This study presents a multi-scale analysis of cross-correlations based on
Haar fluctuations of globally averaged anomalies of precipitation (P),
precipitable water vapor (PWV), surface temperature (T), and atmospheric
radiative fluxes. The results revealed an emergent transition between weak
correlations at sub-yearly timescales (down to ∼5 days) to
strong correlations at timescales larger than about ∼1–2 years (up to
∼1 decade). At multiyear timescales, (i) Clausius–Clapeyron becomes the dominant control of PWV
(ρPWV,T≈0.9), (ii) surface temperature averaged over
global land and over global ocean (sea surface temperature, SST) become strongly correlated (ρTland,SST∼0.6); (iii) globally averaged precipitation
variability is dominated by energetic constraints, specifically the surface
downwelling longwave radiative flux (DLR) (ρP,DLR≈-0.8)
displayed stronger correlations than the direct response to T fluctuations,
and
(iv) cloud effects are negligible for the energetic constraints in (iii),
which are dominated by clear-sky DLR. At sub-yearly timescales, all
correlations underlying these four results decrease abruptly towards
negligible values. Such a transition has important implications for
understanding and quantifying the climate sensitivity of the global hydrological
cycle. The validity of the derived correlation structure is demonstrated by
reconstructing global precipitation time series at 2-year resolution,
relying on the emergent strong correlations (P vs. clear-sky DLR). Such a
simple linear sensitivity model was able to reproduce observed P anomaly
time series with similar accuracy to an (uncoupled) atmospheric model
(ERA-20CM) and two climate reanalysis (ERA-20C and 20CR). The linear
sensitivity breaks down at sub-yearly timescales, whereby the underlying
correlations become negligible. Finally, the relevance of the multi-scale
framework and its potential for stochastic downscaling applications are
demonstrated by deriving accurate monthly P probability density functions (PDFs)
from the reconstructed 2-year P time series based on scale-invariant
arguments alone. The derived monthly PDFs outperform the statistics
simulated by ERA-20C, 20CR, and ERA-20CM in reproducing observations.
Introduction
The precipitation response to changes in increased concentrations of
greenhouse gases is a central topic for the climate science community.
Although its regional variability is essential to determining societal
impacts, globally averaged precipitation is an important first-order climate
indicator, and a measure of the global water cycle, that must be accurately
simulated if robust climate projections are to be obtained across a wide
range of spatial and temporal scales.
However, even the long-term response of globally averaged precipitation is
still poorly understood, constrained, and simulated (Collins et al., 2013;
Allan et al., 2014; Hegerl et al., 2015), largely due to limited
knowledge on the complex interactions between the key components of the
atmospheric branch of the water cycle and its forcing mechanisms. This
problem is tackled here by employing a multi-scale analysis framework to
study globally averaged precipitation variability and its relation to two
key governing mechanisms: the Clausius–Clapeyron relationship and the
constraints imposed by the atmospheric energy balance.
The Clausius–Clapeyron relationship is a well-known mechanism controlling
the variability of the global water cycle. Assuming constant relative
humidity, it implies that fractional changes in globally averaged precipitable
water vapor (ΔPWV / PWV) are linearly related to fluctuations of
globally averaged near-surface air temperature (ΔT) (e.g., Held
and Soden, 2006; Schneider et al., 2010):
ΔPWVPWV≈αPWV,TΔT,
where αPWV,T≈0.07 K-1 at temperatures typical of the
lower troposphere. Numerous studies have provided a robust confirmation for
the Clausius–Clapeyron mechanism at multi-decadal to centennial timescales,
while also reporting an analogous linear response of globally averaged
precipitation to surface temperature fluctuations (see, e.g., Schneider et
al., 2010; Trenberth, 2011; O'Gorman et al., 2012; and Allan et al., 2014
for reviews). In general, these previous investigations agree on the
∼7 % K-1 sensitivity coefficient for precipitable water
vapor. However, there is large spread of the global precipitation
sensitivity coefficient estimates, typically in the 1 % K-1 to 3 % K-1 range.
A widely recognized explanation for the sub-Clausius–Clapeyron sensitivity
of precipitation to temperature fluctuations at long temporal scales comes
from the atmospheric energy balance (Allen and Ingram, 2002; Stephens and
Ellis, 2008; Stephens and Hu, 2010). Specifically, averaging over the
global atmosphere, the latent heat flux associated with precipitation
formation (LVP, with P being the globally averaged precipitation flux and
LV the latent heat of vaporization) should be in balance with the net
atmospheric radiative flux (Ratm) and the surface sensible flux (FSH):
LVP+Ratm+FSH≈0.
Equation (2) represents a general state of radiative convective equilibrium
(Pauluis and Held, 2002), with energy fluxes defined as positive for
atmospheric gain and negative otherwise.
If the Clausius–Clapeyron relationship was the dominant mechanism
controlling the response of atmospheric moisture content and the global
water cycle to temperature fluctuations, then globally averaged precipitable
water vapor and precipitation could be expected to be strongly correlated
with
surface temperature. Previously, Gu and Adler (2011, 2012) found strong
correlations between the interannual variability of globally averaged
precipitable water vapor and surface temperature, in tight agreement with
the Clausius–Clapeyron mechanism. However, they found weaker (yet
significant) correlations between the interannual variability of
globally averaged precipitation and surface temperature, raising doubts
regarding
whether the Clausius–Clapeyron mechanism could be directly extendable to
global precipitation. Note, however, that these results focusing on a
single temporal scale might not represent the entire picture.
A further source of complexity comes from the fact that precipitation and
other relevant atmospheric variables (including temperature, atmospheric
moisture, wind, etc.) display a complex statistical structure, with
significant variability over a wide range of temporal scales and with the
possibility of different mechanisms governing variability at different
timescales (see, e.g., Lovejoy and Schertzer, 2013 for a comprehensive
review). Furthermore, it has been shown that this complex multi-scale
structure plays a role (at least) as important as the large amplitude
periodic components, namely diurnal and seasonal cycles (Lovejoy, 2015;
Nogueira, 2017a). However, our understanding of the underlying governing
mechanisms at different timescales remains largely elusive, representing a
central problem for future improvements to climate simulation and projection.
Recently, Nogueira (2019) analyzed satellite-based observational datasets, a
long global climate model (GCM) simulation, and reanalysis products and found
a tight correlation (∼0.8) between anomaly (deseasonalized)
time series of globally averaged precipitable water vapor and surface
temperature, which emerged at timescales larger than ∼1–2 years.
