Improved low-cloud simulation from a multiscale modeling framework with a third-order turbulence closure in its cloud-resolving model component



[1] In the original multiscale modeling framework (MMF), the Community Atmosphere Model (CAM3.5) is used as the host general circulation model (GCM), and the System for Atmospheric Modeling model with a first-order turbulence closure is used as the cloud resolving model (CRM) for representing cloud physical processes in each grid column of the GCM. This study introduces an upgrade of the MMF in which the first-order turbulence closure scheme is replaced by an advanced third-order turbulence closure in its CRM component. The results are compared between the upgraded and original MMFs, CAM3.5, and observations. The global distributions of low-level cloud amounts in the subtropics in the upgraded MMF show substantial improvement relative to the original MMF when both are compared with observations. The improved simulation of low-level clouds is attributed not only to the representation of subgrid-scale condensation in the embedded CRM but also is closely related to the increased surface sensible and latent heat fluxes, the increased lower tropospheric stability (LTS), and stronger longwave radiative cooling. Both MMF simulations show close agreement in the vertical structures of cloud amount and liquid water content of midlatitude storm-track clouds and subtropical low-level clouds, compared with observations, with the upgraded MMF being better at simulating the low-level cumulus regime. Since the upgraded MMF produces more subtropical low-level clouds and does not produce an excessive amount of optically thick high-level clouds in either the tropics or midlatitudes as the original MMF does, the global mean albedo decreases. The positive bias in albedo and longwave cloud radiative forcing (CRF) and negative bias in shortwave CRF are reduced in the tropical convective regions.

1. Introduction

[2] Cloud topped planetary boundary layers (PBLs) play a key role in the Earth's radiation balance, energy budget, and climate sensitivity. The representation of the low-level clouds in general circulation models (GCMs) is one of the largest sources of uncertainty in climate simulations [e.g., Randall et al., 2007]. Since low-level clouds are associated with turbulent circulations of small spatial scales (meters to kilometers), they cannot be resolved in a GCM with a grid size on the order of 100 km in the horizontal and hundreds of meters in the vertical. (PBL clouds and low-level clouds are interchangeably used in this paper.) These clouds must be represented solely as parameterizations based only on the large-scale variables such as temperature, moisture, and winds prognosticated by the GCM; hereafter, this is referred to as “traditional” cloud parameterization. The mass flux approach has been used to parameterize the PBL clouds in GCMs since 1970s [e.g., Arakawa, 1969; Ooyama, 1971; Albrecht, 1979]. Although more sophisticated formulations have recently been proposed that considered eddy diffusivity combined with mass flux [Neggers et al., 2009] and buoyancy sorting mechanism [e.g., Park and Bretherton, 2009], the mass flux approach of these schemes still has difficulties in representing the dry updraft, moist updraft, and moist downdraft realistically with the limited input large-scale information.

[3] The newly developing multiscale modeling framework model (MMF) partially resolves small-scale processes. In the MMF approach, a cloud-resolving model (CRM) is embedded in each atmospheric grid column of the host GCM to represent cloud physical processes [Grabowski, 2001; Khairoutdinov and Randall, 2001]. This new climate modeling approach is attractive because it allows for an explicit simulation of many cloud processes, including convection, overlapping clouds in both the radiative and microphysical senses, and convectively generated gravity waves, as pointed out by Randall et al. [2003]. MMF still needs to parameterize the subgrid-scale (SGS) processes associated with clouds and large turbulent eddies because circulations associated with boundary layer clouds are unresolved by CRMs with horizontal grid sizes on the order of a few kilometers. That is, processes with spatial scales of meters and temporal scales of seconds, such as turbulence and cloud microphysics, are parameterized in the embedded CRM; those associated with horizontal spatial scales of tens of kilometers and temporal scales of minutes are parameterized in GCMs with traditional cloud parameterizations.

[4] The critical roles that turbulence parameterization plays in CRMs have been gradually recognized. CRMs usually use horizontal grid spacing of a few kilometers, and vertical grid spacing of hundred of meters in order to simulate deep convective cloud systems. The state-of-the-art CRM usually uses a low-order (either first-order or 1.5-order) turbulence closure (LOC) to parameterize the SGS turbulent processes except for Krueger [1988], who implemented a third-order turbulence closure (TOC) in a CRM. Recently, Cheng and Xu [2008] pointed out that cumulus clouds are represented as stratocumulus clouds by the resolved circulations because of insufficient SGS transports of heat and moisture in the LOC approach. On the other hand, TOC schemes are able to parameterize moist processes involving low-level clouds in many case studies [e.g., Bougeault, 1981; Lappen and Randall, 2001; Golaz et al., 2002b; Cheng and Xu, 2008]. The vertical profiles of temperature and moisture, cloud fraction, cloud water and momentum compare well with those from large-eddy simulations, which provide many low- and higher-order statistics, and with limited observations.

[5] Since its inception, it has been shown that the MMF approach can produce the Madden-Julian Oscillation [Madden and Julian, 1972], higher-frequency tropical waves and diurnal cycles of precipitation in a much more realistic manner than a GCM with a traditional cloud parameterization [Khairoutdinov and Randall, 2001; Grabowski, 2003; Randall et al., 2003; Khairoutdinov et al., 2008; Tao et al., 2009] because the embedded CRM is adequate to resolve the large updrafts and downdrafts of cumulonimbus clouds for these convectively active phenomena.

[6] The horizontal CRM grids used in MMF, which are typically 1–4 km, are far too coarse to resolve other kinds of atmospheric turbulence. Over half of the globe at any time is covered by low-lying shallow cumulus and stratocumulus cloud fields associated with turbulent eddies [e.g., Xu et al., 2008] that transport heat and moisture from the surface. These eddies, which are typically a few hundred meters in size or less, are too small to be accurately represented by any embedded CRM. As a result, the MMF does not simulate the climatology of low clouds very well, producing too little in the cool subtropical ocean regions in which stratocumulus clouds are found [Khairoutdinov et al., 2008; Marchand and Ackerman, 2010]. The vertical structure and climatology of low clouds (with tops below 3 km) are simulated as well as those from most conventional climate models [Wyant et al., 2006; Tao et al., 2009]. The latter are, however, dubious for their inability to represent low-level clouds.

