the Creative Commons Attribution 4.0 License.

the Creative Commons Attribution 4.0 License.

# Heuristic estimation of low-level cloud fraction over the globe based on a decoupling parameterization

### Jihoon Shin

Based on the decoupling parameterization of the cloud-topped planetary boundary layer, a simple equation is derived to compute the inversion height. In combination with the lifting condensation level and the amount of water vapor in near-surface air, we propose a low-level cloud suppression parameter (LCS) and estimated low-level cloud fraction (ELF), as new proxies for the analysis of the spatiotemporal variation of the global low-level cloud amount (LCA). Individual surface and upper-air observations are used to compute LCS and ELF as well as lower-tropospheric stability (LTS), estimated inversion strength (EIS), and estimated cloud-top entrainment index (ECTEI), three proxies for LCA that have been widely used in previous studies. The spatiotemporal correlations between these proxies and surface-observed LCA were analyzed.

Over the subtropical marine stratocumulus deck, both LTS and EIS diagnose seasonal–interannual variations of LCA well. However, their use as a global proxy for LCA is limited due to their weaker and inconsistent relationship with LCA over land. EIS is anti-correlated with the decoupling strength more strongly than it is correlated with the inversion strength. Compared with LTS and EIS, ELF and LCS better diagnose temporal variations of LCA, not only over the marine stratocumulus deck but also in other regions. However, all proxies have a weakness in diagnosing interannual variations of LCA in several subtropical stratocumulus decks. In the analysis using all data, ELF achieves the best performance in diagnosing spatiotemporal variation of LCA, explaining about 60 % of the spatial–seasonal–interannual variance of the seasonal LCA over the globe, which is a much larger percentage than those explained by LTS (2 %) and EIS (4 %).

Our study implies that accurate prediction of inversion base height and lifting condensation level is a key factor necessary for successful simulation of global low-level clouds in general circulation models (GCMs). Strong spatiotemporal correlation between ELF (or LCS) and LCA identified in our study can be used to evaluate the performance of GCMs, identify the source of inaccurate simulation of LCA, and better understand climate sensitivity.

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Clouds belong to the most important but uncertain components of the climate
system. Due to their strong shortwave radiative cooling effect on the
Earth, low-level clouds have been the focus of various studies in the past
few decades, both in the observation and modeling communities.
Slingo (1990) estimated that a 4 % increase in the low-level
cloud amount (LCA) has the potential to offset global warming associated with
a doubled CO_{2} concentration. Most low-level clouds exist over the ocean,
mainly due to abundant moisture sources near the surface
(Hahn and Warren, 1999). Among various types of low-level clouds, marine
stratocumulus clouds (MSCs) have received special attention due to their large
spatial coverage and the complexity of physics and dynamic processes
controlling their formation and dissipation (Wood, 2012).
Several planetary boundary layer (PBL) schemes used in general circulation
models (GCMs) have the capability to simulate MSCs and associated feedback
processes in a realistic way (e.g., Lock et al., 2000; Bretherton and Park, 2009; Park and Bretherton, 2009). However, MSCs simulated
by these parameterization schemes or more complex numerical models are the
results of complex interactions among various physics processes. Therefore,
it has not been easy for the climate researchers to understand the feedback
processes of MSCs from the climate perspective. If a simple proxy could diagnose
spatial and temporal variations of MSCs, it would be much easier for the
climate researcher to understand the role of MSCs in future climate, both
qualitatively and quantitatively.

Klein and Hartmann (1993) (KH93 hereafter) showed that a lower-tropospheric
stability (LTS) $\equiv {\mathit{\theta}}_{\mathrm{700}}-{\mathit{\theta}}_{\mathrm{1000}}$ where *θ*_{700} and
*θ*_{1000} are the potential temperatures at 700 and 1000 hPa levels,
respectively, correlates well with seasonal variations of LCA over the
subtropical marine stratocumulus deck. LTS has been widely used as a proxy to
understand the characteristics of MSCs and their impact on future climate. The
success of LTS stems in part from the fact that LTS is correlated with other
factors controlling the formation of MSCs in the subtropical marine
stratocumulus deck such as the sea surface temperature (SST), cold air
advection, free-tropospheric moisture, and subsidence in association with a
subtropical high-pressure system. Using a heuristic Lagrangian MSC model,
Park et al. (2004) (PLR04 hereafter) explored the sensitivity of MSCs to
various environmental conditions in the cold advection regime of the
northeastern subtropical Pacific and to both warm and cold advection regimes
of the eastern equatorial Pacific Ocean. Consistent with
Klein (1997), PLR04 found a positive correlation between the
simulated MSC fraction and strong upstream subsidence, although
Myers and Norris (2013) reported the opposite correlation. PLR04
simulated less MSCs with a drier free atmosphere. However, enhanced longwave
radiative cooling at the top of MSCs capped by a drier free atmosphere (this
process was not included in the PLR04's model) may increase MSCs by enhancing
turbulent vertical moisture transport from the sea surface to overlying MSCs.

Based on the decoupling hypothesis suggested by PLR04 and other proceeding works (e.g., Augstein et al., 1974; Albrecht et al., 1979; Betts and Ridgway, 1988; Bretherton, 1992), Wood and Bretherton (2006) (WB06 hereafter) extended KH93's LTS and suggested an estimated inversion strength (EIS) as a better proxy for LCA in which temperature profiles in the decoupled layer below the inversion and the free troposphere above the inversion are assumed to be close to a moist adiabat that is strongly temperature-dependent. WB06 showed that, compared with LTS, EIS correlates better with LCA over a wide range of stratocumulus regimes, because it captures the lapse rate and boundary layer structure more completely than LTS. Similar to LTS, EIS has been widely used as a good proxy for LCA.

In their derivation of EIS, WB06 assumed that the factor
${z}_{\mathrm{inv}}\cdot ({\mathrm{\Gamma}}_{\mathrm{700}}^{\mathrm{m}}-{\mathrm{\Gamma}}_{\mathrm{LCL}}^{\mathrm{m}})$, where *z*_{inv} is the
inversion height and ${\mathrm{\Gamma}}_{\mathrm{700}}^{\mathrm{m}}$ and ${\mathrm{\Gamma}}_{\mathrm{LCL}}^{\mathrm{m}}$ are the moist
adiabatic lapse rates at 700 hPa and lifting condensation level (LCL) of
near-surface air (*z*_{LCL}), respectively, contributes less to the
correlation relationship between the inversion strength and LTS than the
other terms, so that it was simply set to zero. Although a scaling argument
was provided to justify this simplification, a more fundamental reason for
neglecting this term was associated with the practical difficulty in
estimating and parameterizing *z*_{inv}. PLR04 suggested a heuristic
parameterization for PBL decoupling to simulate stratocumulus cloudiness in
the inversion-capped marine boundary layer. In their study, PBL decoupling is
parameterized as an increasing function of the height difference between
*z*_{inv} and *z*_{LCL}. PLR04's conceptual idea of PBL decoupling was used
in WB06's derivation of EIS and other studies to understand the variation
of cloudiness associated with PBL decoupling (Stevens, 2006; Xiao et al., 2011; Van der Dussen et al., 2014, 2015, 2016; Park, 2014a, b; Dal Gesso et al., 2015a, b; Neggers et al., 2017). In our
study, we suggest a simple heuristic method to estimate *z*_{inv} by
combining PLR04's decoupling parameterization with EIS. By using *z*_{inv},
*z*_{LCL}, and water vapor specific humidity in the surface-based mixed
layer (*q*_{v,ML}), we propose two low-level cloud suppression parameters
(LCS), ${\mathit{\beta}}_{\mathrm{1}}\equiv ({z}_{\mathrm{inv}}+{z}_{\mathrm{LCL}})/\mathrm{\Delta}{z}_{\mathrm{s}}$ and
${\mathit{\beta}}_{\mathrm{2}}\equiv \sqrt{{z}_{\mathrm{inv}}\cdot {z}_{\mathrm{LCL}}}/\mathrm{\Delta}{z}_{\mathrm{s}}$ with Δ*z*_{s}=2750 m, and an estimated low-level cloud fraction, $\text{ELF}\equiv f(\mathrm{1}-{\mathit{\beta}}_{\mathrm{2}})$ with $f\equiv max[\mathrm{0.15},min(\mathrm{1},{q}_{\mathrm{v},\mathrm{ML}}/\mathrm{0.003}\left)\right]$, as new proxy
for the characterization of the spatiotemporal variation of LCA over the
globe. Individual surface and upper-air observations are used to compute
LTS, EIS, LCS, and ELF, and the correlations between these proxies and the
surface-observed seasonal LCA are examined. We also analyzed the recently
proposed estimated cloud-top entrainment index
(ECTEI, Kawai et al., 2017), which is a modified EIS that takes into
account cloud-top entrainment criteria. It will be shown that, compared
with LTS, EIS, and ECTEI, which are mainly designed as proxies for marine
LCA, ELF and LCS are better proxies for the global LCA, applicable over both
the ocean and land.

