Black carbon (BC) has an increased forcing per unit mass when it is located above reflective clouds. To explore sensitivity of forcing to aerosol vertical location, we used a column radiative transfer model to produce globally-averaged values of normalized direct radiative forcing (NDRF) for BC over and under different types of clouds. We developed a simple column-weighting scheme based on the mass fractions of BC that are over and under clouds in measured vertical profiles. The resulting NDRF is in good agreement with global 3-D model estimates, supporting the column-weighted model as a tool for exploring uncertainties due to diversity in vertical distribution. BC above low clouds accounts for about 20% of the global burden but 50% of the forcing. We estimate maximum-minimum spread in NDRF due to modeled profiles as about 40% and uncertainty as about 25%. Because models overestimate BC in the upper troposphere compared with measurements, modeled NDRF might need to be reduced by about 15%. Redistributing BC within the lowest 4 km of the atmosphere affects modeled NDRF by only about 5% and cannot account for very high forcing estimates.
 Absorption of solar radiation by aerosols with high absorption-to-backscatter ratios, such as black carbon (BC), results in a positive direct radiative forcing (RF) [Atwater, 1970]. BC is primarily emitted by incomplete combustion of carbon-based fuels and its positive forcing is estimated to be similar to that of methane [Jacobson, 2001]. Published direct forcing estimates range from +0.34 W/m2 [Forster et al., 2007] to +0.9 W/m2 [Ramanathan and Carmichael, 2008]. The causes of this large range need to be isolated so that measurement strategies can be designed to evaluate them.
 The presence of highly reflective clouds beneath absorbing aerosol layers dramatically enhances RF [Haywood and Shine, 1997]. Liao and Seinfeld  showed that a cloud layer embedded within a broad absorbing aerosol layer increased its overall RF, and that the RF increased with cloud thickness. Haywood and Ramaswamy  found that BC RF increased by 70% when it was raised to 5 kilometers (km) in a global model. Podgorny and Ramanathan  used INDOEX data with a Monte Carlo radiation model and concluded the location of a soot layer with respect to broken low-level clouds contributed significantly to radiative forcing. Chand et al.  recently demonstrated that spatial covariance between cloud cover and aerosol fields could increase regional warming from biomass burning aerosol by a factor of three.
 The foregoing studies indicate that the vertical distribution of BC relative to cloud layers significantly affects its forcing. Because both cloud and BC fields vary widely in magnitude and structure among global aerosol models [Koch et al., 2009], cloud-aerosol location likely contributes to variability in predicted global forcing.
 While global models constrained with observations are the only way to produce realistic aerosol forcing estimates, these complex models are often not susceptible to simple explorations of sensitivity. To estimate uncertainties in global average forcing by BC, we used a column radiative transfer model to derive forcing per mass in different situations. We evaluated this approach by comparing these results to those produced by global climate models (GCM). Finally, we used this model to examine the sensitivity of forcing to plausible BC vertical profiles.
2. Column Model Results
 We used a 1-D, medium spectral resolution radiative transfer model (Streamer [Key and Schweiger, 1998]) to estimate the magnitude of column forcing of BC above and below different cloud types. We calculated forcing per mass of BC for nine conditions, summarized in Table 1. One was a cloudless column, and the other eight contained BC either above or below four types of cloud layers. The four cloud cases considered were: low (LC, cloud top height <2 km), middle (MC, 2–8 km), high cirrus (HC, >8 km), and deep convective (DC, >8 km). Columns were defined as downward-looking, so cloud type and height was defined by the highest cloud in the column. Surface albedo was set to 0.153 [Hummel and Reck, 1979] and solar zenith angle was set to a global average value of 60°. BC optical properties at each model wavelength were taken from OPAC [Hess et al., 1998]. Mass extinction cross-section (MEC) was 9.26 m2/g and single-scatter albedo was 0.21 at 550 nm. While these optical properties are not the latest recommendation and exclude enhanced absorption by mixed BC particles, they are used by many of the models reporting forcing values. We used identical optical properties so we could evaluate comparability between global models and the simple model. It is generally accepted that models neglecting internal mixing tend to underestimate absorption and forcing [Jacobson, 2001] so we emphasize that our results will inherit the same bias. Cloud effective radii (re) and liquid water content (LWC) were taken from Hess et al.  (low and cirrus), Prodi et al.  (middle), and Rosenfeld and Lensky  (convective). All clouds were assumed to be composed of liquid droplets except cirrus clouds which were composed of hexagonal ice columns. Average optical depths (τ) for all four cloud types were provided by Rossow and Schiffer . In all cloud cases, BC with an aerosol optical depth (AOD) of 0.05 at 550 nm was distributed in a layer approximately 1 km completely above or below the cloud. Fluxes were calculated in Streamer's 24 shortwave and 105 longwave bands. Forcing divided by column burden gave normalized direct radiative forcing (NDRF) summarized in Table 1.
