Quantification of Global Cloud Properties With Use of Spherical Harmonic Functions

Spherical harmonic (SH) expansion is a useful tool to study any variable that has valid values at all latitudes and longitudes. The variable can be quantified as a sum of different spherical harmonic components, which are the spherical harmonic functions multiplied by their expansion coefficients. We find that the SH components of cloud radiative effect (CRE) have correlations with El Niño‐Southern Oscillation (ENSO) and the Hadley Circulation (HC). In particular, the expansion degree 2 ( l=2 $l=2$ ) SH power spectrum component anomaly of CRE is strongly correlated with ENSO. The two dipole patterns appearing in the l=2 $l=2$ SH component anomaly map can be reasonably explained by a known mechanism of ENSO's impact on cloud properties. The l=3 $l=3$ and l=5 $l=5$ SH power spectrum components are correlated with HC intensity, whereas the l=6 $l=6$ and l=8 $l=8$ components are correlated with HC latitudinal widths. In ENSO warm and cold phases, the HC‐correlated SH components have opposite anomalies, which suggests the impact of ENSO on HC. This study illustrates that the SH expansion technique provides a different perspective to study the impacts of large‐scale atmospheric circulation on global cloud properties and radiative effects.

2 of 14 and shapes of power spectrum peaks of the CMB temperature fluctuation map are related to many cosmological parameters such as baryon density, dark matter density, and dark energy density. Thus, the variations of the cosmological parameters can be estimated from the power spectra derived from CMB observations. Koppelt et al. (1989) show that the distribution of continents and oceans on the earth can be represented by SH expansion coefficients as low as 34 degrees. The coastlines interpreted from the reconstructed continent-ocean distributions are consistent with the actual coastline.
To date, although SH expansion has seen a few applications as noted above, we believe this method has been underutilized in research involving global datasets and can be successfully extended to additional global geophysical variables. In this study, we demonstrate the applicability of the SH expansion technique for representing cloud property data, such as cloud optical depth (COD), cloud fraction (CF), and cloud effective pressure (CEP) for evaluating global cloud radiative effects (CREs). We also explore how to interpret variations of individual SH expansion components of global cloud properties. Ramanathan et al. (1989) and others (e.g., Hartmann et al., 1992;Stephens et al., 2012;Trenberth et al., 2009) showed that clouds are critical to the earth's radiative budget by reflecting and absorbing solar radiation, and absorbing and emitting terrestrial thermal radiative energy. Based on satellite remote sensing, clouds cover about 56%-73% of the globe (Stubenrauch et al., 2013), depending on the definition of cloud coverage. Clouds and their physical properties have complex spatial and high-frequency temporal variations, but their global distribution and variations are closely related to large-scale atmospheric circulation including ENSO and HC.
Based on surface observations, Eastman et al. (2011) find that tropical cloud cover interannual variations are correlated with ENSO. ENSO can affect both low cloud (e.g., Park & Leovy, 2004;Zhu et al., 2007) and high cloud (e.g., Marchand, 2013;Norris, 2005) amounts via anomalous sea surface temperature and convection variations. Generally, low clouds have negative CRE (radiative cooling), and high clouds have positive CRE (radiative warming). The CRE is thus also affected by ENSO (e.g., Allan et al., 2002;Burleyson et al., 2015;Cess, Zhang, Wang & Wielicki, 2001;Moore & Vonder Haar, 2001). The resulting cloud radiative heating variation can in turn modify cloud properties (Harrop & Hartmann, 2016). As the strongest atmospheric meridional circulation, the HC significantly affects cloud meridional distributions by varying their intensity, positions of ascending and descending branches, and cell widths. Previous studies (e.g., Loeb et al., 2014;Oort & Yienger, 1996;Stachnik & Schumacher, 2011) suggest substantial and complicated relationships between ENSO and HC anomalies and their impacts on global cloud properties and CRE.