[7] To produce more realistic simulations of low clouds and their sensitivity to climate and aerosol perturbations, two parallel strategies should be adopted to improve the MMF approach. One of them is to drastically increase both the horizontal and vertical resolutions of the embedded CRM. Because of 200-fold increase in computing time of a coarse-resolution MMF over a traditional GCM, the potential to further increase the resolutions is very limited. Marchand and Ackerman [2010] improved modestly several aspects of the MMF low-level clouds by increasing both horizontal (4 km to 1 km) and vertical resolutions simultaneously. Another strategy is to improve the CRM with a more sophisticated subgrid turbulence scheme, which has shown promise in previous CRM and single column studies [e.g., Bougeault, 1981; Lappen and Randall, 2001; Golaz et al., 2002b; Cheng and Xu, 2006, 2008]. In the present study, we will adopt the strategy of using a more sophisticated subgrid turbulence scheme rather than using a higher grid resolution to improve the MMF low-cloud simulation. To our knowledge, this study represents the first documented application of TOC schemes to global climate modeling.

[8] The major goal of this study is to improve the simulation of PBL clouds in an MMF. This is achieved by implementing a TOC in the embedded CRM for the MMF. The simulations from the upgraded MMF are compared with satellite and ground-based observations and with those using the original MMF, and the host GCM with the traditional cloud parameterizations. The secondary goal is to understand the physical processes influencing the global distribution of PBL clouds in the MMF simulations.

[9] Section 2 introduces an MMF, a CRM with a TOC scheme, and describes the experiment design. The simulated results from the MMFs and GCM with a mass flux approach are analyzed and compared in section 3. Conclusions and discussion are presented in section 4.

2. Model Description and Experiment Design

[10] The MMF used in this study consists of the Community Atmosphere Model Version 3.5 (CAM3.5) [Collins et al., 2006] and a 2D CRM embedded in each GCM atmospheric grid column [Khairoutdinov and Randall, 2001]. The large-scale atmospheric circulation is represented on the coarse-resolution grid of the atmospheric component of the GCM via a semi-Lagrangian dynamical core; the physical processes such as convection and stratiform cloudiness, usually parameterized in a traditional GCM, are resolved explicitly on the CRM fine grids. Cloud microphysics and radiation are parameterized at the CRM scale. Tendencies of heat and moisture from the CRM scale are communicated to the large scale via the GCM. A wide range of spatial and temporal scales and their interactions are included in the MMF.

[11] The CRM component of the original MMF is the standard System for Atmospheric Modeling (SAM) model [Khairoutdinov and Randall, 2003] with the first-order Smagorinsky turbulence scheme [Smagorinsky, 1963] used to parameterize subgrid-scale (SGS) processes within the CRM and with a relatively simple bulk cloud microphysics package with 15 processes to govern the interactions between hydrometeors and water vapor. Because CRMs with the Smagorinsky scheme have difficulty partitioning kinetic energy into CRM resolved and subgrid scales [Cheng et al., 2010], an intermediately prognostic higher-order turbulence closure (IPHOC) scheme is implemented in SAM [Cheng and Xu, 2008]. The upgraded MMF will hereafter be referred to as SPCAM-IPHOC since the original MMF is known as SPCAM with the CAM as its host GCM. IPHOC assumes a joint double-Gaussian distribution of liquid water potential temperature, total water, and vertical velocity [Cheng and Xu, 2006]. The distribution is inferred from the first-, second-, and third-order moments of the variables given above, and is used to diagnose cloud fraction and grid mean liquid water mixing ratio, as well as the buoyancy terms and fourth-order terms in the equations describing the evolution of the second- and third-order moments. These higher-order moments, which are not available in an LOC, are used to formulate SGS condensation in IPHOC. IPHOC uses the first-order moments predicted by SAM, and the prognostic equations for the second- and third-order moments are shown in detail in the Appendix.

[12] The sub-CRM-grid variability has been partially considered in cloud-radiation interactions because the radiative transfer is calculated for each CRM grid column, and the partial cloudiness and liquid water in each CRM grid column have been passed to the radiation code. Although the SGS distribution of condensate, such as ice and cloud water, is known, microphysical conversion and rainwater collection rates are not derived from the SGS distribution [Cheng and Xu, 2009] in the preliminary testing of SPCAM-IPHOC presented in this paper. These SGS effects will be evaluated in a future study.

[13] Three simulations of a 2 year duration starting from 0000 UT 1 September 1990 were performed using the CAM3.5 climate model, SPCAM, and SPCAM-IPHOC. Khairoutdinov and Randall [2001] and Tao et al. [2009] also performed similar short-term integrations to test initial configurations of the MMF approach. The three models were forced by specifying climatological sea surface temperature (SST) and sea ice with monthly mean annual cycles while coupled with the Community Land Model over land grids [Oleson et al., 2004]. The initial atmospheric conditions were obtained by mapping the high-resolution (T159) European Centre for Medium-Range Weather Forecasts Re-Analysis data (ERA-40) [Uppala et al., 2005] to the coarse-resolution CAM grid in a way that is consistent with the low-resolution topography. All of the three models have the same horizontal and vertical resolutions with T21 in the horizontal and 26 layers in the vertical direction with six layers below 700 hPa level. The embedded CRMs have the same vertical levels as the host GCM, and use homogeneous 4 km horizontal grid spacing and 32 columns in a periodic horizontal domain, as specified by Khairoutdinov and Randall [2001].

[14] The computational cost is an important factor in determining the benefit of a new parameterization. For Khairoutdinov and Randall [2001], SPCAM was 180 times more expensive than CAM at T42 resolution. The cost of a ten model-day run with 256 processors of the NCAR BlueGene Supercomputer System for each model is listed in Table 1. The time steps used in each model are also given. The computational cost of SPCAM-IPHOC at T21 is about twice as expensive as SPCAM, and is about 49 times more than CAM3.5. The cost depends on the total number of CRMs used. In the experiments performed in this study, the total number of CRMs is 2048, which is one fourth of that of Khairoutdinov and Randall [2001]. In order to simulate shallow cumulus realistically with the original SPCAM, the horizontal grid spacing of the embedded CRM should be 100 m or less [Cheng and Xu, 2009]. This requires 40 times more computational cost compared to 4 km grid spacing used in the current SPCAM. This increase of the horizontal resolution is 20 times more expensive than the SPCAM-IPHOC. It should be emphasized that this study presents a preliminary test of the IPHOC scheme. We are attempting to decrease computational cost.