Table 1. Cloud Properties and NDRF for Unmixed BC in Various Types of Cloud Columns and Location Relative to Cloud Layers
 BC NDRF was significantly greater for aerosol located above a cloud layer. For all cloud types, NDRF increased by a factor of 2 to 5 relative to the clear sky (CS) forcing. BC below clouds had reduced forcing; for LC, MC, and DC cases, NDRF was less than one-quarter of the clear sky value. However, BC below optically thinner HC (cirrus) still had a forcing effect about half of that of the clear sky case because of shortwave transmittance.
 We also investigated the assumption that forcing does not depend on height of aerosol above cloud by varying the height of the aerosol layer. Over low and middle clouds, forcing changed by approximately 1–3% through the heights where BC burden is largest (up to 600 millibars (mb) or approximately 4 km). These small differences in forcing likely result from changes in atmospheric transmission above the aerosol layer that alter the radiation incident on the BC [Haywood and Ramaswamy, 1998]. Thus, the underlying cloud properties are orders of magnitude more crucial to NDRF than the aerosol's location relative to the cloud, as long as BC is above the top of the cloud layer. We also ran simulations in each sky coverage case with varying mass absorption cross-section (MAC) and AOD. Forcing scales nearly linearly over the range of values estimated by current models and observed in field data (an order of magnitude centered on our estimates).
 Global average forcing (NDRFavg) may be treated as a weighted sum of the cases in Table 1 (hereafter referred to as the “weighted-column model”):
where NDRFb,i and NDRFa,i are the case-dependent values in Table 1, fsky,i is the sky coverage fraction for clear sky and each cloud type, and γi is the fraction of total column BC that is above cloud top for each type of cloud (zero for clear-sky). To maintain continuity for the remainder of this paper, the estimates of NDRFb,i and NDRFa,i are the same as in Table 1, while fsky and γi will be specific to a given analysis.
 For a first-order estimate of the fractions (fsky), we used satellite-derived sky coverage data from the International Satellite Cloud Climatology Project's (ISCCP) D2 dataset [Rossow et al., 1996]. Our global average consists of five area fractions representing of the column types in Table 1. We also assumed that a single global-average vertical profile could be used to determine γi (the fraction of the burden above cloud top) in each case. This profile (Table S1 and Figure S1 of the auxiliary material) was derived by latitudinally weighting vertical profiles from field data (summarized by Koch et al. ) and the cloud top height for each type of cloud. The profile is similar to those of Corrigan et al. .
 This weighted-column model produces an NDRFavg of 1140 W/g (the “base case”). Figure 1 summarizes the division of burden and forcing for each type of column and aerosol location. Most of the mass was in clear sky columns, but these contribute only 28% of the globally-averaged NDRF. BC above low clouds (e.g., stratus and low cumulus) made the greatest contribution to forcing. Only 21% of the global burden contributed over half of the overall forcing. Underlying cloud was also important for BC above mid-level clouds where 3% of the global BC burden added 8% to overall NDRF. As in Table 1, aerosol above optically deep convective clouds had the highest NDRF; however, the combination of low sky coverage fraction and low BC concentrations at cloud-top altitude resulted in only a 0.1% forcing contribution. Forcing beneath high cirrus clouds represented about 7% of the globally-averaged NDRF value, so BC below optically thin clouds still played a role in forcing, as opposed to BC below all other types of clouds, for which 26% of the burden contributed only 5% of the forcing.