In contrast, at smaller timescales the correlation decreased rapidly
towards negligible values (<0.3). In other words, the
Clausius–Clapeyron relationship is the dominant mechanism of atmospheric
moisture anomalies at multiyear timescales, but not at sub-yearly
timescales. Nogueira (2019) also found that the magnitude of the
correlations between anomaly time series for globally averaged precipitation
and surface temperature was negligible at sub-yearly timescales, while at
multiyear timescales the results showed large spread amongst different
datasets, ranging between negligible (<0.3) and strong (∼0.8)
correlation values. Building on this previous study,
here the multi-scale analysis of the mechanisms governing global
precipitation variability was extended, including the energetic constraints
on precipitation represented in Eq. (2). The paper is organized as
follows: Sect. 2 describes the considered datasets and the multi-scale
analysis framework; the results of multi-scale correlation analysis of
precipitation variability are presented and discussed in Sect. 3; and in
Sect. 4 the validity of the linear sensitivity correlations derived from
the multi-scale analysis is demonstrated by employing a simple linear model
to reconstruct globally averaged precipitation time series from energetic
constraints. At sub-yearly timescales, at which the correlations break down,
it is shown in Sect. 5 how the monthly statistics can be reproduced by
employing a stochastic downscaling algorithm based on scale-invariant
symmetries of precipitation. Finally, the main conclusions are summarized
and discussed in Sect. 6.
Data and methodologyDatasets
Precipitation observations were obtained from the Global Precipitation
Climatology Project (GPCP) version 2.3 monthly precipitation dataset (Adler
et al., 2003), which covers the full globe at 2.5∘
resolution from 1979 to present. Gridded datasets of monthly average surface
temperatures were obtained from the Goddard Institute for Space Studies (GISSTEMP)
analysis (Hansen et al., 2010), which covers the globe at
2∘ resolution from 1880 to present, with the values provided
as anomalies relative to the 1951–1980 reference period. GISSTEMP blends
near-surface air temperature measurements from meteorological stations
(including Antarctic stations) with a reconstructed sea surface temperature (SST)
dataset over oceans. Observations of atmospheric radiative fluxes were
obtained from the National Aeronautics and Space Administration (NASA)
Clouds and the Earth's Radiant Energy System, Energy Balanced and Filled (CERES-EBAF)
edition 4.0 (Loeb et al., 2009), a monthly dataset
covering the full globe at 1∘ resolution from March 2000 to June 2017.
Two state-of-the-art reanalyses of the twentieth century were considered in
the present study. One was the National Oceanic and Atmospheric
Administration Cooperative institute for Research in Environmental Sciences (NOAA-CIRES)
twentieth-century reanalysis (20CR) version 2c (Compo et
al., 2011), which covers the full globe at 2∘ resolution,
spanning from 1851 to 2014. Only surface pressure observations and reports
are assimilated in this reanalysis. SST boundary conditions are obtained
from 18 members of Simple Ocean Data Assimilation with Sparse Input (SODAsi)
version 2, with the high latitudes corrected to the Centennial in
Situ Observation-Based Estimates of the Variability of SST and Marine
Meteorological Variables version 2 (COBE-SST2). Here, global-mean
time series of precipitation, precipitable water vapor, near-surface
temperature, SST, and atmospheric radiative fluxes were obtained from 20CR
at daily resolution for the 1900–2010 period. Note that the net
atmospheric radiative flux cannot be obtained from 20CR because the
incoming solar radiation at the top of the atmosphere is not available for
this dataset due to an error with output processing.
The other reanalysis considered in the present study was the European Centre
for Medium-Range Weather Forecasts (ECMWF) twentieth-century reanalysis
(ERA-20C; Poli et al., 2015), which covers the full globe at
1∘ resolution spanning 1900–2010. It assimilates marine
surface winds from the International Comprehensive Ocean–Atmosphere Data Set
version 2.5.1 (ICOADSv2.5.1) and surface and mean sea-level pressure from
the International Surface Pressure Databank version 3.2.6 (ISPDv3.2.6) and
from ICOADSv2.5.1. SST boundary conditions are obtained from the Hadley
Centre Sea Ice and Sea Surface Temperature dataset version 2.1 (HadISST2.1).
Global-mean time series of precipitation, precipitable water
vapor, near-surface temperature, SST, and atmospheric radiative fluxes were
obtained from ERA-20C at daily resolution for the 1900–2010 period.
Finally, the uncoupled ECMWF twentieth-century ensemble of 10 atmospheric
model integrations (ERA-20CM; Hersbach et al., 2015) was considered, which
uses the same model, grid, initial conditions, and radiative and aerosol
forcings as ERA-20C. However, no observations are assimilated, the
simulation is integrated continuously over the full 1900–2010 period, and
SST is prescribed by an ensemble of realizations from HadISST2.1, including
one control simulation and nine simulations with perturbed SST and sea ice
concentration. A 10-member ensemble of global-mean time series of
precipitation, precipitable water vapor, near-surface temperature, SST, and
atmospheric radiative fluxes was obtained from ERA-20CM at monthly
resolution for the 1900–2010 period. Considering ERA-20CM allowed for the
testing
of the sensitivity of the multi-scale correlation structure derived from
ERA-20C to data assimilation, but different atmospheric evolutions
associated with perturbations to the forcing fields (particularly to SST).
Note that the clear-sky radiative fluxes considered here obtained from
ECMWF datasets are computed for the same atmospheric conditions of
temperature, humidity, ozone, trace gases, and aerosol, but assuming that the
clouds are not there. Clear-sky estimates from ERA-20C and ERA-20CM cover
the full globe area and not just the cloud-free regions at each time
instant. However, they are available for net radiative fluxes, but not for
downwelling or upwelling radiation fluxes.
Multi-scale correlation analysis
Consider two time series, y and y′, with N data points each. Here
the goal is to study the correlation between the fluctuations Δy(Δt)
and Δy(Δt) at different timescales Δt.
Due to the strong yearly cycle present in the considered time series, the
periodic seasonal trend was first eliminated by subtracting the long-term
average (over all the years in the record) of each calendar day (or month,
depending on temporal resolution):
yds(i)=y(i)-〈y〉d,
where yds is the deseasonalized anomalies time series.