Table 1. Computational Cost for 10 Model Days in the NCAR BlueGene Supercomputer System and Time Steps Used in the Three Models
 Total Wallclock (s)Time Step for Dynamic Core (s)Time Step for CRM (s)Time Step for IPHOC (s)

3. Results

[15] Since the primary objective of this study is to improve the simulation of PBL clouds in an MMF, results related to these clouds are extensively presented in sections 3.1 and 3.2, including the global distribution and a vertical cross section in the southeastern (SE) Pacific region. State-of-the-art satellite observations from CloudSat, CALIPSO, MODIS, CERES and reanalysis are used to validate the simulations. Physical processes influencing the global distribution of these clouds in the MMF simulations are explored. Impacts of the upgraded MMF on the radiative transfer and cloud radiative forcing are also evaluated.

3.1. Low-Level Clouds

[16] Low clouds are parameterized and partially resolved in the three climate models in totally different ways. A combination of the mass flux approach [Hack, 1994] and large-scale condensation are used in CAM3.5 to parameterize shallow cumulus and stratocumulus clouds within the grid size of a few hundred kilometers. The detailed explanation of the interactions among the parameterizations is provided by Zhang and Bretherton [2008]. Stratocumulus clouds can be partially resolved in the embedded SAM within SPCAM, but the structure of the turbulence and its intensity cannot be fully represented. The Smagorinsky scheme treats turbulence as eddy diffusivity or viscosity, analogous to the molecular diffusivity or viscosity. The CRM uses the grid mean (taken over a horizontal span of 4 km) temperature and moisture to calculate grid-scale condensation and thus the binary cloudiness (either 1 or 0). The IPHOC parameterizes turbulence by predicting the turbulent kinetic energy (TKE), fluxes, and three third-order moments. The subgrid-scale condensation and partial cloudiness are diagnosed from the low- and higher-order moments and the double Gaussian based probability density function (PDF). This allows clouds to form in relatively dry regions where SPCAM cannot produce any cloud.

[17] In the following analysis, the term “low-level clouds” refers to the clouds produced below 700 hPa, the term “high-level clouds” to those above 400 hPa, and the term “middle-level clouds” to those between 700 hPa and 400 hPa, respectively. The clouds from CAM3.5 are the products of shallow and deep convective parameterizations, large-scale condensation, and cloud microphysics and macrophysics parameterizations. A maximum-random overlap assumption is then used to calculate the cloud fraction within the low, middle, and high categories. The calculation of the cloud fraction for each of these categories from SPCAM is relatively simple. The sum of liquid water and ice water paths within the pressure range is calculated for each column of the CRM. If the sum is larger than a threshold value of 0.001 kg m−2, a 100% cloud fraction is assumed for the specific pressure range of the column, otherwise, the cloud fraction is assigned to be 0%. The cloud fraction for the GCM grid box is the total number of cloudy columns divided by the total number of the embedded CRM columns. Since the cloud faction for each CRM grid is known from the SGS condensation scheme of SPCAM-IPHOC, a maximum-overlap assumption is used to calculate the cloud fraction of the three categories for each CRM column. The cloud fraction for each GCM grid box of SPCAM-IPHOC is the average of the overlapped cloud fraction over all the embedded CRM columns.

[18] The CloudSat-CALIPSO radar-lidar observations provide vertically resolved cloud occurrence frequencies at resolutions higher than those of the CRM models. The CloudSat [Stephens et al., 2002] has a resolution of 1.4 km, and the CALIPSO [Winker et al., 2007] has three shots (samples) of 30 m each within 1 km space. The products from both CloudSat and CALIPSO are carefully synchronized. The cloud fraction inside a GCM column is calculated from the cloud occurrence frequency matrix that contains the vertical cloud profile as a function of the uppermost cloud top [Kato et al., 2010]. The uncertainty of sampling is an issue for accurately calculating the cloud fractions. A longer time span of observations may help decrease this uncertainty substantially. All frequencies for clouds with tops below 700 hPa are maximally overlapped as the low-cloud amount.

[19] Substantial improvement can seen in the cloud amount and location of low-level clouds simulated by SPCAM-IPHOC (Figure 1) when compared to observations. Note that the effect of using different simulation and observation years here is small because there is only a small amount of interannual variability between non–El Niño years. The global annual mean low-cloud amount from the SPCAM-IPHOC is 40.5%, which compares to 48.0% from CloudSat-CALIPSO observations from January 2007 to December 2008. The global means from CAM3.5 and SPCAM are only 36.7% and 34.7%, respectively, which are substantially less than the observations. The strength and location of the low-cloud maxima, in the northeastern (NE) and SE Pacific, SE Indian (west coast of Australia), and NE and SE Atlantic, were well reproduced by SPCAM-IPHOC, while those from CAM3.5 and SPCAM are either too weak or not produced at all, e.g., the SE Indian in SPCAM, as shown in Figure 1b. Furthermore, both CAM3.5 and SPCAM also underestimate low-level clouds in the midlatitude storm track regions, compared with observations and SPCAM-IPHOC, but SPCAM-IPHOC overestimates those clouds in the Southern Hemisphere, compared with observations.

Figure 1.

Global distribution of annual mean low-level (below 700 hPa) cloud amounts (%) from (a) CAM3.5, (b) SPCAM, (c) SPCAM-IPHOC, and (d) the CloudSat and CALIPSO observations, and (e) the frequency of low cloud occurrence in a GCM box from SPCAM-IPHOC (see text for definition). The global mean is printed on the right top corner of each plot.

[20] The annual mean liquid water path (LWP) vertically integrated liquid water content between surface and 700 hPa for low clouds from either MMF is larger in the tropics and some regions of the midlatitude than in the MODIS (Moderate Resolution Imaging Spectroradiometer) observations from March 2000 to August 2004 (Figure 2). The MODIS observations also include contributions from liquid-phase clouds above the 700 hPa level. Thus, a rigorous comparison between the models and observations is difficult as far as LWP is concerned. The resolution of the MODIS sensor is 1 km. Passive satellite retrievals of LWP based upon MODIS measurements (Figure 2d) are expected to bias low when upper-level clouds are present, which largely explains the model's overestimate in the tropics and midlatitude regions at 40–60°S and 50–60°N. The 60°S–60°N oceanic mean LWP is 78.9 g m−2, 86.5 g m−2, and 107.1 g m−2 from the CAM3.5, SPCAM, and SPCAM-IPHOC, respectively, compared to an observed value of 101.2 g m−2. Compared with a 5% increase in global mean low cloud amount, the 60°S–60°N oceanic mean LWP from the SPCAM-IPHOC is about 20% larger than those from either the CAM3.5 or SPCAM. The larger increase in LWP mainly occurs in the subtropical regions such as the NE and SE Pacific, SE Indian, and NE and SE Atlantic, compared to SPCAM, and also can be seen in the other parts of three oceans as compared to CAM3.5, such as the deep convective region south of the equator in the western Pacific. The deep convective parameterization in CAM3.5 does not produce the low cloud liquid water content, which partially explains the underestimate in the tropical regions.