 We tested whether average NDRF could be approximated with globally averaged values of surface albedo and solar zenith angle. In the column model of Chylek and Wong , the expected value of forcing is proportional to the expected value of surface (or underlying) albedo as long as the absorption is much greater than backscattering and albedo is independent of absorption optical depth. We also calculated column forcing for a range of solar zenith angles from 0–85°. The ratio between NDRF for any two globally-averaged column types was within 10% of the ratio predicted by NDRF at 60°. Thus, the weighted-column model appears suitable for exploring relative sensitivities in NDRF.
3. Comparison of Column Model and Three-Dimensional Model Results
 Our estimate of 1140 W/g is in good agreement with previously published estimates for NDRF in 3-D GCMs, calculated by dividing global average forcing by total BC burden. The range from the AeroCom study is 630–2100 W/g, with a median of 1230 W/g [Schulz et al., 2006].
 We used output from the Community Atmosphere Model (CAM version 3.5.07, 1.9° × 2.5° finite-volume [Collins et al., 2004]) to further evaluate the comparison between column-model forcing and forcing predicted by 3-D models. We note that CAM significantly underestimated total cloud cover (47.7%) relative to ISCCP data (68.7%). Underestimates were especially large for mid and low-level clouds. Because of this discrepancy we compared different model-averaging treatments but did not rely on the absolute values.
 The first averaging treatment of interest is a common calculation of NDRF: CAM average forcing divided by the average global burden, or 1210 W/g. A second treatment uses clouds and vertical profiles from one year of monthly-average CAM output in the weighted-column model, along with NDRF from Table 1. The cloud fractions from CAM were used to derive fsky,i in equation (1). Values of γi came from integrated mass fractions of BC above high (cloud ptop < 400 mb), medium (ptop > 700 mb and < = 400 mb), and low (ptop = > 700 mb) cloud fractions in each gridbox. Since NDRF is used, there is no need to consider individual column burdens in the model, only the structure of the vertical profile. The calculated forcing was 1290 W/g. One cause of the small (6%) discrepancy between the weighted-column method and the ratio of the global values could be the difference in cloud optical properties.
 We also explored the possibility that global averages of cloud fractions and vertical profiles are insufficient to reproduce forcing, and covariance between aerosol and cloud must be considered. We calculated weighted-column forcing using the cloud and the vertical profile in each gridbox, along with the NDRF in Table 1, and area-weighted the resulting NDRF values to obtain a global average of 1245 W/g. The difference of less than 4% from the estimate using global averages indicated that horizontal covariance between clouds and aerosol has little effect on average forcing, and that use of the simple weighted-column model was a reasonable way to explore diversity in forcing estimates on large scales. This finding does not contradict the results of Chand et al. , who found that covariance on sub-grid scales did matter; their spatial and temporal scales were much smaller than those examined here.
 CAM overestimated the BC burden above clouds relative to measured profiles. This phenomenon is consistent with other AeroCom models, which tend to overpredict BC concentrations in the mid-to-upper troposphere [Koch et al., 2009], and therefore overpredict above-cloud fractions. This high-altitude BC leads to NDRF values that are too high, but the bias is offset by CAM's underestimate of total cloud cover.
4. Sensitivity to Changes in Profile
 Because the weighted column model produced reasonable results, we used it to explore the sensitivity of forcing to BC vertical distribution. Table 2 summarizes the experiments described here as well as those in the preceding section. For most of these experiments, we assumed the ISCCP cloud distribution. We then explored how vertical location affected NDRF by altering profiles in the weighted-column model.
Table 2. Summary of Weighted-Column Model BC NDRFs for the Combinations of Vertical Profile and Cloud Inputs Explored in This Studya
Wtd-Column NDRF (W/g)
Diff From Base case
These NDRF values were determined for unmixed aerosol, so relative values are more important than absolute values.
