region where the dynamically important length scales can become unresolved is near the surface; accordingly, the current criterion may not apply within the surface layer.

### b. Numerical dissipation

For all the present simulations the advection discretization for momentum and scalars is nondissipative, with the exception of the rain mass and raindrop number advection in the precipitating shallow cumulus runs. That is, all dissipation is solely furnished by the SGS model. Many previous LES investigations include some form of numerical dissipation, especially for scalar variables, but its effects have not been easy to quantify (e.g., Brown et al. 2000), whereas some studies used numerical dissipation as a physical model for turbulent dynamics (e.g., Margolin et al. 1999; Savic-Jovcic and Stevens 2008).

To demonstrate that in the current simulation all dissipation is provided by the SGS model, we take an instance of the free convection LES run (Δ*x* = 10 m) and continue the simulation without the SGS model. The top panel of Fig. 18 shows the initial potential temperature field, whereas the bottom panel shows the same field after 360 s (443 iterations) with the SGS terms switched off. Because there is no dissipation mechanism, the energy that cascades to smaller scales accumulates on the grid as fine-grained noise. The large-scale structures are still discernible on the bottom panel of Fig. 18 because they have larger time scales compared to the time interval between the two panels. Eventually all variables will become meaningless, but unless the Courant–Friedrichs–Lewy (CFL) numerical stability condition is violated, the simulation will not “blow up” because the advection scheme conserves kinetic energy and scalar variance.

### c. Computational cost considerations

Execution time is always a practical consideration for numerical models. Thus, we provide some basic estimates of the computational cost of BASVM in order to have a complete account of the model’s performance aspects. The code profiling measures are calculated for the shallow cumulus (BOMEX) run during an interval of about 300 iterations after *t* = 3 h. The profiling statistics correspond to a serial (single CPU) run and exclude the overhead of parallel execution.

A run with BASVM compared to the Smagorinsky takes about 4.2 times more time to complete. The calculation of the SGS terms is the most time consuming operation in the LES, accounting for 23.5% of the time with the Smagorinsky and 64.7% with BASVM. The Poisson solver for the pressure is the second most expensive operation accounting for 23.2% and 5.6% of the time when the Smagorinsky and BASVM are used, respectively.

Although BASVM appears, at first instance, more computationally expensive than the Smagorinsky model, LES runs using BASVM are typically more accurate at lower resolutions. For instance, in the stable boundary layer results, the better performance of BASVM with respect to resolution outweighs the increase in computational cost because a doubling in resolution increases the computational cost by a factor of 16.

## 6. Conclusions

In this second part of the large-eddy simulation (LES) modeling of stratified turbulence, the central theme of the investigation is the performance of the buoyancy-adjusted stretched-vortex subgrid-scale (SGS) model (BASVM) in LES of the atmospheric boundary layer. However, the overarching aim of Part II is to consider the potential of the LES technique in predicting diverse atmospheric conditions without any change (or tuning) of the model setup. Thus, the main finding of the study is the demonstration that such an LES model setup, which can accurately capture diverse conditions without any flow-adjustable parameters, is feasible. The LES implementation is based on BASVM and a fourth-order, nondissipative fully conservative discretization of the momentum and scalar advection terms.

BASVM is assessed using cases that were studied extensively. The LES runs are based on the corresponding model intercomparisons, and the results are in agreement with the intercomparison studies and observations. The good performance of the SGS model is attributed to the physics-based construction of the turbulence closure. The stability correction for stratified flows is faithful to the physics of stratified turbulence and accounts for the increasing flow anisotropy as stratification increases.

For all cases, grid convergence of the mean fields, including the liquid water vertical distribution and basic precipitation statistics, is achieved. Flow statistics converge for resolutions that are typically considered coarse, such as Δ*x* = 80 m for shallow cumulus convection and Δ*x* = 8 m for moderately stably stratified boundary layers. Moreover, no spurious laminarization is observed in the stably stratified case when resolution is coarsened. Second-order statistics (variances and turbulent kinetic energy) require finer grid resolutions to achieve convergence and only for the free convection and stable boundary layer cases convergence is attained.