Traditionally, fluctuations are defined by the difference Δy(Δt)=y(t+Δt)-y(t). However, it has been
shown that such definition is only appropriate for fluctuations increasing
with timescale (Lovejoy and Schertzer, 2013). Consequently, the traditional
a definition is not useful for the present study, since the fluctuations for
most atmospheric variable time series (including temperature, rain, wind,
and water vapor, amongst others) decrease with increasing timescale over the
tens of days to tens of years range (e.g., Lovejoy and Schertzer, 2013;
Lovejoy, 2015; Lovejoy et al., 2017; Nogueira, 2017a, b, 2019). In this
sense, here the fluctuations were defined using the Haar wavelet, which is
appropriate for the full range of timescales and all atmospheric variables
considered, in cases in which fluctuations both increase and decrease with
timescale. Furthermore, correlations computed from Haar fluctuation
time series also avoid the low-frequency biases that emerge in standard
correlation analysis due to climate variability (see Lovejoy et al., 2017,
for a detailed description of the Haar fluctuations and correlations of Haar fluctuations).
The Haar fluctuations are simply defined as the difference of the means from
t+Δt/2 to t+Δt/2 and from t to t+Δt/2, i.e.,
(Δy(Δt))Haar=2Δt∫t+Δt/2t+Δty(t)dt-2Δt∫tt+Δt/2y(t)dt.
For the sake of simplicity, henceforth the fluctuation notation Δy(Δt)
will be employed to refer to Haar fluctuations (i.e., Δy(Δt)≡(Δy(Δt))Haar). A Haar
fluctuation time series was computed by employing Eq. (4) at each instant
of the deseasonalized anomaly time series for each variable considered.
Finally, at each timescale, Δt, the correlation coefficient,
ρ, of the corresponding Haar fluctuations time series was computed for each
pair of variables considered.
Note that, in computing correlations at timescales larger than 2 times
the original time series resolution, there is overlap of the data points
considered in computing the Haar fluctuations. While this could introduce
spurious effects in the computed correlations, previous works have shown the
robustness of the Haar-fluctuation-based correlation methodology used here
(e.g., Lovejoy et al., 2017). Additionally, the analogous method of detrended
cross-correlation analysis has also been employed on overlapping windows and
demonstrated to provide accurate correlation estimates at different
timescales using overlapping windows (see, e.g., Podobnik and Stanley, 2008;
Podobnik et al., 2011; Piao and Fu, 2016). In fact, in Sect. 3 below it is
shown that identical correlation structures are obtained between
correlations of Haar fluctuations and detrended cross-correlation analysis.
Since the multi-scale cross-correlation structure obtained with Haar
fluctuations is identical to the results using detrended cross-correlation
analysis, it is assumed that critical points for the 95 % significance
level of Haar fluctuation correlations are identical to the ones
demonstrated by Podobnik et al. (2011) for detrended cross-correlation
analysis using overlapping windows, whereby the significant values can vary
between values below 0.1 and up to about 0.4, depending on the time series
length, the considered timescale, and the power-law exponents of both
time series. In this sense, here it is assumed that correlation magnitudes
below 0.3 are nonsignificant and that magnitudes in the 0.3 to 0.4 range
should be interpreted with care.
Analysis of the mechanisms governing P variability across timescalesMulti-scale structure of the atmospheric water cycle response to surface temperature fluctuations
The correlations between Haar fluctuation time series revealed strong
correlations (∼0.9) between deseasonalized anomaly
time series for globally averaged precipitable water vapor and near-surface
temperature (or, alternatively, SST) at multiyear timescales (Fig. 1a).
However, as the timescale decreases there is a transition in the
correlation structure, and negligible correlations (<0.3) emerge at
sub-yearly timescales. This result suggested that the Clausius–Clapeyron
relationship (see Eq. 1) holds to a very good approximation at multiyear
timescales, but not at sub-yearly timescales. Interestingly, Lovejoy et
al. (2017) computed the Haar fluctuation correlations for GISSTEMP
surface temperatures and found a similar transition in the multi-scale
correlation structure of SST against globally averaged surface temperature,
with low correlations at timescales below a few months and strong
correlations (∼0.8) at multiyear timescales. Note that
the latter strong correlations were not surprising, since SST was a major
component in their definition of globally averaged surface temperature (which
for GISSTEMP uses SST over ocean pixels and 2 m air temperature over
land pixels). Nonetheless, Lovejoy et al. (2017) also found a similar
transition for the correlation between SST and near-surface air temperature
averaged over global land, with maximum correlation values ∼0.6
at multiyear timescales. The transition in the correlation structure
between SST and global land temperature was confirmed here for ERA-20C,
ERA-20CM, 20CR, and GISSTEMP (Fig. 1b). Thus, the present results support
the Lovejoy et al. (2017) argument that these abrupt correlation changes
correspond to a fundamental behavioral transition, whereby the atmosphere and
the oceans start to act as a single coupled system. Furthermore, the results
presented here suggest that precipitable water vapor anomalies at multiyear
resolution can be derived, to a very good approximation, from SST alone.
Cross-correlation coefficients against temporal scale computed from
Haar fluctuations for global-mean time series of (a) PWV vs. T2m
(solid) and PWV vs. SST (dashed); (b) SST vs. Tland; and
(c)LvP vs. T2m (solid) and LvP
vs. SST (dashed). Red lines represent results from ERA-20C, blue lines are from
ERA-20CM, pink lines are from 20CR, and black lines are estimated from observational products.
Nogueira (2019) also reported a transition in the multi-scale correlation
structure between deseasonalized anomaly time series of globally averaged
precipitation and surface temperature (considering SST over the oceans and
2 m air temperature over land), with negligible values at sub-yearly
timescales, but with large spread in the magnitude of the multiyear
correlations ranging between ∼0.3 and ∼0.8.
Here, a similar result was found for the multi-scale correlation structure
between globally averaged precipitation and surface temperature and also
globally averaged precipitation and SST (Fig. 1c), with large spread in
correlation magnitude at multiyear timescales (∼0.7 in
ERA-20C and ERA-20CM, ∼0.6 in 20CR, and ∼0.4 in
observations). Furthermore, considering different time lags in computing the
cross-correlations between precipitation and surface temperature did not
reveal the presence of significant lagged correlations over the daily to
decadal timescale range.