Figure 2.

Nearly global oceanic distribution of annual mean liquid water paths (LWPs; g m−2) for low clouds (between surface and 700 hPa) from (a) CAM3.5, (b) SPCAM, (c) SPCAM-IPHOC, and (d) MODIS (Moderate Resolution Imaging Spectroradiometer) observations. LWPs at high latitudes are not plotted due to large uncertainties there in MODIS retrieval. Note that the MODIS LWP is not restricted to the lower troposphere.

[21] There are several reasons for the more realistic representation of low-level clouds in the SPCAM-IPHOC simulation. The primary reason is the representation of subgrid-scale variability and turbulent transports within CRM grids. The more realistic cloud distributions in the subtropical regions feed back to the large-scale circulations, which create more productive environments to the maintenance of these clouds. The details are explained below.

[22] According to resolution sensitivity tests [Cheng and Xu, 2008; Cheng et al., 2010], shallow cumulus clouds are simulated as scattered stratocumulus clouds in SPCAM owing to the low-order turbulence closure and its lack of SGS condensation in the CRM. In SPCAM, the GCM column is filled with clouds of high water content over a small horizontal area, as opposed to the column with clouds of low water content over a large horizontal area in SPCAM-IPHOC. Thus, the infrared radiation cooling and reflected solar radiation can be significantly different between these two cloud fields and radiative feedbacks to large-scale circulations are likely different. This will be discussed in section 3.2. Besides the differences in cloud fields when both MMFs can produce clouds, the SGS condensation scheme used by the IPHOC can predict partial cloudiness in a relatively dry region where the CAM3.5 and SPCAM cannot produce a cloud. This is because the subgrid-scale variations in thermodynamic and dynamic variables are considered to determine what fraction of a CRM grid can form a cloud. In CAM3.5 [Collins et al., 2006], the specification of low-level cloud amount is based upon low-level stability, which represents a shortcut of physical processes controlling low-level clouds.

[23] To quantitatively show the SGS condensation scheme producing clouds in a GCM grid box, the frequency of the low-level cloud occurrence in a GCM grid box is plotted in Figure 1e. If any cloud is produced in a CRM grid below 700 hPa, the entire CRM grid column is regarded as cloudy. The frequency of the low-cloud occurrence is defined as the total number of cloudy columns divided by the total number of the CRM columns. According to this definition, the cloud fraction and cloud occurrence frequency is the same for SPCAM, but the former is smaller than the latter for SPCAM-IPHOC. The difference in the global mean cloud fraction between SPCAM-IPHOC and SPCAM is about 5%, but that in frequency is 17%. One can see that most of the low-level oceanic clouds produced by SPCAM-IPHOC are partially cloudy, especially near the warm pool region of the Pacific by comparing Figure 1c with Figure 1e. The stratocumulus clouds in the storm track region in the Southern Hemisphere at 120°W is an exception since the cloud fraction and cloud frequency are nearly equal there.

[24] Another reason for the more realistic low-level clouds near the west coast of the continents is due to the efficient transports of latent (moisture) and sensible heat by the IPHOC scheme. According to Cheng and Xu [2008], the vertical transports of sensible and latent heats within the GCM column are solely due to SGS transports parameterized by IPHOC, but are mostly due to CRM-resolved transports in SPCAM at the 4 km CRM grid spacing. The intensity of turbulence by SPCAM-IPHOC within the boundary layer is much larger than that from the default first-order turbulence scheme in SAM, which increases the SGS transports and further impacts the surface fluxes. The surface fluxes are tightly coupled to the intensity of turbulence although they are calculated from the bulk formula. On the other hand, the resolved vertical transports by the CRM with IPHOC are much less than the CRM with the default first-order turbulence scheme because the higher-order turbulence SGS model more realistically absorbs the subgrid energy. The surface sensible heat flux from SPCAM-IPHOC is larger by a few W m−2 in the subtropical regions, as compared with SPCAM and CAM3.5 (Figure 3). Local maximum centers of the sensible heat fluxes in the JRA-25 reanalysis [Kazutoshi et al., 2007], which are more pronounced than in the three simulations, can be seen in the NE and SE Pacific, SE Indian, and NE and SE Atlantic regions (Figure 3d), corresponding well to the observed large cloudiness and liquid water centers in those regions (Figures 1c and 2c).

Figure 3.

Global oceanic and annual mean surface sensible heat flux (W m−2) from (a) CAM3.5, (b) SPCAM, (c) SPCAM-IPHOC, and (d) Japanese Reanalysis Project.

[25] The surface sensible and latent heat fluxes and radiative cooling provide the source of TKE for the low-level clouds. The global oceanic and annual mean vertically integrated SGS TKE below 700 hPa, diagnosed from the embedded CRMs of SPCAM and SPCAM-IPHOC, are shown in Figure 4. The vertically integrated SGS TKE from SPCAM-IPHOC is about twice as large as that from SPCAM. In addition to the maximum values over 300 kg s−2 in the middle latitude storm track regions, local maximum centers with SGS TKE larger than 180 kg s−2 are seen slightly to the west of the maximum cloudiness centers in the NE and SE Pacific, SE Indian, and NE and SE Atlantic regions. The slight shift suggests that the TKE maxima correspond to cumulus and stratocumulus, instead of stratus. The maximum TKE centers correspond well to the large sensible heat centers in those regions from SPCAM-IPHOC. SPCAM, however, lacks the large TKE in the cool subtropical ocean regions (Figure 4a).

Figure 4.

Global oceanic and annual mean vertically integrated SGS TKE (kg s−2) below 700 hPa from (a) SPCAM and (b) SPCAM-IPHOC.

