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  • View in gallery
    Fig. 1.

    Snapshots of LES clouds, showing liquid water specific humidity (gray to white, low to high) and ice water specific humidity (red to white, low to high). Subplots to the right show vertical profiles of cloud fraction for each case. The thick line shows the profile for the specific snapshot in the 3D rendering, the thin lines show all other snapshots, and the shading shows the range. (a),(b) Shallow convective clouds. (c) Stratocumulus clouds. (d) Deep convective clouds. Note that the domain sizes vary between the cases.

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    Fig. 2.

    Bias (ICA-3D) in (a) TOA reflected flux and (b) albedo as a function of zenith angle for ShCu (BOMEX and RICO), Sc (DYCOMS-II RF01), and Cb (TRMM-LBA and TRMM-LBA agg.). For each cloud type, average fluxes (with shaded 1σ error bars) are computed over the individual snapshots. Positive bias means the ICA approximation is reflecting more incoming flux than in the 3D RT calculation.

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    Fig. 3.

    Mean TOA reflected flux bias across all solar zenith angles computed for different resolutions of the same cloud fields. The horizontal axis shows the horizontal resolution; the vertical resolution is kept fixed. The four cases of ShCu, Sc, and Cb are shown in the same colors as Fig. 2. For each case, three snapshots from the original ensemble are used and the spread is shown by the shading.

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    Fig. 4.

    Surface irradiance bias (ICA-3D) as function of zenith angle for ShCu (BOMEX and RICO), Sc (DYCOMS-II RF01), and Cb (TRMM-LBA). (a) The total surface irradiance bias is split into the (b) direct and (c) diffuse components, which largely compensate each other, especially at larger zenith angles. For each cloud type, average fluxes (with shaded 1σ error bars) are computed over the individual snapshots. Positive bias means the ICA approximation has more downwelling radiation at the surface than the 3D calculation.

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    Fig. 5.

    Bias (ICA-3D) in surface irradiance as predicted by bias in TOA reflected flux via Eq. (5) compared to the computed surface irradiance bias. Cloud type and zenith angle are indicated by marker shape and color, respectively. The 1:1 line is shown for reference.

  • View in gallery
    Fig. 6.

    Daily mean bias (ICA-3D) as a function of latitude and day of year assuming the globe is covered by (a)–(d) ShCu (BOMEX and RICO), (e),(f) Sc (DYCOMS-II RF01), (g),(h) Cb (TRMM-LBA), and (i),(j) more aggregated Cb (TRMM-LBA agg.). (left) Flux bias; (right) albedo bias. Note the color scales vary between LES cases. Inset panels on the left show annual average biases with shaded error bars that denote the spread across the LES ensembles as described in the text. Only latitudes from 60°S to 60°N are shown because the simulated clouds are not representative of the high-latitude regions.

  • View in gallery
    Fig. 7.

    Scatterplot of in-cloud cloud water path (CWP) from LES domain against flux bias at zenith angles (a) 20°, (b) 40°, (c) 60°, and (d) 80°. LES ensemble members are plotted with the same color convention as in Fig. 2. The gray lines show the regression fit constrained to go through the origin. The gray shaded areas show the 68% and 95% confidence intervals. The rms error of the regression is indicated at the top of each panel.

  • View in gallery
    Fig. 8.

    Predicted albedo bias from Eq. (6) compared to the computed albedo bias for convective cloud scenes (BOMEX, RICO, and TRMM-LBA) with different surface albedos. The predicted surface albedo is calculated from the simulations using a surface albedo of αo = 0.06 corresponding to an ocean surface. Each point represents the albedo bias at integer solar zenith angle from 0° to 90° for five ensemble members of each LES case. The colors denote the different cases and the symbol shapes denote the surface albedo. The 1:1 line is shown for reference.

  • View in gallery
    Fig. 9.

    (right) Map of annual-mean flux bias inferred from ISCCP in-cloud CWP. (left) Zonally averaged flux bias in the black line and 1σ error bars in the gray shading that are derived from the linear regression in Fig. 7. As in Fig. 6, we show only 60°S–60°N because the clouds we have modeled are not representative of high latitudes.

  • View in gallery
    Fig. A1.

    Normalized histogram of total precipitable water from the TRMM-LBA simulations in a 20-km domain vs 40-km domain, which we use as a less and more aggregated case of deep convection. The variance across the ensemble, shown by the width of the histogram, is representative of the degree of convective aggregation.

  • View in gallery
    Fig. B1.

    TOA reflected bias across zenith angles for different ice parameterizations in one TRMM-LBA cloud snapshot.

  • View in gallery
    Fig. C1.

    Regression of flux bias on several different cloud properties (a) in-cloud cloud water path (CWP), (b) cloud top height, (c) cloud cover, and (d) cc(1cc). All panels show the flux bias for a zenith angle of 40°. The plot in (a) is the same as Fig. 7b. The rms error is shown for each panel and is smallest for the in-cloud CWP case (a).

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Top-of-Atmosphere Albedo Bias from Neglecting Three-Dimensional Cloud Radiative Effects

Clare E. SingeraDepartment of Environmental Science and Engineering, California Institute of Technology, Pasadena, California

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Ignacio Lopez-GomezaDepartment of Environmental Science and Engineering, California Institute of Technology, Pasadena, California

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Xiyue ZhangaDepartment of Environmental Science and Engineering, California Institute of Technology, Pasadena, California

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Tapio SchneideraDepartment of Environmental Science and Engineering, California Institute of Technology, Pasadena, California
bJet Propulsion Laboratory, California Institute of Technology, Pasadena, California

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Abstract

Clouds cover on average nearly 70% of Earth’s surface and regulate the global albedo. The magnitude of the shortwave reflection by clouds depends on their location, optical properties, and three-dimensional (3D) structure. Due to computational limitations, Earth system models are unable to perform 3D radiative transfer calculations. Instead they make assumptions, including the independent column approximation (ICA), that neglect effects of 3D cloud morphology on albedo. We show how the resulting radiative flux bias (ICA-3D) depends on cloud morphology and solar zenith angle. We use high-resolution (20–100-m horizontal resolution) large-eddy simulations to produce realistic 3D cloud fields covering three dominant regimes of low-latitude clouds: shallow cumulus, marine stratocumulus, and deep convective cumulonimbus. A Monte Carlo code is used to run 3D and ICA broadband radiative transfer calculations; we calculate the top-of-atmosphere (TOA) reflected flux and surface irradiance biases as functions of solar zenith angle for these three cloud regimes. Finally, we use satellite observations of cloud water path (CWP) climatology, and the robust correlation between CWP and TOA flux bias in our LES sample, to roughly estimate the impact of neglecting 3D cloud radiative effects on a global scale. We find that the flux bias is largest at small zenith angles and for deeper clouds, while the albedo bias is most prominent for large zenith angles. In the tropics, the annual-mean shortwave radiative flux bias is estimated to be 3.1 ± 1.6 W m−2, reaching as much as 6.5 W m−2 locally.

Significance Statement

Clouds cool Earth by reflecting sunlight back to space. The amount of reflection is determined by their location, details of their 3D structure, and the droplets or ice crystals they are composed of. Global models cannot simulate the 3D structure of clouds because computational power is limited, so they approximate that clouds only scatter sunlight in a 1D vertical column. In this study, we use local models to directly simulate how clouds scatter sunlight in 3D and compare with a 1D approximation. We find the largest bias for overhead sun and for deeper clouds. Using satellite observations of bulk cloud properties, we estimate the tropical annual-mean bias introduced by the 1D approximation to be 3.1 ± 1.6 W m−2.

Zhang’s current affiliation: Department of Earth and Planetary Sciences, The Johns Hopkins University, Baltimore, Maryland.

© 2021 American Meteorological Society. For information regarding reuse of this content and general copyright information, consult the AMS Copyright Policy (www.ametsoc.org/PUBSReuseLicenses).