The structure of this paper is as follows. Section 2.1 provides a detailed
explanation on the conceptual framework used to compute *z*_{inv}, LCS,
ELF, and other related proxies for LCA. Section 2.2 describes the data and
analysis method. The correlations between various proxies and LCA in spatial
and temporal domains over the globe are presented in Sect. 3. A summary
and conclusion are provided in Sect. 4.

## 2.1 Conceptual framework

Following PLR04 and WB06, we assume that the lower troposphere below
700 hPa
consists of four regimes (see Fig. 1): a surface-based mixed layer (ML)
topped at *z*_{ML} with the potential temperature, *θ*_{ML}, and water
vapor specific humidity, *q*_{v,ML}, specified at the reference height,
*z*_{ref} or *p*_{ref} (i.e., *θ*_{ML}=*θ*_{ref},
${q}_{\mathrm{v},\mathrm{ML}}={q}_{\mathrm{v},\mathrm{ref}}$); a decoupled cloud layer (DL) with a vertical gradient
of *θ* approximated by the moist *θ* adiabat at *z*_{ML}
(${\mathrm{\Gamma}}_{\mathrm{DL}}^{\mathrm{m}}>\mathrm{0}$); an inversion at the DL top (*z*_{inv}); and the free
atmosphere with a vertical gradient of *θ* approximated by the moist
*θ* adiabat at *p*=700 hPa (${\mathrm{\Gamma}}_{\mathrm{700}}^{\mathrm{m}}>\mathrm{0}$). The moist adiabatic
lapse rate of *θ* used in our study (${\mathrm{\Gamma}}_{\mathrm{DL}}^{\mathrm{m}}$ and
${\mathrm{\Gamma}}_{\mathrm{700}}^{\mathrm{m}}$ in unit of K m^{−1}) is a function of *T* and *p*;
it increases as *p* increases or *T* increases. The inversion strength at
*z*_{inv}, IS $\equiv {\mathit{\theta}}_{\mathrm{inv}}^{+}-{\mathit{\theta}}_{\mathrm{inv}}^{-}$, becomes

where ${\mathit{\theta}}_{\mathrm{inv}}^{+}$ is the potential temperature just above the
inversion, ${\mathit{\theta}}_{\mathrm{inv}}^{-}$ is the potential temperature just below the
inversion, and LTS $\equiv {\mathit{\theta}}_{\mathrm{700}}-{\mathit{\theta}}_{\mathrm{ML}}$ is the lower-tropospheric stability. By assuming that the contribution of ${z}_{\mathrm{inv}}\cdot ({\mathrm{\Gamma}}_{\mathrm{700}}^{\mathrm{m}}-{\mathrm{\Gamma}}_{\mathrm{DL}}^{\mathrm{m}})$ to variability in the relationship between
IS and LTS is negligibly small due to the opposite variations of *z*_{inv}
and ${\mathrm{\Gamma}}_{\mathrm{700}}^{\mathrm{m}}-{\mathrm{\Gamma}}_{\mathrm{DL}}^{\mathrm{m}}$ with LTS, WB06 derived the so-called
estimated inversion strength, EIS:

which was shown to be a better proxy for LCA than LTS.
Because it does not contain *z*_{inv}, which is hard to estimate, EIS has been used as a convenient proxy for LCA.

If *z*_{inv} can be reasonably estimated instead of being neglected, it may
be possible to construct a better proxy than EIS for LCA. In this study, we
suggest an approach to estimate *z*_{inv} and other related proxies in a
heuristic way based on the decoupling hypothesis suggested by PLR04. From
the analysis of sounding data, PLR04 showed that the decoupling parameter
*α* can be parameterized as an increasing function of the decoupled
layer thickness, $\mathrm{\Delta}{z}_{\mathrm{DL}}\equiv {z}_{\mathrm{inv}}-{z}_{\mathrm{ML}}$,

where the scale height, Δ*z*_{s}≈2750 m, is obtained from the
analysis of a set of sounding data over the ocean (see PLR04). Originally,
PLR04 defined *α* for the condensate potential temperature
${\mathit{\theta}}_{\mathrm{c}}\equiv \mathit{\theta}-({L}_{\mathrm{v}}/{C}_{\mathrm{p}})\cdot {q}_{\mathrm{l}}-({L}_{\mathrm{s}}/{C}_{\mathrm{p}})\cdot {q}_{\mathrm{i}}$ with *γ*≈1.1–1.3, where *q*_{l} and *q*_{i} are the cloud liquid and ice
contents, respectively, and *θ*_{c} is a conserved scalar with respect to
the phase change. In our study, however, *α* is defined for *θ*
and, accordingly, *γ* is slightly reduced to 1 to account for
${\mathit{\alpha}}_{\mathit{\theta}}>{\mathit{\alpha}}_{{\mathit{\theta}}_{\mathrm{c}}}$ in the cloud-topped decoupled layer.
The choice of *γ*=1 allows us to obtain analytical expressions for
various proxies as will be shown below. The small (large) *α* indicates
that the environmental air at the inversion base, ${z}_{\mathrm{inv}}^{-}$, is well
connected to (decoupled from) the surface air property with abundant
moisture, providing more (less) favorable conditions for the formation of LCA
at ${z}_{\mathrm{inv}}^{-}$. It should be noted that *α* only measures the degree
of vertical decoupling of thermodynamic properties between *z*_{ML} and
${z}_{\mathrm{inv}}^{-}$. That is, *α* does not provide information regarding
the amount of surface moisture.

By combining Eq. (3) with ${\mathit{\theta}}_{\mathrm{inv}}^{+}={\mathit{\theta}}_{\mathrm{700}}-{\mathrm{\Gamma}}_{\mathrm{700}}^{\mathrm{m}}\cdot ({z}_{\mathrm{700}}-{z}_{\mathrm{inv}})$ and ${\mathit{\theta}}_{\mathrm{inv}}^{-}={\mathit{\theta}}_{\mathrm{ML}}+{\mathrm{\Gamma}}_{\mathrm{DL}}^{\mathrm{m}}\cdot ({z}_{\mathrm{inv}}-{z}_{\mathrm{ML}})$, we can derive the following expressions for the inversion height,

which can be also written as ${z}_{\mathrm{inv}}=\mathit{\alpha}\cdot \mathrm{\Delta}{z}_{\mathrm{s}}+{z}_{\mathrm{ML}}$ from Eq. (3). Then, the inversion strength, IS, becomes

and decoupling strength, DS $\equiv {\mathit{\theta}}_{\mathrm{inv}}^{-}-{\mathit{\theta}}_{\mathrm{ML}}$, becomes

and $\text{LTS}=\text{IS}+\text{DS}+{\mathrm{\Gamma}}_{\mathrm{700}}^{\mathrm{m}}\cdot ({z}_{\mathrm{700}}-{z}_{\mathrm{inv}})$, as shown in
Fig. 1. Once *θ*_{ML}=*θ*_{ref}, ${q}_{\mathrm{v},\mathrm{ML}}={q}_{\mathrm{v},\mathrm{ref}}$, and
*z*_{ML} are obtained, we can consecutively compute $\text{LTS}={\mathit{\theta}}_{\mathrm{700}}-{\mathit{\theta}}_{\mathrm{ref}}$ and *z*_{inv} using the first expression of Eq. (4), *α*
from Eq. (3), IS from Eq. (5), and DS from Eq. (6). Note that IS is identical
to the sum of EIS and ${z}_{\mathrm{inv}}\cdot ({\mathrm{\Gamma}}_{\mathrm{700}}^{\mathrm{m}}-{\mathrm{\Gamma}}_{\mathrm{DL}}^{\mathrm{m}})$, the
neglected term in the original formulation of WB06.

Following previous studies (e.g., PLR04 and WB06), we will assume that
*z*_{ML}≈*z*_{LCL} over the ocean. However, due to insufficient moisture
at the surface, it is likely that *z*_{ML}<*z*_{LCL} over most land areas
unless strong buoyancy or shear production near the surface sufficiently
deepens the surface-based mixed layer. It may be possible to parameterize
*z*_{ML} as a function of turbulent kinetic energy within the PBL; however,
for simplicity, we assume that *z*_{ML}≈*z*_{LCL} over the entire
globe.