In this study, we use SH expansion to investigate the impacts of ENSO and HC on CRE, which may provide a different perspective on the impact of atmospheric circulation on clouds and their radiative effects. Section 2 describes methods and data used in this study, including illustrating the accuracy of representing cloud properties using finite numbers of SH functions. Section 3 presents the results and discussions, and Section 4 summarizes the main findings.

Spherical Harmonic Expansion
The cloud properties considered in this study are COD (or ), CF, CEP, and CRE. Hereafter, CRE is defined as the net TOA radiative flux difference between cloudy and clear-sky conditions. Net radiative flux is computed as the difference between the downward flux and upward flux. Theoretically, a global cloud property can be expanded in an infinite number of SH functions. Practically, we can only expand cloud properties in terms of finite SH functions, and retain acceptable accuracy. For example, the global COD is expanded as where ( ) is a SH function with degree and order , and and are zenith and azimuth angles (equivalent to latitude and longitude) at the surface of the earth. The expansion coefficients are computed by With the truncation of the coefficient series to degree , a finite series of SH terms can be used to reconstruct the global COD distribution in the form Because the cloud properties involved in this study are mathematically real quantities, the SH expansion coefficients satisfy = − . The number of independent coefficients from degree 0 to is Each cloud property SH expansion is evaluated by comparing CRE computations using the original COD data (e.g., ) and reconstructed data (e.g., ).
To evaluate the contribution of SH expansion components of individual degrees, the SH power spectrum SH is defined as According to the SH addition theorem and Equation 2, for ( ) , we have where is the Legendre polynomial of degree , is a unit vector corresponding to the angular coordinates and , and Ω is the solid angle. ′ and Ω ′ are defined in the same manner. Equation 6 suggests that SH measures the prevalence of cloud spatial variations with a specific angular separation consistent with the widths of peaks which are roughly equal to ΔΘ = 180 • ∕ . The SH power spectrum of clouds can be used to analyze the variation of cloud spatial distributions of specific scales.

Cloud Radiative Effect Computation
The differences between computed CRE with original and reconstructed cloud properties are used to evaluate the accuracy of the SH expansion of selected cloud properties. In particular, the net radiative flux at the TOA is computed by using the Rapid Radiative Transfer Model for general circulation model applications (RRTMG) (Clough et al., 2005;Iacono et al., 2008), in which SW and LW components are treated separately by RRTMG_ SW and RRTMG_LW.
The global COD, CF, and CEP data in the RRTMG computation are obtained from the CERES Energy Balanced and Filled (EBAF) data set (Kato et al., 2018;Loeb, Doelling, et al., 2018). The CERES EBAF data set contains monthly-averaged global longwave and shortwave flux as well as CRE data at both the top of the atmosphere (TOA) and the surface with 1 • spatial resolution. The single-layer monthly-averaged COD, CF, and CEP data are available in EBAF and are used in a radiative transfer model to compute CRE. The CF and CEP data cover both day and night, while COD data are only available for daytime.
The monthly-averaged COD, CF, and CEP data from the CERES EBAF data set are the inputs to RRTMG. The fractions of liquid and ice clouds are estimated from the cloud water content data of the European Centre for Medium Range Weather Forecasts (ECMWF) Reanalysis fifth Generation (ERA5) global reanalysis data set (Hersbach et al., 2020). The global-averaged cloud effective radius (CER) from the MODIS Collection 6 cloud product (Platnick et al., 2017) is 32 μm for ice clouds and 12 μm for liquid clouds (Yi et al., 2017). For simplicity, we use the above-mentioned global-averaged CER at all locations. The COD data are given in a visible spectral 4 of 14 band. By assuming the extinction efficiency in a visible band to be 2, the COD data can be converted to liquid water path (LWP) and ice water path (IWP). LWP and IWP can be used to obtain CODs in different RRTMG bands together with the CER values.