Grid convergence of the mean fields is achieved when about 90% of the turbulent kinetic energy (TKE) is resolved. The present results suggest that about 95% of the TKE must be resolved for the convergence of second-order statistics. A prerequisite for the proposed convergence criterion is that all dynamically significant length scales must be resolved. This is exemplified in the stratocumulus simulations where although the mixed layer is very well resolved (about 97% resolved TKE fraction at Δ*x* = 5 m), the liquid water path does not converge because the small-scale radiation–turbulence interactions near the cloud top are not adequately resolved.

The current simulations corroborate previous findings that for moderate grid resolutions the choice of SGS closure and advection scheme has a significant impact on the LES prediction. Comparisons between BASVM and the constant coefficient Smagorinsky–Lilly SGS model show that the two models converge to identical flow statistics for sufficiently fine grid resolutions. It appears that the coveted outcome of previous LES model intercomparison studies—that the prediction is independent of the choice of the SGS model—can be achieved as the grid is refined. In turn, this can suggest a metric of SGS model prediction skill, that is, the coarsest resolution at which the correct solution is attained.

Finally, the current simulations demonstrate that numerical dissipation is not a necessary component of an LES and that accurate predictions can be achieved by solely relying on the SGS model for all dissipation.

## Acknowledgments

We would like to acknowledge discussions with J. Teixeira (JPL) and P. Dimotakis (Caltech). We thank B. Stevens (Max Planck Institute for Meteorology) for making the UCLALES code available and for his assistance. We acknowledge M. Inoue (JPL) for contributions to the present LES code. The 3D radiative transfer calculations were performed by A. Davis and Z. Qu (JPL). Computational resources supporting this work were provided by the JPL Office of the Chief Information Officer and the NASA High-End Computing (HEC) Program through the NASA Advanced Supercomputing (NAS) Division at Ames Research Center. We acknowledge the support provided by the Office of Naval Research, Marine Meteorology Program under Awards N0001411IP20087 and N0001411IP20069, the NASA MAP Program, and the NOAA/CPO MAPP Program. This research was carried out at the Jet Propulsion Laboratory, California Institute of Technology, under a contract with the National Aeronautics and Space Administration.

## APPENDIX A

### Governing Equations

*ϕ*, where

*ρ*is the density. The overbar indicates the filtering operationwith a convolution kernel

*G*(

**x**) (Leonard 1974).

*f*plane and neglecting resolved-scale viscous terms, are, respectively,The thermodynamic variables are decomposed into a constant potential temperature basic state, denoted by subscript 0, and a dynamic component. Accordingly,

*θ*

_{0}is the constant basic state potential temperature, and

*ρ*

_{0}(

*z*) is the density. The subgrid terms

*τ*

_{ij},

*σ*

_{θ}, and

*σ*

_{q}represent the subgrid stress tensor and subgrid

*θ*

_{l}and

*q*

_{t}flux, respectively. The quantities

*u*

_{i}and

*u*

_{g,i}are the Cartesian components of the velocity vector and geostrophic wind, respectively, and

*f*= [0, 0,

*f*

_{3}] is the Coriolis parameter. Buoyancy is proportional to deviations of the virtual potential temperature

*θ*

_{υ}from its instantaneous horizontal average 〈

*θ*

_{υ}〉. The virtual potential temperature is given bywhere

*q*

_{υ}and

*q*

_{c}denote vapor and water condensate mixing ratios, respectively. In the momentum Eq. (A4),

*π*

_{2}denotes the dynamic part of the Exner function

*π*, that is,which is used to enforce the anelastic constraint Eq. (A3). The thermodynamic pressure

*p*at each grid point is computed from Eq. (A8), the sum of the basic-state Exner

*π*

_{0}(

*z*) plus a contribution due to the deviation of the horizontal mean from the basic state

*π*

_{1}(

*t*,

*z*), and the dynamic

*π*

_{2}(

*t*,

*x*,

*y*,

*z*) (Clark 1979).

