PDF Parameterization of Boundary Layer Clouds in ... - AMS Journals

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LARSON ET AL.

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PDF Parameterization of Boundary Layer Clouds in Models with Horizontal Grid Spacings from 2 to 16 km VINCENT E. LARSON AND DAVID P. SCHANEN University of Wisconsin–Milwaukee, Milwaukee, Wisconsin

MINGHUAI WANG, MIKHAIL OVCHINNIKOV, AND STEVEN GHAN Pacific Northwest National Laboratory, Richland, Washington (Manuscript received 26 November 2010, in final form 25 May 2011) ABSTRACT Many present-day numerical weather prediction (NWP) models are run at resolutions that permit deep convection. In these models, however, the boundary layer turbulence and boundary layer cloud features are still grossly underresolved. Underresolution is also present in climate models that use a multiscale modeling framework (MMF), in which a convection-permitting model is run in each grid column of a global general circulation model. To better represent boundary layer clouds and turbulence in convection-permitting models, a parameterization was developed that models the joint probability density function (PDF) of vertical velocity, heat, and moisture. Although PDF-based parameterizations are more complex and computationally expensive than many other parameterizations, in principle PDF parameterizations have several advantages. For instance, they ensure consistency of liquid (cloud) water and cloud fraction; they avoid using separate parameterizations for different cloud types such as cumulus and stratocumulus; and they have an appropriate formulation in the ‘‘terra incognita’’ in which updrafts are marginally resolved. In this paper, an implementation of a PDF parameterization is tested to see whether it improves the simulations of a state-of-the-art convection-permitting model. The PDF parameterization used is the Cloud Layers Unified By Binormals (CLUBB) parameterization. The host cloud-resolving model used is the System for Atmospheric Modeling (SAM). SAM is run both with and without CLUBB implemented in it. Simulations of two shallow cumulus (Cu) cases and two shallow stratocumulus (Sc) cases are run in a 3D configuration at 2-, 4-, and 16-km horizontal grid spacings. Including CLUBB in the simulations improves some of the simulated fields—such as vertical velocity variance, horizontal wind fields, cloud water content, and drizzle water content—especially in the two Cu cases. Implementing CLUBB in SAM improves the simulations slightly at 2-km horizontal grid spacing, significantly at 4-km grid spacing, and greatly at 16-km grid spacing. Furthermore, the simulations that include CLUBB exhibit a reduced sensitivity to horizontal grid spacing.

1. Introduction Convection-permitting simulations have a resolution that permits explicit simulation of deep convection but does not adequately resolve it. Often such simulations have horizontal grid spacings of 2 to 4 km. As computer power has increased in recent years, convection-permitting simulations have become more widely used in practical

Corresponding author address: Vincent E. Larson, Dept. of Mathematical Sciences, University of Wisconsin—Milwaukee, P.O. Box 413, Milwaukee, WI 53211. E-mail: [email protected] DOI: 10.1175/MWR-D-10-05059.1 Ó 2012 American Meteorological Society

applications because they avoid the need to use deep convection parameterizations, which are inherently uncertain. Currently, two popular applications of convectionpermitting simulations are 1) day-ahead numerical weather prediction (NWP); and 2) climate simulations using a multiscale modeling framework (MMF). Convection-permitting resolutions are feasible for NWP forecasts of relatively limited area and short duration. The convection-permitting forecasts of Kain et al. (2008) and Weisman et al. (2008) parameterize subgrid turbulence using the Yonsei University parameterization (Noh et al. 2003), but they do not parameterize the subgrid cloud

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fraction or shallow cloud features. However, shallow clouds are important, in part because they are a precursor to deep convection. MMF simulations embed a (convection-permitting) cloud-resolving model (CRM) in each column of a general circulation model (GCM; Khairoutdinov and Randall 2001; Khairoutdinov et al. 2005; Wyant et al. 2006; Khairoutdinov et al. 2008; Tao et al. 2009; Wang et al. 2011; Arakawa et al. 2011). The CRMs typically have a horizontal grid spacing of 4 km and therefore permit deep convection. The MMF simulations cited above do prognose subgrid turbulence kinetic energy (TKE; Tao et al. 2009) or use a Smagorinsky closure for subgrid fluxes, but they do not include parameterizations of sub-CRM-grid cloud fraction or shallow clouds. Because of their coarse vertical and horizontal resolution, standard MMF models may misrepresent low-cloud feedbacks (Blossey et al. 2009). Unfortunately, although convection-permitting resolutions may permit deep convection, they do not come close to resolving shallow clouds, which have important updrafts on the scale of O(100 m). This raises the following questions: Which aspects of convection-permitting simulations benefit most from a shallow-cloud parameterization? At what horizontal grid spacings is parameterization of shallow clouds still beneficial? The answer depends on the type of cloud to be studied and the kind of shallow cloud parameterization used. In this paper, we address these questions by implementing a parameterization of clouds and turbulence in a CRM and testing it for four idealized simulations: two of marine stratocumulus (Sc), and two of shallow cumulus (Cu). The parameterization that we will explore is based on the subgrid probability density function (PDF) of vertical velocity, moisture, and heat content (Golaz et al. 2002; Larson and Golaz 2005). It is called Cloud Layers Unified By Binormals (CLUBB). Earlier work in this area was done by Cheng and Xu (2008), who implemented in a CRM a PDF parameterization originally derived from Golaz et al. (2002). They used a vertical grid spacing of 25 or 40 m. Here, we use a coarser vertical grid spacing, and our time step for CLUBB is two minutes, thereby reducing the computational cost significantly. Furthermore, we include plots of scalar variances and horizontal winds, in addition to those for horizontal means and vertical turbulent fluxes.