Corresponding author: Clare E. Singer, csinger@caltech.edu

Abstract

Clouds cover on average nearly 70% of Earth’s surface and regulate the global albedo. The magnitude of the shortwave reflection by clouds depends on their location, optical properties, and three-dimensional (3D) structure. Due to computational limitations, Earth system models are unable to perform 3D radiative transfer calculations. Instead they make assumptions, including the independent column approximation (ICA), that neglect effects of 3D cloud morphology on albedo. We show how the resulting radiative flux bias (ICA-3D) depends on cloud morphology and solar zenith angle. We use high-resolution (20–100-m horizontal resolution) large-eddy simulations to produce realistic 3D cloud fields covering three dominant regimes of low-latitude clouds: shallow cumulus, marine stratocumulus, and deep convective cumulonimbus. A Monte Carlo code is used to run 3D and ICA broadband radiative transfer calculations; we calculate the top-of-atmosphere (TOA) reflected flux and surface irradiance biases as functions of solar zenith angle for these three cloud regimes. Finally, we use satellite observations of cloud water path (CWP) climatology, and the robust correlation between CWP and TOA flux bias in our LES sample, to roughly estimate the impact of neglecting 3D cloud radiative effects on a global scale. We find that the flux bias is largest at small zenith angles and for deeper clouds, while the albedo bias is most prominent for large zenith angles. In the tropics, the annual-mean shortwave radiative flux bias is estimated to be 3.1 ± 1.6 W m−2, reaching as much as 6.5 W m−2 locally.

Significance Statement

Clouds cool Earth by reflecting sunlight back to space. The amount of reflection is determined by their location, details of their 3D structure, and the droplets or ice crystals they are composed of. Global models cannot simulate the 3D structure of clouds because computational power is limited, so they approximate that clouds only scatter sunlight in a 1D vertical column. In this study, we use local models to directly simulate how clouds scatter sunlight in 3D and compare with a 1D approximation. We find the largest bias for overhead sun and for deeper clouds. Using satellite observations of bulk cloud properties, we estimate the tropical annual-mean bias introduced by the 1D approximation to be 3.1 ± 1.6 W m−2.

Zhang’s current affiliation: Department of Earth and Planetary Sciences, The Johns Hopkins University, Baltimore, Maryland.

© 2021 American Meteorological Society. For information regarding reuse of this content and general copyright information, consult the AMS Copyright Policy (www.ametsoc.org/PUBSReuseLicenses).

Corresponding author: Clare E. Singer, csinger@caltech.edu

1. Introduction

Earth’s average albedo is roughly 29%, with clouds accounting for about half of the solar radiative energy fluxes reflected back to space (Stephens et al. 2015). Accurately simulating clouds is crucial for modeling Earth’s albedo. However, Earth system models (ESMs) struggle to accurately represent the albedo’s magnitude, spatial patterns, and seasonal variability (Bender et al. 2006; Voigt et al. 2013; Engström et al. 2015). Simulating clouds is difficult for several reasons, but one major factor is their wide range of spatial scales. Clouds have complex three-dimensional (3D) morphologies created by turbulent motions at length scales down to tens of meters or smaller. However, the typical resolution of an ESM is around only 10–100 km in the horizontal and 100–200 m in the vertical in the lower troposphere (Schneider et al. 2017). This discrepancy means that clouds are not explicitly resolved in ESMs. Instead, they are represented by parameterizations and, for purposes of radiative transfer (RT) calculations, are approximated as broken plane-parallel structures within grid cells (Marshak and Davis 2005).

The plane-parallel approximation (PPA) leads to important biases in RT calculations (Cahalan and Wiscombe 1992). Over the past 20 years, RT solvers have made significant progress in reducing some of these biases, either by making use of semiempirical deterministic parameterizations of cloud heterogeneity (Shonk and Hogan 2008) or through stochastic sampling of plane-parallel cloudy columns based on assumed distributions and characteristics of cloud structural properties (Pincus et al. 2003). These approximate solvers are likely to become even more accurate in the future, as dynamical parameterizations provide increasingly detailed cloud statistics (e.g., Cohen et al. 2020). Moreover, the PPA bias may be reduced in ESMs by using embedded cloud-resolving models (Cole et al. 2005b; Kooperman et al. 2016), albeit at great additional computational expense, in an approach known as cloud superparameterization (Khairoutdinov and Randall 2001).

This progress has led to a renewed interest in another source of bias that was, until recently, overshadowed by biases due to the PPA: the treatment of horizontal radiative fluxes in ESMs (Cahalan et al. 1994; Schäfer et al. 2016; Hogan et al. 2019). ESMs make the independent column approximation (ICA) when performing RT calculations. This approximation neglects horizontal radiative fluxes, decoupling the RT calculation between atmospheric columns to make the problem computationally tractable. Three-dimensional radiative transfer will remain too expensive to run in ESMs in the foreseeable future, making the ICA a necessary simplification (Hogan and Bozzo 2018). For this reason, it is important to quantify and document biases due to the ICA.

In this context, the effect of cloud structure on horizontal radiative transfer has gained attention, enabled by advances in computation that make 3D RT feasible at high spectral resolution (Mayer and Kylling 2005; Emde et al. 2016; Villefranque et al. 2019; Gristey et al. 2020; Veerman et al. 2020). The structural differences between ICA and a full 3D RT calculation have been documented before (Marshak et al. 1995b; O’Hirok and Gautier 1998, 2005; Barker et al. 2003, 2012), and many alternatives to ICA have been proposed to minimize their mismatch (e.g., Marshak et al. 1995a; Várnai and Davies 1999; Frame et al. 2009; Hogan and Shonk 2013; Wissmeier et al. 2013; Okata et al. 2017; Oreopoulos and Barker 1999; Klinger and Mayer 2016, 2020; Hogan et al. 2019).

Nevertheless, most studies have been focused on theoretical cases, small spatial and temporal domains, or improving satellite retrieval algorithms. Some notable exceptions are Cole et al. (2005a), who calculate the ICA bias from two-dimensional (2D) RT in a superparameterized cloud resolving model at 4-km horizontal resolution, and Barker et al. (2015, 2016), who calculate the ICA bias using 2D cloud fields retrieved from CloudSat and CALIPSO.

Here we discuss the magnitude of the bias that results from neglecting the 3D cloud radiative effects by making the ICA. We use large-eddy simulations (LES) to generate 3D cloud fields representing three canonical cloud regimes: shallow cumulus convection, stratocumulus, and deep convection. These cloud regimes are representative of the clouds typically found in the tropics. Previous studies that quantify 3D cloud radiative effects globally have used 2D cloud fields retrieved from satellites or superparameterized models, or inferred 3D fields using some stochastic generator (O’Hirok and Gautier 1998, 2005; Barker et al. 2015, 2016). These approaches can better represent the spatial distribution of cloud types but are restricted to the coarse resolution of satellite footprints or rely on assumptions to generate 3D fields. We instead use high-fidelity models to generate realistic 3D cloud fields at very high resolution, sacrificing some ability to generalize beyond the tropics from our limited number of LES cases. All of these methods present different challenges, either in generalizing to global scales, or in representing the details of small scales, but the simplifications are necessary because 3D cloud retrievals from satellite are not yet available. However, recent progress in stereoscopic observations is bringing us closer to having global high-resolution observations of 3D cloud structure (e.g., Romps and Öktem 2018; Castro et al. 2020).

We calculate the bias between the true reflected flux and the flux approximated by ICA using a Monte Carlo RT code. The shortwave radiative flux bias is shown to vary with solar zenith angle and cloud type. Because the solar zenith angle varies with the diurnal and seasonal cycle, we quantify the effect of the 3D bias on these time scales. Finally, using global satellite observations of cloud climatology, we estimate the spatiotemporal bias that would result in global models that resolve clouds but still make the ICA. As stated earlier, most ESMs make the ICA and use some cloud heterogeneity parameterization to reduce the PPA bias, so the bias associated with only the ICA is an underestimate of the total bias. Because of the diversity of assumptions made by global models to account for phenomena such as cloud overlap, and the fundamental resolution dependence of cloud heterogeneity emulators, in this study we focus on the bias resulting from RT using only the ICA on fully resolved 3D cloud structures from LES.