[26] The lower tropospheric stability (LTS; defined as the difference in potential temperature between 700 hPa and surface [Klein and Hartmann, 1993]) is closely related to the production of low-level clouds. Klein and Hartmann [1993] found a good correlation between LTS and low-level cloud amount in monthly mean data. The LTS is also an indicator for the difference in subsidence between the simulations because strong subsidence warms the free atmosphere. Figure 5 shows that the annual mean LTS increases in the NE and SE Pacific and Atlantic Oceans where large cloud amounts are produced in SPCAM-IPHOC compared with SPCAM, especially the areas with LTS values being larger than 14 K. The LTS values averaged over the 60°S–60°N belt is 12.6 K, 12.9 K, and 13.4 K for CAM3.5, SPCAM, and SPCAM-IPHOC, respectively. The observed LTS is 13.9 K according to the NCEP/NCAR reanalysis for the period from 1979 to 1998 [Kistler et al., 2001]. Although the LTS spatial patterns are generally similar among the three simulations, the larger LTS in the SPCAM-IPHOC simulation may be related to more low-level clouds in the tropical and subtropical oceanic regions compared to those in SPCAM and CAM3.5.

Figure 5.

Nearly global distribution of annual mean LTS (K) from (a) CAM3.5, (b) SPCAM, (c) SPCAM-IPHOC, and (d) NCEP reanalysis.

3.2. Vertical Cross Sections in Subtropical Stratocumulus to Trade Cumulus Transition

[27] In order to further understand the differences in the simulated low-level clouds, a vertical cross section along 15°S from the South American coast (∼80°W) to 120°W is selected from the most persistent PBL cloud region in the world, i.e., the SE Pacific. This cross section represents a stratocumulus to trade cumulus transition. The large-scale dynamics in this region are well represented in CAM3.5, so model biases are due mainly to the moist physics parameterizations [Park and Bretherton, 2009]. The vertical structure of the cloud amount exhibits large differences among the three models (Figure 6). SPCAM-IPHOC produced the largest overall cloud amount among the three models. The highest simulated cloud fractions are located between 90°W and 100°W, compared with the observed maximum located between 80°W and 90°W. The observational data from CALIPSO, CloudSat and MODIS are constrained by CERES (Clouds and Earth's Radiant Energy System) [Wielicki et al., 1996; Loeb et al., 2009] with an average footprint size of approximately 20 km × 20 km and called C3M [Kato et al., 2010]. The C3M data are available from January 2007 to December 2008. The maximum cloudiness centers from both MMFs contributed mainly by stratocumulus clouds, are located farther west than the observation and CAM3.5. Although the maximum center from CAM3.5 is located closer to the coast, CAM3.5 underestimates the height of the stratocumulus clouds near 80°W. The most likely reason for the misplaced stratocumulus maxima in the MMFs is the coarse vertical resolution, which cannot resolve the thin coastal stratocumulus clouds in the embedded CRMs. In the longitudes farther from the coast, thick cumulus layers were produced in all three models. But SPCAM produced the least overall cloud amount among the three models as discussed earlier, due mainly to the lack of subgrid cloudiness parameterization in its embedded CRM and inadequate amount of subgrid-scale transports.

Figure 6.

Cross-sectional plots of annual mean cloud fraction (%, shaded) and cloud liquid water (mg kg−1, contoured) along 15°S over the southeast Pacific from (a) CAM3.5, (b) SPCAM, (c) SPCAM-IPHOC, and (d) C3M observations. The thick solid line in each plot indicates the PBL height diagnosed from GCM grid variables based upon the formula presented by Holtslag and Boville [1993]. The contour interval is 20 mg kg−1 for the cloud liquid water.

[28] There are significant differences in the magnitudes of liquid water content among the three models and in the vertical structures between the MMFs and CAM3.5. For the shallow cumulus cloud regime west of 100°W, SPCAM-IPHOC produced the largest amount of liquid water content (Figure 6), which is at least twice as large as that produced by SPCAM although their vertical structures are rather similar. On the other hand, CAM3.5 does not produce much liquid water content, whose peaks are located at the PBL top. One can deduce that the shallow cumulus clouds tend to contain more liquid water in two MMFs than in CAM3.5 on average. This result suggests that the positive feedback between cloud water and longwave cooling might also play a role in increasing the liquid water content, as seen from the high correlation between the longwave cooling and the cloud liquid water (Figure 7). The magnitudes of both liquid water content and radiative cooling are larger in the two MMFs than in CAM3.5. The vertical thickness of the cooling rates larger than 3 K d−1 from two MMFs are much larger than in CAM3.5 west of 100°W, implying a stronger positive feedback between the longwave radiative cooling and cloud water in MMFs. The positive feedback process is also beneficial to the production of partially cloudy shallow cumulus clouds by the SGS condensation scheme.

Figure 7.

Same as Figure 6 but for longwave heating rate only (K d−1).

[29] For the stratocumulus cloud regime east of 95°W, the location of the liquid water content for SPCAM-IPHOC agree relatively well with the C3M observations, but the cloud amount is underestimated. This may be caused by the coarse vertical grid spacing below 700 hPa. SPCAM tends to produce stratocumulus clouds that are too close to the ocean surface and very thin, consistent with an MMF study by Marchand and Ackerman [2010], while CAM3.5 produces rather thin stratocumulus layer around the PBL top height. The longwave radiative cooling also plays an important role in the stratocumulus and transition areas east of 110°W (Figure 7), especially for the stratocumulus clouds at 85°W for SPCAM and CAM3.5 where the longwave radiative cooling rate is more than 10 K d−1. The shortwave radiative heating rate is near 2 K d−1, about five times less than the longwave cooling rate (not shown), and may become important for a short period during daytime.

3.3. Middle- and High-Level Clouds and Precipitation

[30] Since clouds at low, middle, and high levels are interconnected in most regions, it is informative to show the global distributions of middle- and high-level clouds (Figures 8 and 9). As previously shown, the SGS condensation and turbulence have positive effects on simulation of low-level clouds in midlatitudes. But the middle-level cloud amount does not increase in the storm track regions from the SPCAM-IPHOC compared to SPCAM and CAM3.5. The differences in the global mean middle-level cloud amount among the three models come mainly from the clouds in the high latitude of the Southern Hemisphere, for example, the maximum centers located between 0 and 120°E from CAM3.5, and between 60°W and 100°W from SPCAM, respectively. Overall, all three models underestimate the middle-level clouds compared to the CloudSat and CALIPSO observations (Figure 8d).

Figure 8.

Global distribution of annual mean middle-level (between 400 hPa and 700 hPa) cloud amounts (%) from (a) CAM3.5, (b) SPCAM, (c) SPCAM-IPHOC, and (d) the CloudSat and CALIPSO product.