The liquid water cloud optical properties assumed in the MODIS Collection 6 cloud property retrievals (Platnick et al., 2017) are parameterized and used in RRTMG computations. The two-habit ice cloud optical property model (Loeb, Yang, et al., 2018) is parameterized and used in RRTMG computations involving ice clouds. The cloud layers are assumed to follow the maximum-random overlap with the Monte Carlo Independent Column Approximation (Pincus et al., 2003). All clouds are single-layered and homogeneous in the computation. The RRTMG computations in longwave bands explicitly consider cloud multiple scattering effects. Previous studies (e.g., Gu et al., 2021;Kuo et al., 2017) have shown the significance of accounting for cloud multiple scattering in longwave radiative flux computations. The 2/4-stream approximation method (Fu et al., 1997;Kuo et al., 2020;Toon et al., 1989) is incorporated into the RRTMG_LW runs to account for cloud multiple scattering.
Other input data are averaged over the month to be consistent with the cloud property data. The atmospheric profile data including temperature, water vapor concentration, and ozone concentration are from the ERA5 global reanalysis data mentioned above. Concentrations of absorptive gases CO 2 , CH 4 , and CO are taken from the Copernicus Atmosphere Monitoring Services (CAMS) reanalysis of atmospheric composition (Inness et al., 2019). The present RRTMG_SW and RRTMG_LW runs use ERA5 monthly averaged surface albedos and surface emissivities for different surface types from ECMWF (2020), respectively. Solar zenith angles are specified to be monthly averaged values computed according to Gupta et al. (2001). Aerosols are not considered in the present RRTMG runs.

El Niño-Southern Oscillation Index and Hadley Circulation Parameters
We consider the impact of ENSO and HC on CRE. The CRE data are derived from the CERES EBAF data set from the year 2000-2021. The overlapping bi-monthly Multivariate ENSO Index (MEI) (Wolter & Timlin, 1998 time series are used to quantify the ENSO. The HCs are characterized by the zonal-mean meridional streamfunc tion ( ), which is computed using the meridional wind field from ERA5 global reanalysis data. is a function of pressure and latitude , and is calculated as (Nguyen et al., 2013) where is the averaged radius of the earth, is the gravitational acceleration, is the surface pressure and ( ) is the zonal-mean meridional wind at pressure level and latitude .
The HC parameters include HC intensity and latitudinal widths. The HC intensity is defined as the maximum absolute value of the streamfunction max . The edges of HC cells are denoted by the latitudes of zero isolines averaged between pressure level 400 hPa and 600 hPa (Nguyen et al., 2013). The edge closest to the equator is defined as the intersection between the northern and southern HC cells. The latitudinal width of an HC cell is defined as the latitudinal difference between its poleward edge and the intersection.

Spherical Harmonic Power Spectra
Figures 3a-3c show the SH power spectra ( SH ) of annual mean SW, LW, and net CRE from 2001 to 2020 (Note that each panel in Figure 3 contains 20 curves). The CRE data are from the CERES EBAF data set. Over the 20 years, the positions of the spectral peaks are consistent. The SW CRE power spectra have strong peaks at = 4 and = 6 , the LW CRE power spectra have strong peaks at = 2 and = 6 , and the net CRE power spectra have strong peaks at = 4 and = 6 . = 2 , = 4 , and = 6 correspond to angular scales 90 • , 45 • , and 30 • , as explained in Section 2.1. As shown in Figures 3d-3f at half period 30 • is consistent with the local maxima of LW CRE SH at = 6 . The above analysis suggests that the CRE SH power spectrum is associated with the latitudinal variation of CRE. According to Equation 6, the SH power spectrum also accounts for variations that are not along the latitudinal direction. Figures 4a and 4b show the SH power spectra of annual mean low and midlevel (>450 hPa), and high (<450 hPa) cloud