As mentioned above, *α* only measures the degree of vertical decoupling
of thermodynamic properties between the inversion base and surface air, not
the surface moisture itself. Conceptually, however, the formation of LCA at
the inversion base is likely to be influenced by both surface moisture and
*α*. As a simple but practical proxy representing surface moisture, we
select *z*_{LCL}. From the simple conceptual argument that small (large)
*z*_{LCL} and *α* are likely to be associated with large (small) LCA at
the inversion base, we define the first low-level cloud suppression
parameter (LCS), *β*_{1}, as

where the second equality is obtained by assuming *z*_{ML}≈*z*_{LCL}
and *μ*=2. In principle, *μ* can be estimated in an empirical way using
multiple linear regression analysis of LCA fitted to *z*_{inv} and *z*_{LCL}
such that *β*_{1} explains the maximum fraction of the variance of LCA. We
performed a multiple linear regression analysis of LCA on *z*_{inv} and
*z*_{LCL} using individual seasonal data in each 2.5^{∘}
latitude × 5^{∘} longitude grid box over the globe. Except over some portions, the
values of *μ* were mostly between 0 and 4 with an approximate average
value, *μ*=2 (not shown). The resulting *β*_{1} is a non-dimensional sum
of *z*_{inv} and *z*_{LCL}. By simply extending *β*_{1}, we also define the
second LCS, *β*_{2}, as a non-dimensional product of *z*_{inv} and
*z*_{LCL},

which, similar to *β*_{1}, increases as the surface air becomes drier and
the PBL deepens. In the case of *z*_{ML}≈*z*_{LCL}, it becomes
${\mathit{\beta}}_{\mathrm{1}}\approx ({z}_{\mathrm{LCL}}/\mathrm{\Delta}{z}_{\mathrm{s}})\cdot [\mathrm{2}+({z}_{\mathrm{inv}}-{z}_{\mathrm{ML}})/{z}_{\mathrm{ML}}]$ and
${\mathit{\beta}}_{\mathrm{2}}\approx ({z}_{\mathrm{LCL}}/\mathrm{\Delta}{z}_{\mathrm{s}})\cdot \sqrt{\mathrm{1}+({z}_{\mathrm{inv}}-{z}_{\mathrm{ML}})/{z}_{\mathrm{ML}}}$,
where the first factor in both formulae represents the degree of
subsaturation of near-surface air, and the second factor represents the
decoupling strength. Note that *z*_{LCL} oppositely controls the surface
moisture parameter and decoupling strength: small *z*_{LCL} decreases the
first factor but increases the second factor and vice versa. Large (small)
values of *β*_{1} and *β*_{2} likely favor the dissipation (formation) of
LCA at the inversion base.

Finally, we define the estimated low-level cloud fraction (ELF) as

where *f* is the freeze-dry factor (Vavrus and Waliser, 2008),

designed to reduce the parameterized cloud fraction in the extremely cold and
dry atmospheric conditions typical of polar and high-latitude winter. Cloud
fraction parameterization based on the grid-mean relative humidity (RH)
assumes that there is a certain amount of subgrid variability of
thermodynamic scalars (e.g., Park et al., 2014), which allows the
formation of cloud fraction even when the grid-mean RH is smaller than 1. In
the very stable Arctic and high-latitude atmosphere during winter, however,
there is little subgrid variability (Jones et al., 2004). Thus, any
grid-mean RH-based cloud fraction parameterization (e.g., our LCS based on
*z*_{LCL}) is likely to predict too much LCA there. Vavrus and Waliser (2008)
showed that the implementation of the freeze-dry factor into GCMs substantially
reduced the simulated LCA in the Arctic and high-latitude regions during
winter and improved the simulation. Compared to *β*_{2}, ELF diagnoses
smaller LCA in the high-latitude region during winter where the amount of
water vapor within the ML is often smaller than 3 g kg^{−1} (see Fig. 2e, f).
If *z*_{ML}≈*z*_{LCL}, it becomes $\text{ELF}=f\cdot [\mathrm{1}-({z}_{\mathrm{LCL}}/\mathrm{\Delta}{z}_{\mathrm{s}}\left)\sqrt{\mathrm{1}+({z}_{\mathrm{inv}}-{z}_{\mathrm{ML}})/{z}_{\mathrm{ML}}}\right]$, where *f* denotes
the amount of water vapor in the surface-based ML air, *z*_{LCL} represents
the degree of subsaturation of near-surface air, and
$({z}_{\mathrm{inv}}-{z}_{\mathrm{ML}})/{z}_{\mathrm{ML}}$ quantifies the degree of thermodynamic decoupling
of the inversion base air from the surface-based ML air. Our ELF predicts
that LCA increases as the near-surface air becomes more saturated with enough
water vapor and as the PBL becomes more vertically coupled. When
the inversion base air is fully coupled with saturated near-surface air
containing enough water vapor, ELF approaches its upper bound of 1.
Considering that low-level clouds usually form at *z*_{inv}, the conceptual
cloud formation processes embedded in ELF are consistent with what is expected
to happen in nature. We defined ELF using *β*_{2} instead of *β*_{1},
because *β*_{2} has a better global performance than *β*_{1} (see Tables 1, 3, 4, 6). Note that our ELF can have small negative values, which,
however, can be reset to zero for more complete parameterization. In our
study, we do not reset ELF to zero.

In reality, the key factor controlling the formation of clouds is the
relative humidity. According to our conceptual framework, most of the clouds
are likely to form at the inversion base. To compute the relative humidity at
the inversion base, ${\text{RH}}_{\mathrm{inv}}^{-}$, we follow PLR04 and assume that the
decoupling parameter, *α*, defined in Eq. (3) also describes the
decoupling of the water vapor specific humidity, *q*_{v}. Then, the water vapor
specific humidity (${q}_{\mathrm{v},\mathrm{inv}}^{-}$) and potential temperature
(${\mathit{\theta}}_{\mathrm{inv}}^{-}$) at the inversion base can be computed as
${q}_{\mathrm{v},\mathrm{inv}}^{-}=\mathit{\alpha}\cdot {q}_{\mathrm{v},\mathrm{inv}}^{+}+(\mathrm{1}-\mathit{\alpha})\cdot {q}_{\mathrm{v},\mathrm{ML}}$ and
${\mathit{\theta}}_{\mathrm{inv}}^{-}=\mathit{\alpha}\cdot {\mathit{\theta}}_{\mathrm{inv}}^{+}+(\mathrm{1}-\mathit{\alpha})\cdot {\mathit{\theta}}_{\mathrm{ML}}$.
The lapse rate of *q*_{v} in the free atmosphere ${\mathrm{\Gamma}}_{\mathrm{700}}^{{q}_{\mathrm{v}}}$, which
is required to compute ${q}_{\mathrm{v},\mathrm{inv}}^{+}={q}_{\mathrm{v},\mathrm{700}}-{\mathrm{\Gamma}}_{\mathrm{700}}^{{q}_{\mathrm{v}}}\cdot ({z}_{\mathrm{700}}-{z}_{\mathrm{inv}})$, is obtained from the linear slope
of *q*_{v} at the 700 and 750 hPa levels. Using ${q}_{\mathrm{v},\mathrm{inv}}^{-}$ and
${\mathit{\theta}}_{\mathrm{inv}}^{-}$, we compute ${\text{RH}}_{\mathrm{inv}}^{-}$ as an additional proxy for LCA.
We note that only ${\text{RH}}_{\mathrm{inv}}^{-}$ needs the information on ${q}_{\mathrm{v},\mathrm{inv}}^{-}$;
the other proxies do not need information on the vertical profile of *q*_{v}
other than *q*_{v,ML} to compute *z*_{LCL}.

## 2.2 Data and analysis

The surface observation data used in our study are from the Extended Edited
Cloud Report Archive (EECRA, Hahn and Warren, 1999) for January 1956 to
December 2008 over the ocean (without sea ice) and January 1971 to December 1996
over land. The EECRA compiles individual ship and land observations of
clouds (e.g., cloud amount and cloud type for each low, middle, and
upper level), present weather (e.g., fog, rain, snow, thunderstorm shower,
and drizzle), and other coincident surface meteorologies (e.g., sea level
pressure, sea surface temperature, ship-deck air temperature, dew-point
depression, and wind speed and direction) at every 3 or 6 h based on the
strict hierarchy of the World Meteorological Organization (WMO,
WMO, 1975). Following Park and Leovy (2004), we filtered out the
observations obtained under poor illumination conditions (i.e., the moonlight
screening criteria, Hahn et al., 1995), from the WMO's Historical Sea
Surface Temperature (HSST) Data Project (identified by card deck numbers
150–156), or with any missing surface meteorologies or cloud information. In
this study, we focus on the analysis of LCA of all cloud types including
cumulus as well as stratus. The upper-level meteorologies (e.g., *p*,
*θ*, and *q*_{v}) are from the ERA-Interim reanalysis products (ERAI,
Simmons et al., 2007) from January 1979 to December 2008 at 6-hourly time
intervals. Spatial and temporal interpolations are performed to compute the
upper-level meteorologies at the exact time and location at which the EECRA
surface observers reported LCA. Because both EECRA and ERAI are necessary,
our analysis only uses the data from January 1979 to December 2008 (30 years)
over the ocean and January 1979 to December 1996 over land (18 years). Using
the 6-hourly ERAI vertical profiles of *θ* and *q*_{v} interpolated to
individual EECRA surface observations, we computed the 12 proxies for
LCA (LTS, EIS, ECTEI, IS, DS, *α*, *z*_{LCL}, *z*_{inv}, ${\text{RH}}_{\mathrm{inv}}^{-}$,
*β*_{1}, *β*_{2}, ELF) and averaged them into 5^{∘}
latitude × 10^{∘} longitude seasonal data for each year. Thus, if there is no missing
grid value, a total of 120 and 72 seasonal datapoints is available in each
grid box over the ocean and land, respectively. To reduce the impact of
random noise in association with the small observation number, the year with
an observation number smaller than 10 in each season and grid box is not used
in our analysis.