Figure 9.

Global distribution of annual mean high-level (<400 hPa) cloud amounts (%) from (a) CAM3.5, (b) SPCAM, (c) SPCAM-IPHOC, (d) the CloudSat and CALIPSO product, and (e) ISCCP D2 product.

[31] The global mean high-level cloud amounts are 32.7%, 26.1%, and 24.9% from the CAM3.5, SPCAM, and SPCAM-IPHOC, respectively, compared to the ISCCP D2 [Rossow et al., 1996] observed value of 21.26% and a CloudSat and CALIPSO observed value of 40.1%. Note that the reason for the difference in the global mean cloud amount between ISCCP D2 and C3M is that CALIPSO observations used in C3M can detect thin high-level clouds with optical depth less than 0.3 (Figure 9d), which the passive sensors of ISCCP cannot (Figure 9e). Both MMFs seem to have difficulties to capture the abundance of optically thin high-level clouds (Figures 9b and 9c), which is a challenging work left for future.

[32] To further understand the interactions among clouds at different levels in the storm track region, the vertical structures of the storm-track clouds in the Southern Hemisphere are examined utilizing a vertical cross section along 60°S (Figure 10). Although there are longitudinal variations, the following discussion will be focused mainly on the differences in the vertical structures of the storm-track clouds among the three models. The vertical extents are rather different among the three models, for example, the annual mean maximum heights in which cloud fraction exceeds 1% are 13–15 km in CAM3.5, 11 km in SPCAM, and 10 km in SPCAM-IPHOC, respectively (Figures 10a10c). Overall, the SPCAM-IPHOC produces the smallest amounts of high-level clouds in the middle-latitude storm-track regions among the three simulations. The reason is that the shallow turbulent mixing produced by the IPHOC scheme may consume part of the CAPE needed for some parcels to reach high altitudes. According to a series of offline tests, Cheng and Xu [2008] found that a CRM with a low-order turbulence closure lacks of SGS transport of surface moisture and sensible heat, resulting in an accumulation of CAPE. Once the CAPE is released after excessive accumulation, resolved scale circulations with much larger scales than those in reality are produced. The clouds can reach much higher levels with larger sizes than from a CRM with an IPHOC scheme.

Figure 10.

Cross-sectional plots of annual mean cloud fraction (%, shaded) and the sum of cloud liquid water and cloud ice (mg kg−1, contoured) along 60°S from (a) CAM3.5, (b) SPCAM, (c) SPCAM-IPHOC, and (d) C3M observations.

[33] In addition to the maximum around the 1 km height, the CAM3.5 has a secondary maximum with cloud fraction being larger than 20% that is located between 6 km to 9 km, which may be closely related to moisture transport by deep convection as will be discussed shortly. This feature implies that the high-level clouds from CAM3.5 deep convective parameterization reach a much higher level than the two MMFs, with those from SPCAM-IPHOC having the smallest vertical extent.

[34] Despite the large difference in the vertical structures of cloud fraction between the three models discussed above, the maximum cloud liquid water is located below 1.5 km (850 hPa) in both MMFs and C3M observations, with cloud liquid water decreasing slightly more rapidly with height in SPCAM-IPHOC (Figure 10). This result is reasonable because of the rapid decrease of water vapor with height and the lower freezing heights in midlatitudes than in the tropics. On the other hand, the maximum cloud liquid water extends into much higher altitudes in CAM3.5 than in either MMF. This implies that large-scale condensation may not be the dominant processes for producing these clouds in CAM3.5, but they are produced by condensation on CRM scales in MMFs.

[35] An analysis of cumulus mass flux shows that the middle- and high-level clouds and liquid water from CAM3.5 (not shown) are mainly produced by the deep convective parameterization since the mass flux produced from the Zhang and McFarlane [1995] deep convective scheme is one order of magnitude larger than the mass flux diagnosed from the two MMFs (not shown). The cumulus detrainment is the main source for producing these clouds in CAM3.5. This explains why the maximum liquid water extends into higher altitudes in CAM3.5 than in either MMF.

[36] The SPCAM-IPHOC has the same global distribution of the annual mean precipitation as SPCAM (Figure 11). There are only some minor improvements. For example, the precipitation along the ITCZ in the central and eastern Pacific is too weak in both the SPCAM and CAM3.5, while it is relatively strong in SPCAM-IPHOC. This may be related to the slightly stronger Hadley circulation, which also improves the simulation of low-level clouds in their subsidence branch, e.g., the large liquid water content as shown in Figure 2.

Figure 11.

Global distribution of annual mean surface precipitation rate from (a) CAM3.5, (b) SPCAM, (c) SPCAM-IPHOC, and (d) the Legates observation [Legates and Willmott, 1990].

3.4. Impact on Radiative Transfer and Global Energy Balance

[37] The radiative heating rates are calculated on the CRM grids for both MMFs with the same radiation transfer scheme as in CAM3.5. Partial cloudiness from IPHOC is used as an input parameter in the radiation transfer scheme for each CRM column of SPCAM-IPHOC. This results in a global mean top of the atmosphere (TOA) albedo of 0.29, its spatial correlation with CERES of 0.9649, and its root mean square (rms) error of 0.0424 (Figure 12), which matches with CERES observations from 2000 to 2005. Note that the TOA albedo is calculated from global mean outgoing and incoming shortwave fluxes, not from the global mean of albedo. The TOA albedo from SPCAM and its RMS are slightly higher at 0.31 and 0.0522, respectively, while its spatial correlation with CERES (0.9501) is lower than SPCAM-IPHOC. Because there are fewer high-level clouds produced by the SPCAM-IPHOC from Indian ocean to western Pacific in the tropical region, the albedo is smaller in this region from SPCAM-IPHOC than from SPCAM, and compared relatively well with CERES. On the other hand, we can see an increase of albedo relative to SPCAM in the SE and NE Pacific, and SE Atlantic oceans due to the large amount of low-level clouds produced there by SPCAM-IPHOC.

Figure 12.

Global distribution of annual mean top of the atmosphere albedo from (a) CAM3.5, (b) SPCAM, (c) SPCAM-IPHOC, and (d) CERES. The global annual mean value, the root-mean-square (rms) error, and the spatial correlation between the model and the CERES observation are shown on the top of each plot.