2. Our PDF parameterization: CLUBB and its underlying methodology CLUBB is based on the assumed probability density function method (Golaz et al. 2002; Larson and Golaz

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2005; Larson and Griffin 2011, manuscript submitted to Quart. J. Roy. Meteor. Soc.; Griffin and Larson 2011, manuscript submitted to Quart. J. Roy. Meteor. Soc.). From the viewpoint of the assumed PDF method, the job of a cloud parameterization is primarily to predict the subgrid joint PDF of vertical velocity, heat content, and moisture content. The joint PDF is the probability that the array (vertical velocity, heat content, moisture content) takes on a particular value at a particular location and time. PDFs in the atmosphere evolve with time and space, and they contain a wealth of information. Because CLUBB’s PDF includes both vertical velocity and thermodynamic scalars, we call it a velocityscalar PDF (e.g., Randall et al. 1992; Anand et al. 1997; Lappen and Randall 2001). The steps in the PDF method may be outlined as follows (e.g., Sommeria and Deardorff 1977; Tompkins 2002). We first write down standard higher-order moment equations based on the equations of fluid flow (Navier–Stokes and advection–diffusion equations). This includes equations for turbulent fluxes and variances of heat and moisture. These equations contain unclosed, higher-order terms. But since we predict the joint PDF of vertical velocity, heat, and moisture, we can close any moments or correlations of these variables. Predicting an arbitrarily shaped subgrid PDF has been explored theoretically (e.g., Jeffery and Reisner 2006), but such a method would be complex and computationally expensive to implement. Therefore we assume a functional form for the PDF, namely a mixture (or sum) of Gaussians. This double Gaussian shape defines a functional form whose width, center, and skewness may vary. Such a form captures well the observed PDF structure in the marine boundary layer (Larson et al. 2001). Within this functional form, we need to determine the particular PDF for each grid box and time step. To do so, we use the higher-order moments that are prognosed by the model. Once the subgrid PDF is determined, the unclosed moments can be diagnosed directly from the PDF. For details of the methodology, please see Golaz et al. (2002) and Larson and Golaz (2005). Next we discuss two algorithmic changes to CLUBB that have not been reported in the prior literature: 1) a limitation on the turbulent length scale in order to make CLUBB scale aware, and 2) the separate prognosis of horizontal and vertical velocity variances.

a. Limiting CLUBB’s turbulent length scale and scale awareness It is advantageous for a parameterization to be able to perform accurate simulations over a range of horizontal grid spacings. Such insensitivity to horizontal grid spacing is useful, for instance, in 3D simulations that

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contain nested grids or variable-resolution grids. It is also useful when one wishes to perform a suite of separate 3D simulations, each with a different (constant) grid spacing. In contrast, if a parameterization is sensitive to the grid spacing, the parameterization must be retuned for each grid spacing. When a parameterization accounts for grid spacing in order to reduce the sensitivity of the results to grid spacing, the parameterization is said to be ‘‘scale aware’’ (Ringler et al. 2011). In a PDF parameterization, the grid spacing is of special relevance in computing variances. Typically, a large grid box encompasses more variability than a small grid box. That is, the subgrid-scale variance is typically larger in larger grid boxes (De Roode et al. 2004). CLUBB is scale aware in the sense that it limits the turbulent length scale, L, according to the horizontal grid spacing: L 5 min(L, aDx),

(1)

where Dx is the horizontal grid spacing of the host model, a 5 0.25 is a constant, and ‘‘5’’ denotes an assignment, not equality. Physically, L represents a typical size of large, coherent eddies. We limit L to a fraction of the grid box size because doing so helps the host model develop resolved circulations. When the eddy size L is larger than the grid box size, the eddy is at least partly resolved and ought to be handled at least partly by the host model. The length scale L is formulated as in Golaz et al. (2002) and appears in various terms throughout CLUBB (see Golaz et al. 2002). The length scale L is an important part of CLUBB’s turbulent dissipation time scale t, which is defined as t 5 L/ e,

(2)

where e is the subgrid-scale turbulence kinetic energy. Limiting L is of special importance in the prognostic equations for scalar variances. Those equations contain a term that models the dissipation of subgrid-scale scalar variance: ›x92 x92 5 2 . ›t t

(3)

When the grid boxes are small, Eq. (1) limits L, which in turn limits t [see Eq. (2)]. A reduced dissipation time scale, in turn, increases the modeled dissipation [see Eq. (3)]. Thereby, the subgrid variance, x92 , tends to be reduced, and the host model is allowed to resolve more of the variance. As the grid scale becomes very small,

CLUBB’s effects are greatly reduced, and the host model generates almost all the small-scale variability. For the shallow cloud layers that we study here, the limitation on L has a significant effect only at a grid box spacing of 2 km (or finer). Limiting L tends to make the simulated profiles look more like the SAM simulations and less like CLUBB, as expected (see the figures below).

b. Prognosis of horizontal velocity variances in CLUBB The horizontal and vertical components of turbulence can differ significantly from each other near the ground or stable layers, where vertical velocity is suppressed. For this reason, it is advantageous to parameterize horizontal and vertical velocity variances separately. Here we list CLUBB’s (prognostic) equation for horizontal velocity variance. First, we review CLUBB’s formulation of the east– west component of mean horizontal wind (u) and vertical turbulent momentum flux (u9w9). The formulation of the north–south components is analogous. The mean wind is computed as (Bougeault 1981) ›u ›u 1 ›ru9w9 5 2w 2 f (y g 2 y) 2 . ›t ›z r ›z |fflfflffl{zfflfflffl} |fflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflffl} |fflfflfflfflfflfflffl{zfflfflfflfflfflfflffl} ma

(4)

ta

gf

Here w is the mean vertical velocity, y is the mean north–south velocity, yg is the mean north–south geostrophic velocity, r is air density, z is altitude, and t is time. Here ma denotes the mean vertical advection term, gf denotes the geostrophic departure term, and ta denotes the vertical turbulent advection term. The momentum fluxes are closed using downgradient diffusion, u9w9 5 2Km

›u , ›z

(5)

where the eddy diffusivity coefficient Km is given by Km 5 cK Le1/2 .

(6)

Here cK 5 0.2, and turbulence kinetic energy, e, is the sum of the vertical velocity variance w92 , north–south wind variance y92 , and east–west wind variance u92 : 1 e 5 (w92 1 u92 1 y92 ). 2

(7)

To predict wind profiles accurately using Eq. (5), it is necessary to predict Km accurately, which in turn requires the accurate prediction of u92 [see Eqs. (6) and (7)]. CLUBB predicts u92 via (Bougeault 1981):

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! ›u92 ›u92 1 ›rw9u92 ›u 2 e 2 g ›u ›y 5 2w 2 2 (1 2 C5 )2u9w9 2 C14 1 C5 w9u9y 2 u9w9 2 y9w9 r ›z ›z 3 t 3 ›z ›z ›t ›z uy |fflfflfflfflffl{zfflfflfflfflffl} |fflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflffl} |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl} |fflfflfflfflffl{zfflfflfflfflffl} |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl} ma

ta

tp

pr1

C 2 › › (cK9 Km 1 n9 ) u92 . 2 4 u92 2 e 1 3 ›z ›z t |fflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflffl} |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl} dp1

pr2

(8)

dp2

Here g is the acceleration due to gravity, uy is mean virtual potential temperature, w9u9y is the turbulent flux of uy, and y9w9 is the turbulent flux of y. The adjustable constants have been set to the following values: C5 5 0.3, C14 5 1.0, C4 5 5.2, CK9 5 0.25, and n9 5 20 m2 s21. The terms represent, from left to right, vertical mean advection (ma), turbulent advection (ta), turbulent production (tp), pressure covariance terms (pr1 and pr2), and dissipation terms (dp1 and dp2). The tp term generates u92 in layers with vertical wind shear. This term does not appear in the equation for w92 (Golaz et al. 2002). However, the dp1 term tends to make u92 similar to the other two components of velocity variance. In CLUBB, Eqs. (4)–(8) are applied at all times, regardless of whether Cu, Sc, or neither are present. Hence there is no need for a separate Cu momentum flux parameterization.