Three separate correlation analyses were performed between LCA and the 12 proxies: spatial–seasonal correlation analysis using climatological seasonal (i.e., DJF, MAM, JJA, SON) grid data (Table 1 and Figs. 3–6; Table 4 and Figs. 11, 12) or climatological seasonal data averaged over selected regions (Figs. 7, 8, 12), seasonal–interannual and interannual correlation analysis using seasonal grid data from each year (Figs. 9–10, 13) or seasonal data averaged over selected regions from each year (Tables 2, 5), and combined spatial–seasonal–interannual correlation analysis (Tables 3, 6) using seasonal grid data over the globe in each year. For the spatial–seasonal correlation analysis, only the seasons and grid boxes with a climatological seasonal observation number equal to or larger than 100 are used. For any temporal correlation analysis containing interannual variations, we only used the years and seasons with a seasonal observation number equal to or larger than 10 for each year within the grid box or selected regions. Only the grid boxes with the number of effective seasonal data points equal to or larger than 50 out of the maximum 120 over the ocean (30 years) and 72 over land (18 years) are used for the temporal correlation analysis.

As the reference height near the surface, we tested both *p*_{ref}=*p*_{1000}
and *p*_{ref}=*p*_{sfc}. In both cases, the density of the atmosphere within
the lower troposphere below 700 hPa was set to *ρ*=1 kg m^{−3}. When
*p*_{ref}=*p*_{1000}, all thermodynamic properties at *p*_{1000} (e.g., *p*,
*θ*, and *q*_{v}) are obtained from 6-hourly ERA-Interim data interpolated
into the time and location of individual EECRA observations. The geometric
height of *p*_{1000} from the surface (*z*_{1000}) is computed using a
hydrostatic equation. If the air at *p*_{1000} is saturated, *z*_{LCL} is set
to *z*_{1000}. When *p*_{ref}=*p*_{sfc}, a set of thermodynamic properties at
*p*_{sfc} (e.g., *θ*, *q*_{v}) is obtained from two different data
sources: one is from the 6-hourly ERA-Interim data interpolated into the
time and location of individual EECRA observations similar to the case of
*p*_{ref}=*p*_{1000} and the other is from EECRA surface observations. When the
surface air is saturated, we set *z*_{LCL}=*z*_{sfc}. Three separate analyses
were performed using individual datasets (e.g., *p*_{ref}=*p*_{1000},
*p*_{ref}=*p*_{sfc} with ERA-Interim surface data, and *p*_{ref}=*p*_{sfc} with
EECRA surface observations). In the next section, we will show the results
based on *p*_{ref}=*p*_{sfc} with EECRA surface observations of *p*_{sfc},
*θ*_{sfc}, and *q*_{v,sfc}. The analyses using the other two choices
produced similar results. It is not shown here, but the same analysis using
the NCEP/NCAR reanalysis product (Kalnay et al., 1996) instead of the
ERA-Interim data also produced similar results.

A fundamental assumption of our decoupling hypothesis is that thermodynamic
scalars at the inversion base (${\mathit{\theta}}_{\mathrm{inv}}^{-}$) are bounded by the ML
(*θ*_{ML}) and inversion top (${\mathit{\theta}}_{\mathrm{inv}}^{+}$) properties. That is,
the decoupling parameter *α* is between 0 and 1 (see Eq. 3). However,
the inversion height, *z*_{inv}, estimated from Eq. (4) can have any numerical
values, such that *α* parameterized by *z*_{inv} (i.e., $\mathit{\alpha}=({z}_{\mathrm{inv}}-{z}_{\mathrm{ML}})/\mathrm{\Delta}{z}_{\mathrm{s}}$) can be smaller than 0 or larger than 1. To be
consistent with our decoupling hypothesis, we reset *z*_{inv}=*z*_{ML}
whenever *z*_{inv} estimated from Eq. (4) is smaller than *z*_{ML} (i.e., we
wrap all *α*<0 to *α*=0), and, similarly, we reset ${z}_{\mathrm{inv}}=\mathrm{\Delta}{z}_{\mathrm{s}}+{z}_{\mathrm{ML}}$ whenever the estimated *z*_{inv} is larger than Δ*z*_{s}+*z*_{ML} (i.e., we wrap all *α*>1 to *α*=1). As shown in Fig. 2,
the cases of *α*=0 after wrapping frequently occur in stably stratified
regimes over the Arctic and northern continents in winter, desserts during
the night, the northwestern Pacific and southern-hemispheric (SH) circumpolar regions during
boreal summer when warm air is advected into cold SST, and along the west
coast of major continents where cold SST exists due to coastal upwelling. On
the other hand, the cases of *α*=1 after wrapping occur in unstable
regimes over the tropical SST warm pools in JJA and along the midlatitude
storm tracks during winter. In the next section, we will first present the
results based on the data of $\mathrm{0}<\mathit{\alpha}<\mathrm{1}$ and then all data of
$\mathrm{0}\le \mathit{\alpha}\le \mathrm{1}$ (e.g., the sum of $\mathrm{0}<\mathit{\alpha}<\mathrm{1}$, *α*=0, and
*α*=1).

## 3.1 Spatial–seasonal correlation

Figures 3 and 4 show the seasonal climatology of LCA (solid lines; also see
Fig. 7 for the color maps of LCA in the 2.5^{∘} latitude × 5^{∘} longitude grid
box) and 12 proxies for LCA defined in the previous section during JJA
and DJF, respectively. The spatial–seasonal correlation coefficients
between these variables are summarized in Table 1. The scatter plots between
the 12 proxies and LCA over the ocean and land are shown in Figs. 5
and 6, respectively.

Consistent with KH93 and WB06, both LTS and EIS are strong in subtropical
marine stratocumulus decks west of major continents but weak in the tropical
deep convection regime in which LCA is small. Although the overall pattern of
EIS looks similar to that of LTS, several notable differences exist between
them. For example, in the vicinity of polar and high-latitude regions in both
hemispheres where LCA is large, EIS is strong but LTS is weak, which seems
to contribute to a low spatial–seasonal correlation between LTS and EIS
over the globe (see Table 1). According to Eq. (2), EIS here should be
weaker than in other regions, because *z*_{LCL} is low in these cold
regions (see Figs. 3g and 4g). However, probably due to the decreases
in ${\mathrm{\Gamma}}_{\mathrm{700}}^{\mathrm{m}}$ and *z*_{700} in the cold region, EIS becomes strong
here, resulting in a better spatial–seasonal correlation between EIS and
LCA (*r*=0.62) than between LTS and LCA ($r=-\mathrm{0.05}$) over the ocean.
Table 1 shows that the spatial–seasonal correlation between LTS and LCA is
very weak over the ocean and land. This is somewhat surprising, because
KH93 reported that LTS is significantly positively correlated with LCA. It
should be noted that KH93's finding was based on the analysis over the marine
stratocumulus deck, while our result is based on global analysis. Because of
this potential sensitivity of the relationship between LTS and LCA to the
analysis domain, we need to be careful when using LTS as a global proxy for
LCA in climate sensitivity studies.