water path (CWP) from 2001 to 2020. The power spectrum curves of the low and midlevel cloud CWP and high cloud CWP resemble those of the SW CRE and LW CRE, respectively. The similarity in the SH power spectrum curve shapes indicates similarity in spatial distribution. The SW CRE spatial distribution is more relevant to the low and midlevel cloud, and the LW CRE spatial distribution is more relevant to the high cloud. The SW CRE results from clouds reflecting incident solar radiation. Global low clouds greatly modify the albedo of the earth, and their reflecting effect (radiative cooling) dominates their thermal emission effect toward the surface (radiative warming) (Hartmann et al., 1992). In LW, clouds absorb thermal emission from the atmosphere and emit LW radiation to space at a lower temperature than the surface. High clouds, which usually have a lower cloud top temperature, contribute more to LW CRE than low clouds. The SW CRE (Figure 3a), net CRE (Figure 3c) One interesting feature in the CRE and CWP SH power spectra (Figures 3 and 4) is that the power spectrum components at even degree is systematically stronger than at odd . This can be explained by the symmetry of clouds in the northern and southern hemispheres. The Legendre polynomial is even at even and is odd at odd . According to Equation 6, at an odd , if (̂) = (−̂) , the values (̂) (̂′) (̂⋅̂′) and (−̂) (̂′) (−̂⋅̂′) will cancel out in the integration calculation. Therefore, if the cloud is symmetric to some extent with respect to the equator, which can be seen in the zonally averaged CRE in Figure 3, the SH power spectrum components at odd tend to be smaller than at even . Figure 5 shows the correlation coefficients ( ) between MEI and anomalies of CRE SH (Δ SH ) at various expansion degrees. To be consistent with the bi-monthly MEI, the CRE SH are computed using bi-monthly averaged global CRE data from the CERES EBAF data set spanning from March 2000 to December 2021. The anomaly of a SH is the difference between the bi-monthly SH for a specific 2 months in a given year and the averaged bi-monthly SH for the same 2 months in all available years. As a comparison, the dotted lines in Figure 5 also show between MEI and the anomalies of global mean bi-monthly averaged SW,  Figure 6 shows the time series of the MEI plotted together with global mean CRE anomalies Δ CRE and CRE Δ SH 2 . In some strong El Niño years such as 2015-2016 and La Niña years such as 2010-2011, the global mean SW CRE anomalies are strongly correlated with the MEI, but the correlation is not obvious in other years. The correlation between global mean LW CRE anomalies and the MEI is weak in all the considered years. In contrast, the LW and SW Figure 5. The correlation coefficients between anomalies of CRE SH power spectrum components (Δ SH ) and MEI. The CRE SH power spectra are computed using bi-monthly averaged global CRE data from the CERES EBAF data set. The data span from March 2000 to December 2021. The anomaly of a power spectrum is the difference between the bi-monthly power spectrum for specific two months at a given year and the averaged bi-monthly power spectrum for the two months at all available years. The blue square, red circle and cyan triangle symbols are for SW, LW, and net CRE, respectively. The blue, red and cyan dotted lines show the correlation coefficients between MEI and anomalies of global mean bi-monthly averaged SW, LW, and net CRE (Δ CRE ), respectively. 10.1029/2022EA002718 9 of 14 CRE Δ SH 2 have strong correlation with the MEI in all the considered years. MEI ≥ 0.5 corresponds to an ENSO warm phase (El Niño-like), MEI ≤ −0.5 corresponds to an ENSO cold phase (La Niña-like), and −0.5 < MEI < 0.5 is an ENSO neutral phase (Zhang et al., 2019). In most of the considered years, a positive (negative) MEI corresponds to a negative (positive) Δ SH 2 for both SW and LW CRE. In the ENSO warm (cold) phase, the contribution of the = 2 SH component decreases (increases) relative to the neutral phase for both SW and LW CRE.