Two types of source terms are present in the momentum and scalar transport equations. The terms *H*_{q} parameterize processes that cannot be captured by the periodic boundary conditions of the computational domain, such as large-scale subsidence and humidity and temperature advection. Radiative warming/cooling is also included in *H*_{θ}. The terms *S*_{θ} and *S*_{q} impart the effects of microphysics, that is, warm rain precipitation in the current LES, on *q*_{t} and *θ*_{l}.

*r*

_{r}and mass specific number of rainwater drops

*n*

_{r}are also integrated in addition to Eqs. (A3)–(A6). These are of the formwhere

*w*

_{s}is the sedimentation velocity, and

*σ*

_{n,j}has the same form as the one for temperature and humidity. This is a crude approximation, as large drops are not expected to follow the motion of the surrounding gas. However, the dominant terms in the evolution of rain mass and number in each grid cell are the ones corresponding to sedimentation, thermodynamics, and droplet interactions.

## APPENDIX B

### Surface Fluxes

A requirement for the grid resolution studies is that all boundary conditions and forcings remain unchanged as the grid is refined. In two of the cases considered, the surface temperature is specified rather that the heat flux. Although setting the temperature is a well-posed boundary condition, the calculation of the surface heat flux can impact the simulation if the heat flux depends on the grid resolution. Figures B1 and B2 show the mean sensible and latent heat fluxes for the shallow precipitating cumulus (RICO) and stratocumulus (DYCOMS-II RF01) runs for variable resolution. The cumulus case uses a bulk aerodynamic formula (vanZanten et al. 2011), but the coefficients were not corrected for the change in height of the first grid cell. However, because the dependence is logarithmic with height (e.g., Stevens et al. 2001), the fluxes do not change significantly. The Monin–Obukhov similarity theory is used to compute the fluxes for the stratocumulus case (Fig. B2), and, as expected, there are only minor variations with the grid resolution.

## REFERENCES

Arakawa, A., , and V. R. Lamb, 1977: Computational design of the basic dynamical processes of the UCLA general circulation model.

*General Circulation Models of the Atmosphere,*J. Chang, Ed., Vol. 17,*Methods of Computational Physics,*Academic Press, 173–265, doi:10.1016/B978-0-12-460817-7.50009-4.Batchelor, G. K., 1953: The condition for dynamical similarity of motions of a frictionless perfect-gas atmosphere.

,*Quart. J. Roy. Meteor. Soc.***79**, 224–235, doi:10.1002/qj.49707934004.Beare, R. J., and et al. , 2006: An intercomparison of large-eddy simulations of the stable boundary layer.

,*Bound.-Layer Meteor.***118**, 247–272, doi:10.1007/s10546-004-2820-6.Betts, A. K., 1973: Non-precipitating cumulus convection and its parameterization.

,*Quart. J. Roy. Meteor. Soc.***99**, 178–196, doi:10.1002/qj.49709941915.Brasseur, J. G., , and T. Wei, 2010: Designing large-eddy simulation of the turbulent boundary layer to capture law-of-the-wall scaling.

*Phys. Fluids,***22,**021303, doi:10.1063/1.3319073.Brost, R. A., , D. H. Lenschow, , and J. C. Wyngaard, 1982: Marine stratocumulus layers. Part I: Mean conditions.

,*J. Atmos. Sci.***39**, 800–817, doi:10.1175/1520-0469(1982)039<0800:MSLPMC>2.0.CO;2.Brown, A. R., 1999: The sensitivity of large-eddy simulations of shallow cumulus convection to resolution and subgrid model.

,*Quart. J. Roy. Meteor. Soc.***125**, 469–482, doi:10.1002/qj.49712555405.Brown, A. R., , M. K. MacVean, , and P. J. Mason, 2000: The effects of numerical dissipation in large eddy simulations.