The spatial pattern of the inversion strength (IS, Figs. 3d and 4d),
which is about 1–3 K higher than EIS, is roughly similar to that of EIS.
However, in contrast to EIS but similar to LTS, IS tends to be small in
high-latitude regions, which results in a weaker spatial–seasonal
correlation between IS and LCA over the ocean (*r*=0.39) than between EIS
and LCA. Table 1 shows that the decoupling strength (DS, Figs. 3e and 4e)
is almost perfectly correlated with EIS over the globe ($r=-\mathrm{0.98}$), with a
correlation with LCA very similar to that of EIS. However, the named EIS has a weaker correlation with inversion strength
than anti-correlation with decoupling strength. In other words, a strong EIS
indicates that the air at the inversion base is coupled well with the
surface-based ML air. The decoupling parameter (*α*, Figs. 3f and 4f)
also has a near-perfect correlation with EIS ($r=-\mathrm{0.98}$), supporting our
interpretation of EIS. When ${\mathrm{\Gamma}}_{\mathrm{700}}^{\mathrm{m}}\approx {\mathrm{\Gamma}}_{\mathrm{DL}}^{\mathrm{m}}$, it
becomes $\mathit{\alpha}=\text{DS}/({\mathrm{\Gamma}}_{\mathrm{DL}}^{\mathrm{m}}\cdot \mathrm{\Delta}{z}_{\mathrm{s}})\approx \mathrm{1}-\text{EIS}/({\mathrm{\Gamma}}_{\mathrm{DL}}^{\mathrm{m}}\cdot \mathrm{\Delta}{z}_{\mathrm{s}})$. As will be shown
later, stronger correlation among EIS–DS–*α* than EIS–IS also exists in
the combined spatial–seasonal–interannual correlation statistics (Table 3)
and in the analysis using all of the observation data (Tables 4 and 6).

The inversion height (${z}_{\mathrm{inv}}=\mathit{\alpha}\cdot \mathrm{\Delta}{z}_{\mathrm{s}}+{z}_{\mathrm{LCL}}$, Figs. 3h
and 4h), which is strongly correlated with EIS, DS, and *α*, has a
high spatial–seasonal correlation with LCA ($r=-\mathrm{0.69}$ over the ocean and
$r=-\mathrm{0.77}$ over land), even better than EIS (*r*=0.62 over the ocean and
*r*=0.07 over land). The superiority of *z*_{inv} over EIS as a proxy for
LCA is more pronounced over land. Why does *z*_{inv} characterize LCA better
than EIS over land? If ${\mathrm{\Gamma}}_{\mathrm{700}}^{\mathrm{m}}\approx {\mathrm{\Gamma}}_{\mathrm{DL}}^{\mathrm{m}}$, ${z}_{\mathrm{inv}}=\mathrm{\Delta}{z}_{\mathrm{s}}-\text{EIS}/{\mathrm{\Gamma}}_{\mathrm{DL}}^{\mathrm{m}}+{z}_{\mathrm{LCL}}$ such that the different correlation
characteristics of *z*_{inv} and EIS with LCA are likely associated with
*z*_{LCL}. In the case that surface air is saturated and very low-level
clouds are formed (e.g., fog with a low *z*_{LCL}), small *z*_{LCL} causes
EIS to decrease ($\text{EIS}\approx \text{LTS}-{\mathrm{\Gamma}}_{\mathrm{DL}}^{\mathrm{m}}\cdot ({z}_{\mathrm{700}}-{z}_{\mathrm{LCL}})$
from Eq. 2). This contributes to the weaker, negative EIS–LCA correlation
seen over land compared to the stronger, positive correlation observed in the
marine stratocumulus deck. Because *z*_{inv} is defined as the addition of
*z*_{LCL} to EIS, the undesirable negative impact of *z*_{LCL} to the
correlation with LCA is removed from EIS compared to *z*_{inv}. Over the
western continental United States during summer, both *z*_{LCL} and *z*_{inv} are
at their maximum where LCA is at its minimum.
In this region, similar to the ocean
case, *z*_{inv} is negatively correlated with LCA; however, opposite to the
ocean case, EIS is not positively correlated with LCA. Consequently, EIS
has a weaker global spatial–seasonal correlation with LCA (*r*=0.22)
than *z*_{inv} ($r=-\mathrm{0.67}$). It is very interesting to note that a simple
proxy, *z*_{LCL}, shows a stronger spatial–seasonal (and also
spatial–seasonal–interannual) correlation with LCA than LTS, EIS, and
*z*_{inv} over both the ocean and land.

According to our conceptual framework, low-level clouds likely form just
below *z*_{inv} (see Fig. 1). If that is the case, it is likely that the
relative humidity at the base of *z*_{inv} (${\text{RH}}_{\mathrm{inv}}^{-}$) is a good proxy for
LCA. Over both the ocean and land, ${\text{RH}}_{\mathrm{inv}}^{-}$ shows a stronger correlation
with LCA than LTS and EIS but a weaker correlation than *z*_{inv} and
*z*_{LCL}. We speculate that this relatively poor performance of
${\text{RH}}_{\mathrm{inv}}^{-}$ compared to *z*_{inv} and *z*_{LCL} is due in part to the
poor estimation of ${q}_{\mathrm{v},\mathrm{inv}}^{-}$ rather than indicating that ${\text{RH}}_{\mathrm{inv}}^{-}$
is a poor proxy for LCA. Because ${\text{RH}}_{\mathrm{inv}}^{-}$ is estimated using the roughly
estimated ${\mathit{\theta}}_{\mathrm{inv}}^{-}$ and ${q}_{\mathrm{v},\mathrm{inv}}^{-}$ without performing any
saturation adjustment, the absolute values of ${\text{RH}}_{\mathrm{inv}}^{-}$ in our analysis
can be unreasonably larger than 100 % in some regions (Figs. 3i and 4i).

The spatial patterns of two LCS parameters,
${\mathit{\beta}}_{\mathrm{1}}=({z}_{\mathrm{inv}}+{z}_{\mathrm{LCL}})/\mathrm{\Delta}{z}_{\mathrm{s}}$ and ${\mathit{\beta}}_{\mathrm{2}}=\sqrt{{z}_{\mathrm{inv}}\cdot {z}_{\mathrm{LCL}}}/\mathrm{\Delta}{z}_{\mathrm{s}}$, are roughly similar to those of *z*_{inv} and
*z*_{LCL}. However, both LCS parameters have a better spatial–seasonal
correlation with LCA than *z*_{inv} and *z*_{LCL}, with *β*_{2} showing a
slightly better performance than *β*_{1} (see Table 1 and Figs. 5, 6). The
estimated low-level cloud fraction, $\text{ELF}=f(\mathrm{1}-{\mathit{\beta}}_{\mathrm{2}})$, shows a very similar
correlation to *β*_{2} because the freeze-dry factor, *f*, is approximately 1
in the regime of $\mathrm{0}<\mathit{\alpha}<\mathrm{1}$ (see Fig. 2). Among all 12 proxies,
*β*_{2} and ELF have the highest spatial–seasonal correlation with LCA over
both the ocean and land. ECTEI is equivalent if slightly better than EIS for
the ocean, equivalent over land, and between LTS and EIS over the globe.

Figure 8 shows the scatter plots between the seasonal LCA and 12
proxies in several regions shown in Fig. 7. LTS over land shows a stronger
interregional–seasonal correlation with LCA than that over the ocean;
however, EIS (and ECTEI, DS, and *α*) over the ocean has a stronger
correlation than that over land. Over both the ocean and land, *z*_{inv} and
*z*_{LCL} are strongly correlated with LCA. ${\text{RH}}_{\mathrm{inv}}^{-}$ is strongly
correlated with LCA over the ocean and land; however, the correlation
over the globe is weak. Each of *β*_{1} and *β*_{2}–ELF explains about
80 % of the interregional–seasonal variations of the seasonal LCA, which is a
much larger percentage than those explained by LTS (25 %), EIS (25 %), and
ECTEI (7 %). Except LTS, the difference between the regression slopes for the
ocean and land in each proxy shown in Fig. 8 is generally larger than the
seasonal differences between the regression slopes shown in Figs. 5 and 6,
indicating a need to incorporate additional factors characterizing the
contrast between the ocean and land in the future.

## 3.2 Seasonal–interannual correlation

Figure 9 shows the global distribution of seasonal–interannual correlation
coefficients between the seasonal LCA and 12 proxies. Over the North
Pacific and subtropical marine stratocumulus deck west of major continents,
LTS and LCA are positively correlated. However, a negative correlation also
exists, for example, over Europe and the northwestern Atlantic. Similar to LTS,
EIS and LCA are positively correlated over marine stratocumulus deck; but
interestingly, the correlation signs over the northwestern Atlantic and Europe
and several continents (e.g., Southeast Asia, Australia, South America, and
Central Africa) are reversed compared with LTS. As mentioned before,
*z*_{LCL} mainly contributes to the different correlation characteristics
between EIS and LTS with LCA in these regions (see Eq. 2). Although LTS and
EIS are good proxies for LCA over the marine stratocumulus deck, they have a
limitation as a global proxy due to the spatial changes of temporal
correlation signs. ECTEI shows very similar correlation characteristics to
EIS. With an opposite sign, the overall correlation patterns of DS and
*α* are quite similar to that of EIS. Compared with the other proxies,
LCA tends to be more homogeneously correlated with *z*_{LCL}, *z*_{inv},
${\text{RH}}_{\mathrm{inv}}^{-}$, *β*_{1}, and *β*_{2} and ELF over the globe, without the
spatial reversal of the correlation sign. Figure 10 shows a similar
temporal correlation analysis to that shown in Fig. 9 but only for interannual
variations without the seasonal cycle. Overall, the spatial pattern of
interannual correlation is similar to that of seasonal–interannual
correlation; however, the magnitude is reduced, particularly over the marine
stratocumulus deck. Both in terms of seasonal–interannual and interannual
correlations, *β*_{2} and ELF perform better than LTS and EIS over the entire
globe without the spatial changes of temporal correlation signs; thus, they
are well suited as global proxies for LCA. *β*_{1} performs nearly as well
if not equivalently to *β*_{2} and ELF.