[38] Cloud radiative effects are best illustrated by shortwave cloud radiative forcing (SWCRF) and longwave cloud radiative forcing (LWCRF), which is the difference between the clear-sky and total radiative fluxes at TOA. The global distributions of the 5 year averaged SWCRF and LWCRF from the CERES observational estimate data set and the three models are shown in Figures 13 and 14, respectively. The large negative SWCRF in the tropical convective regions and in North Pacific is greatly reduced in SPCAM-IPHOC, compared to SPCAM. As a result, the global mean SWCRF is smaller (−48.8 W m−2) in SPCAM-IPHOC and more comparable to the CERES observation (−46.6 W m−2) than in SPCAM (−54.5 W m−2) and CAM3.5 (−51.3 W m−2). Its RMS decreases by 4 W m−2, while its spatial correlation with CERES increases by 0.05 compared with SPCAM. On the other hand, the insufficiently negative SWCRF from CAM3.5 and SPCAM caused by the underestimates of the low cloud amount west of subtropical continents where stratocumulus clouds frequently occur are increased in magnitude in SPCAM-IPHOC.

Figure 13.

Global distribution of annual mean SWCRF (W m−2) from (a) CAM3.5, (b) SPCAM (c), SPCAM-IPHOC, and (d) CERES. The global annual mean value, the root-mean-square (rms) error, and the spatial correlation between the model and the CERES observation are shown on the top of each plot.

Figure 14.

Same as Figure 13 but for LWCRF (W m−2).

[39] The biases of LWCRF are also highly correlated with the representation of the cloud fields in the three models. The positive bias of LWCRF is well correlated with the overestimates of thick high clouds by SPCAM (Figure 14). On the other hand, the underestimate of the thick high-level cloud in the storm track region located at 60°N is largely responsible for the underestimate in the global mean LWCRF (23.9 W m−2) in SPCAM-IPHOC compared to CERES observations (29.5 W m−2). The LWCRF from the low-level cloud is relatively small, but can still be seen from the NE and SE Pacific and SE Atlantic oceans by comparing SPCAM and SPCAM-IPHOC. On the whole, the RMS of the LWCRF from SPCAM-IPHOC increases by 0.7 W m−2 compared with that of SPCAM, but the spatial correlation with CERES is almost unchanged.

[40] Of note, the SPCAM-IPHOC was not tuned to achieve the global TOA and surface energy balance. For the short 2 year simulation, an imbalance of −1.5 W m−2 at the surface (Table 2), and 6.16 W m−2 at TOA (Table 3) were produced. The amounts of imbalances are slightly larger than those in a similar MMF also not tuned for the radiation balance [Tao et al., 2009], but the imbalance is smaller than CAM3.5 and close to SPCAM.

Table 2. Energy Budget at Surface (W m−2)
Table 3. Energy Budget at TOA (W m−2)

4. Conclusions and Discussion

[41] In this study, we have presented preliminary results from a multiscale modeling framework model (MMF) with an advanced third-order turbulence closure (IPHOC) in its cloud-resolving model (CRM) component. In the original MMF or SPCAM, CAM3.5 is used as the host general circulation model (GCM), and the System for Atmospheric Modeling model with a first-order turbulence closure is used as the CRM for representing cloud physical processes in each grid column of the GCM. Despite the relatively coarse horizontal and vertical resolution used for both the GCM and embedded CRM, we have shown that the application of a TOC to the CRM component of MMF results in improved representation of the global distributions of low-level clouds and an increase in the amounts of low-level clouds in the subtropics.

[42] The improved simulation of low-level clouds is attributed not only to the representation of subgrid-scale condensation in the embedded CRM, but also closely related to the increased surface sensible and latent heat fluxes, the lower tropospheric stability (LTS) and stronger long wave radiative cooling. The subgrid-scale condensation scheme in SPCAM-IPHOC considers the SGS variabilities in thermodynamic and dynamic variables within a CRM grid, which are ignored in SPCAM, to determine whether or not a fraction of the grid can form a cloud. The global mean sensible and latent heat fluxes increase due to the efficient SGS transports parameterized by the IPHOC. The locations of the maximum centers of the sensible heat flux near the west coast of the continents where the low-levels clouds prevail are well reproduced by SPCAM-IPHOC. The increased surface sensible heat flux provides one of the energy sources of TKE for the production of low-level clouds. In SPCAM-IPHOC, the lower tropospheric stability (LTS) in the regions where low-level clouds prevail is higher than that of either SPCAM or CAM3.5. The higher LTS is strongly related to stronger radiative longwave cooling and large-scale subsidence.

[43] The vertical structures of the low-level cloud amount and liquid water content exhibit large differences among the three models for a vertical cross section representative of the stratocumulus to trade cumulus transition in the SE Pacific. The SPCAM-IPHOC simulation produced the largest overall cloud amount in the cumulus regions among the three models, but still underestimated the shallow and especially the stratocumulus clouds near the coast compared to C3M observations. On the other hand, the SPCAM simulation produced the smallest amount of low-level clouds in the cumulus regions and was unable to simulate the right amount of thin stratocumulus clouds near the coast. The coastal stratocumulus is, however, adequately simulated by CAM3.5 except for lower altitudes, due perhaps to an imposed relationship between cloud amount and LTS.

[44] More vigorous turbulent mixing at low levels produced by the IPHOC tends to decrease the CAPE thereby reducing the strength or vertical extent of deep convection. The large turbulence mixing produced by the IPHOC scheme may consume part of the CAPE needed for some parcels to reach high altitudes [Cheng and Xu, 2008]. The annual mean highest level that clouds can reach in the storm track region of the southern hemisphere from SPCAM-IPHOC is about 2 km less than from SPCAM. Because there are fewer clouds above 250 hPa, the global mean high-level clouds produced by SPCAM-IPHOC is less than in SPCAM. In the tropics, the shallower clouds sufficiently alleviate the excessive albedo produced by overestimated high-level clouds from SPCAM between the tropical Indian and western Pacific regions. The low-level clouds in SPCAM-IPHOC, on the other hand, contribute to the increase of albedo in the west coast of the continents, such as the SE Pacific and Atlantic oceans. As a result, the global mean albedo becomes more reasonable than that of SPCAM when compared with CERES observations. The negative SWCRF bias of SPCAM and CAM3.5 is reduced in SPCAM-IPHOC due to the more reasonable representation of the low- and high-level clouds. The regional positive LWCRF bias of SPCAM is also reduced, but become a large negative bias. The similar negative LWCRF bias occurred when a mass flux scheme was implemented in CAM3.5 [Park and Bretherton, 2009].