,*J. Atmos. Sci.***57**, 3337–3348, doi:10.1175/1520-0469(2000)057<3337:TEONDI>2.0.CO;2.Buras, R., , and B. Mayer, 2011: Efficient unbiased variance reduction techniques for Monte Carlo simulations of radiative transfer in cloudy atmospheres: The solution.

,*J. Quant. Spectrosc. Radiat. Transf.***112**, 434–447, doi:10.1016/j.jqsrt.2010.10.005.Charnock, H., 1955: Wind stress over a water surface.

,*Quart. J. Roy. Meteor. Soc.***81**, 639–640, doi:10.1002/qj.49708135027.Chow, F. K., , and P. Moin, 2003: A further study of numerical errors in large-eddy simulations.

,*J. Comput. Phys.***184**, 366–380, doi:10.1016/S0021-9991(02)00020-7.Chung, D., , and D. I. Pullin, 2010: Direct numerical simulation and large-eddy simulation of stationary buoyancy-driven turbulence.

,*J. Fluid Mech.***643**, 279–308, doi:10.1017/S0022112009992801.Chung, D., , and G. Matheou, 2012: Direct numerical simulation of stationary homogeneous stratified sheared turbulence.

,*J. Fluid Mech.***696**, 434–467, doi:10.1017/jfm.2012.59.Chung, D., , and G. Matheou, 2014: Large-eddy simulation of stratified turbulence. Part I: A vortex-based subgrid-scale model.

,*J. Atmos. Sci.***71,**1863–1879, doi:10.1175/JAS-D-13-0126.1.Clark, T. L., 1979: Numerical simulations with a three-dimensional cloud model: Lateral boundary condition experiments and multicellular severe storm simulations.

,*J. Atmos. Sci.***36**, 2191–2215, doi:10.1175/1520-0469(1979)036<2191:NSWATD>2.0.CO;2.Cuijpers, J. W. M., , and P. G. Duynkerke, 1993: Large eddy simulation of trade wind cumulus clouds.

,*J. Atmos. Sci.***50**, 3894–3908, doi:10.1175/1520-0469(1993)050<3894:LESOTW>2.0.CO;2.Cullen, M. J. P., , and A. R. Brown, 2009: Large eddy simulation of the atmosphere on various scales.

,*Philos. Trans. Roy. Soc. London***A367**, 2947–2956, doi:10.1098/rsta.2008.0268.Ebert, E., , U. Schumann, , and R. Stull, 1989: Nonlocal turbulent mixing in the convective boundary layer evaluated from large-eddy simulation.

,*J. Atmos. Sci.***46**, 2178–2207, doi:10.1175/1520-0469(1989)046<2178:NTMITC>2.0.CO;2.Frisch, U., 1995:

*Turbulence: The Legacy of A. N. Kolmogorov.*Cambridge University Press, 296 pp.Ghosal, S., 1996: An analysis of numerical errors in large-eddy simulations of turbulence.

,*J. Comput. Phys.***125**, 187–206, doi:10.1006/jcph.1996.0088.Holland, J. Z., , and E. M. Rasmusson, 1973: Measurements of the atmospheric mass, energy, and momentum budgets over a 500-kilometer square of tropical ocean.

,*Mon. Wea. Rev.***101**, 44–55, doi:10.1175/1520-0493(1973)101<0044:MOTAME>2.3.CO;2.Huang, J., , and E. Bou-Zeid, 2013: Turbulence and vertical fluxes in the stable atmospheric boundary layer. Part I: A large-eddy simulation study.

,*J. Atmos. Sci.***70**, 1513–1527, doi:10.1175/JAS-D-12-0167.1.Khanna, S., , and J. G. Brasseur, 1998: Three-dimensional buoyancy- and shear-induced local structure of the atmospheric boundary layer.

,*J. Atmos. Sci.***55**, 710–743, doi:10.1175/1520-0469(1998)055<0710:TDBASI>2.0.CO;2.Kosovic, B., , and J. Curry, 2000: Large-eddy simulation of a quasi-steady, stably stratified atmospheric boundary layer.