Multi-scale structure of the energetic constraints to precipitation variability
A study of the circulation component of the precipitation response to
temperature fluctuations requires a detailed representation of several
spatially heterogeneous variables and their nonlinear interactions. An
alternative path towards understanding globally averaged precipitation
temporal variability was taken in the present investigation, focusing on the
constraints imposed by the atmospheric energy balance represented in
Eq. (2). Figure 2a (solid lines) shows that the estimated multi-scale
correlation coefficients between the deseasonalized anomaly time series for
precipitation and net atmospheric radiative fluxes were strongly
(negatively) correlated at multiyear timescales (|ρ|=0.8 in
ERA-20C, ERA-20CM, and observations), in agreement with the balance in
Eq. (2). In contrast, at sub-yearly timescales the correlation
magnitude decreased rapidly, changed sign around monthly timescales, and
reached values ∼0.4 at timescales below about 10 days.
Cross-correlation coefficients against temporal scale computed from
Haar fluctuations of (a)LvP vs. Ratm (solid),
LvP vs. (Ratm+FSH) (dashed), and LvP
vs. FSH (dot-dashed); (b)LvP vs. Ratm
(solid), LvP vs. RLW,net (dashed), and LvP
vs. RSW,net (dot-dashed); (c)LvP vs. Ratm
(solid), LvP vs. RLW,SFC (dashed), and LvP
vs. RLW,TOA (dot-dashed); and (d)LvP
vs. Ratm (solid), LvP vs. DLR (dashed), and LvP
vs. RLW,SFC,UP (dot-dashed). Red lines are computed from ERA-20C,
blue lines are from ERA-20CM, pink lines are from 20CR, and black lines are
computed from GPCP and CERES-EBAF observational products. Note that
Ratm and RSW,net were not available from 20CR, while
sensible heat flux was not available from observations.
Considering the combined effect of the net atmospheric radiative fluxes and
sensible heat flux in Eq. (2) slightly increased the (positive)
correlations at sub-monthly timescales (Fig. 2a, dashed lines), although
the absolute changes are essentially below 0.1. More importantly, Fig. 2a
shows that the magnitude of the correlation at multiyear timescales
between globally averaged precipitation and net atmospheric radiative fluxes
is significantly larger than when the combined effect of net atmospheric
radiative fluxes and sensible heat flux was considered. Indeed, the
correlation between globally averaged precipitation and sensible heat flux
displayed values up to about 0.5 at sub-monthly timescales, but
essentially <0.4 at multiyear timescales (Fig. 2a, dot-dashed lines). Given
the results in Fig. 2a, the following linear relation was hypothesized:
LVΔP≈c1×(-ΔRatm)+c2,
where c1 and c2 are arbitrary constants, and Δ represents
fluctuations taken as deseasonalized anomalies at multiyear resolutions. At
sub-yearly timescales this simplification is not adequate, since the
correlations between globally averaged precipitation and net atmospheric
radiative fluxes become negligible. In other words, the energy balance
represented in Eq. (2) does not represent the dominant constraint on
precipitation variability at sub-yearly timescales, most likely due to
non-negligible changes in atmospheric heat storage.
The analysis was extended by decomposing the net atmospheric radiative flux
into its net atmospheric longwave and shortwave radiative flux components,
i.e., Ratm=RLW,net+RSW,net. On the one hand, the correlation
between globally averaged precipitation and net atmospheric radiative fluxes
is nearly identical to the correlation between globally averaged precipitation
and net atmospheric longwave radiative fluxes (i.e., ρP,Ratm≈ρP,RLW,net) over the full range of
timescales considered (Fig. 2b). On the other hand, ρP,RSW,net
also displayed significant values (∼0.6) at multiyear
timescales, but the latter magnitude was nearly 0.2 lower when compared to
ρP,Ratm and ρP,RLW,net (Fig. 2b). Consequently, the
above linear relationship for multi-scale P anomalies was further refined as
LVΔP≈c1×(-ΔRatm)+c2≈c3×(-ΔRLW,net)+c4,
where c3 and c4 are arbitrary constants.
Subsequently, the net atmospheric longwave radiative flux was further
decomposed into the top-of-atmosphere (TOA) and surface net longwave fluxes,
i.e., RLW,net=RLW,TOA+RLW,SFC. At multiyear timescales,
ρP,Ratm≈ρP,RLW,SFC (Fig. 2c), suggesting that the
surface net longwave radiative fluxes provide the main constraint to
globally averaged precipitation variability. The correlation between
globally averaged precipitation and TOA longwave radiative fluxes also
displayed significant values at multiyear timescales, up to ∼-0.6
in ERA-20C and ERA-20CM datasets, but much lower in
20CR in which the magnitude of the correlation was <0.4 at multiyear
timescales. Nonetheless, the former correlations (in ERA-20C and ERA-20CM)
were in better agreement with observations, suggesting that significant
(negative) correlations existed between globally averaged precipitation and
net longwave fluxes for TOA anomalies at multiyear timescales. However, for
all datasets, the magnitude of ρP,RLW,TOA at multiyear
timescales was nearly 0.2 lower than for ρP,RLW,SFC.
Consequently, a further approximation was considered on the linear model for
precipitation fluctuations at multiyear timescales:
LVΔP≈c1×(-ΔRatm)+c2≈c3×(-ΔRLW,net)+c4≈c5×(-ΔRLW,SFC)+c6.
Finally, the surface net longwave radiative flux can be further decomposed
into its upwelling and downwelling (henceforth denoted downwelling longwave
radiation, DLR) components. Figure 2d shows that, at multiyear timescales,
the differences in the correlations of globally averaged precipitation against
DLR (ρP,DLR) or against net atmospheric radiative fluxes
(i.e., ρP,Ratm) were within 0.1 in observations, ERA-20C, and ERA-20CM
(Ratm is unavailable for 20CR). Thus, at multiyear timescales, the
fluctuations in downwelling surface longwave radiative fluxes are, to a good
approximation, linearly related to precipitation fluctuations:
LVΔP≈c7×(-ΔDLR)+c8. Note that the
correlation structure of globally averaged precipitation against upwelling
surface radiative fluxes or against net atmospheric radiative fluxes is
nearly identical in observations. However, significant differences emerged
between these two quantities (∼0.2) in ERA-20CM and ERA-20C.