Table 2 summarizes the temporal correlation coefficients between LCA and
12 proxies for various regions shown in Fig. 7. The first six regions in
our analysis are the subtropical marine stratocumulus decks. Although less
well known than the other regions, stratocumulus and fog also exist over the
Arabian Sea during summer and thus are included in our analysis (see
Park and Leovy, 2004; Schubert et al., 1979; and also Fig. 7a). The
next three regions are the midlatitude marine stratocumulus decks which have
large LCA and undergo frequent passages of synoptic storms (North Pacific,
North Atlantic, SH circumpolar). The last seven regions are over the
continents and have small LCA. China is a unique continental stratocumulus
deck with high LCA. Over the marine stratocumulus decks, except in the SH
circumpolar region, LTS has a significant positive seasonal–interannual
correlation with LCA. In all regions except the Arabian Sea, both EIS and
ECTEI show strong correlations similar to that of LTS. Although mainly
designed as a proxy for the marine stratocumulus fraction, LTS is also
positively correlated with some continental LCA in a statistically
significant way. On the other hand, as is also shown in Fig. 9, EIS and ECTEI
are strongly negatively correlated with the continental LCA over India, South
America, and the southwestern Sahara. Similar to the spatial–seasonal
correlations, the temporal correlation characteristics of DS and *α*
with LCA are very similar to that of EIS. Over the stratocumulus decks,
*z*_{inv} is a proxy that is as good as or better than LTS and EIS. Notably,
*z*_{LCL} shows a very good performance over land (and also in the
midlatitude marine stratocumulus decks), indicating that surface moisture
(regardless of whether it is locally originated or advected) substantially
contributes to the temporal variations of the continental LCA. In most areas,
${\text{RH}}_{\mathrm{inv}}^{-}$ has a significant positive seasonal–interannual correlation with
LCA. In general, the interannual correlation tends to be weaker than the
seasonal–interannual correlation. In the subtropical marine stratocumulus
decks, except the Namibian and Australian ones, the seasonal cycle dominantly
contributes to the temporal correlation between various proxies and LCA.
Regardless of whether the seasonal cycle is included or not, two LCS
parameters (*β*_{1},*β*_{2}) and ELF have better temporal correlations
with LCA than any other proxies over almost all regions, including the marine
stratocumulus decks.

Table 3 summarizes the combined spatial–seasonal–interannual correlations
between the seasonal LCA and 12 proxies. Over the ocean, EIS and
ECTEI show better correlations with LCA than LTS. EIS correlates better
with DS and *α* than with IS. As a global proxy for LCA, both
*z*_{LCL} and *z*_{inv} perform better than LTS, EIS, and ECTEI because of
their better performance over land. A very simple proxy, *z*_{LCL} performs
better than *z*_{inv} and explains almost 50 % of the spatiotemporal
variations of LCA over the globe. Among all 12 proxies, *β*_{2} and ELF
explain the largest fraction of spatial and temporal variations of the
seasonal LCA (more than 56 % over the entire globe), much more than the ones
explained by LTS (3 %) and EIS (4 %). Over the ocean, *β*_{2} performs
slightly better than *β*_{1} with a similar performance over land.

## 3.3 Extended analysis using all data

Until now, we have presented results based on the analysis of the data
satisfying $\mathrm{0}<\mathit{\alpha}<\mathrm{1}$ (i.e., ${z}_{\mathrm{LCL}}<{z}_{\mathrm{inv}}<\mathrm{\Delta}{z}_{\mathrm{s}}+{z}_{\mathrm{LCL}}$) when our
decoupling hypothesis can be applied without any conceptual ambiguity.
However, for a fair comparison with the previous proxies of LTS, EIS, and
ECTEI, which can be defined in any of the cases, it is necessary to examine the
performance of various proxies in general situations. This section provides
an extended analysis using all observation data of $\mathrm{0}\le \mathit{\alpha}\le \mathrm{1}$ (i.e.,
the sum of $\mathrm{0}<\mathit{\alpha}<\mathrm{1}$, *α*=0, and *α*=1). Figure 2 shows the
frequency of occurrence of *α*=0 and *α*=1 among all observations
during JJA and DJF. The cases of *α*=0 frequently occur when the
lower troposphere is highly stable (i.e., *z*_{inv}≤*z*_{LCL}) in the
vicinity of the Arctic area north of 60^{∘} N, the northern continents and
desserts during boreal winter, the northwestern Pacific and SH circumpolar
regions during boreal summer, and along the west coast of major continents.
On the other hand, the cases of *α*=1 occur when the troposphere is
highly unstable (i.e., ${z}_{\mathrm{inv}}\ge \mathrm{\Delta}{z}_{\mathrm{s}}+{z}_{\mathrm{LCL}}$) in the tropical warm
pool regions during JJA and the midlatitude storm tracks during boreal
winter. The differences between the results presented in this section and
previous sections are due to the inclusion of the data obtained from these
highly stable (*α*=0) and unstable regimes (*α*=1), mostly from
stable regimes. This comparison allows us to obtain insights into the impact
of extreme vertical stratification in the lower troposphere on the
relationship between various proxies and LCA.

Figure 11 shows the seasonal climatology of LCA (solid lines) and six key
proxies (LTS, EIS, ECTEI, *β*_{1}, *β*_{2}, and ELF) during JJA and DJF,
respectively, for all observation data. The overall pattern of the
climatological proxies is similar to those with $\mathrm{0}<\mathit{\alpha}<\mathrm{1}$ (Figs. 3, 4), but
the magnitude in the high-latitude regions is amplified due mainly to the
contribution of the data in the very stable regimes. In particular, over the
northern continents during DJF, the values of LTS, EIS, and ECTEI
(*β*_{1}/*β*_{2}) are substantially higher (lower) than those of
$\mathrm{0}<\mathit{\alpha}<\mathrm{1}$. The spatial–seasonal correlation between LTS and EIS (*r*=0.85
in Table 4) is now much stronger than the previous case of $\mathrm{0}<\mathit{\alpha}<\mathrm{1}$
($r=-\mathrm{0.08}$ in Table 1), but, in contrast to EIS, LTS still tends to decrease
with latitude in the SH high-latitude regions.

Figure 12 shows the scatter plots between LCA and six key proxies over the
ocean, land, globe, and several regions defined in Fig. 7. This is the
reproduction of Figs. 5, 6, and 8 using all observation data. Except ELF, the
correlations between various proxies and LCA are degraded by the inclusion of
the data from the very stable and unstable regimes. In particular, EIS and
ECTEI over the ocean and two LCS parameters (*β*_{1} and *β*_{2}) show
substantial degradation of the correlation with LCA, due mainly to the
enhanced scatters induced by the data in the very stable regimes where LCA is
small (the light-colored data points of $\mathrm{0}\le \mathit{\alpha}<\mathrm{0.01}$ in Fig. 12). Very
surprisingly, however, ELF continues to maintain a strong correlation
relationship with LCA over both the ocean and land, explaining 70 % of
spatial–seasonal variations of LCA over the entire globe, which is a much
larger percentage than those explained by *β*_{1} (19 %), *β*_{2} (30 %),
LTS (5 %), EIS (6 %), and ECTEI (4 %). In comparison with the scatter plots
of individual 5^{∘} latitude × 10^{∘} longitude grid point data, the scatter plots of the
selected regions (the last column of Fig. 12) show weaker degradation of
correlations because the selected regions are relatively free from the
occurrence of *α*=0 and *α*=1 (compare Fig. 7 with Fig. 2a, b). If
the analysis is performed with stratiform LCA only, EIS and ECTEI show better
performance than the ones shown in Fig. 12 (see Supplement).