2. Methods

a. Large-eddy simulations of clouds

We generate three-dimensional cloud fields from high-resolution LES using the anelastic solver PyCLES (Python Cloud Large Eddy Simulation; Pressel et al. 2015, 2017). The LES are run in three dynamical regimes to simulate shallow cumulus (ShCu), stratocumulus (Sc), and deep-convective cumulonimbus clouds (Cb). Figure 1 shows volume renderings of each cloud regime alongside profiles of cloud fraction; key properties of the different cloud regime simulations can be found in Table 1 with more details in appendix A. In general, LES are capable of reproducing observed cloud characteristics (e.g., cloud cover, liquid water path, cloud depth) including aspects of 3D structures (Griewank et al. 2020).

Fig. 1.
Fig. 1.

Snapshots of LES clouds, showing liquid water specific humidity (gray to white, low to high) and ice water specific humidity (red to white, low to high). Subplots to the right show vertical profiles of cloud fraction for each case. The thick line shows the profile for the specific snapshot in the 3D rendering, the thin lines show all other snapshots, and the shading shows the range. (a),(b) Shallow convective clouds. (c) Stratocumulus clouds. (d) Deep convective clouds. Note that the domain sizes vary between the cases.

Citation: Journal of the Atmospheric Sciences 78, 12; 10.1175/JAS-D-21-0032.1

Table 1.

LES case properties: name, type of cloud (shallow cumulus: ShCu; stratocumulus: Sc; deep convective cumulonimbus: Cb), domain size, resolution, cloud cover, in-cloud cloud water path (CWP), cloud-top height (CTH), and thermodynamic phase. Shown are ensemble means and standard deviations not accounting for spatial variance within a single ensemble member.

Table 1.

ShCu clouds are convective clouds with typical cloud cover of 10%–20% and cloud-top height (CTH) around 2 km. They occur frequently over tropical oceans, covering 20% on average but up to 40% of their surface (Cesana et al. 2019). In this study, ShCu are represented by two LES case studies, Barbados Oceanographic and Meteorological Experiment (BOMEX) and Rain in Cumulus over the Ocean (RICO), which represent nonprecipitating and precipitating shallow convection over tropical oceans, respectively (Siebesma et al. 2003; vanZanten et al. 2011). Sc clouds are shallower, with CTH only around 1 km. They have near 100% cloud cover and typically blanket subtropical oceans off the west coast of continents (Cesana et al. 2019). Sc are represented by the Second Dynamics and Chemistry of Marine Stratocumulus field study (DYCOMS-II) RF01 LES case of a Sc deck off the coast of California (Stevens et al. 2005). Cb clouds are deep convective thunderstorm clouds that occur frequently over midlatitude continents in summer and in the tropics, e.g., in the intertropical convergence zone (ITCZ). Their CTH can reach up to 15 km or higher, they often contain ice, and anvils at the top contribute to a cloud cover around 30%. Cb clouds are represented in this paper by the TRMM Large Scale Biosphere–Atmosphere Experiment in Amazonia (TRMM-LBA) LES case, based on measurements of convection over land in the Amazon (Grabowski et al. 2006).

An ensemble of snapshots is used to estimate the mean and variance of the bias for each cloud type. The snapshots are chosen to be at least one convective turnover time apart (1 h for BOMEX and RICO, 30 min for DYCOMS-II RF01, and 90 min for TRMM-LBA). For ShCu and Sc, we take snapshots evenly spaced in time starting once the simulation has reached a statistically quasi-steady state, after an initial spinup period. For the Cb case we take snapshots from an initial-condition ensemble at several time points representative of transient and fully developed deep convection at 4, 5.5, and 7 h into the simulation (1000, 1130, and 1300 local time). We also analyze the effect of convective aggregation in Cb (Jeevanjee and Romps 2013; Wing et al. 2017; Patrizio and Randall 2019) by analyzing snapshots from an initial-condition ensemble run over a larger domain [(40 km)2, compared to the original (20 km)2]. In both cases, we use only the snapshots at 1300 local time of fully developed deep convection, characterized by stable liquid and ice water paths, for the cloud-type specific calculations. The rest of the snapshots are used in our estimate of the tropical shortwave flux bias. We choose ensemble sizes that capture the natural variability of morphology in each LES case: 20 for ShCu (10 each of BOMEX and RICO) and 5 for Sc; for Cb we take 15 snapshots from each time point (45 in total) from the (20 km)2 TRMM-LBA simulations and 5 snapshots of fully developed, more aggregated deep convection from the (40 km)2 TRMM-LBA aggregated (agg.) simulations. The smaller ensemble is determined to sufficiently capture the dynamical variability for the larger domain.

The increase in convective aggregation for the larger domain Cb simulations can be seen in typical measures such as the variance of the column relative humidity or total precipitable water (Wing et al. 2017) (see appendix A, Fig. A1). The domain-mean cloud cover, cloud top height, and cloud water path from the two sets of Cb simulations are similar, indicating that the difference in radiative flux bias is being driven by a change in the aggregation or domain size. Larger domains may lead to even more aggregation (Patrizio and Randall 2019); however, synoptic noise may become important and disrupt the self-aggregation of convection on large scales in reality (Bretherton and Khairoutdinov 2015). The Sc and ShCu results are unchanged for larger domain sizes (not shown), but we do see an expected reduction in variance across the ShCu ensemble due to the larger dynamical variability captured in each snapshot of the larger domain.

b. Radiative transfer computations

The RT calculations were done using the libRadtran software package with the MYSTIC Monte Carlo solver (Mayer and Kylling 2005; Mayer 2009; Emde et al. 2016). Details of the setup can be found in appendix B. The MYSTIC solver requires 3D fields of liquid and ice water content and particle effective radius as input. We use MYSTIC to do the full 3D RT and ICA calculations. The LES uses simple microphysics schemes that do not explicitly compute the effective radius. To compute the effective radius, we follow the parameterization from Ackerman et al. (2009) and Blossey et al. (2013) for liquid and Wyser (1998) for ice (appendix B). For the RT calculation, MYSTIC finds the scattering phase function from precomputed lookup tables. In the case of liquid droplets, which are assumed spherical, the full Mie phase function is used. For the case of ice clouds, a parameterization of the habit-dependent scattering must be used. We use the hey parameterization with “general habit mixture” (Yang et al. 2013; Emde et al. 2016). The results are insensitive to the choice of ice parameterization (Fig. B1) because the reflected flux signal is dominated by the liquid droplets for the clouds we simulated.

c. Observations of cloud climatology

The LES cloud fields allow for precise calculation of the 3D cloud radiative effect on small domains. To estimate the global impact of the 3D cloud radiative effect, we use the results from LES along with satellite observations of cloud climatology and surface albedo to scale up from these few cases to a global picture. We find that in-cloud cloud water path (CWP), defined as the domain-mean cloud water path divided by cloud cover, is a simple but robust predictor of the flux bias (will be shown in section 5). We use the International Satellite Cloud Climatology Project (ISCCP) D2 dataset of CWP (Rossow et al. 1999; Rossow and Duenas 2004; Marchand et al. 2010; Stubenrauch et al. 2012, 2013). The ISCCP D2 cloud product is a monthly climatological mean with spatial resolution of 1° × 1° constructed from measurements during the period 1984–2007. These data are collected by a suite of weather satellites that are combined into a 3-hourly global gridded product at the D1 level and are averaged, including a mean diurnal cycle, into the D2 product we use.

We also account for the observed surface albedo that varies seasonally and spatially and affects the flux bias. We use observations of surface albedo from the Global Energy and Water Exchanges Project’s surface radiation budget product version 3.0, which is aggregated to a monthly mean climatology for the period 1984–2007 and gridded to 1° × 1°.