c. Disadvantages and advantages of CLUBB The assumed PDF method differs markedly from the mass-flux or eddy diffusivity schemes that have been used in large-scale models (e.g., Arakawa 2004; Kain 2004; Park and Bretherton 2009; Soares et al. 2004; Siebesma et al. 2007; Neggers et al. 2009; Neggers 2009; Pergaud et al. 2009). As compared to those schemes, the assumed PDF method has both disadvantages and advantages. We now enumerate some of these.

1) DISADVANTAGES OF CLUBB (i) Computational expense. Because CLUBB prognoses nine higher-order moments, it is more expensive than mass-flux or eddy diffusivity schemes. However, in convection-permitting simulations, the cost can be mitigated by calling CLUBB only once every 12th time step of the host model. In our tests, this reduces CLUBB’s cost to about 18% of the total model cost (see section 4 below). The cost of CLUBB is far less than the cost of halving the horizontal grid spacing, and CLUBB’s cost is reasonable when compared to the cost of parameterizations of other physical processes, such as radiative transfer, aerosol dynamics, and atmospheric chemistry.

(ii) Software size. CLUBB is a unified methodology that is designed to handle all cloud and boundary layer types. To contain enough physics to represent all cloud types, the code is larger than that of other individual schemes. Currently, the core of CLUBB contains about 37 000 lines of FORTRAN code. However, many of these lines of code consist of comments, output statistics, little-used options, and so forth. Furthermore, CLUBB’s code size is approximately the collective size of the parameterizations that CLUBB could replace in a general circulation model.

2) ADVANTAGES OF CLUBB (i) Problem: Artificial categorization of cloud types. Although cumulus and stratocumulus clouds are inextricably interconnected, these two cloud types are often treated using distinct parameterizations. Then the consistent interfacing of these two parameterizations with each other (and with a deep convective parameterization) becomes problematic (e.g., Arakawa 2004). CLUBB’s approach: Using ‘‘separate schemes for separate regimes’’ does not necessarily reduce overall complexity but instead transfers it to the interactions between schemes (e.g., Bretherton 2007). In contrast, CLUBB has one scheme, which contains a single joint PDF that is general enough to model both cumulus and stratocumulus clouds. Therefore no interface between schemes is needed. (ii) Problem: Artificial separation of physical processes. Typically, in large-scale models, microphysics and cloud variability interact directly with the resolved fields of the model, but not directly with each other on the subgrid scales. Instead, Arakawa (2004) argues that we need a parameterization that acts as a ‘‘physics coupler.’’ Schemes that contain a univariate PDF (e.g., Tompkins 2002; Neggers 2009) but not a multivariate velocity-scalar PDF are hindered in their ability to model subgrid interactions between turbulence and microphysics on the subgrid scale. This, in turn, hinders the driving of aerosol activation by updrafts or the formation of drizzle by cloud variability.

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CLUBB’s approach: The subgrid PDF itself can, in principle, serve as a physics coupler such as the one that Arakawa (2004) recommends. In CLUBB, the variability embodied in the PDF can drive the various physical processes, and the physical processes, in turn, can modify the PDF. Since the PDF contains information on subgrid variability, the physical processes can interact directly at the subgrid scale. Because CLUBB’s PDF is multivariate, it can, in theory, activate aerosol using the joint marginal PDF of vertical velocity and moisture (e.g., Guo et al. 2010), and it can drive drizzle processes using the joint marginal PDF of droplet numbers, liquid (cloud) water, and rainwater (Cheng and Xu 2009; Larson and Griffin 2011, manuscript submitted to Quart. J. Roy. Meteor. Soc.; Griffin and Larson 2011, manuscript submitted to Quart. J. Roy. Meteor. Soc.). These aspects are described in the aforementioned references and are not explored here. (iii) Problem: Internal inconsistency. Parameterizations often predict cloud fraction and cloud water using separate formulas, without guaranteeing consistency between them. Such parameterizations may predict ‘‘empty clouds,’’ which have nonzero cloud fraction but zero cloud water or ice (e.g., Bretherton 2007). CLUBB’s approach. Use of a single, joint (multivariate) PDF allows us to diagnose many closures (e.g., for cloud fraction, cloud water, and buoyancy flux) that are consistent with one another. (iv) Problem: Inconsistent microphysics parameterizations. Cloud parameterizations often call separate microphysics parameterizations for shallow cumulus and stratocumulus clouds (Neggers 2009; Park and Bretherton 2009). This is undesirable because the fundamental equations of microphysics are the same in Cu and Sc clouds. We suspect that the reason that two different microphysics schemes are needed is that the subgrid variability in Cu and Sc is quite different. CLUBB’s approach: Because CLUBB uses a single joint PDF that is general enough to model both Cu and Sc, it drives a single microphysics scheme. (v) Problem: Inappropriate formulation in the ‘‘terra incognita.’’ In convection-permitting simulations at, say, grid spacings between 1 and 10 km, convection is partly resolved and partly subgrid scale. Therefore, these grid spacings are terra incognita for parameterizations (Wyngaard 2004; Gerard 2007). Many parameterizations assume that cloudy updrafts occupy a small fraction of a grid column, but this assumption is no longer true at finer grid spacings (e.g., Arakawa et al. 2011).

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CLUBB’s approach: Because CLUBB does not employ a mass-flux scheme, it needs to make no assumption about the fraction of a grid box that is occupied by updrafts or environmental air. From a PDF perspective, the key difference between small and large grid boxes is that in nature, smaller volumes encompass less variance of cloud and turbulence. This can taken into account by making CLUBB’s turbulent length scale a function of grid box size.