Thus, a similar linear relationship between ΔP and ΔRLW,SFC,UP
might also hold to a good approximation, although the results
are less robust than for ΔP against ΔDLR.
The correlation between globally averaged precipitation and clear-sky net
radiative atmospheric heating (i.e., ρP,Ratm,cs) was nearly
identical to ρP,Ratm at multiyear timescales (Fig. 3a). This
suggested that the cloud effects on the multiyear linear dependence between
precipitation variability and net atmospheric radiative fluxes may be
neglected. But the same is not true at timescales below a few months, at
which
significant differences emerge between ρP,Ratm,cs and
ρP,Ratm. The clear-sky approximation holds at multiyear timescales
for correlations of globally averaged precipitation against net atmospheric
longwave radiative fluxes and also against the globally averaged net
surface longwave fluxes (Fig. 3b). Based on these results, it was further
hypothesized that cloud effects are also negligible for the correlation
between globally averaged precipitation and DLR at multiyear temporal scales.
This hypothesis could not be tested directly because clear-sky DLR
time series were not available for the ECMWF datasets. Nonetheless, the
results in Sect. 4 based on an empirical algorithm for DLR estimation
under a clear-sky approximation provided support for this hypothesis.
Cross-correlation coefficients against temporal scale computed from
Haar fluctuations of (a)LvP vs. Ratm (solid)
and LvP vs. Ratm,CS (dashed); (b)LvP
vs. RLW,SFC (solid) and LvP vs. RLW,SFC,CS
(dashed). Red lines are computed from ERA-20C and blue lines are from ERA-20CM.
At this point, it is important to note that the existence of strong
correlations does not necessarily imply causality between two variables.
However, the atmospheric energy balance in Eq. (2) provides a physical
basis for the obtained strong (negative) correlation values between
precipitation and atmospheric radiative fluxes. In fact, the multi-scale
analysis presented here provided further robustness to previous
investigations on the importance of energetic constraints to global
precipitation, the dominant role of surface longwave fluxes, namely DLR, and
the negligible cloud effects in these relationships (e.g., Stephens and Hu,
2010; Stephens et al., 2012a, b). More importantly, a clear transition
emerged between robust correlations at multiyear timescales and negligible
correlations at sub-yearly timescales, which was found for globally averaged
precipitation against atmospheric radiative fluxes (particularly total net,
net longwave, and DLR), globally averaged precipitable water vapor against
surface temperature (and SST), global SST against global near-surface
air temperature, and, less robustly, globally averaged precipitation
against surface temperature (or SST).
Note that the correlation structure derived from Haar fluctuations of
different combinations of variables presented in the present section is
identical to the correlation structure obtained by employing detrended
cross-correlation analysis (DCCA; see Figs. S1–S3 in the Supplement).
DCCA has been previously shown to robustly quantify correlations at
different timescales (Podobnik and Stanley, 2008; Piao and Fu, 2016;
Nogueira, 2017b, 2019, where detailed descriptions of DCCA methodology are
also provided). This result provides one of the first empirical
verifications for the multi-scale correlation obtained from Haar
fluctuations recently introduced by Lovejoy et al. (2017).
Evaluation of the multiyear linear relationships between globally averaged precipitation, clear-sky DLR, and surface temperature
The strong correlations between globally averaged precipitation and
atmospheric longwave radiative fluxes imply that a simple linear model should
be able to reproduce the variability in precipitation anomalies at multiyear
timescales. This hypothesis is tested in the present section, aiming to
provide robustness to the strong multiyear correlations presented in
Sect. 3. Specifically, the robustness of the tight correlation between
globally averaged precipitation and clear-sky DLR at multiyear timescales is
tested. Additionally, whether the more robust correlation
between globally averaged precipitation and clear-sky DLR at multiyear
timescales compared to globally averaged precipitation against surface
temperature results in a better reconstruction of precipitation variability
by such a linear model is tested.
The clear-sky DLR can be derived, to a good approximation, from the globally
averaged near-surface temperature alone (e.g., Stephens et al., 2012b).
Furthermore, given the tight coupling between globally averaged temperature
over land and SST at multiyear timescales (Fig. 1b), it is hypothesized
that clear-sky DLR variability could be obtained, to a good approximation,
directly from the SST forcing. This hypothesis is also supported by the
nearly identical correlations between globally averaged precipitable water
vapor against surface temperature or against SST (Fig. 1a).
Here two different algorithms to estimate clear-sky DLR are tested: the
Dilley–O'Brien model (Dilley and O'Brien, 1998) and the Prata model
(Prata, 1996). In the Dilley–O'Brien model,
DLR2y,DO=a1+a2SST2ySSTc65+a3ΔPWV2y+PWVcPWVc1/2,
where a1=59.38 W m-2, a2=113.7 W m-2, and
a3=96.96 W m-2 are the model parameters, and PWVc=22.5 kg m-2
is the climatological value for precipitable water vapor. The
subscript “2 y” (e.g., DLR2y) indicates a fluctuation for
Δt=2 years. Note that DLRc,DO=a1+a2+a3 is obtained by
replacing the climatological values of PWV and SST in Eq. (8).
The Prata model for ΔDLR2y,Pr is based on the Stefan–Boltzmann equation:
DLR2y,Pr=εclrσSBSST2y4,
where σSB=5.67×10-8 W m-2 K-4 is the
Stefan–Boltzmann constant and
εclr=1-1+PWV2yexp-1.2+3PWV2y1/2.
The anomaly time series is computed from ΔDLR2y,Pr=DLR2y,Pr-DLRc,Pr, where DLRc,Pr is obtained
by replacing the climatological values of PWV and SST in Eqs. (9) and (10).
The strong correlation between globally averaged precipitable water vapor and
SST at multiyear timescales (Fig. 1a) allowed for the removal of the PWV dependence
in Eqs. (8) and (11) by replacing PWV2y≈αPWV,SSTΔSST2y+PWVc. The forcing ΔSST2y
time series were obtained by coarse-graining the deseasonalized (using
Eq. 3) globally averaged SST obtained from the GISSTEMP dataset. The
sensitivity coefficient, αW,SST≈0.08 K-1, was
estimated by least-squares regression of ΔPWV2y/PWVc
against ΔSST2y, pooling together all datasets (ERA-20C,
ERA-20CM, and 20CR). The αPWV,SST estimates are summarized in
Table 1, including for each individual dataset, ranging between 0.07 and
0.10 K-1. Note that the obtained values are close to the typical
0.07 K-1 value predicted by the Clausius–Clapeyron relationship.
Linear regression coefficient αW,SST estimated from
ΔPWV / PWVc against ΔSST at 2-year resolution, assuming
a relationship as given by Eq. (1). The respective coefficient of determination
is also provided. The αW,SST values are computed for ERA-20C, 20CR,
and for the ensemble of ERA-20CM simulations. Additionally, the coefficient is
estimated by pooling together ERA-20C, ERA-20CM (ensemble), and 20CR datasets.