3. Radiative flux bias dependence on zenith angle

a. Top-of-atmosphere

The top-of-atmosphere (TOA) radiative flux bias is measured (in W m−2) as the difference in reflected irradiance between the ICA and 3D RT calculations averaged over the full domain. A positive bias means that, under the ICA, clouds reflect more energy back to space than in reality (i.e., 3D), implying that Earth’s surface is artificially dimmed (cooled) in a model that uses the ICA. The albedo bias (Δα) is computed as the flux bias (ΔF = FICAF3D) divided by the total incoming solar flux (Fin),
Δα=ΔFFin×100%.
Figure 2 shows the flux and albedo biases (ICA–3D) for the five cases of ShCu, Sc, and Cb clouds. The solid lines show the ensemble mean bias, and the shading denotes one standard deviation (σ). The combined variance (σ2) is computed as
σ2=1NLESi=1NLES[(σi,ICA2+σi,3D2)+(ΔFiΔF)2],
where NLES is the number of ensemble members, σi,ICA and σi,3D are the standard deviations from the MYSTIC solver photon tracing, ΔFi is the TOA flux bias of each ensemble member, and 〈⋅〉 denotes a mean over the LES ensemble. This variance includes both the statistical noise from the Monte Carlo RT and the dynamical variability of the cloud field (which are assumed to be uncorrelated). The Monte Carlo noise is proportional to 1/n where n = 104 is the number of photons used for the RT simulation, and is in fact ∼0.7% for these calculations. The variance between cloud scenes is much larger than the Monte Carlo error, by more than an order of magnitude.
Fig. 2.
Fig. 2.

Bias (ICA-3D) in (a) TOA reflected flux and (b) albedo as a function of zenith angle for ShCu (BOMEX and RICO), Sc (DYCOMS-II RF01), and Cb (TRMM-LBA and TRMM-LBA agg.). For each cloud type, average fluxes (with shaded 1σ error bars) are computed over the individual snapshots. Positive bias means the ICA approximation is reflecting more incoming flux than in the 3D RT calculation.

Citation: Journal of the Atmospheric Sciences 78, 12; 10.1175/JAS-D-21-0032.1

Sc show negligible deviation between ICA and 3D reflected fluxes. For convective clouds (ShCu and Cb), the bias from the ICA is positive, except for ShCu at very large solar zenith angles. At large zenith angles, ShCu show a large negative flux and albedo bias for ICA. ShCu scatter far fewer photons than Cb due to the low cloud cover and their smaller optical thickness, corresponding to small vertical extent. Cb exhibit the largest reflected irradiance and also the largest bias between the ICA and 3D RT calculations. While the mean flux bias is similar, the structure of the bias with zenith angle is markedly different for the two domain sizes (Fig. 2). For the small-domain simulations with a lesser degree of aggregation, the bias is approximately linear with zenith angle (as seen by Barker et al. 2015, 2016). For the more aggregated case, the flux bias is nearly uniform up until a solar zenith angle of 60° and then decreases rapidly toward zero; this translates to an albedo bias that peaks at large zenith angles (around 70°).

The convective clouds show much more variation than the stratiform clouds between snapshots due to the variability in cloud cover even in a statistically steady state. The less aggregated Cb clouds have the largest variability, which is expected since the domain size is small relative to the scale of the clouds, i.e., in each snapshot we capture only approximately one deep convective cloud, compared to many small cumulus clouds; therefore, we are effectively averaging over fewer realizations even though we take our ensemble size to be larger. Similarly, for the more aggregated Cb clouds, since we use a 4-times-larger domain, a smaller ensemble (NLES = 5 compared to 15) is large enough to capture the variability.

In the ICA, the horizontal photon fluxes between neighboring columns are ignored. For the Sc clouds that uniformly cover the whole domain (Fig. 1c), this assumption has little effect: the flux bias is near zero for all zenith angles. However, for cumulus clouds, making the ICA has two effects that are described in detail by Hogan et al. (2019).

  1. The long-recognized “cloud-side illumination” effect in 3D radiative transfer. This describes how horizontally propagating photons can encounter the side of a cloud and can be scattered by it, rather than being restricted to hit the top of a cloud in the ICA. Side illumination happens when photons travel across columns at slant angles, brightening the cloud sides and enhancing cloud reflectance; it also creates larger shadows, or larger effective cloud cover. This effect acts to enhance reflectance in 3D, and thus would appear as a negative ICA flux bias in our terminology.

  2. The newer “entrapment” effect that Hogan et al. (2019) presented. This mechanism is similar to the “upward trapping” mechanism discussed by Várnai and Davies (1999). It describes how in 3D a scattered photon may be intercepted by another cloud, or the same cloud, in a different column higher in the domain and scattered back down to the surface. In the ICA by contrast, when a photon travels through clear sky and is scattered by a cloud, it will necessarily travel back through the same column of clear sky to the TOA. The entrapment mechanism acts to decrease cloud reflectance in 3D, i.e., it creates a positive flux bias.

The calculated 3D effects we show in Fig. 2 are a combination of these competing mechanisms. At large solar zenith angles, cloud shadowing, by which clouds can shade each other, clear-sky regions, and the surface when photons are coming in at slant angles, can be important for surface irradiance and surface fluxes (Frame et al. 2009; Veerman et al. 2020).

For small zenith angles, when the sun is overhead, the convective clouds (ShCu and Cb) produce a positive flux bias because entrapment is dominant over cloud-side illumination. For large zenith angles, the flux and albedo bias from ShCu is negative because cloud-side illumination becomes the dominant effect. In the mean, the solar zenith angle at which the flux bias becomes negative is around 70°, but for the individual ensemble members this ranges from around 45° to 75°. This has been seen before for ShCu by Barker et al. (2015, 2016) and Hogan et al. (2019). For Cb clouds, however, even at large zenith angles, the flux and albedo biases remain positive, indicating that the entrapment mechanism continues to dominate over cloud-side illumination. This is not the case for every scene in the Cb ensemble, but it is true in the mean, in agreement with the results from Hogan et al. (2019). This difference between ShCu and Cb is related to the aspect ratio of the clouds; the cloud-side illumination mechanism can only become dominant if the aspect ratio is small (clouds are not too deep). Furthermore, in the case of the more aggregated Cb clouds, a greater degree of aggregation decreases the surface area to volume ratio of the clouds, or what Schäfer et al. (2016) call the length of cloud edge, or cloud perimeter. A smaller cloud perimeter will decrease the cloud side illumination as well as the entrapment efficiency of the cloud (Hogan et al. 2019). The uncertainty in flux bias due to the degree of aggregation of deep convection is much larger than the spread across the LES ensemble and represents a structural uncertainty, which is more challenging to quantify.

These 3D cloud effects can be understood from Fig. 1, which shows illustrations of the clouds from the four LES cases. The scattered cumulus in the BOMEX and RICO cases are shallow and spaced apart, which allows for cloud-side illumination at large zenith angles to dominate over the entrapment mechanism. The DYCOMS-II RF01 stratocumulus clouds are quite horizontally homogeneous over this small domain, therefore, ICA biases are small. As discussed in Hogan et al. (2019), when in-cloud heterogeneity is larger, e.g., for open-celled marine stratocumulus, the entrapment effect is larger. Finally, for the deep TRMM-LBA clouds, the entrapment mechanism remains dominant even for large zenith angles because the clouds at higher levels can intercept and trap outgoing photons that are able to escape to TOA in the ICA.