3. Description and configuration of the cloud-resolving model The CRM that we use is the System for Atmospheric Modeling (SAM). It is described in Khairoutdinov and Randall (2003). Here we merely note the most salient features. SAM is a nonhydrostatic cloud-resolving model that has been implemented as part of a MMF model (Khairoutdinov et al. 2008) but also can be run at fine resolutions [O(10 m)], that is, as a large-eddy simulation (LES) model. SAM prognoses conservative moisture and heat variables and uses a monotonic flux limiter in order to avoid spurious numerical oscillations. To parameterize subgrid-scale turbulent fluxes, we choose a Smagorinsky option in the code (Khairoutdinov and Randall 2003). Cloud water is diagnosed in CLUBB using saturation adjustment; that is, instantaneous condensation and evaporation within small air parcels. This is a reasonable assumption when the Damkohler number—that is, the ratio of mixing to evaporative time scales—is large. The Damkohler number is large for typical droplet sizes (e.g., 10 m) and the grid box sizes (>2 km) that this paper considers (Jeffery and Reisner 2006). For nonsedimenting droplets with prescribed number concentration, the primary effect of a nonzero time scale of evaporation is a short delay in the associated release of latent heating or cooling. Additionally, for sedimenting droplets, the droplets may fall a short distance with respect to the embedding air parcel before evaporating. Including explicit supersaturation calculations in PDF methods is a challenge, but theoretical progress has been made by Jeffery and Reisner (2006). To simulate drizzle, we choose the microphysics scheme of Morrison et al. [2009; see also Morrison and Pinto (2005)]. In general, the Morrison microphysics scheme is double moment in both rainwater and cloud water, but for this paper we choose to prescribe cloud droplet number concentration. This choice avoids the complexity of droplet formation processes and allows us to examine, for a prescribed droplet number, how drizzle amount is influenced by the presence of CLUBB’s subgrid parameterizations of moisture and heat content

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variability. The droplet number concentration is set to 100 cm23 for the cumulus cases and 55 cm23 for the stratocumulus cases. We choose the Morrison microphysics scheme in part because it is similar to the microphysics schemes used in current-generation weather and climate models. More detailed (but more computationally expensive) microphysics schemes exist, such as bin microphysics schemes (Jeffery and Reisner 2006), which explicitly compute supersaturation and the size spectra of hydrometeor species, and Lagrangian particle models (Andrejczuk et al. 2008), which track the location of individual cloud particles or particle clusters. However, the goal of this paper is to compare CLUBB’s representation of subgrid variability to LES, rather than assess the accuracy of microphysics parameterizations. The key requirement for assessing CLUBB’s representation of subgrid variability is that identical microphysics schemes are used in LES and CLUBB. One difference between convection-permitting NWP forecasts and MMF CRM simulations is that NWP forecasts use a 3D mesh of grid points, whereas MMF simulations typically use a 2D (vertical–horizontal) grid. In this study, we choose to use a 3D grid because it simulates more realistic horizontal winds. We choose a 90-level stretched vertical grid that is used by the European Centre for Medium-Range Weather Forecasts (ECMWF, http://www.ecmwf.int/products/data/ technical/model_levels/index.html). The grid extends to 70 km and has a grid spacing of ;140 m at 1 km altitude. We chose this grid because it approximates the resolution that we believe will be commonly used in high-resolution NWP models in the near future. In the horizontal, we use either 2-, 4-, or 16-km grid spacing, which spans the range of grid spacings used in many day-ahead NWP forecasts. We include the 16-km grid spacing in order to examine results from a grid spacing that suppresses convection to a large degree. The horizontal domain of all simulations is 128 km 3 128 km. SAM’s computational time step is set to 10 s. We will compare simulations in which CLUBB has been implemented in SAM to handle subgrid variability versus those in which CLUBB has not been implemented. When CLUBB is implemented, for partly cloudy grid boxes, the Morrison microphysics is fed within-cloud averages and the microphysical tendencies are weighted

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TABLE 1. Computational cost for BOMEX Cu case using 4-km horizontal grid spacing. Advection and diffusion are called once per each of 10 scalars in SAM, leading to a total of 3600 calls. Process computed by subroutine

No. of times called

% wall clock time

Advection of non-CLUBB scalars Microphysics CLUBB Scalar diffusion of non-CLUBB scalars Time interpretation of CLUBB and hole-filling Subgrid TKE Pressure calculations Other processes Total

3600

25.80

360 30 3600

29.67 14.83 4.84

360

3.09

360 360

2.96 2.40 16.41 100.00

appropriately according to cloud fraction. When CLUBB is implemented, it is called every 12th SAM time step in order to reduce computational expense. Therefore, CLUBB’s time step is 2 min. When CLUBB is called, its tendencies are divided into 12 equal parts and applied equally to all 12 time steps before the next call to CLUBB. Occasionally, during time steps for which CLUBB is not called, the combined tendencies of SAM and CLUBB yield negative values of positive semidefinite quantities such as cloud water or water vapor. When this happens, a hole-filling algorithm is applied in order to remove the negative values without changing the column-integrated values. Calling CLUBB every 12th time step does not appear to significantly degrade CLUBB’s performance. Finally, we perform a reference high-resolution LES of each case using SAM. Each case is configured according to Global Energy and Water Cycle Experiment (GEWEX) Cloud System Study (GCSS) specifications.

4. Computational cost of CLUBB In CLUBB, nine higher-order moments are prognosed, which entails computational expense. To help assess whether the improvement in results due to CLUBB is worth the extra expense, we summarize and categorize the expense for hours 5 and 6 of the Barbados Oceanographic and Meteorological Experiment (BOMEX) Cu

TABLE 2. Grid spacing and number of grid boxes for our four LES.

Vertical grid spacing (m) Horizontal grid spacing (m) Number grid boxes (x 3 y 3 z)

DYCOMS-II RF01

DYCOMS-II RF02

BOMEX

ARM

5 35 96 3 96 3 320

Refined to 5 m near cloud top 50 128 3 128 3 96

40 100 64 3 64 3 75

40 67 96 3 96 3 130

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FIG. 1. RF01 profiles averaged over hour 5. (upper left) Mean liquid water potential temperature ul, (upper right) mean total water mixing ratio rt, (lower left) turbulent flux of ul, and (lower right) turbulent flux of rt is shown. The plot shows a reference 3D SAM LES (thick red solid line) and simulations with coarser horizontal resolutions of 2 (blue), 4 (purple), and 16 km (green). SAM with CLUBB simulations are denoted by thin solid lines and SAM without CLUBB by dashed–dotted lines. The dark gray shading denotes the central half of the distribution of LES in Stevens et al. (2005); the light gray shading spans the full range of LES. Both [SAM with CLUBB] and [SAM without CLUBB] produce adequate simulations of the means and fluxes at 2- and 4-km horizontal grid spacing.

case (Table 1). This simulation used 4-km horizontal grid spacing and 32 3 32 3 90 grid points. Otherwise, it is configured as described in section 3. The tests were performed on a single processor in order to avoid communication costs. The computer is a Sunfire X4600 M2 from Sun Computers (now Oracle). The processor is an Opteron 8220. The compiler is PGI FORTRAN 10.8–0, but similar results were obtained for the Intel FORTRAN 11.1.064 compiler. Table 1 shows that CLUBB uses 18% of the total runtime: 15% for CLUBB itself, and 3% for hole filling during the time steps in which CLUBB is not called. In

judging this time percentage, recall that CLUBB is called only every 12th time step, whereas the microphysics is called every time step. However, the BOMEX case does not call a radiation parameterization, which in many simulations entails considerable expense outside of CLUBB.