,*J. Atmos. Sci.***57**, 1052–1068, doi:10.1175/1520-0469(2000)057<1052:ALESSO>2.0.CO;2.Lenschow, D. H., , J. C. Wyngaard, , and W. T. Pennell, 1980: Mean-field and second-moment budgets in a baroclinic, convective boundary layer.

,*J. Atmos. Sci.***37**, 1313–1326, doi:10.1175/1520-0469(1980)037<1313:MFASMB>2.0.CO;2.Lenschow, D. H., and et al. , 1988: Dynamics and Chemistry of Marine Stratocumulus (DYCOMS) experiment.

,*Bull. Amer. Meteor. Soc.***69**, 1058–1067, doi:10.1175/1520-0477(1988)069<1058:DACOMS>2.0.CO;2.Leonard, A., 1974: Energy cascade in large-eddy simulations of turbulent fluid flows.

*Advances in Geophysics,*Vol. 18, Academic Press, 237–248, doi:10.1016/S0065-2687(08)60464-1.Lilly, D. K., 1962: On the numerical simulation of buoyant convection.

,*Tellus***14**, 148–172, doi:10.1111/j.2153-3490.1962.tb00128.x.Lumley, J. L., , and H. A. Panofsky, 1964:

*The Structure of Atmospheric Turbulence*. Interscience Publishers, 239 pp.Margolin, L. G., , P. K. Smolarkiewicz, , and Z. Sorbjan, 1999: Large-eddy simulations of convective boundary layers using nonoscillatory differencing.

,*Physica D***133**, 390–397, doi:10.1016/S0167-2789(99)00083-4.Mason, P. J., , and D. J. Thomson, 1992: Stochastic backscatter in large-eddy simulations of boundary layers.

,*J. Fluid Mech.***242**, 51–78, doi:10.1017/S0022112092002271.Mason, P. J., , and A. R. Brown, 1999: On subgrid models and filter operations in large eddy simulations.

,*J. Atmos. Sci.***56**, 2101–2114, doi:10.1175/1520-0469(1999)056<2101:OSMAFO>2.0.CO;2.Matheou, G., , A. M. Bonanos, , C. Pantano, , and P. E. Dimotakis, 2010: Large-eddy simulation of mixing in a recirculating shear flow.

,*J. Fluid Mech.***646**, 375–414, doi:10.1017/S0022112009992965.Matheou, G., , D. Chung, , L. Nuijens, , B. Stevens, , and J. Teixeira, 2011: On the fidelity of large-eddy simulation of shallow precipitating cumulus convection.

,*Mon. Wea. Rev.***139**, 2918–2939, doi:10.1175/2011MWR3599.1.Mayer, B., 2009: Radiative transfer in the cloudy atmosphere.

*Eur. Phys. J. Conf.,***1,**75–99, doi:10.1140/epjconf/e2009-00912-1.Mellado, J. P., 2012: Direct numerical simulation of free convection over a heated plate.

,*J. Fluid Mech.***712**, 418–450, doi:10.1017/jfm.2012.428.Meyers, J., , B. J. Geurts, , and M. Baelmans, 2003: Database analysis of errors in large-eddy simulation.

,*Phys. Fluids***15**, 2740–2755, doi:10.1063/1.1597683.Misra, A., , and D. I. Pullin, 1997: A vortex-based subgrid stress model for large-eddy simulation.

,*Phys. Fluids***9**, 2443–2454, doi:10.1063/1.869361.Moeng, C.-H., 1986: Large-eddy simulation of a stratus-topped boundary layer. Part I: Structure and budgets.

,*J. Atmos. Sci.***43**, 2886–2900, doi:10.1175/1520-0469(1986)043<2886:LESOAS>2.0.CO;2.Moeng, C.-H., , and J. C. Wyngaard, 1988: Spectral analysis of large-eddy simulations of the convective boundary layer.