LW and net CRE (Δ CRE
A degree SH component is defined as where is a variable that has a valid value at each arbitrary latitude and longitude, is the degree SH component, and ( = − − + 1 . . . 0 . . . − 1 ) are SH expansion coefficients of at . Figures 7a(7c) and 7b(7d) show the mean anomalies of the = 2 SH component of SW (LW) CRE in ENSO warm and cold phases. The anomalies of the SH component in ENSO warm or cold phases are the mean anomalies over months with MEI ≥0.5 or ≤−0.5, respectively. The CRE data are bi-monthly averaged to be consistent with the MEI data. The anomaly of an SH component is the difference between the bi-monthly SH component for any specific 2 months in a given year and the averaged bi-monthly SH component for the same 2 months in all available years. Figure 7, the mean anomalies of the = 2 CRE SH component have a dipole pattern over the Indian and Pacific oceans. In the ENSO warm phase, the LW (SW) CRE dipole pattern is negative (positive) over Indonesia and the western Pacific, and positive (negative) over the central and eastern Pacific. In the ENSO cold phase, the dipole pattern is reversed and the magnitude is weaker than the warm phase. The variation of the dipole pattern is consistent with previous studies on the impact of ENSO on clouds over the tropics. In the ENSO warm phase, the low cloud fraction increases over Indonesia and decreases in the eastern Pacific (Zhu et al., 2007), whereas the high cloud fraction decreases over Indonesia and the western tropical Pacific, and increases over the central tropical Pacific (Marchand, 2013;Norris, 2005). The dipole patterns of SW and LW CRE have opposite signs over the Indian and Pacific oceans. Climatologically, the SW CRE value is negative (radiative cooling) and the LW CRE value is positive (radiative warming) in the dipole region. Although SW and LW CRE have different sensitivities to low and high clouds, the overall ability of cloud reflecting incoming solar radiation and reducing outgoing longwave emission is weakened (strengthened) over Indonesia and the western tropical Pacific, and is strengthened (weakened) over central and eastern tropical Pacific in the ENSO warm (cold) phase.  Figure 7, there is another dipole pattern centered over South America and Africa. In the ENSO warm phase, the SW (LW) CRE anomaly dipole pattern is negative (positive) over Africa, and positive (negative) over South America. In the ENSO cold phase, the dipole pattern is reversed and the magnitude is weaker than the warm phase. A modeling study by Yang et al. (2016) calculates the correlation coefficient between CRE and the Niño 3.4 index. The SW (LW) CRE over Africa have weak negative (positive) correlations with the Niño 3.4 index, and the signs of correlations are largely reversed over South America. This is consistent with the dipole pattern of the = 2 CRE SH component anomalies over Africa and South America.

As shown in
The above results and analysis indicate that the CRE Δ SH 2 can be an alternative parameter to measure the phase and strength of ENSO. The impact of ENSO on CRE is clearly present in the = 2 CRE SH component anomalies. Figure 8 shows the values between the monthly averaged HC parameters and LW CRE SH at various expansion degrees. The SW CRE SH has similar but weaker correlations with the HC parameters. At expansion degrees ≤ 8 , SH is strongly correlated with HC intensity and width. In particular, between SH of odd degrees and HC intensity is up to 0.767. The absolute value of between SH of even degrees, and the northern and southern branch widths can be more than 0.8. values are positive for the northern branch width and negative for the southern branch width. Figure 9 shows the time series of the HC parameters plotted together with LW CRE SH . The variations of SH 3 and SH 5 are consistent with the HC intensity in most months. The oscillation of SH 6 and SH 8 time series is in phase with the  Using upper-air wind data, Oort and Yienger (1996) observed the correlation between ENSO and HC. The impact of ENSO on HC is also revealed in various reanalysis datasets (Nguyen et al., 2013;Stachnik & Schumacher, 2011). ENSO can cause strengthening and weakening of the local HC (Klein et al., 1999). Figure 10 shows the mean  anomalies of the = 3 and = 6 LW CRE SH components in ENSO warm and cold phases. The signs of the SH component anomalies in ENSO warm and cold phases are opposite. Because the = 3 and = 6 components are strongly correlated with HC as shown in Figure 9, ENSO's impact on the HC can be observed in the anomalous SH components in different ENSO phases. For example, in Figures 10a and 10c, the LW CRE has positive anomalies over the equatorial central and eastern Pacific in the ENSO warm phase, which corresponds to increased cloud cover in those regions and indicates an enhanced local HC ascending branch. Klein et al. (1999) also found an enhanced HC in those regions during El Niño periods using upper-tropospheric relative humidity data.