5. Comparison of SAM without CLUBB versus SAM with CLUBB We now discuss results from simulations with and without CLUBB. We analyze two marine stratocumulus cases and two shallow cumulus cases. All cases have

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FIG. 2. As in Fig. 1, but for RF02 profiles averaged over hour 6. The gray shading denotes the full range of LES in Ackerman et al. (2009).

been intercompared by GCSS. Our 2-, 4-, and 16-km configurations are the same as in the GCSS intercomparisons, except for differences (e.g., in grid spacing or time step) noted in section 3. The four cloud cases are (i) a marine Sc layer observed off the coast of California during Research Flight 1 (RF01) of the Second Dynamics and Chemistry of Marine Stratocumulus (DYCOMS-II) field experiment (Stevens et al. 2005); (ii) a heavily drizzling marine Sc observed during Research Flight 2 (RF02) of the same field experiment (DYCOMS-II; Ackerman et al. 2009); (iii) a trade-wind cumulus layer that is loosely based on observations during the BOMEX field experiment (Siebesma et al. 2003); and

(iv) a case of diurnally evolving shallow cumulus over land observed by the Atmospheric Radiation Measurement (ARM) facility in Oklahoma (Brown et al. 2002). We perform LES of these four cases with grid spacings that are determined by the aforementioned references and are listed in Table 2. In all our plots, thick red lines denote reference LES. Gray shading denotes the spread in LES results in the GCSS intercomparisons, where available. Our goal is to match these fine-resolution LES as closely as possible using coarser-resolution simulations. The dashed–dotted lines denote SAM simulations with 2-, 4-, or 16-km horizontal grid spacings that do not use CLUBB. The solid lines denote SAM simulations of the same grid spacings that do use CLUBB. Our

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FIG. 3. As in Fig. 1, but for BOMEX profiles averaged over hours 5–6. The gray shading is a measure of spread in the LES of Siebesma et al. (2003). Both [SAM with CLUBB] and [SAM without CLUBB] produce adequate simulations of the means and fluxes at 2- and 4-km horizontal grid spacing, except for SAM’s flux of rt at 4-km grid spacing (purple dashed–dotted line).

question throughout this section will be: Which matches LES (thick red lines) more closely—SAM without CLUBB (dashed–dotted lines), or SAM with CLUBB (solid lines)?

a. Means and turbulent fluxes of moisture and heat First we present profiles, from all four cases, of horizontally averaged (i.e., mean) total water mixing ratio (rt ), cloud water potential temperature (ul ), vertical turbulent flux of rt (w9r9t ), and vertical turbulent flux of ul (w9u9l ).

1) STRATOCUMULUS CASES Figure 1 shows the RF01 Sc case. With the exception of the SAM profile of rt at 16-km grid spacing, which is

not well mixed, all profiles are simulated with satisfactory accuracy. (In this paper, an ‘‘accurate’’ profile will mean one that has qualitative agreement with the shape and magnitude of the corresponding LES profile.) The means and fluxes from the RF02 drizzling Sc case (Fig. 2) show similar characteristics to RF01. However, at all grid spacings, the cloud top is too low, presumably because there is insufficient turbulent entrainment of clear air into the cloud layer itself.

2) CUMULUS CASES For the BOMEX trade-wind Cu case (Fig. 3), SAM’s 16-km solution is poor, as expected, with almost no turbulent fluxes above 500 m. SAM’s 4-km solution produces accurate profiles of w9u9l and ul , but w9r9t is

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FIG. 4. As in Fig. 1 but for ARM profiles averaged over hour 9.

too small, and consequently rt accumulates near the ground and is not transported upward strongly enough. For the ARM continental Cu case (Fig. 4), the 16-km SAM simulation exhibits poor fluxes, but SAM’s 4-km results are reasonably accurate. The simulations at all grid spacings that include CLUBB look similar to each other and match the LES satisfactorily except that the turbulent fluxes do not extend to sufficiently high altitudes. Overall, SAM simulates rt , ul , w9r9t , and w9u9l well at 4-km grid spacing, even though turbulent drafts are severely underresolved. At 4 km, SAM’s simulated fluxes in the interior are produced mostly by resolved flow rather than the Smagorinsky subgrid parameterization. To achieve this at 4-km grid spacing, SAM simulates a flow field in which the draft width is larger than in LES but in which the turbulent fluxes have the same magnitude as in LES.

b. Variances of w, ul, and rt We now plot and discuss the variances of vertical velocity (w92 ), heat content (u9l 2 ), and moisture (r9t 2 ). The variances that we compute are spatial variances at a grid level. For instance, w92 5

1 N

å(wij 2 w)2,

(9)

ij

where the sum is taken over the ith row and jth column of a horizontal grid level, and N is the number of grid boxes at a level. The variance is large if the spatial variability among points at the grid level is large.

1) STRATOCUMULUS CASES For the RF01 marine Sc case (Fig. 5), SAM underpredicts w92 moderately at 2-km grid spacing, greatly at

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4 km, and severely at 16 km. This is unsurprising, given that simulations at these grid spacings underresolve the drafts. At such grid spacings, generating explicit vertical motion requires overturning an unrealistically wide column of air. Including CLUBB partially mitigates the underprediction and leads to more consistent results at different grid spacings. Although SAM underpredicts w92 , it overpredicts the scalar variances (i.e., the variances of rt and ul) at 16- and 4-km grid spacings below cloud top. The overprediction appears to be related to time intermittency in the scalar fields; we discuss this more in the conclusions. Including CLUBB mitigates the overprediction below cloud top. However, both [SAM with CLUBB] and [SAM without CLUBB] underpredict the spike in scalar variances at cloud top; SAM without CLUBB underpredicts the spike by at least a factor of 10 in the RF01 case. The results for the RF02 heavily drizzling marine Sc case (Fig. 6) look similar to those for RF01. SAM still underpredicts w92 and overpredicts r9t 2 and u9l 2 . Including CLUBB mitigates but does not eliminate the underprediction of w92 . Including CLUBB largely eliminates the overprediction of r9t 2 and u9l 2 below cloud top but leads to an underprediction in RF02 of the spike of r9t 2 and u9l 2 near cloud top by at least a factor of 10.