In this way, two reconstructed anomaly time series for globally averaged
precipitation were obtained using the Dilley–O'Brien and the Prata
algorithms. The climatological globally averaged precipitation
Pc≈2.7 mm d-1 was estimated from the GPCP dataset. The sensitivity coefficient
αP,DLR≈0.004 (W m2)-1 was estimated by
least-squares regression of ΔP2y/Pc against
ΔDLR2y, pooling together all available datasets (ERA-20C, ERA-20CM,
20CR, and GPCP against CERES-EBAF). Note that, in estimating
αP,DLR, clear-sky DLR time series were used where available (i.e., for
ERA-20C and ERA-20CM), but they were replaced by (full-sky) DLR
otherwise (i.e., for 20CR and CERES-EBAF). The αP,DLR estimates
are summarized in Table 2, including values obtained from each dataset (no
estimate was made for GPCP against CERES-EBAF due to the limited duration of
the latter), ranging between 0.003 (W m2)-1 and 0.005 (W m-2)-1.
Linear regression coefficient αP,DLR estimated from
ΔP/Pc against ΔDLR at 2-year resolution, assuming a
relationship as given by Eq. (11). The respective coefficients of determination
are also provided. The αP,DLR values are computed for ERA-20C,
20CR, and for the ensemble of ERA-20CM simulations. Additionally, the coefficient
is estimated by pooling together all datasets, including GPCP observations
against DLR from CERES-EBAF.
Another simple linear model for the reconstruction of multiyear globally averaged
precipitation anomaly time series was tested based on the direct response
(correlations) of P to SST fluctuations,
i.e., P2y,SST≈αP,SSTΔSST2yPc+Pc.
Again, ΔSST2y was obtained from the GISSTEMP dataset. The sensitivity coefficient,
αP,SST≈0.02 K-1, was estimated by least-squares
regression of ΔP2y/Pc against ΔSST2y, pooling
together all datasets (ERA-20C, ERA-20CM, 20CR, and GPCP against GISSTEMP).
The αP,SST estimates are summarized in Table 3, including for
each individual dataset, ranging between 0.02 and 0.04 K-1. Note that
the obtained values are close to the 0.01 to 0.03 K-1 range reported in
the relevant literature (e.g., Schneider et al., 2010; Trenberth, 2011;
O'Gorman et al., 2012; Allan et al., 2014).
Error estimates from simulated anomaly time series for P at 2-year
resolution against GPCP computed for the 1979–2010 period, including
(a) mean bias (Bias), (b) root mean square error after bias
correction (RMSEbc), (c) model standard deviation normalized by observed
standard deviation (σn), and (d) Pearson correlation
coefficient (r). For the ERA-20CM dataset the range for all ensemble members is
shown, while “x” marks their mean value. The p value for all correlations
shown in (d) is <0.05.
Linear regression coefficient αP,SST estimated from
ΔPPc against ΔSST at 2-year resolution. The respective
coefficients of determination are also provided. The αP,SST
values are computed for ERA-20C, 20CR, for the ensemble of ERA-20CM simulations,
and for GPCP against ERA-20CM control SST forcing. Additionally, the coefficient
is estimated by pooling together all datasets.
When compared against ΔP2y directly derived from GPCP for the
1979 to 2010 period, the errors in the proposed linear ΔP2y
reconstructions were generally close to those for atmospheric-model-based
products (Fig. 4). ΔP2y,Pr displays the highest mean bias,
somewhat higher than for atmospheric-model-based datasets, but also higher
than the mean bias for ΔP2y,DO and ΔP2y,SST (Fig. 4a).
Note that all atmospheric-model-based products considered here also
display a positive bias. While this may be due to a negative bias of GPCP
(e.g., Gehne et al., 2016), this observational dataset represents the longest
reliable dataset for global precipitation studies and was thus considered
here as “the truth”. More importantly, the mean bias is easy to correct by
simply subtracting its value from the time series. This correction was
implemented here for all atmospheric-model-based and linear-model-based
ΔP2y time series. Figure 4c shows that the normalized standard
deviation (σn=σ2y,model/σ2y,obs) estimated from
ΔP2y,DO (∼0.4) and, particularly, from
ΔP2y,SST (∼0.3) was lower than the values estimated from
atmospheric-model-based products (∼0.5–0.9). In contrast,
σn estimated from ΔP2y,Pr was nearly 0.8, which was
higher than 20CR and most members in the ERA-20CM ensemble, and was only
outperformed by the ERA-20C dataset. The root mean square error after
bias correction (RMSEbc) estimated from ΔP2y,Pr and
ΔP2y,DO was well within the range of values obtained from
atmospheric-model-based products (Fig. 4b), with the Prata model slightly
overperforming the Dilley–O'Brien model. RMSEbc estimated from
ΔP2y,SST was on the high end of the atmospheric-model-based range of
values and somewhat worse than for the DLR-based linear models. Finally,
the Pearson correlation coefficient between models and observations (Fig. 4d)
was similar amongst all linear-based models and well within the range of
values estimated from the atmospheric-model-based products. The latter
result was expected since all linear models were forced by the same SST time series.
Overall, these results suggested that ΔP2y,Pr (after bias
correction) reproduced the observations with similar accuracy to atmospheric-model-based products, including similar RMSEbc, variability
amplitude,
and phase of the signal. ΔP2y,DO displayed a similar performance
for RMSEbc and for the phase, but not for the variability amplitude.
Finally, ΔP2y,SST had the worst performance concerning
RMSEbc, but also in capturing the variability amplitude, while it
displayed a similar ability as the other linear models in reproducing the
phase. The overall weakest performance of ΔP2y,SST was coherent
with the less robust correlations underlying this model. Additionally, the
results indicate that the nonlinear transformations on SST employed in the
Prata and the Dilley–O'Brien algorithms improved the linear models.