Conclusions
The spherical harmonic (SH) expansion technique can quantify any variable which has valid values at each latitude and longitude in terms of different SH components. In this study, we investigate the impact of large-scale atmospheric circulation, including El Niño-Southern Oscillation (ENSO) and Hadley Circulation (HC), on SH components of global cloud radiative effect (CRE). This provides a different perspective about the impact of ENSO and HC on cloud properties and radiative effects.
We first validate the accuracy of using SH functions to represent global cloud properties, including cloud optical depth (COD), cloud fraction (CF), and cloud effective pressure (CEP). The accuracy of using a limited number of SH functions to represent COD, CF and CEP is evaluated concerning CRE computation. Based on year 2019 CERES EBAF cloud property data, we examined the CRE global distribution. SH expansion of cloud properties with maximum degree 100 in total 5,151 coefficients, can model CRE with errors less than 4.3 W m −2 in SW and 1.7 W m −2 in LW in more than 95% regions.
The SH power spectrum of a variable quantifies the contributions of individual SH components to the global distribution of the variable, and can be computed using SH expansion coefficients. We find that the SH power spectra of annual mean CRE have similar shapes in all years from 2001 to 2020. However, the power spectra for different variables at the same expansion degree have some variations. The SW CRE SH power spectrum shape is similar to that of the low and midlevel cloud water path (CWP), and the LW CRE SH power spectrum shape is similar to that of the high CWP. This difference of SW and LW SH power spectra is consistent with the previous studies on the impact of cloud height on SW and LW CRE.
The variation of the CRE SH power spectrum is correlated with large-scale atmospheric circulation such as ENSO and HC. In particular, the anomalies of the CRE = 2 power spectrum component are strongly correlated with the Multivariate ENSO Index (Wolter & Timlin, 1998. The = 2 SH component of SW (LW) CRE anomalies are positive (negative) over Indonesia, the western Pacific and South America, and negative (positive) over the central and eastern Pacific and Africa. The anomaly patterns can be explained by the mechanism of ENSO's impact on clouds suggested by previous studies. The CRE = 3 and = 5 SH power spectrum components are found to be strongly correlated with the HC intensity, whereas the = 6 and = 8 components are correlated with variations in the HC latitudinal widths. The impact of ENSO on the HC can be observed in opposite anomaly patterns of those SH components in ENSO warm versus cold phases.
In summary, the SH expansion technique can be a useful tool to study a variable such as CRE that has global valid values, by decomposing it into different SH components. As we illustrated in this study, the effects of large-scale atmospheric circulation phenomena such as ENSO and HC can be interpreted from the SH power spectrum and individual SH components of CRE. The SH expansion technique provides an alternative perspective to study the variability of global geophysical parameters such as cloud properties and cloud radiative effects.

Data Availability Statement
The CERES EBAF datasets used to obtain cloud property data are publicly available via the CERES project website https://ceres.larc.nasa.gov/data/. The ERA5 datasets used to obtain atmospheric profile and surface albedo are publicly available via the Climate Data Store from https://doi.org/10.24381/cds.6860a573 and https:// doi.org/10.24381/cds.f17050d7. The CAMS datasets used to obtain CO 2 , CH 4 , and CO concentrations are publicly available via the Atmosphere Data Store (https://ads.atmosphere.copernicus.eu/). The MEI time series is publicly available via the website https://psl.noaa.gov/enso/mei/. The HEALPix software used to compute spherical harmonic expansion is publicly available via https://healpix.sourceforge.net/ and https://github.com/ healpy/healpy/. The RRTMG software used to compute radiative flux is publicly available via https://github.com/ AER-RC/RRTMG_LW and https://github.com/AER-RC/RRTMG_SW.