2) CUMULUS CASES In the BOMEX trade-wind Cu case (Fig. 7), SAM again underpredicts w92 , but furthermore, SAM does not produce the bimodal profile of w92 , which the LES produces, with one turbulence maximum in the subcloud layer, a distinct maximum in the cloud layer, and a minimum within the more stable layer between. Including CLUBB leads to a bimodal w92 profile as in the LES. Although CLUBB still underpredicts the cloudlayer w92 , it improves the subcloud w92 . SAM underpredicts r9t 2 and u9l 2 at 16-km grid spacing in BOMEX because that simulation produces little cloud (see Fig. 15 below), but SAM overpredicts the variances at 2- and 4-km grid spacing. When CLUBB is included, the scalar variances for BOMEX are greatly improved at all grid spacings. SAM’s simulations of the ARM continental Cu case (Fig. 8) are similar to those for BOMEX. SAM fails to produce a bimodal profile of w92 and underpredicts w92 at the 4- and 16-km grid spacings. SAM overpredicts the

FIG . 5. RF01 profiles averaged over hour 5: (top) w92 , (middle) u9l 2 , and (bottom) rt92 . Line styles and colors are as in Fig. 1.

The error bars denote variability in aircraft observations (Stevens et al. 2003). SAM without CLUBB underpredicts w92 but overpredicts r9t 2 .

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FIG. 6. As in Fig. 5, but for RF02 profiles averaged over hour 6. Line styles and colors are as in Fig. 1. SAM without CLUBB underpredicts w92 but overpredicts r9t 2 .

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FIG. 7. As in Fig. 5, but for BOMEX profiles averaged over hours 5–6. Line styles and colors are as in Fig. 1. SAM without CLUBB at 2and 4-km grid spacing underpredicts w92 but overpredicts r9t 2 and u9l 2 .

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scalar variances at all grid spacings. With CLUBB included, the results are improved.

c. Mean, vertical flux, and variance of horizontal wind Here we present profiles of the mean (u), vertical flux (u9w9), and variance (u92 ) of the east–west component of horizontal wind. [The results from the north–south wind component are similar (not shown).] Plotted are wind fields from RF01 (Fig. 9), RF02 (Fig. 10), BOMEX (Fig. 11), and GCSS ARM (Fig. 12). SAM without CLUBB tends to underpredict the magnitude of u9w9, particularly at 4- and 16-km grid spacing, except for the RF02 case. The underprediction of ju9w9j, in turn, tends to lead to excessively sharp gradients of u near the ocean or ground surface. The inclusion of CLUBB helps mitigate these problems, but it leads to overly smoothed profiles of u in RF01 and BOMEX. As was the case for SAM’s simulations of scalar variances, SAM greatly overpredicts the east–west velocity variances u92 . The exceptions are RF01 and BOMEX at 16-km grid spacing, where almost no turbulence develops. Including CLUBB in the SAM simulations reduces the magnitude of u92 , as desired, but also leads to underprediction of the local maximum of u92 near the cloud-top inversion in the Sc cases.

d. Cloud fraction and cloud water mixing ratio A primary motivation for including a cloud parameterization in a large-scale model is to better estimate partial cloudiness and cloud water mixing ratio (rc) in order to calculate their effects on microphysics. We now analyze the quality of the simulations of cloud fraction (cf) and rc from simulations with and without CLUBB included.

1) STRATOCUMULUS CASES First we present cf and rc from the RF01 case (Fig. 13). Except for the 16-km simulation, SAM predicts relatively accurate profiles of cf and rc. Including CLUBB does not improve the RF01 simulations of cf or rc at 2 or 4 km, but does improve the results at 16 km. The RF02 heavily drizzling marine Sc case is shown in Fig. 14. At 2-km grid spacing, SAM predicts cf and rc accurately; at 4 km, it predicts rc accurately but underpredicts cf; and at 16 km, it underpredicts both cf and rc. CLUBB improves the prediction of rc at 4 km, but otherwise does not improve the results at 2 or 4 km. CLUBB does improve the results at 16 km.

2) CUMULUS CASES FIG. 8. As in Fig. 5, but for ARM profiles averaged over hour 9. Line styles and colors are as in Fig. 1. SAM without CLUBB at 4- and 2 16-km grid spacing underpredicts w92 but overpredicts rt9 and u9l 2 .

For the BOMEX trade-wind Cu case (Fig. 15), SAM’s 16-km simulation produces a fog layer because it fails to

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FIG. 9. RF01 profiles averaged over hour 5: (top) u, (middle) u9w9, and (bottom) u92 . Line styles and colors are as in Fig. 1. SAM without CLUBB overpredicts u92 .

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FIG. 10. As in Fig. 9, but for RF02 profiles averaged over hour 6. Line styles and colors are as in Fig. 1. SAM without CLUBB overpredicts u92 .

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FIG. 11. As in Fig. 9, but for BOMEX profiles averaged over hours 5–6. Line styles and colors are as in Fig. 1. SAM without CLUBB overpredicts u92 .

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FIG. 12. As in Fig. 9, but for ARM profiles averaged over hour 9. Line styles and colors are as in Fig. 1. SAM without CLUBB overpredicts u92 .

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FIG. 13. RF01 profiles averaged over hour 5. Shown is (top) cloud fraction and (bottom) liquid (cloud) water mixing ratio. Line styles and colors are as in Fig. 1. SAM at 2- and 4-km grid spacings produces relatively accurate profiles of cloud fraction and cloud water.

transport moisture upward. SAM’s 2- and 4-km simulations predict accurate cf profiles but overpredict rc. Including CLUBB yields a cloud top that is too low but improves the prediction of rc at all grid spacings and reduces the sensitivity to grid spacing. The ARM continental Cu results (Fig. 16) are similar to those for BOMEX, except that SAM’s 16-km simulation does not produce fog but does overpredict cf and rc. SAM’s 2- and 4-km simulations predict cf accurately

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FIG. 14. As in Fig. 13, but for RF02 profiles averaged over hour 6. Line styles and colors are as in Fig. 1.

but overpredict rc. Including CLUBB yields a cloud top that is too low but mitigates the overprediction of rc and exhibits less sensitivity to grid spacing.

e. Drizzle As mentioned above, one motivation for including a cloud parameterization in a large-scale model is to improve the driving of microphysical processes, such as drizzle formation. An accurate microphysics scheme will produce inaccurate results if it is fed inaccurate cloud water fields from a cloud parameterization. In

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FIG. 15. As in Fig. 13, but for BOMEX profiles averaged over hours 5–6. Line styles and colors are as in Fig. 1. SAM with CLUBB underpredicts cloud top height but has a more accurate prediction of cloud water for all grid spacings.