Table 4 is the reproduction of Table 1 using all the data of $\mathrm{0}<\mathit{\alpha}<\mathrm{1}$,
*α*=0, and *α*=1. Except the IS over the ocean and ELF, the
correlations between the proxies and LCA are generally degraded in comparison
to the previous case of $\mathrm{0}<\mathit{\alpha}<\mathrm{1}$. The spatial–seasonal correlations
between LTS (and EIS, ECTEI) and LCA are very weak and even tend to be
negative. Similar to the previous case, EIS is more strongly correlated with
DS and *α* than with IS. Although it shows a weaker correlation than
that in Table 1, *z*_{LCL} still remains as a good proxy for LCA. Due mainly
to the rapid decrease in the correlation between *z*_{inv} and LCA, the
correlations between two LCS parameters and LCA are substantially weaker than
the previous case of $\mathrm{0}<\mathit{\alpha}<\mathrm{1}$. However, ELF successfully maintains a very
strong spatial–seasonal correlation with LCA over both the ocean and land.

Figure 13 shows the global distribution of seasonal–interannual (upper) and
interannual (lower) correlation coefficients between the seasonal LCA and six
key proxies. The overall correlation patterns over the ocean are similar to
the previous cases of $\mathrm{0}<\mathit{\alpha}<\mathrm{1}$ (see Figs. 9 and 10). However, over Asia,
LTS, EIS, and ECTEI show very strong negative seasonal–interannual correlations,
resulting in the opposite correlations between the ocean and land. Over Asia,
with similar contributions from *z*_{inv} and *z*_{LCL} (not shown but more
strongly from *z*_{inv} for seasonal–interannual and more strongly from
*z*_{LCL} for interannual correlations), both *β*_{1} and *β*_{2} show
undesirable positive seasonal–interannual correlations with LCA. This
indicates that LCS, similar to EIS, LTS, and ECTEI, is not appropriate as a global
proxy for diagnosing seasonal–interannual variations of LCA. However, LCS is
still a useful global proxy for diagnosing interannual variations of LCA.
Among all the proxies examined, ELF remains the best proxy in terms of
diagnosing both the seasonal–interannual and interannual variations of LCA
over the globe, including the marine stratocumulus deck.

Table 5 is the reproduction of Table 2 using all data of $\mathrm{0}\le \mathit{\alpha}\le \mathrm{1}$.
Similar to Table 2, LTS remains a good proxy for diagnosing the
seasonal–interannual variations of LCA in the subtropical marine
stratocumulus deck. However, its performance over several land areas (e.g.,
South America, Australia, southwestern Sahara, India, and China) is degraded,
but its performance over Europe is improved. Both EIS and ECTEI show similar
correlation characteristics to those in Table 2, but the undesirable negative
correlation in the western US is changed to a desirable strong positive
correlation. LTS outperforms EIS and ECTEI over the Arabian marine
stratocumulus deck and India. LTS, EIS, and ECTEI all show undesirable
significant negative correlations with LCA in the SH circumpolar region. In
the stratocumulus deck (and also over most land areas), *z*_{inv} remains as
a proxy that is as good as or better than LTS and EIS. Over the land and
midlatitude stratocumulus deck, including the Arabian region, *z*_{LCL} tends
to work better than *z*_{inv}. Similar to Table 2, ELF and two LCS parameters
perform better than LTS and EIS in most regions, including the marine
stratocumulus deck. Overall, among the 12 proxies, ELF shows the best
performance in diagnosing the seasonal–interannual variations of the seasonal
LCA in a statistically significant way. The only exception is continental
Australia in which both *z*_{LCL} and *z*_{inv} are weakly correlated with
LCA. In addition to the combined seasonal–interannual variations, ELF also
diagnoses the interannual variations of LCA well. However, similar to LTS and
EIS, ELF still does not show good performance in diagnosing the interannual
variations of LCA in several subtropical marine stratocumulus decks in a
statistically significant way.

Table 6 summarizes the combined spatial–seasonal–interannual correlations
between the seasonal LCA and 12 proxies in general cases with all
observation data. LTS is poorly correlated with LCA. Over the ocean, EIS and
ECTEI show slightly better performance than LTS, but the improvement seems to
be marginal. EIS correlates much better with DS and *α* than with IS.
Among all 12 proxies, ELF shows the best performance in diagnosing the
spatiotemporal variations of LCA, explaining almost 60 % of the
spatial–seasonal–interannual variance of the seasonal LCA over the entire
globe (50 % and 62 % over the ocean and land, respectively), which is a much
larger percentage than those explained by LTS (2 %), EIS (4 %), and ECTEI
(2 %).

Based on the decoupling parameterization of the cloud-topped PBL suggested
by PLR04, a simple heuristic equation is derived to compute the inversion
height, *z*_{inv}. As an attempt to find a simple heuristic proxy for
diagnosing the spatiotemporal variations of LCA over the globe, we defined
two low-level cloud suppression parameters (LCS),
${\mathit{\beta}}_{\mathrm{1}}=({z}_{\mathrm{inv}}+{z}_{\mathrm{LCL}})/\mathrm{\Delta}{z}_{\mathrm{s}}$ and ${\mathit{\beta}}_{\mathrm{2}}=\sqrt{{z}_{\mathrm{inv}}\cdot {z}_{\mathrm{LCL}}}/\mathrm{\Delta}{z}_{\mathrm{s}}$, by combining *z*_{inv} with the lifting condensation
level of near-surface air, *z*_{LCL}, and normalizing with a constant scale
height, Δ*z*_{s}=2750 m. To better diagnose LCA in extremely cold and
dry atmospheric conditions, we also defined an estimated low-level cloud
fraction, $\text{ELF}\equiv f(\mathrm{1}-{\mathit{\beta}}_{\mathrm{2}})$, with a freeze-dry factor,
$f=max[\mathrm{0.15},min(\mathrm{1},{q}_{\mathrm{v},\mathrm{ML}}/\mathrm{0.003}\left)\right]$, where *q*_{v,ML} is the water vapor
specific humidity in the surfaced-based mixed layer that is assumed to be
topped by *z*_{LCL}, for simplicity. If *z*_{ML}≈*z*_{LCL} where
*z*_{ML} is the height of the surface-based mixed layer, then ${\mathit{\beta}}_{\mathrm{1}}\approx ({z}_{\mathrm{LCL}}/\mathrm{\Delta}{z}_{\mathrm{s}})\cdot [\mathrm{2}+({z}_{\mathrm{inv}}-{z}_{\mathrm{ML}})/{z}_{\mathrm{ML}}]$ and ${\mathit{\beta}}_{\mathrm{2}}\approx ({z}_{\mathrm{LCL}}/\mathrm{\Delta}{z}_{\mathrm{s}})\cdot \sqrt{\mathrm{1}+({z}_{\mathrm{inv}}-{z}_{\mathrm{ML}})/{z}_{\mathrm{ML}}}$, where the first
factor in both formulae represents the degree of subsaturation of
near-surface air, and the second factor represents the decoupling strength.
ELF predicts that LCA increases as the inversion base air is
thermodynamically coupled with the moist near-surface air containing enough
water vapor. Considering that low-level clouds usually form at the
inversion base, the conceptual cloud formation processes embedded in ELF are
consistent with what is expected to happen in nature.

Using the near-surface *θ* and *q*_{v} obtained from individual EECRA
surface observations and the 6-hourly ERA-Interim profile of *θ*, we
computed the IS (inversion strength), DS (decoupling strength), *α*
(decoupling parameter), *z*_{LCL}, *z*_{inv}, ${\text{RH}}_{\mathrm{inv}}^{-}$, *β*_{1},
*β*_{2}, and ELF for January 1979–December 2008 over the ocean and January 1979–December 1996
over land, respectively, which were then averaged into
5^{∘} latitude × 10^{∘} longitude seasonal grid data. Spatial and
temporal correlations between these proxies and surface-observed seasonal LCA
were computed over the globe and compared with those of LTS, EIS, and ECTEI,
all of which have been widely used as proxies for LCA in previous studies.
To obtain insights into the impact of extreme vertical stratification in the
lower troposphere (i.e., very stable or unstable regimes) on the correlation
relationship between LCA and various proxies, we first analyzed the results
only using the data satisfying $\mathrm{0}<\mathit{\alpha}<\mathrm{1}$ (i.e., ${z}_{\mathrm{LCL}}<{z}_{\mathrm{inv}}<\mathrm{\Delta}{z}_{\mathrm{s}}+{z}_{\mathrm{LCL}}$) and then the analysis was extended to include all of the
observation data. Here, we provide a summary of the results for $\mathrm{0}<\mathit{\alpha}<\mathrm{1}$
and then for all data.