Below all clouds
Above all clouds
Simple Profile With Covariance
 Extreme profiles: If all BC was located above 250 mb (10–12 km), forcing would be about 2250 W/g. Conversely, if all BC was compressed below 850 mb, the value would be 525 W/g. The range of 525–2250 W/g is an absolute limit on NDRF uncertainty due to vertical distribution. Observational data does not support such extreme profiles.
 Upper-bound profiles: We estimated more realistic, but still uncommon, boundaries using monthly CAM concentration fields. We selected two profiles reflecting extremes found over small areas, which we will call highly-lofted and near-surface. These profiles represent upper and lower limits on vertical distributions given the dynamical characteristics of the model. For the highly-lofted column burden, 95% of the BC burden resided above 825 mb, 50% above 500 mb, and 5% above 320 mb. A near-surface BC profile had only 15% of BC above 825 mb and none above 500 mb. The highly-lofted and near-surface simulations resulted in NDRFs of 1490 and 630 W/g, respectively (+31 and −45% compared with the base case).
 Model diversity: We explored how differences in modeled vertical profiles might contribute to variance in model forcing. We compiled global profiles from concentration fields archived under the AeroCom initiative for four models in addition to CAM (Table S2). These were MPI_HAM [Stier et al., 2005], UMI [Liu et al., 2005], SPRINTARS [Takemura et al., 2002], and LSCE [Textor et al., 2006]. Table 2 summarizes the results. MPI_HAM and LSCE were assumed to bound calculated NDRF from the AeroCom initiative since they represent the nearest-surface (least BC above clouds) and most lofted (most BC above clouds) profiles, respectively [Koch et al., 2009]. UMI and SPRINTARS typified profiles closer to the average. The two bounding models show a total spread in NDRF of about 40%, with LSCE 26% higher than our base case and MPI 11% lower. These differences are caused by globally-averaged vertical distribution alone; individual model features not considered here, such as cloud fields and aerosol optical properties, may also account for large variations in NDRF. A better understanding of diversity in modeled vertical profiles and clouds will likely be a topic of further research [Schulz et al., 2006].
 Relative sensitivities: Table 3 summarizes sensitivities of global-average NDRF to aerosol location. Recall that the absolute value of aerosol absorption used in our study is too low because we ignore aerosol mixing, so relative sensitivities are more robust. The upper portion of Table 3 shows changes in NDRF as BC is moved to a different cloud-column type (changes in fsky). For instance, if 10% of the global burden is moved from columns topped by high-cirrus (HC) to columns with low cloud (LC) only, global average NDRF increases by 15%. The greatest forcing increase comes from moving aerosol to low-cloud columns from any other column.
Table 3. Sensitivities of Global-Average BC NDRF to Changes in Cloud-Aerosol Locationa
The top portion of the table reflects a move of 10% burden between column types. Percentages do not sum to one because a clear-sky column is not given. The lower portion shows change in global-average NDRF if 10% of BC column burden is moved from below to above cloud in the same type of column. Fraction of above-cloud aerosol assumed in the base case is bolded for reference.
Cloud-Aerosol Horizontal Collocation
Base case fraction
10% to CS
10% to LC
10% to MC
10% to HC
Aerosol Vertical Location
Base case above
10% below to above
 The lower portion of Table 3 tests moving BC from below to above cloud in the same sky coverage column (changes in γi). For example, if 10% of the aerosol in low cloud (LC) columns is moved above the cloud, forcing would increase by 6%. Moving aerosol from below to above cloud results in changes of only a few percent.
 In summary, exploration with the column-weighted model for global forcing yielded the following ranges for NDRF: −55% to 100% for BC at the upper and lower boundaries of the troposphere, which is unrealistic; −45% to 30% for extreme profiles found in a single model (CAM); and −10% to 25% for variability among models.