FIG. 16. As in Fig. 13, but for ARM profiles averaged over hour 9. Line styles and colors are as in Fig. 1. SAM with CLUBB underpredicts cloud top height but has a smaller overprediction of cloud water.

particular, because drizzle processes are often nonlinear, feeding them grid-mean cloud fields can produce incorrect grid-mean drizzle amount. For instance, accounting for variability in cloud water is important for modeling autoconversion of cloud water to drizzle water because the autoconversion rate is proportional to a large power of cloud water mixing ratio (Khairoutdinov and Kogan 2000). In the simulations presented here, CLUBB accounts for subgrid variability in rt (vapor plus cloud water), ul,

and w. In the simulations that include CLUBB, when there is partial cloudiness, we calculate the microphysics only within the cloudy areas, inputting within-cloud fields and the saturation value of vapor mixing ratio. Other than this, the simulations do not account for subgrid variability in drizzle fall speeds or drizzle mixing ratio. Excluding within-cloud subgrid variability of drizzle allows us to isolate the effects of using CLUBB’s more realistic cloud water values in order to drive drizzle formation.

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FIG. 17. Profiles of (upper left) rain mixing ratio for RF01 Sc, (upper right) RF02 heavily drizzling Sc, (lower left) BOMEX trade-wind Cu, and (lower right) ARM continental shallow Cu. Averaging periods are the same as in Figs. 13–16. The plot shows a reference 3D SAM LES (thick red solid line) and simulations with coarser horizontal resolutions of 2 (blue), 4 (purple), and 16 km (green). SAM with CLUBB simulations are denoted by thin solid lines; SAM without CLUBB, by dashed–dotted lines. Gray shading denotes the range of LES in Ackerman et al. (2009). The addition of CLUBB mitigates the overprediction of drizzle and leads to results that are less sensitive to horizontal grid spacing.

Figure 17 presents profiles of drizzle mixing ratio from all four cases. Usually, SAM overpredicts drizzle mixing ratio, sometimes grossly. The major exception is the BOMEX case at 16-km grid spacing, which produces no rain (drizzle) because it produces little cloud or turbulence. SAM with CLUBB included produces drizzle profiles that are closer to the LES profiles. Furthermore, including CLUBB yields more uniform results with changes in grid spacing. Why do the profiles of drizzle in SAM without CLUBB vary so greatly? Fig. 18 shows time series of drizzle. Drizzle often onsets suddenly and increases dramatically. These bursts of drizzle are usually associated

with increased amounts of cloud water, as can be seen by comparing the drizzle profiles (Fig. 17) with the corresponding cloud water profiles (Figs. 13–16), or by comparing the drizzle time series (Fig. 18) with the cloud water time series (Fig. 19). SAM’s overprediction of cloud water is related to SAM’s overprediction of r9t 2 and u9l 2 (Figs. 5–8). The large scalar variances are related to both temporal and spatial variations in moisture. Cloud water and drizzle can form readily in the moist regions, and because of the nonlinearity of drizzle processes, this can cause net overprediction of drizzle. In submitted manuscripts (Larson and Griffin 2011, manuscript submitted to Quart. J. Roy. Meteor. Soc.;

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FIG. 18. Time series of (upper left) rainwater path from RF01 Sc, (upper right) RF02 heavily drizzling Sc, (lower left) BOMEX trade-wind Cu, and (lower right) ARM continental shallow Cu. The line styles and colors are as in Fig. 17. The addition of CLUBB reduces excessive time variations in drizzle.

Griffin and Larson 2011, manuscript submitted to Quart. J. Roy. Meteor. Soc.), CLUBB has analytically accounted for subgrid variability of drizzle fall speed and mixing ratio by including drizzle mixing and number concentration in CLUBB’s multivariate PDF. This was feasible because of the simple formulation of the microphysics scheme used in those manuscripts (Khairoutdinov and Kogan 2000). The result of accounting for subgrid variability was to increase drizzle mixing ratio, as expected. However, the increases did not lead to the gross overpredictions with respect to LES that were exhibited by coarse-resolution SAM simulations. In the present article, including variability in drizzle fields is more complex because of the more complicated mathematical form of the Morrison microphysics. However, doing so in future work may improve the accuracy of

the drizzle mixing ratio profiles in simulations that include CLUBB.

6. Conclusions In this study, we have implemented a parameterization of clouds and turbulence, CLUBB, into a cloudresolving model, SAM. Then, in order to compare [SAM with CLUBB] to [SAM without CLUBB], we have simulated two marine Sc cases (RF01 and RF02) and two shallow Cu cases (BOMEX and ARM). The four cases were run at various horizontal grid spacings. Our goal is to address the following two questions: Which aspects of the simulations benefit most from the inclusion of CLUBB? What is the finest horizontal grid spacing at which CLUBB is still beneficial?

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FIG. 19. Time series of (upper left) cloud (liquid) water path from RF01 Sc, (upper right) RF02 heavily drizzling Sc, (lower left) BOMEX trade-wind Cu, and (lower right) ARM continental shallow Cu. The line styles and colors are as in Fig. 17. SAM’s large variations in drizzle water path (see Fig. 18) are associated with variations in cloud water path, especially in the cumulus cases.