,*J. Atmos. Sci.***45**, 3573–3587, doi:10.1175/1520-0469(1988)045<3573:SAOLES>2.0.CO;2.Moeng, C.-H., , S. Shen, , and D. A. Randall, 1992: Physical processes within the nocturnal stratus-topped boundary layer.

,*J. Atmos. Sci.***49**, 2384–2401, doi:10.1175/1520-0469(1992)049<2384:PPWTNS>2.0.CO;2.Moeng, C.-H., and et al. , 1996: Simulation of a stratocumulus-topped planetary boundary layer: Intercomparison among different numerical codes.

,*Bull. Amer. Meteor. Soc.***77**, 261–278, doi:10.1175/1520-0477(1996)077<0261:SOASTP>2.0.CO;2.Morinishi, Y., , T. S. Lund, , O. V. Vasilyev, , and P. Moin, 1998: Fully conservative higher order finite difference schemes for incompressible flow.

,*J. Comput. Phys.***143**, 90–124, doi:10.1006/jcph.1998.5962.Neggers, R. A. J., , H. J. J. Jonker, , and A. P. Siebesma, 2003: Size statistics of cumulus cloud populations in large-eddy simulations.

,*J. Atmos. Sci.***60**, 1060–1074, doi:10.1175/1520-0469(2003)60<1060:SSOCCP>2.0.CO;2.Neggers, R. A. J., , M. Koehler, , and A. C. M. Beljaars, 2009: A dual mass flux framework for boundary layer convection. Part I: Transport.

,*J. Atmos. Sci.***66**, 1465–1487, doi:10.1175/2008JAS2635.1.Nieuwstadt, F. T. M., , P. J. Mason, , C.-H. Moeng, , and U. Schumann, 1993: Large-eddy simulation of the convective boundary layer: A comparison of four computer codes.

*Turbulent Shear Flows 8,*F. Durst et al., Eds., Springer-Verlag, 344–367.Nuijens, L., , B. Stevens, , and A. P. Siebesma, 2009: The environment of precipitating shallow cumulus convection.

,*J. Atmos. Sci.***66**, 1962–1979, doi:10.1175/2008JAS2841.1.Ogura, Y., , and N. A. Phillips, 1962: Scale analysis of deep and shallow convection in the atmosphere.

,*J. Atmos. Sci.***19**, 173–179, doi:10.1175/1520-0469(1962)019<0173:SAODAS>2.0.CO;2.Pantano, C., , D. I. Pullin, , P. Dimotakis, , and G. Matheou, 2008: LES approach for high Reynolds number wall-bounded flows with application to turbulent channel flow.

,*J. Comput. Phys.***227**, 9271–9291, doi:10.1016/j.jcp.2008.04.015.Phillips, N. A., 1956: The general circulation of the atmosphere: A numerical experiment.

,*Quart. J. Roy. Meteor. Soc.***82**, 123–164, doi:10.1002/qj.49708235202.Pope, S. B., 2000:

*Turbulent Flows*. Cambridge University Press, 802 pp.Pope, S. B., 2004: Ten questions concerning the large-eddy simulation of turbulent flows.

,*New J. Phys.***6**, 35, doi:10.1088/1367-2630/6/1/035.Pullin, D. I., 2000: A vortex-based model for the subgrid flux of a passive scalar.

,*Phys. Fluids***12**, 2311–2316, doi:10.1063/1.1287512.Rauber, R. M., and et al. , 2007: Rain in shallow cumulus over the ocean: The RICO campaign.

,*Bull. Amer. Meteor. Soc.***88**, 1912–1928, doi:10.1175/BAMS-88-12-1912.Richardson, L. F., 1922:

*Weather Prediction by Numerical Process*. Cambridge University Press, 236 pp.Riechelmann, T., , Y. Noh, , and S. Raasch, 2012: A new method for large-eddy simulations of clouds with Lagrangian droplets including the effects of turbulent collision.

*New J. Phys.,***14,**065008, doi:10.1088/1367-2630/14/6/065008.Savic-Jovcic, V., , and B. Stevens, 2008: The structure and mesoscale organization of precipitating stratocumulus.