Exploring scale invariance for stochastic downscaling of precipitation to monthly resolution
At sub-yearly timescales, the magnitude of the correlations between
globally averaged precipitable water vapor and SST, precipitation and DLR, and
precipitation and SST decreases abruptly to negligible values (see Sect. 3).
Additionally, the cloud effects on the energetic constraints of
precipitation variability become non-negligible (Fig. 3). Consequently, the
linear relationships underlying the above simple linear reconstructions of
globally averaged precipitation at 2-year resolution are no longer appropriate
at sub-yearly timescales. Previous investigations have suggested that this
transition should be related to a fundamental transition in the stochastic
scale-invariant behavior of climate variables, which separates a
high-frequency weather regime that extends up to about the synoptic scales
(around 10 days to 1 month in the atmosphere and around 1 year in the
oceans) from a low-frequency weather (or macroweather) regime that extends
up to a few decades (see, e.g., Lovejoy et al., 2017; Nogueira, 2019). In the
weather regime the amplitude of the fluctuations tends to increase with
timescale, while in the macroweather regime the amplitude of the
fluctuations tends to decrease with increasing timescale, hence implying a
convergence toward the “climate normal” at timescales of a few decades (Lovejoy, 2015).
In the present section, it is shown that the multi-scale analysis framework
can also be taken advantage of to perform stochastic downscaling from
multiyear to monthly resolution. This exercise allows for the demonstration of the
relevance of understanding and characterizing the multi-scale structure of
atmospheric variables and its remarkable potential for stochastic
downscaling applications.
Building on the strong scale-invariant symmetries present in the variability
of global and regional precipitation across wide ranges of timescales
(e.g., Lovejoy and Schertzer, 2013; Nogueira et al., 2013; Nogueira and Barros,
2014, 2015; Nogueira, 2017a, b, 2019), an algorithm was proposed here to derive
the sub-yearly statistics from the multiyear information alone. The
physical basis for this algorithm is that while the atmosphere is governed
by continuum mechanics and thermodynamics, it simultaneously obeys
statistical turbulence cascade laws (e.g., Lovejoy and Schertzer, 2013;
Lovejoy et al., 2017).
Conveniently, precipitation (and many other atmospheric variables) is
characterized by low spectral slopes β<1, with quasi-Gaussian and
quasi-non-intermittent statistics, at timescales between ∼10 days
and a few decades (Lovejoy and Schertzer, 2013; de Lima and Lovejoy,
2015; Lovejoy et al., 2015, 2017; Nogueira, 2017b, 2019). Grounded by these
scale-invariant properties, fractional Gaussian noise was used here to
generate multiple realizations of downscaled ΔP at monthly
resolution from each member of each ΔP2y time series:
ΔP1m(t)=fGn1m(t)ΔP2y(t)fGn2y(t),
where fGn1m is a fractional Gaussian noise, which was computed by
first generating a random Gaussian noise (g), then taking its Fourier
transform (g̃), multiplying by k-β/2, and finally taking
the inverse transform to obtain fGn1m. The mean of fGn1m was
forced to be equal to the number of data points of ΔP2y. Then
fGn2y was obtained by coarse-graining fGn1m using 24-point
(i.e., 2 years) length boxes. In this way, ΔP1m,DO,
ΔP1m,Pr, and ΔP1m,SST ensembles are respectively generated from
the bias-corrected ΔP2y,DO, ΔP2y,Pr, and
ΔP2y,SST time series. A total of 100 plausible realizations are generated
for each ensemble, corresponding to 100 different realizations of
fGn1m. Based on recent investigations of P scale invariance
properties, a spectral exponent β≈0.3 is assumed (de Lima and
Lovejoy, 2015; Nogueira, 2019). In Eq. (11), the 2-year resolution
time series were assumed to have a constant value for every month inside
each 2-year period.
Note that a resolution limit should exist for the proposed stochastic
downscaling algorithm, namely at timescales below ∼10 days
when a fundamental transition occurs in the scaling behavior of most
atmospheric fields (including globally averaged precipitation; see,
e.g., Lovejoy and Schertzer, 2013; Lovejoy, 2015; de Lima and Lovejoy, 2015;
Nogueira, 2017a, b, 2019). At faster timescales intermittency becomes
non-negligible and the quasi-Gaussian approximation to the statistics is no longer robust.
The proposed downscaling methodology corresponds to treating the sub-yearly
frequencies as random “weather noise”, which is characterized, to a good
approximation, by scale-invariant behavior with quasi-Gaussian statistics
(Vallis, 2009; Lovejoy et al., 2015). A similar downscaling methodology has
been previously demonstrated to reproduce the spatial sub-grid-scale
variability of topographic height (Bindlish and Barros, 1996),
precipitation (Bindlish and Barros, 2000; Rebora et al., 2006; Nogueira et
al., 2013; Nogueira and Barros, 2015), and clouds (Nogueira and Barros, 2014).
Figure 5a shows that the PDFs computed from ΔP1m,DO,
ΔP1m,Pr, and ΔP1m,SST were in remarkable agreement with PDFs
obtained from the GPCP observational dataset for the 1979–2010 period,
representing a significant improvement compared to all atmospheric-model-based products. This improved PDF accuracy was quantified using the
Perkins skill score, S score (Perkins et al., 2007), defined as:
Sscore=100×∑i=1Mminfmod(i),fobs(i)
where fmod(i) and fobs(i) are
respectively the frequency of the modeled and observed P anomaly values in
bin i, M is the number of bins used to compute the PDF (here M=15), and
min[x, y] is the minimum between the two values. The S score is a measure of
similarity between modeled and observed PDFs such that if a model
reproduces the observed PDF perfectly then S score = 100 %.
PDFs estimated from monthly anomaly time series for P from ERA-20C
(red), ERA-20CM (dark blue), 20CR (pink), GPCP (black), ΔP1m,DO
(dark green), ΔP1m,Pr (light green), and ΔP1m,SST
(light blue). In (a) the PDFs are estimated for the 1979–2010 period,
and in (b) the PDFs are estimated for the 1979–1990 period (solid)
and the 1999–2010 period (dashed).