In addition to the LES ensembles described previously, we run one additional set of tests to quantify the dependence of the flux bias calculations on the LES resolution (Fig. 3). We take the original LES and systematically coarse-grain the 3D fields to lower resolution. Doing so ensures that we do not change the dynamics of the clouds so that we can test the effect of resolution on only the radiative transfer. We are not able to bridge the gap all the way to ESM scales (10–100-km horizontal resolutions) due to computational limits on running the LES, but we show results across a range of horizontal scales. When coarse-graining, we keep the vertical resolution fixed to better represent the very large aspect ratio grid boxes found in ESMs compared to the relatively isotropic grid boxes in LES. The mean TOA flux bias is nearly constant across resolutions for the shallow clouds (Sc and ShCu). For Cb, the mean TOA flux bias decreases with larger grid spacing, as expected, from around 17 W m−2 at the original resolution and down to 6 W m−2 for 2-km horizontal resolution. Since the bias does not asymptote as we move toward smaller horizontal grid spacing, we expect that if the LES were run at even higher resolutions, we would find an even larger bias between the ICA and 3D. We conclude that our estimated bias is a lower bound in this regard.

Fig. 3.
Fig. 3.

Mean TOA reflected flux bias across all solar zenith angles computed for different resolutions of the same cloud fields. The horizontal axis shows the horizontal resolution; the vertical resolution is kept fixed. The four cases of ShCu, Sc, and Cb are shown in the same colors as Fig. 2. For each case, three snapshots from the original ensemble are used and the spread is shown by the shading.

Citation: Journal of the Atmospheric Sciences 78, 12; 10.1175/JAS-D-21-0032.1

b. Surface

Using the same radiative transfer calculations, we also quantify the bias in downwelling surface irradiance. This bias as a function of solar zenith angle is shown for each cloud type in Fig. 4. The total bias (Fig. 4a) is the sum of a direct component and diffuse component, shown in Figs. 4b and 4c, respectively. These components largely offset each other, as has been found in previous studies (Gristey et al. 2020). The direct surface irradiance bias is always positive and the diffuse always negative, resulting from side illumination (shadowing) and entrapment, respectively. Note that these two mechanisms have opposite effects in the surface irradiance bias and the TOA reflected flux bias.

Fig. 4.
Fig. 4.

Surface irradiance bias (ICA-3D) as function of zenith angle for ShCu (BOMEX and RICO), Sc (DYCOMS-II RF01), and Cb (TRMM-LBA). (a) The total surface irradiance bias is split into the (b) direct and (c) diffuse components, which largely compensate each other, especially at larger zenith angles. For each cloud type, average fluxes (with shaded 1σ error bars) are computed over the individual snapshots. Positive bias means the ICA approximation has more downwelling radiation at the surface than the 3D calculation.

Citation: Journal of the Atmospheric Sciences 78, 12; 10.1175/JAS-D-21-0032.1

For the total surface irradiance bias (Fig. 4), we observe a similar pattern to the TOA bias (Fig. 2), except with the opposite sign. This has been noted before (Barker et al. 2015), and is to be expected given that when the ICA produces anomalous extra TOA reflectance, it simultaneously decreases the surface irradiance with respect to the 3D calculation. We can quantify this by considering in a simple way how the TOA reflected flux and surface irradiance depend on the incoming flux, surface albedo (αs), cloud albedo (αc), and cloud cover (fc). The total scene albedo stems from scattering by the clouds and scattering by the surface. Considering up to two scattering events, we can write
α=fcαc+(1fc)αs+2fc(1fc)(1αc)αs.
The first term comes from reflection directly from the clouds, the second from reflection directly from the surface, and the third from reflection of diffuse radiation from the surface. The albedo bias is therefore
Δα=fcΔαc[12αs(1fc)],
where Δα = αICA − α3D. From Eq. (3) we see that the albedo bias will decrease with surface albedo because when the surface accounts for a larger fraction of the total albedo the cloud bias is less pronounced. For the downwelling surface irradiance (I), we can do the same and consider up to two scattering events,
I=[(1fc)+fc(1αc)+fc(1fc)αcαs]Fin,
where the first term comes from direct irradiance, the second from forward scattering through the cloud, and the third from multiple scattering first off the surface and then back down off the cloud. To first order the surface irradiance does not depend on the surface albedo, but including higher-order terms we see that the surface irradiance increases with surface albedo. The surface irradiance bias (ΔI = IICAI3D) can be written as
ΔI=fcΔαc[1αs(1fc)]Fin.
Finally, with these approximations, and noting that ΔF = ΔαFin, we can relate the surface irradiance bias to the TOA reflected flux bias by
ΔI=ΔF[1αs(1fc)12αs(1fc)].

Figure 5 shows the calculated surface irradiance bias compared to the predicted bias calculated from the TOA bias via Eq. (5). The relation between surface irradiance bias and TOA reflected flux bias is robust and predictable, with an r2 = 0.92. Equation (5) considers up to two scattering events, but considering only one scattering event, which simplifies the equation to ΔI = −ΔF, does not significantly change the quality of the prediction (r2 = 0.91 in this case).

Fig. 5.
Fig. 5.

Bias (ICA-3D) in surface irradiance as predicted by bias in TOA reflected flux via Eq. (5) compared to the computed surface irradiance bias. Cloud type and zenith angle are indicated by marker shape and color, respectively. The 1:1 line is shown for reference.

Citation: Journal of the Atmospheric Sciences 78, 12; 10.1175/JAS-D-21-0032.1

While the surface irradiance bias is critically important for surface fluxes, which may have dynamical feedbacks on the clouds, or for impacts on vegetation and carbon uptake (Veerman et al. 2020), the relevant quantity for the overall climate system energetics is the TOA energy balance. The remainder of this paper is devoted to analyzing the TOA reflected flux bias and the relevance of 3D cloud radiative effects for climate.

4. Seasonal cycle of radiative flux bias

The solar zenith angle varies over the course of the day from sunrise to sunset, and therefore the dependence of the bias on zenith angle manifests itself as a diurnally varying bias. But the zenith angle also varies on seasonal time scales for different latitudes.

To assess the climate impact of the ICA bias, we consider the flux and albedo bias for each cloud type as a function of day of year and latitude. This calculation is done by assuming that the LES-generated cloud field is present at any given latitude circle on any given day of the year. This exercise is done without a claim to be realistic, but to demonstrate the impact each cloud type might have on Earth given the spatiotemporal variations of solar zenith angle. For any location and time, including a diurnal cycle, the solar zenith angle is calculated and the flux bias is estimated based on the results presented in Fig. 2. The flux and albedo biases are computed hourly and averaged to show the daily mean bias.

Figure 6 shows the annual mean and seasonal cycle of TOA flux and albedo biases for each cloud type. To estimate the uncertainties of the annual-mean bias, we calculate the LES ensemble spread as follows. For each hour in the year and each latitude, the solar zenith angle is calculated, and we interpolate between integer zenith angles in the flux bias calculations to find the mean flux bias. This is done individually for each LES cloud scene in the ensemble. The ensemble mean for each latitude and day of the year is shown (colored contour plots in Fig. 6), as well as the annual mean of the ensemble (black lines on Fig. 6). The spread across the ensemble in the annual mean is shown as one standard deviation (gray shading in Fig. 6).

Fig. 6.
Fig. 6.

Daily mean bias (ICA-3D) as a function of latitude and day of year assuming the globe is covered by (a)–(d) ShCu (BOMEX and RICO), (e),(f) Sc (DYCOMS-II RF01), (g),(h) Cb (TRMM-LBA), and (i),(j) more aggregated Cb (TRMM-LBA agg.). (left) Flux bias; (right) albedo bias. Note the color scales vary between LES cases. Inset panels on the left show annual average biases with shaded error bars that denote the spread across the LES ensembles as described in the text. Only latitudes from 60°S to 60°N are shown because the simulated clouds are not representative of the high-latitude regions.