,*J. Atmos. Sci.***65**, 1587–1605, doi:10.1175/2007JAS2456.1.Schmidt, H., , and U. Schumann, 1989: Coherent structure of the convective boundary layer derived from large-eddy simulations.

,*J. Fluid Mech.***200**, 511–562, doi:10.1017/S0022112089000753.Schumann, U., 1985: Algorithms for direct numerical simulation of shear-periodic turbulence.

*Lecture Notes in Physics,*Vol. 218, Springer, 492–496, doi:10.1007/3-540-13917-6_187.Seifert, A., , and K. D. Beheng, 2001: A double-moment parameterization for simulating autoconversion, accretion and selfcollection.

,*Atmos. Res.***59–60**, 265–281, doi:10.1016/S0169-8095(01)00126-0.Siebesma, A. P., and et al. , 2003: A large eddy simulation intercomparison study of shallow cumulus convection.

,*J. Atmos. Sci.***60**, 1201–1219, doi:10.1175/1520-0469(2003)60<1201:ALESIS>2.0.CO;2.Smagorinsky, J., 1963: General circulation experiments with the primitive equations. I. The basic experiment.

,*Mon. Wea. Rev.***91**, 99–164, doi:10.1175/1520-0493(1963)091<0099:GCEWTP>2.3.CO;2.Snodgrass, E. R., , L. Di Girolamo, , and R. M. Rauber, 2009: Precipitation characteristics of trade wind clouds during RICO derived from radar, satellite, and aircraft measurements.

,*J. Appl. Meteor.***48**, 464–483, doi:10.1175/2008JAMC1946.1.Spalart, P. R., , R. D. Moser, , and M. M. Rogers, 1991: Spectral methods for the Navier–Stokes equations with one infinite and two periodic directions.

,*J. Comput. Phys.***96**, 297–324, doi:10.1016/0021-9991(91)90238-G.Stevens, B., 2002: Entrainment in stratocumulus-topped mixed layers.

,*Quart. J. Roy. Meteor. Soc.***128**, 2663–2690, doi:10.1256/qj.01.202.Stevens, B., , and A. Seifert, 2008: Understanding macrophysical outcomes of microphysical choices in simulations of shallow cumulus convection.

,*J. Meteor. Soc. Japan***86A**, 143–162, doi:10.2151/jmsj.86A.143.Stevens, B., , G. Feingold, , W. R. Cotton, , and R. L. Walko, 1996: Elements of the microphysical structure of numerically simulated nonprecipitating stratocumulus.

,*J. Atmos. Sci.***53**, 980–1006, doi:10.1175/1520-0469(1996)053<0980:EOTMSO>2.0.CO;2.Stevens, B., , C.-H. Moeng, , and P. P. Sullivan, 1999: Large-eddy simulations of radiatively driven convection: Sensitivities to the representation of small scales.

,*J. Atmos. Sci.***56**, 3963–3984, doi:10.1175/1520-0469(1999)056<3963:LESORD>2.0.CO;2.Stevens, B., and et al. , 2001: Simulations of trade wind cumuli under a strong inversion.

,*J. Atmos. Sci.***58**, 1870–1891, doi:10.1175/1520-0469(2001)058<1870:SOTWCU>2.0.CO;2.Stevens, B., and et al. , 2003a: Dynamics and chemistry of marine stratocumulus—DYCOMS-II.

,*Bull. Amer. Meteor. Soc.***84**, 579–593, doi:10.1175/BAMS-84-5-579.Stevens, B., and et al. , 2003b: On entrainment rates in nocturnal marine stratocumulus.

,*Quart. J. Roy. Meteor. Soc.***129**, 3469–3493, doi:10.1256/qj.02.202.Stevens, B., and et al. , 2005a: Evaluation of large-eddy simulations via observations of nocturnal marine stratocumulus.