5. Sensitivity Using Global Burden Distributions
 Most BC atmospheric burden lies near the surface, and the column model indicated that between 75–80% of BC forcing comes either from clear-sky conditions or BC above low clouds. Therefore, low cloud fraction and the structure of the BC vertical profile within the first 3–5 km of the atmosphere play a crucial role in its overall forcing. We suggested earlier that cloud-aerosol covariance at macroscopic scales (i.e., among model grid boxes) did not produce large changes in aerosol forcing. We again consider this horizontal covariance using CAM burdens and cloud fields as we examine the role of dynamics in the lower troposphere. Again, the cloud fields in this version of CAM are poor, so we compare only relative changes caused by altering vertical distribution. Holding the CAM-derived column burden in each gridbox constant, we made different vertical profile assumptions, calculated the partitioning into column types in each gridbox, and then used the weighted-column model to obtain an average global forcing. This is the method discussed earlier in testing BC/cloud covariance that produced a forcing of 1245 W/g for CAM.
 Constant Mixing (ConstMix): We investigated the vertical profile used by Ramanathan and Carmichael , a study that produced one of the highest published forcing estimates for BC (+0.9 W/m2). Our purpose was to determine whether the vertical profile assumed in that study was responsible for a significant fraction of the high forcing. That study assumed a constant BC mixing ratio from the surface to 3.4 km in the tropics and 2.0 km elsewhere, with an exponential decay above the constant layer [Chung et al., 2005].
 Low-Level CAM (LLCAM): We also examined another profile that maintains the relative vertical profile produced by CAM below 650 mb, but removes BC burden above that level. This profile has similar vertical extent to ConstMix, but preserves the shape of the modeled profile through 4 km altitude.
 Both ConstMix and LLCAM profiles are closer to field campaign data than the unaltered CAM. The ConstMix case has a mass concentration peak about 50 mb higher than the LLCAM. This peak results in a higher global burden fraction above low-level clouds (15% for ConstMix versus 13% for LLCAM). See Table S1 and Figure S1.
 Average forcing is 1105 W/g for the ConstMix case, and 1060 W/g for the LLCAM case. Considering only forcing of BC below 4 km, the ConstMix assumption used by Ramanathan and Carmichael  could result in a small increase in NDRF (5%) relative to the low-level dynamics in CAM, because there is slightly more BC above low clouds.
 However, the vertical profile predicted by CAM – and by most other global models, according to Koch et al. —places more BC at high altitudes (above 5 km) than the ConstMix case. As shown previously, the same calculation with the unmodified CAM profile results in 1245 W/g, a 15% increase above both low-level profiles explored here. This is approximately the bias in forcing caused by overestimating BC in the upper troposphere.
 This study explored the contribution to global-average NDRF from enhanced forcing by BC lofted above clouds. Even small changes in BC distribution at cloud interfaces may result in a 5–10% change in global forcing. Profile diversity among current models results in a 40% maximum-to-minimum spread in BC NDRF. Maximum NDRF comes from profiles with large BC burdens in the upper troposphere, in disagreement with observations. NDRF in individual model columns may vary by about a factor of two from smallest to largest, but these are extremely localized cases. Unless average vertical profiles differ significantly from those observed to date, an upper bound in global-average NDRF uncertainty due to vertical distribution is probably closer to 25%.
 While the vertical profile chosen by Ramanathan and Carmichael  leads to a small increased forcing in the lower troposphere through elevated BC loading at low cloud tops, it cannot explain why their estimate of RF (+0.9 W/m2) is more than double that of most global models. Based on the results of this study, their NDRF should actually be lower than that of most global models, which overpredict high-level BC concentrations.
 Model diversity in vertical distribution and cloud fields influences estimated direct forcing values for BC. In this study, we used a simple model to explore sensitivities and showed that this effect is not unbounded. To refine these broad assumptions and reduce the uncertainty in NDRF below our estimate of 25%, a rigorous analysis using 3-D cloud and aerosol fields from multiple GCMs, as well as in-situ and remote sensing of cloud and aerosol vertical and horizontal locations is required.
 This research was funded by the U.S. EPA Climate Office through a subcontract to RTI, Intl. by NASA under grant NNG04GL91G, and by the National Science Foundation under ATM0852775. Model runs were accomplished at the National Center for Atmospheric Research, sponsored by NSF. We thank Joyce Penner, Xiaohong Liu, Philip Stier, Michael Schulz, and Toshi Takemura for use of their model data.