Citation: Journal of the Atmospheric Sciences 78, 12; 10.1175/JAS-D-21-0032.1

Both ShCu cases show similar patterns of flux bias with latitude and time (Figs. 6a,c). As seen in Fig. 2, these cases both have a small positive bias for small solar zenith angles, transitioning to a small negative bias for larger zenith angles, which is manifest here as a positive bias at low latitudes, transitioning to a negative bias only in midlatitude winters. The albedo bias for both ShCu cases is near zero in the tropics and becomes more negative at higher latitudes (Figs. 6b,d). Sc show a very small flux (and albedo) bias for all solar zenith angles due to their high cloud cover and horizontal homogeneity, but they do exhibit a small positive flux bias (∼0.2 W m−2) during winter in midlatitudes (Fig. 6e). For Cb, the flux bias is comparatively large and always positive (Fig. 2). In the less aggregated state, the flux bias is nearly linear in zenith angle, which gives rise to a bias pattern that roughly mimics the insolation pattern with latitude and day of year (Fig. 6g). In the more aggregated state, the flux bias is roughly constant in the tropics and overall larger than in the less aggregated case (Fig. 6i). The albedo bias for Cb is largest and positive during summer, though seasonal variations are less pronounced for aggregated convection (Figs. 6h,j). In particular, deep convective clouds are frequently found in the ITCZ, which migrates with the insolation maxima and therefore results in a tropical TOA reflected flux bias that peaks in each hemisphere during their respective summers and is smallest during the shoulder seasons (Figs. 6g,i).

5. Implications for climate models

To assess the effect that 3D radiative transfer through cloud fields has on climate simulated with ESMs, we must account for the climatological occurrence of different cloud types in space and time. A simple parameter that can account for much of the flux bias variability in our LES ensemble is in-cloud cloud water path (CWP), defined as domain-mean cloud water path divided by cloud cover. By regressing the flux bias against CWP for integer solar zenith angles between 0° and 90°, constraining the regression lines to pass through the origin because there is no flux bias in clear-sky conditions (CWP = 0), we observe a robust positive correlation between CWP and flux bias (Fig. 7). The best fit line and confidence intervals are estimated with Gaussian process regression; we use a dot product kernel, with the intercept constrained to zero. We apply regularization by specifying the “nugget” (the values added to the diagonal of the correlation matrix) as the empirically calculated variance scaled by a constant factor. The variance is calculated as the sample variance in a 100 g m−2 CWP interval around each point. The positive correlation between CWP and flux bias, though not perfect, allows us to approximate TOA flux biases using CWP on the global scale. We choose CWP as our proxy for flux bias because it is robustly observed by satellite and, among the other cloud properties we explored (e.g., cloud top height, see appendix C for details), the best predictor for flux bias (Fig. C1). Despite the fact that the radiative flux bias certainly depends on more than just CWP, we use it here as a first approximation to model the flux bias.

Fig. 7.
Fig. 7.

Scatterplot of in-cloud cloud water path (CWP) from LES domain against flux bias at zenith angles (a) 20°, (b) 40°, (c) 60°, and (d) 80°. LES ensemble members are plotted with the same color convention as in Fig. 2. The gray lines show the regression fit constrained to go through the origin. The gray shaded areas show the 68% and 95% confidence intervals. The rms error of the regression is indicated at the top of each panel.

Citation: Journal of the Atmospheric Sciences 78, 12; 10.1175/JAS-D-21-0032.1

Using this relationship between CWP and flux bias for a series of zenith angles, we can take the observed climatological CWPs from ISCCP to infer the resulting flux bias that would be associated with using the ICA for RT calculations in place of 3D RT. The monthly temporal resolution is not inherently an issue for this analysis given that we use a linear relationship between CWP and flux bias.

Additionally, we can account for spatial variations in surface albedo. In the RT calculations previously shown, we assume a constant surface albedo of αO = 0.06, corresponding to an ocean surface. The surface albedo affects the computed TOA and surface flux biases as shown in Eqs. (3) and (4). The albedo bias, written in Eq. (3), scales with the factor [1 − 2αs(1 − fc)]. We can therefore correct for the effect of the surface albedo by multiplying our computed flux or albedo bias by the ratio of the surface absorptions:
Δα|αs=[12αs(1fc)12αO(1fc)]Δα|αo.

To test this scaling, we run additional RT calculations for the convective cloud cases (BOMEX, RICO, and TRMM-LBA) and vary the specified surface albedo from αs = 0.1–0.25 (spanning the range for subpolar land surfaces). Figure 8 shows the correlation between the predicted TOA albedo bias using Eq. (6) and the explicitly calculated TOA albedo bias. The prediction based on Eq. (6) is robust with an r2 value of 0.94. For stratocumulus clouds, since the cloud cover is nearly 1, the albedo bias depends very little on the surface albedo and is not shown. We use Eq. (6) globally to account for variations in the observed surface albedo without the need to run additional RT calculations and interpolate between discrete values.

Fig. 8.
Fig. 8.

Predicted albedo bias from Eq. (6) compared to the computed albedo bias for convective cloud scenes (BOMEX, RICO, and TRMM-LBA) with different surface albedos. The predicted surface albedo is calculated from the simulations using a surface albedo of αo = 0.06 corresponding to an ocean surface. Each point represents the albedo bias at integer solar zenith angle from 0° to 90° for five ensemble members of each LES case. The colors denote the different cases and the symbol shapes denote the surface albedo. The 1:1 line is shown for reference.

Citation: Journal of the Atmospheric Sciences 78, 12; 10.1175/JAS-D-21-0032.1

To construct the annual-mean flux bias map shown in Fig. 9, we first calculated the solar zenith angle for each location on Earth and each hour of the year. Then, we obtained the flux bias given the observed CWP from the linear regression at the given zenith angle (Fig. 7). Finally, we made a correction using Eq. (6), based on the ratio of the observed surface absorption to the assumed ocean surface absorption used in the MYSTIC RT calculations. The resulting flux bias is an estimate of the bias that would be present in an ESM that is able to resolve the relevant dynamical scales of clouds, but makes the ICA during radiative transfer. This bias is smaller than the bias present in current ESMs, which also contains the biases due to PPA and cloud parameterizations, given their very coarse horizontal resolution (Cole et al. 2005b).

Fig. 9.
Fig. 9.

(right) Map of annual-mean flux bias inferred from ISCCP in-cloud CWP. (left) Zonally averaged flux bias in the black line and 1σ error bars in the gray shading that are derived from the linear regression in Fig. 7. As in Fig. 6, we show only 60°S–60°N because the clouds we have modeled are not representative of high latitudes.

Citation: Journal of the Atmospheric Sciences 78, 12; 10.1175/JAS-D-21-0032.1

We focus on the tropics (30°S–30°N, dotted box on Fig. 9), where our estimation of flux bias based on the LES cases is most robust and relevant; for higher latitudes, we do not capture all the relevant cloud regimes with our sample of LES clouds, and so our flux bias estimate needs to be interpreted with caution. Shown in the left inset plot is the zonal-mean flux bias. The shading represents 1σ error from the regression of flux bias on CWP shown in Fig. 7 (as opposed to spatial or temporal variability).

The largest bias occurs in the ITCZ region and the storm track regions, especially over eastern Asia where the climatological CWP is maximal (Fig. 9). It corresponds to locations where the tallest clouds on Earth exist and where the mean zenith angle is smallest. The region of maximum bias migrates seasonally following the location of the ITCZ (and maximum insolation). Seasonal variations in cloud cover and cloud type are also manifest in the seasonal cycle of the 3D flux bias. In the annual mean, the zonal-mean tropical flux bias is estimated to be 3.1 ± 1.6 W m−2, and the maximum local flux bias in the annual mean is around 6.5 W m−2 (99th percentile). The annual-mean, zonal-mean tropical albedo bias is 0.7% ± 0.4% and is locally as large as 1.5% (99th percentile).

Our results are of the same order as those reported in Cole et al. (2005a) and Barker et al. (2015, 2016). Cole et al. (2005a) also found the largest flux bias occurring over the ITCZ region, with a maximum bias of 5 W m−2 and tropical zonal-average bias of 1.5 W m−2 during the boreal winter. The larger value reported here is likely due to the fact that Fig. 9 averages over the shoulder seasons and the regression is based on higher-resolution cloud scenes, as quantified in Figs. 3 and 6.