[45] To summarize the overall performance of SPCAM-IPHOC, the Taylor diagram [Taylor, 2001] is plotted for annual mean surface pressure, surface precipitation, SWCRF, LWCRF, latent heat surface flux, sensible heat surface flux, low-level clouds, middle-level clouds, and high-level clouds between 30°S and 30°N for the three models, reanalysis data and observations (Figure 15). The reference data are the ECMWF ERA40 [Uppala et al., 2005] surface pressure for the period of 1980–2001, the Legates and Willmott [1990] surface precipitation for the period of 1920–1980, CERES SWCRF and LWCRF for the period of 2000–2005, the JRA-25 (1979–2004) surface latent and sensible heat fluxes, and the CloudSat and CALIPSO (2007–2008) low-, middle-, and high-level clouds. Overall, the SPCAM-IPHOC outperforms both SPCAM and CAM3.5. Most noticeably, the low-level clouds from SPCAM-IPHOC are slightly more highly correlated with observations, but they have much more spatial variability (standard deviation) than those from the other two models and compare well with the observations. The large spatial variabilities of LWCRF and SWCRF from the other two models have also been decreased in SPCAM-IPHOC.

Figure 15.

Taylor diagram for annual mean surface pressure (PS), surface precipitation (PRECT) [Legates and Willmott, 1990], SWCRF, LWCRF, latent heat surface flux (LHFLX), sensible heat surface flux (SHFLX), low-level clouds (CLDLOW), middle-level clouds (CLDMED), and high-level clouds (CLDHGH) between 30°S and 30°N for the three models, reanalysis data, and observations. The reanalysis and observation are denoted by the REF point.

[46] There is still room to improve the SPCAM-IPHOC, such as the overestimation of low-level clouds in the storm track region of the Southern Hemisphere and the west coast of Africa, and underprediction of low-level cloud amount over most of the Pacific. Further improvement can be obtained from increased resolution in both directions of the host GCM, for example, using the T42 version of CAM3.5 and doubling the number of levels in the vertical direction for better resolving large-scale dynamics [Marchand and Ackerman, 2010]. Increasing vertical resolution will result in a more reasonable representation of entrainment, detrainment, and turbulence for stratocumulus clouds in the embedded CRM, especially near the inversion layer. The additional layers in the middle troposphere will also help to better resolve congestus clouds in the tropics, which are not simulated in MMFs, although improvement in cloud microphysics parameterization is also critically important.

[47] This study provides a starting point for exciting new research due to the fact that the majority of radiation and microphysics parameterizations currently used in CRMs and climate models neglect most SGS variations. For example, the information from the SGS probability density function can be used to produce more accurate and sophisticated radiative transfer calculation [Pincus and Evans, 2009], so that a better global radiative energy balance can be achieved. This may lead to a reduced bias in precipitation. This may also lead to improved long-term climate simulations, particularly when ocean-atmosphere coupling is implemented [Stan et al., 2010], because the interactions of physical processes on finer scales will be more realistic.

Appendix A

[48] The first-order moments of vertical velocity (w), liquid water potential temperature (θl), and total water (qt) are predicted by the host CRM (SAM). The following second- and third-order moments are forecasted by the IPHOC scheme:

equation image
equation image
equation image
equation image
equation image
equation image

where α can be θl or qt, τ is the dissipation time scale, and ∇z2 = ∂2/∂z2. The terms on the right-hand side of (A1)(A6) are the mean and turbulence transports, the shear and/or buoyancy productions, the diffusion, and the dissipation, respectively. Note that the pressure redistribution (terms associated with c5, c7, and c11) is usually proportional to the anisotropic part of the shear and buoyancy productions and has been parameterized based on work by Rotta [1951] and Launder [1975]. The equation image and equation image are simply parameterized using K diffusion, where K = 0.3lequation image, and l is the dissipation length scale. The set of constants ci and ki are as follows: c1 = 1.7, c2 = 1.04, c5 = 0.3, c6 = 2.8, c7 = 0.8, c8 = 2.73, c10 = 3.12, c11 = 0.3, k2 = 15 m2 s−1, and k3 = 25 m2 s−1. The constants for the dissipation of equation image (c1), equation image (c2) and equation image (c8), and c7 are the same as in work by Golaz et al. [2002a]. André et al. [1976] chose c5 = 1/3, compared with our c5 = 0.3. Our c11 is between 0.2 from André et al. [1976] and 0.4 from Bougeault [1981]. The constants for the dissipation of equation image and equation image are somewhat smaller than c6 = 4.85, and c10 = 3.9 from Bougeault [1981]. The diffusion coefficients are also smaller than k2 = 20 m2 s−1, and k3 = 30 m2 s−1 used by Golaz et al. [2002a].

[49] In (A1) to (A6), the buoyancy terms are parameterized similarly to those of Bougeault [1981] as equation image, where p0 is the reference pressure of 105 Pa, p is the pressure, Rd is the gas constant of dry air, Cp is the specific heat for moist air, L is the latent heat of condensation of water, equation image is the mean potential temperature, θv is the virtual potential temperature, and ql is the liquid water content. ϕ′ can be w′, θ′l, qt, or w2. The fourth-order moment terms can be calculated by

equation image
equation image

where subscripts 1 and 2 represent the first Gaussian and the second Gaussian pdf or plume, respectively. Here a is the magnitude of the first plume, σ is the standard deviation, and r is the within-Gaussian correlation. Any normalized variable is calculated by equation imagei = (χiequation image)/σχi, where χ can be w, θl, and qt, and i is either 1 or 2. Equations (A7) and (A8) can be derived by integrating the fourth-order moments over the double-Gaussian pdf. It is obvious that the fourth-order moments are closely related to the variance of each Gaussian and their correlation.


[50] This work has been supported by the NSF Science and Technology Center for Multiscale Modeling of Atmospheric Processes (CMMAP), managed by Colorado State University under cooperative agreement ATM-0425247. This work was also partially supported by NASA Modeling, Analysis and Prediction program managed by David Considine. The computation resources from NCAR BlueGene supercomputer were provided by the Teragrid organization. Special thanks go to Marat Khairoutdinov of Stony Brook University for providing SPCAM, Seiji Kato for providing the C3M data set, and Kirk Ayers and Zachary Eitzen of SSAI for reading drafts of this paper. Helpful discussions with David Randall of Colorado State University are appreciated. The editor and three anonymous reviewers are thanked for their constructive comments and suggestions.