In contrast to previous studies, the spatial–seasonal correlation between
LTS and LCA is very weak, reflecting the sensitivity of the LTS–LCA relationship
to the analysis domain. However, EIS and ECTEI show a significant positive
correlation with LCA over the ocean (Table 1 and Figs. 5, 6). It is shown
that EIS is more strongly anti-correlated with *α* and DS than with
IS, implying that EIS may measure the magnitude of decoupling strength
more strongly than the inversion strength. As a proxy for diagnosing
spatial–seasonal variations of LCA, *z*_{LCL} performs better than *z*_{inv}
and ${\text{RH}}_{\mathrm{inv}}^{-}$, which work better than LTS. The LCS parameter,
*β*_{2}, performs slightly better than *β*_{1}, which is better than
*z*_{LCL}. Among all 12 proxies, ELF shows the best performance. A
similar interregional–seasonal correlation analysis over several selected
regions (Fig. 8) also showed that *β*_{1} and *β*_{2}–ELF are the best
proxies for LCA, explaining about 80 % of interregional–seasonal variance
of the seasonal LCA, which is much higher than the ones explained by LTS and EIS
(25 %).

In addition to the spatial–seasonal correlation, we also examined the
seasonal–interannual and interannual correlations between the proxies and
LCA (Table 2 and Figs. 9, 10). Consistent with previous studies, both LTS
and EIS are good proxies characterizing the seasonal–interannual variation
of LCA over subtropical marine stratocumulus decks. Although mainly designed
as a proxy for marine stratocumulus, LTS is also positively correlated with
some continental LCA. Due to the spatial changes of temporal correlation
signs, however, the use of LTS and EIS as global proxies for LCA is limited.
In contrast to LTS and EIS, the proxies of *z*_{inv}, *z*_{LCL},
${\text{RH}}_{\mathrm{inv}}^{-}$, *β*_{1}, and *β*_{2} and ELF tend to be more homogeneously
correlated with LCA all over the globe, without the reversal of correlation
signs (Fig. 9). Similar to the spatial–seasonal correlations, the temporal
correlation characteristics of DS and *α* with LCA are very similar to
that of EIS. The interannual correlation between the proxies and LCA tends
to be weaker than the seasonal–interannual correlation, and all proxies over
the subtropical marine stratocumulus decks except the Namibian ones have weak
interannual correlations smaller than 0.5 with LCA (Table 2 and Fig. 10).
Regardless of whether the seasonal cycle is included or not, *β*_{2} and ELF
show the best temporal correlation with LCA in almost all regions. The
combined spatial–seasonal–interannual correlation analysis reveals that EIS
and ECTEI over the ocean are significantly correlated with LCA but LTS is
poorly correlated with LCA (Table 3). Among all 12 proxies,
*β*_{2} and ELF explain the largest fraction of spatial and temporal
variations of the seasonal LCA over the globe.

For a fair comparison with the previous proxies of LTS, EIS, and ECTEI that
can be defined in any situations, we repeated our analysis by using all
observation data, including the ones in very stable and unstable regimes
(Tables 4–6 and Figs. 11–13). In general, in comparison with the previous
case of $\mathrm{0}<\mathit{\alpha}<\mathrm{1}$, the correlations between the proxies and LCA are
degraded mainly due to the enhanced scatters induced by the data in the very
stable regimes (Fig. 12). However, ELF continues to maintain a very strong
correlation with LCA. LTS, EIS, and ECTEI remain good proxies for diagnosing
the seasonal–interannual variations of LCA in the subtropical marine
stratocumulus deck with better performance of LTS than EIS and ECTEI over the
Arabian and Canarian regions (Table 5). However, they show undesirable
negative correlations with LCA in the SH circumpolar region and several
continental regions, particularly Asia (Fig. 13). Over the stratocumulus
deck and most land areas except Asia, *z*_{inv} remains a proxy that is as
good as or better than LTS and EIS, while *z*_{LCL} generally works better
than *z*_{inv} over land except the eastern United States EIS anti-correlates much
better with DS and *α* than with IS. Among all 12 proxies, ELF shows
the best performance in diagnosing the spatiotemporal variations of LCA
(Table 6), explaining almost 60 % of spatial–seasonal–interannual variances
of the seasonal LCA over the entire globe (50 % and 62 % over the ocean and
land, respectively), which is a much larger percentage than those explained
by LTS (2 %), EIS (4 %), and ECTEI (2 %). However, similar to LTS and EIS,
ELF still has a weakness in diagnosing interannual variation of LCA in
several subtropical marine stratocumulus decks in a statistically significant
way (Table 5).

We have shown that ELF and two LCS parameters are superior to the previously
proposed LTS, EIS, and ECTEI in diagnosing the spatial and temporal
variations of the seasonal LCA over both the ocean and land, including the
marine stratocumulus deck. However, there are a couple of aspects that need
to be addressed in future research. First, mainly due to insufficient surface
moisture, the physical processes controlling the formation and dissipation of
LCA over land are likely to be different from those over the ocean
(e.g., Ek and Mahrt, 1994; Ek and Holtslag, 2004; Zhang and Klein, 2010, 2013; Gentine et al., 2013). In contrast to ocean
areas where *z*_{ML}≈*z*_{LCL}, *z*_{ML} over desserts during the night
is likely to be lower than *z*_{LCL} due to the radiative stabilization of
near-surface air. In addition, the mixed layer developed during the daytime
may reside above the stable nocturnal surface layer; therefore, the idealized
vertical structure depicted in Fig. 1 may not be valid. At this stage, we are
not aware of any comprehensive ways to handle these complicated cases over
land within the degree of complexity suitable for a heuristic proxy or simple
parameterization. One very simple way is to impose a certain upper limit on
*z*_{ML} (e.g., ${z}_{\mathrm{ML}}=min({z}_{\mathrm{LCL}},\mathrm{1500}\phantom{\rule{0.125em}{0ex}}\mathrm{m})$), but it would be more
desirable to parameterize *z*_{ML} as a function of the buoyancy flux and
shear production within the PBL. Second, although named an estimated
low-level cloud fraction due mainly to the existence of an upper bound of 1
(when near-surface air is saturated with enough water vapor), our
ELF has systematic biases against LCA. For example, as seen in Figs. 5l, 6l,
8l, and 12u–x, the dashed gray lines representing LCA = ELF are offset from the
thick black regression lines, and the regression slope over the ocean is
different from that over land. As a result, our ELF tends to overestimate LCA
over land and to underestimate it over the marine stratocumulus deck (Fig. 2g, h).
This feature may be addressed by using different scale heights
(Δ*z*_{s}) over the ocean and land, respectively. Finally, a strong
correlation relationship between LCA and ELF (and two LCS parameters)
identified in our study can be used to evaluate the realism of simulated LCA
in GCMs, as was done by Park et al. (2014) using LTS. Because the
freeze-dry factor is derived from the analysis of observation data, and the
definitions of the observed and simulated LCA may differ, it might be better
to use the LCS parameters (*β*_{1} or *β*_{2}) instead of ELF to evaluate
the simulated LCA, unless a GCM has a cloud fraction parameterization
incorporating the freeze-dry factor. It was not shown here, but we checked
that the observed significant correlations between LCS and LCA were also
simulated by the Community Atmosphere Model version 5 (CAM5,
Park et al., 2014) and the Seoul National University Atmosphere
Model version 0 with a Unified Convection Scheme (SAM0-UNICON,
Park et al., 2019, 2017; Park, 2014a, b), which will be reported in a separate paper with
additional observational analysis by cloud types (e.g., cumulus,
cumulonimbus, stratus, stratocumulus, and fog).

Our study implies that, regardless of other properties, accurate prediction of inversion base height and lifting condensation level is a key factor necessary for successful simulation of global low-level clouds in weather prediction models and GCMs. Strong spatiotemporal correlation between ELF (or LCS) and LCA identified in our study can be used to evaluate the performance of GCMs, identify the source of inaccurate simulation of LCA, and better understand climate sensitivity.

The EECRA cloud data used in our study are available at https://rda.ucar.edu/datasets/ds292.2/, last access: 28 April 2019 (Fleet Numerical Meteorology and Oceanography Center/U.S. Navy/U. S. Department of Defense, Department of Atmospheric Science/University of Washington, Department of Atmospheric Sciences/University of Arizona, and National Centers for Environmental Prediction/National Weather Service/NOAA/U.S. Department of Commerce, 2006).

The supplement related to this article is available online at: https://doi.org/10.5194/acp-19-5635-2019-supplement.

SP has led the overall research and JS helped to conduct the analysis under the supervision of SP.

The authors declare that they have no conflict of interest.

The first author expresses his deepest thanks to the late Conway B. Leovy at the University of Washington, the first author's PhD advisor, who developed the concept of PLR04's decoupling parameterization on which our derivation of ELF and LCS is based. This work was supported by the Creative–Pioneering Researchers Program of Seoul National University (SNU; 3345-20180017) and Research Resettlement Fund for new faculty of Seoul National University (SNU; 3345-20150014).

This paper was edited by Hailong Wang and reviewed by Hideaki Kawai and Timothy Myers.

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