6. Summary and conclusions

In this paper we estimated the TOA flux and albedo biases that result from neglecting 3D radiative transfer through cloudy atmospheres. Although TOA shortwave radiative flux biases in current ESMs are predominantly due to deficiencies of subgrid-scale dynamical parameterizations that generate cloud cover biases, as convection parameterizations improve and model resolution increases, the relative contribution of 3D radiative effects to the total model error will increase. We have quantified the radiative flux and albedo biases that result from making the ICA by using a 3D Monte Carlo radiative transfer scheme applied to LES-generated 3D cloud fields. The flux and albedo biases were assessed across different cloud regimes and solar zenith angles. We took our findings from four canonical LES cases and applied them to observed climatological cloud occurrence to infer the spatially and temporally resolved flux and albedo biases.

We find that the largest flux bias comes from deep convective clouds at small solar zenith angles. The albedo bias is large and negative for shallow cumulus clouds at large solar zenith angles. These results quantitatively agree with previous studies using LES clouds to assess 3D effects (Hogan et al. 2019). There is room for future work considering a larger ensemble of cloud morphologies, which could be generated again by LES or alternatively could be retrieved from satellite observations. Our inferred global flux bias is based on only four tropical/subtropical LES cases and therefore does not represent the full diversity of extratropical cloud morphologies. This methodology cannot fully capture the effects of midlatitude storms, for instance, which is why we do not emphasize our results outside of the tropics.

We use the observed positive correlation between CWP and TOA flux bias from our LES ensemble to estimate the global spatiotemporal bias from neglecting 3D radiative transfer in a high-resolution ESM. We choose a simple linear model to map from satellite observations of climatological CWP to TOA flux bias. The deviations in our regression fit suggest that there is potential for a more robust mapping from cloud properties to radiative flux bias. Future work is necessary to explore this path toward a parameterization of 3D radiative effects in ESMs.

The large flux bias for Cb clouds at small solar zenith angles translates into a seasonal bias that peaks just off the equator in the summer hemisphere, tracking the position of the ITCZ. We estimate the annual-mean tropical-mean flux bias to be 3.1 ± 1.6 W m−2. The flux bias computed here is small compared to the TOA shortwave flux errors typical for CMIP5 and CMIP6 models, which are on the order of 10 W m−2 in the mean (Zhao et al. 2018; Hourdin et al. 2020) and can reach 50 W m−2 in stratocumulus regions (Brient et al. 2019). However, the 3D bias is still comparable to the signal of anthropogenic greenhouse gas emissions for the coming decades, which is on the order of 2.5–3.1 W m−2 (Myhre et al. 2013). These results highlight the importance of considering the 3D radiative fluxes through clouds for Earth’s radiation budget and Earth system modeling.

Acknowledgments

C.E.S. acknowledges support from NSF Graduate Research Fellowship under Grant DGE-1745301. I.L. is supported by a fellowship from the Resnick Sustainability Institute at Caltech. This research was additionally supported by the generosity of Eric and Wendy Schmidt by recommendation of the Schmidt Futures program. Part of this research was carried out at the Jet Propulsion Laboratory, California Institute of Technology, under a contract with the National Aeronautics and Space Administration. We thank three anonymous reviewers for helpful comments on an earlier version of this paper.

Data availability statement.

All code or data used in this paper are freely available online. The LES were run using the PyCLES code (https://climate-dynamics.org/software/#pycles). The radiative transfer computations were done using the libRadtran code (http://www.libradtran.org). Post-processed LES 3D fields used as input files for libRadtran computations are available in Singer et al. (2020). The ISCCP data were downloaded from https://climserv.ipsl.polytechnique.fr/gewexca/ and the GEWEX albedo measurements were downloaded from https://eosweb.larc.nasa.gov.

APPENDIX A

LES Model Setup

LES are performed using the anelastic fluid solver PyCLES (Pressel et al. 2015). Subgrid-scale fluxes are treated implicitly by the WENO scheme used in the numerical discretization of the equations (Pressel et al. 2017).

For each case, the characteristic time scale of convection is evaluated and taken to be representative of the dynamical decorrelation time τ. Snapshots are taken at least one dynamical decorrelation time apart, so that the cloud samples can be treated as independent in a statistical analysis of the flux biases. The decorrelation time scale is calculated as
τ=zblw*+dcw¯u,
where zbl is the mixed layer height, w*=(zblwb¯|s)1/3 is the Deardoff convective velocity, dc is the cloud depth, and w¯u is the mean updraft velocity within the cloud.

a. Shallow cumulus (ShCu) convection, BOMEX

The BOMEX LES case study is described in Siebesma et al. (2003). Surface boundary conditions wqt¯|s and wθl¯|s are prescribed, resulting in sensible and latent heat fluxes of about 10 and 130 W m−2, respectively. The atmospheric column is forced by clear-sky longwave radiative cooling, neglecting radiative cloud effects. A prescribed subsidence profile induces mean vertical advection of all fields, and specific humidity is further forced by large-scale horizontal advective drying in the lower 500 m. The liquid-water specific humidity is diagnosed through a saturation adjustment procedure. For BOMEX, the characteristic time scale of convection is τ ≈ 40 min, where zbl = 500 m, w*=0.66 m s1, dc = 1300 m, and w¯u=0.85 m s1, and snapshots are taken every 1 h. The domain size is set to 6.4 km in the horizontal and 3 km in the vertical. Results are reported for an isotropic resolution of Δxi = 20 m.

b. Shallow cumulus (ShCu) convection, RICO

The RICO LES case study is described in vanZanten et al. (2011). The surface sensible and latent heat fluxes are modeled using bulk aerodynamic formulae with drag coefficients as specified in vanZanten et al. (2011), resulting in fluxes of around 6 and 145 W m−2, respectively. The atmospheric column is forced by prescribed profiles for subsidence and large-scale heat and moisture forcings that are a combination of radiative and advective forcings. The two-moment cloud microphysics scheme from Seifert and Beheng (2006) is used with cloud droplet concentration set to Nd = 70 cm−3. For RICO, the characteristic time scale of convection is τ ≈ 50 min, where zbl ≈ 500 m, w*0.62 m s1, dc = 2500 m, and w¯u1.2 m s1, and snapshots are taken every 1 h. The domain size is set to 12.8 km in the horizontal and 6 km in the vertical. Results are reported for an isotropic resolution of Δxi = 40 m.

c. Stratocumulus-topped marine boundary layer (Sc), DYCOMS-II RF01

The simulation setup for DYCOMS-II RF01 follows the configuration of Stevens et al. (2005). The initial state consists of a well-mixed layer topped by a strong inversion in temperature and specific humidity, with Δθl = 8.5 K and Δqt = −7.5 g kg−1. Surface latent and sensible heat fluxes are prescribed as 115 and 15 W m−2, respectively. In addition, the humidity profile induces radiative cooling above cloud top and warming at cloud base. As in BOMEX, the liquid-water specific humidity is diagnosed through a saturation adjustment procedure. For the stratocumulus clouds, without strong updrafts and a thin cloud layer, the characteristic convective time scale is taken to be just the first term of Eq. (A1), which evaluates to τ ≈ 20 min, with zbl ≈ 850 m and w*=0.8 m s1. Snapshots taken every 30 min are used in the analysis. The domain size is set to 3.36 km in the horizontal and 1.5 km in the vertical. Results are reported for a resolution of Δz = 5 m in the vertical and Δx = Δy = 35 m in the horizontal.