We find that including CLUBB in SAM benefits four main aspects of the simulations: 1) it improves the prediction of variances, both of w and u, and of rt and ul; 2) it improves the prediction of drizzle mixing ratio; 3) it improves the mean and turbulent flux of the horizontal wind; and 4) it leads to results that are less sensitive to variations in horizontal grid spacing. We find a significant improvement in the results at 4-km grid spacing, particularly for the Cu cases, and an improvement in certain fields at 2-km grid spacing. We now elaborate on these conclusions. In making the following remarks, we do not intend to imply that SAM is inherently defective in any way; all CRMs suffer a degradation in accuracy at coarse resolution, and SAM performs relatively well at coarse grid spacing. Our goal

is merely to explore the possible benefits of implementing CLUBB in SAM. First consider the results with 4-km horizontal grid spacing. At 4-km grid spacing, SAM produces accurate simulations of the vertical turbulent flux of rt, w9r9t , in all cases except BOMEX; SAM accurately predicts the vertical turbulent flux of ul, w9u9l in all cases. But the inclusion of CLUBB improves, partially at least, the simulation of the horizontally averaged variance of vertical velocity w92 , total water mixing ratio r9t 2 , liquid water potential temperature u9l 2 , and east–west velocity variance, u92 . SAM underpredicts w92 , as expected, because updrafts and downdrafts are grossly underresolved at 4-km grid spacing. In contrast, SAM overpredicts other variances, for example, r9t 2 (Figs. 5–8). Note that a flux,

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such as w9r9t , is closely related to the product of the relevant variances, such as w92 and r9t 2 . This suggests that SAM predicts w9r9t accurately through an underprediction in w92 and a compensating overprediction in r9t 2 . A similar situation holds for ul and u. Accurate simulation of the variances is important in part because variances drive microphysics. For instance, variations in rt and ul are related to variations in cloud water mixing ratio, rc, and water vapor mixing ratio. These, in turn, influence various processes that form drizzle, such as autoconversion of cloud droplets to form drizzle drops, accretion of cloud droplets onto drizzle drops, and evaporation of drizzle drops below cloud. Given SAM’s overprediction of r9t 2 and u9l 2 , it is perhaps unsurprising that SAM overpredicts drizzle mixing ratio (Fig. 17). This occurs presumably because SAM produces excessively moist pockets that, through the nonlinearities in autoconversion and accretion, produce excessively large drizzle rates. The inclusion of CLUBB helps ameliorate the problem, but does not entirely remove it, perhaps in part because these simulations do not account for within-cloud variability of drizzle fields. In the preceding three paragraphs, we have discussed results at 4-km horizontal grid spacing. At 16-km grid spacing, SAM produces poor results, as expected, and CLUBB helps mitigate the coarse resolution. This result is unsurprising—3D models are rarely run at 16-km grid spacing without a cloud parameterization. However, CLUBB helps improve some of the fields even at 2-km grid spacing. CLUBB does not improve the turbulent fluxes at 2 km, but it does improve the shape of the w92 profile, and, in the cumulus cases, it does improve rc (Figs. 15–16), the scalar variances, and drizzle mixing ratio. Therefore, we conclude that the finest horizontal grid spacing at which CLUBB significantly improves results is between 2 and 4 km. The inclusion of CLUBB also improves the simulated wind fields (Figs. 9–12). CLUBB reduces the horizontal wind variance (u92 ), as desired. Furthermore, in most cases, CLUBB improves the turbulent fluxes (u9w9) and mean wind (u). Finally, we note that the inclusion of CLUBB helps produce results that are more consistent with respect to changes in grid spacing. Insensitivity to grid spacing is an important practical virtue because it reduces the amount of model retuning that needs to be done when the grid spacing is changed. In most cases, the profiles that include CLUBB are similar at 2-, 4-, and 16-km grid spacings, except that the 2-km profile takes on some of the characteristics of the corresponding profile from SAM without CLUBB. This behavior is reasonable. At 2-km grid spacing, more of the motions and variability

are resolved, and SAM stamps its character on the solutions, as desired. Acknowledgments. The authors are grateful to Dr. Marat Khairoutdinov for the use of his numerical model, SAM. The research was supported by the NASA Interdisciplinary Science Program under Grant NNX07AI56G. The Pacific Northwest National Laboratory is operated for DOE by Battelle Memorial Institute under Contract DE-AC06-76RLO 1830. V. Larson and D. Schanen were supported by Contract 47164 from Battelle Memorial Institute. REFERENCES Ackerman, A. S., and Coauthors, 2009: Large-eddy simulations of a drizzling, stratocumulus-topped marine boundary layer. Mon. Wea. Rev., 137, 1083–1110. Anand, M. S., A. T. Hsu, and S. B. Pope, 1997: Calculations of swirl combustors using joint velocity-scalar probability density function method. AIAA J., 35, 1143–1150. Andrejczuk, M., J. M. Reisner, B. Henson, M. K. Dubey, and C. A. Jeffery, 2008: The potential impacts of pollution on a nondrizzling stratus deck: Does aerosol number matter more than type? J. Geophys. Res., 113, D19204, doi:10.1029/ 2007JD009445. Arakawa, A., 2004: The cumulus parameterization problem: Past, present, and future. J. Climate, 17, 2493–2525. ——, J.-H. Jung, and C.-M. Wu, 2011: Toward unification of the multiscale modeling of the atmosphere. Atmos. Chem. Phys. Discuss., 11, 3181–3217. Blossey, P. N., C. S. Bretherton, and M. C. Wyant, 2009: Subtropical low cloud response to a warmer climate in a superparameterized climate model. Part II. Column modeling with a cloud resolving model. J. Adv. Model. Earth Syst., 1, doi:10.3894/JAMES.2009.1.8. Bougeault, P., 1981: Modeling the trade-wind cumulus boundary layer. Part II: A higher-order one-dimensional model. J. Atmos. Sci., 38, 2429–2439. Bretherton, C. S., 2007: Challenges in numerical modeling of Tropical circulations. The Global Circulation of the Atmosphere, T. Schneider and A. H. Sobel, Eds., Princeton University Press, 302–330. Brown, A. R., and Coauthors, 2002: Large-eddy simulation of the diurnal cycle of shallow cumulus convection over land. Quart. J. Roy. Meteor. Soc., 128, 1075–1093. Cheng, A., and K.-M. Xu, 2008: Simulation of boundary-layer cumulus and stratocumulus clouds using a cloud-resolving model with low- and third-order turbulence closures. J. Meteor. Soc. Japan, 86, 67–86. ——, and ——, 2009: A PDF-based microphysics parameterization for simulation of drizzling boundary layer clouds. J. Atmos. Sci., 66, 2317–2334. De Roode, S. R., P. G. Duynkerke, and H. J. J. Jonker, 2004: Largeeddy simulation: How large is large enough? J. Atmos. Sci., 61, 403–421. Gerard, L., 2007: An integrated package for subgrid convection, clouds and precipitation compatible with meso-gamma scales. Quart. J. Roy. Meteor. Soc., 133, 711–730. Golaz, J.-C., V. E. Larson, and W. R. Cotton, 2002: A PDF-based model for boundary layer clouds. Part I: Method and model description. J. Atmos. Sci., 59, 3540–3551.

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