,*Mon. Wea. Rev.***133**, 1443–1462, doi:10.1175/MWR2930.1.Stevens, B., , G. Vali, , K. Comstock, , R. Wood, , M. C. vanZanten, , P. H. Austin, , C. S. Bretherton, , and D. H. Lenschow, 2005b: Pockets of open cells and drizzle in marine stratocumulus.

,*Bull. Amer. Meteor. Soc.***86**, 51–57, doi:10.1175/BAMS-86-1-51.Stevens, D. E., , A. S. Ackerman, , and C. S. Bretherton, 2002: Effects of domain size and numerical resolution on the simulation of shallow cumulus convection.

,*J. Atmos. Sci.***59**, 3285–3301, doi:10.1175/1520-0469(2002)059<3285:EODSAN>2.0.CO;2.Stoll, R., , and F. Porté-Agel, 2008: Large-eddy simulation of the stable atmospheric boundary layer using dynamic models with different averaging schemes.

,*Bound.-Layer Meteor.***126**, 1–26, doi:10.1007/s10546-007-9207-4.Sullivan, P. G., , and E. G. Patton, 2011: The effect of mesh resolution on convective boundary layer statistics and structures generated by large-eddy simulation.

,*J. Atmos. Sci.***68**, 2395–2415, doi:10.1175/JAS-D-10-05010.1.Tiedtke, M., 1989: A comprehensive mass flux scheme for cumulus parameterization in large-scale models.

,*Mon. Wea. Rev.***117**, 1779–1800, doi:10.1175/1520-0493(1989)117<1779:ACMFSF>2.0.CO;2.Van Leer, B., 1977: Towards the ultimate conservative difference scheme III. Upstream-centered finite-difference schemes for ideal compressible flow.

,*J. Comput. Phys.***23**, 263–275, doi:10.1016/0021-9991(77)90094-8.vanZanten, M. C., and et al. , 2011: Controls on precipitation and cloudiness in simulations of trade-wind cumulus as observed during RICO.

*J. Adv. Model. Earth Syst.,***3,**M06001, doi:10.1029/2011MS000056.Voelkl, T., , D. I. Pullin, , and D. C. Chan, 2000: A physical-space version of the stretched-vortex subgrid-stress model for large-eddy simulation.

,*Phys. Fluids***12**, 1810–1825, doi:10.1063/1.870429.Vreman, B., , B. Geurts, , and H. Kuerten, 1996: Comparision of numerical schemes in large-eddy simulation of the temporal mixing layer.

,*Int. J. Numer. Methods Fluids***22**, 297–311, doi:10.1002/(SICI)1097-0363(19960229)22:4<297::AID-FLD361>3.0.CO;2-X.Waite, M. L., 2011: Stratified turbulence at the buoyancy scale.

*Phys. Fluids,***23,**066602, doi:10.1063/1.3599699.Wang, S., , Q. Wang, , and G. Feingold, 2003: Turbulence, condensation, and liquid water transport in numerically simulated nonprecipitating stratocumulus clouds.

,*J. Atmos. Sci.***60**, 262–278, doi:10.1175/1520-0469(2003)060<0262:TCALWT>2.0.CO;2.Wood, R., 2012: Stratocumulus clouds.

,*Mon. Wea. Rev.***140**, 2373–2423, doi:10.1175/MWR-D-11-00121.1.Yamaguchi, T., , and D. A. Randall, 2008: Large-eddy simulation of evaporatively driven entrainment in cloud-topped mixed layers.

,*J. Atmos. Sci.***65**, 1481–1504, doi:10.1175/2007JAS2438.1.Yamaguchi, T., , and G. Feingold, 2012: Technical note: Large-eddy simulation of cloudy boundary layer with the Advanced Research WRF model.

*J. Adv. Model. Earth Syst.,***4,**M09003, doi:10.1029/2012MS000164.Zheng, X., and et al. , 2011: Observations of the boundary layer, cloud, and aerosol variability in the southeast Pacific near-coastal marine stratocumulus during VOCALS-REx.

,*Atmos. Chem. Phys.***11**, 9943–9959, doi:10.5194/acp-11-9943-2011.