d. Deep convection (Cb), TRMM-LBA

Deep convective clouds are generated using the TRMM-LBA configuration detailed in Grabowski et al. (2006), based on observations of the diurnal cycle of convection in the Amazon during the rainy season. The diurnal cycle is forced by the surface fluxes, which are prescribed as a function of time. The magnitude of the fluxes maximizes 5.25 h after dawn, with a peak latent and sensible heat fluxes of 554 and 270 W m−2, respectively. The radiative cooling profile is also prescribed as a function of time. We use the one-moment microphysics scheme based on Kaul et al. (2015) with modifications described in Shen et al. (2020). Since this case study is not configured to reach a steady state, the simulation is run up to t =7 hours. Deep convection is considered to be fully developed after 5 h, when the liquid-water and ice-water paths stabilize (Grabowski et al. 2006). The ensemble of cloud snapshots is formed by sampling after t= 4, 5.5, and 7 h from a set of simulations with different initial conditions. For the idealized case (Figs. 2 and 6) only the 15 snapshots from t= 7 h are used. The characteristic convective time scale is given by just the second term of Eq. (A1), τ=0zctwu1dz80 min, where zct and wu are the cloud-top height and updraft vertical velocity averaged over the last two hours, respectively. The random perturbations used in the initialization ensure that all cloud snapshots in the ensemble are uncorrelated. The domain size is set to 20 km in the horizontal and 22 km in the vertical. Results are reported for a resolution of Δz = 50 m in the vertical and Δx = Δy = 100 m in the horizontal.

For the large-domain simulations, we double the domain size to 40 km in the horizontal and run a smaller ensemble of NLES = 5 simulations. The mean cloud cover, cloud top heights, and cloud water path in the large and small domain ensembles are comparable at 0.30 and 0.32, 12.2 and 10.0 km, and 360 and 290 g m−2, respectively. The large-domain simulations show a higher degree of aggregation as measured by the variance in total precipitable water, 4.3 mm2, compared to 3.7 mm2 in the original 20-km domain. Figure A1 shows histograms of the total precipitable water for each of the TRMM-LBA simulations at 7 h (NLES = 15 for the 20-km domain, and NLES = 5 for the 40-km domain). The wider histograms for the large-domain simulations illustrate the larger variance in this field, which is indicative of a higher degree of convective aggregation.

Fig. A1.
Fig. A1.

Normalized histogram of total precipitable water from the TRMM-LBA simulations in a 20-km domain vs 40-km domain, which we use as a less and more aggregated case of deep convection. The variance across the ensemble, shown by the width of the histogram, is representative of the degree of convective aggregation.

Citation: Journal of the Atmospheric Sciences 78, 12; 10.1175/JAS-D-21-0032.1

APPENDIX B

Radiative Transfer Details

We use the libRadtran MYSTIC Monte Carlo solver for the 3D and ICA radiative transfer calculations with no variation reduction techniques applied. The ICA is done using the mc_ipa setting which horizontally averages the fluxes from the full 3D calculation. The RT is done using n = 104 photons using the atlas_plus_modtran solar spectrum. The atmospheric molecular absorption is done using the kato2 correlated-k parameterization (Kato et al. 1999; Mayer and Kylling 2005). The atmospheric profile (pressure, temperature, density, and specific humidity) defined for the radiative transfer comes from the LES output and aerosols are neglected in these calculations. Because we only consider the flux bias and not absolute values, the LES are not embedded in a clear-sky atmosphere with a profile extension to a fixed height, although this is typical for radiative transfer calculations. The surface is treated as a Lambertian scatterer and the albedo was set to αO = 0.06 for all RT calculations. The observed surface albedo is accounted for through the approximation described in section 5. Both the LES and RT assume doubly periodic horizontal boundary conditions and the spatial resolution, which can be found in Table 1 for each case, is the same.

The MYSTIC solver from libRadtran requires 3D fields of liquid and ice water content and particle effective radius as input. The LES uses bulk microphysics schemes and does not explicitly compute the effective radius. For liquid-only clouds, the parameterization from Ackerman et al. (2009) and Blossey et al. (2013) with assumed droplet number of Nd = 108 m−3 is used. The full Mie scattering phase function is taken from the libRadtran lookup tables. Because the lookup tables are only valid for droplets with radius greater than 1 µm, smaller calculated effective radii were rounded to this minimum value.

For ice clouds, the parameterization from Wyser (1998) is used. The hey parameterization from Yang et al. (2013) and Emde et al. (2016) with habit type set to general habit mixture (ghm) is used. The hey parameterization uses the complete scattering phase function as calculated from single scattering models for ice crystals in (Yang et al. 2013), rather than employing an approximation like Henyey–Greenstein phase function, which has been shown to be another source of error in RT (Barker et al. 2015). The results are not dependent on the exact choice for ice crystal shape or roughness (Fig. B1). Note that the hey ice parameterization is only valid for radii less than 90 µm, and larger calculated effective radii were rounded to this maximum value.

Fig. B1.
Fig. B1.

TOA reflected bias across zenith angles for different ice parameterizations in one TRMM-LBA cloud snapshot.

Citation: Journal of the Atmospheric Sciences 78, 12; 10.1175/JAS-D-21-0032.1

Deep convective clouds, reaching upward of 10 km, nearly always contain ice crystals in addition to liquid water. Optical properties of ice crystals depend on their size, shape (or habit), and surface smoothness. Two different parameterizations, with three and four habit choices, respectively, were tested. The differences between these parameterization variants are negligible; they are much smaller than the variability stemming from the cloud dynamics (statistical spread between snapshots) and also much smaller than the magnitude of the 3D effects (Fig. B1).

The hey parameterization with ghm is used in the main text (Yang et al. 2013; Emde et al. 2016). This parameterization is valid for a spectral range from 0.2 to 5 μm, and for ice effective radii from 5 to 90 μm. hey assumes smooth crystals and allows for four choices of habit: ghm, solid column (col), rough aggregate (agg), and plate.

The other parameterization tested was baum_v36 (Heymsfield et al. 2013; Yang et al. 2013; Baum et al. 2014). This parameterization is valid over a wider spectral range (0.2–99 μm), but a narrower effective radius range (5–60 μm). Particles with effective radius outside of the accepted range were rounded to the maximum allowed value. The baum_v36 parameterization assumes severely roughened particles. It allows for three choices of habit: ghm, col, and agg.

These seven variants are compared in Fig. B1 for one cloud snapshot from the TRMM-LBA case and they show very similar results. Also shown in Fig. B1 is a RT calculation done on the same cloud field, but only including the liquid droplets and ignoring the ice particles. We use the full Mie scattering phase function without any parameterization for the liquid portion of the cloud in all cases. The difference between the liquid-only and liquid and ice simulated absolute fluxes can be up to 20% depending on the parameterization used (not shown), but the flux bias (ICA-3D) is very similar for the liquid-only and all ice parameterizations.

APPENDIX C

Cloud Property Proxy for Flux Bias

We explored several different cloud properties to use as a proxy for the flux bias. Our limited study concluded that the in-cloud cloud water path (CWP) was the best proxy because it shows a strong positive, linear correlation with flux bias. Other cloud scene properties we examined included cloud top height (CTH), cloud cover (cc), and the geometric mean of covered area and uncovered area, cc(1cc). The linear regression fits are shown in Fig. C1. The rms error for CWP is the smallest. Although CTH (or cloud depth) are also reasonable proxies, they are more difficult to measure from satellite, and therefore we use CWP in this study. An important extension to this work would be to allow for multiple cloud properties and a more complex model than a linear fit to describe the flux bias. However, with our limited data from only four LES cases in this present study, we do not feel justified to use a more complex model.

Fig. C1.
Fig. C1.

Regression of flux bias on several different cloud properties (a) in-cloud cloud water path (CWP), (b) cloud top height, (c) cloud cover, and (d) cc(1cc). All panels show the flux bias for a zenith angle of 40°. The plot in (a) is the same as Fig. 7b. The rms error is shown for each panel and is smallest for the in-cloud CWP case (a).

Citation: Journal of the Atmospheric Sciences 78, 12; 10.1175/JAS-D-21-0032.1

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