The linear-based models showed S score values around 95 %, which were
significantly higher than the ∼80 % found for the
atmospheric-model-based products (Fig. 6). Furthermore, the stochastic model
captured the change in the PDFs between two separate periods (1979–1990 and
1999–2010; Fig. 5b), while preserving the remarkably high (≥90 %)
S scores (Fig. 6, blue and red markers). Indeed, the S scores for
linear-based models were consistently better than the S scores obtained from
atmospheric-model-based products (∼80 %). Despite some
differences between PDFs obtained from ΔP1m,DO,
ΔP1m,Pr, and ΔP1m,SST, their similar performance in
reproducing observations was somewhat unexpected given the distinct
performances in reproducing the observed time series at multiyear
resolutions. While the error analysis here was based on a limited sample
(mainly due to the short duration of the satellite period), these results
suggested that the proposed stochastic downscaling mechanism is quite robust
in reproducing the monthly statistics of globally averaged precipitation, with
only moderate sensitivity to the coarse-resolution forcing.
S score computed from the different P simulations against GPCP.
The values estimated for the full satellite period (1979–2010) are presented
in black, for the 1979–1990 period are presented in red, and for
the 1990–2010 period are presented in blue. For the ERA-20CM dataset, the S score
is estimated from the 10-member ensemble PDF.
Discussion and conclusions
Atmospheric variables display significant variability over a wide range of
temporal scales due to changes in external forcings (including surface
fluxes, changes to greenhouse gases and aerosol concentrations, solar
forcing, and volcanic eruptions), but also due to intrinsic modes of
atmospheric variability. Additionally, external and internal atmospheric
processes interact nonlinearly amongst themselves, resulting in an intricate
multi-scale structure, which is still ill understood and responsible for
significant uncertainties in climate models. Here a multi-scale analysis
framework was employed, aiming to disentangle the complex structure of
globally averaged precipitation variability.
The multi-scale correlation structure obtained from Haar fluctuations
suggested that global-mean precipitation variability at multiyear
timescales is linearly related to the net atmospheric radiative fluxes,
corresponding to the dominant effect of energetic constraints on
precipitation variability. Furthermore, this linear relationship is
dominated by its longwave component and, more specifically, by the surface
longwave radiative fluxes, particularly DLR. The results also suggest that
clouds play a negligible role in these linear correlations at multiyear scales.
Building on the previous works of Lovejoy et al. (2017) and Nogueira (2019), the
present paper highlights a critical transition in the multi-scale
correlation structure at timescales of ∼1 year, revealing a
change in the control mechanisms of several atmospheric variables, including
precipitation. Specifically, at multiyear timescales the following is true: (i) globally averaged
precipitation becomes tightly correlated with the net atmospheric radiative
fluxes (|ρ|0.8), and this correlation is dominated by the
downwelling longwave radiative flux at the surface; (ii) the cloud effects
on the atmospheric radiative fluxes in (i) can be neglected; and
(iii) globally averaged precipitable water vapor becomes tightly correlated
(ρ∼0.9) with surface temperature. The respective sensitivity coefficient
for multiyear fluctuations of precipitable water vapor to surface
temperature is estimated here to be 0.07 % K-1, close to the value predicted
by the Clausius–Clapeyron relationship. (iv) Globally averaged SST and
near-surface air temperature over land become strongly correlated
(ρ∼0.7), implying a strong atmosphere–ocean coupling in agreement with and
extending the results from Lovejoy et al. (2017) based on one observational
dataset. In contrast, at sub-yearly timescales, the magnitude of all these
correlations decreases abruptly towards negligible values, and cloud effects
are no longer negligible in the correlations between atmospheric radiative
fluxes and precipitation. Hints of a similar, but less robust, transition
also emerged for the correlation structure between globally averaged
precipitation and surface temperature, with negligible correlations at
sub-yearly timescales, which tend increase at multiyear timescales,
although the latter displayed significant spread amongst different datasets
(ranging between ∼0.4 and ∼0.7).
The transition found here also contributes to sharpening the picture from
previous studies reporting a “slow” response, in which globally averaged
precipitation increases due to increasing surface temperature, and a “fast”
response, in which the atmosphere rapidly adjusts to changes in top-of-atmosphere radiative forcing, and that is independent of temperature
fluctuations (Allen and Ingram, 2002; Bala et al., 2010; Andrews et al.,
2010; O'Gorman et al., 2012; Allan et al., 2014). The correlation structure
found here suggests that the slow component corresponds to multiyear
timescales and that radiative constraints (particularly surface longwave
fluxes) are the key mechanism controlling precipitation variability rather
than temperature, while cloud effects are negligible. On the other hand, the
correlations here confirm the breakdown of the linear relation between
temperature fluctuations at fast (sub-yearly) timescales, but the dominant
effect of top-of-atmosphere radiative forcing is not evident and, most
likely, the situation is much more complex (for example, surface sensible
heat fluxes seem to become relevant at these timescales).
The robustness of this emergent multi-scale correlation structure is
demonstrated by proposing simple models for the reconstruction of
globally averaged precipitation at multiyear timescales. Anomaly time series for
globally averaged precipitation at 2-year resolution were derived from SST
observations alone, either directly based on precipitation vs. SST
correlation structure or by combining the more robust energetic constraints
of globally averaged precipitation (namely the precipitation vs. clear-sky DLR
correlation) with an empirical algorithm for clear-sky DLR estimation and the
Clausius–Clapeyron relationship. After correcting for their systematic mean
bias, the highly idealized model for ΔP2y based on clear-sky DLR
estimated from the Prata algorithm displayed similar accuracy in reproducing
observations as atmospheric-model-based products, as measured by
RMSEbc, the Pearson correlation coefficient, and normalized standard
deviation. Finally, the model based on precipitation vs. SST correlation
showed the weakest performance, which agrees with the less robust
correlations underlying this idealized model.
The proposed linear models cannot be extended to sub-yearly timescales
because all the correlations upon which they rely become negligible. This
abrupt transition in the multi-scale correlation structure implies that at
sub-yearly timescales the tight linear coupling between atmospheric and
ocean temperature, the Clausius–Clapeyron relationship, and the atmospheric
energy balance are no longer dominant linear constraints for
globally averaged precipitation. Nonetheless, the multi-scale analysis framework provides
another path for the reconstruction of precipitation statistics at
sub-yearly resolution. A stochastic downscaling algorithm based on
scale-invariant symmetries of precipitation was applied to ΔP2y
reconstructed time series, resulting in monthly globally averaged
precipitation PDFs. Remarkably, the reconstructed PDFs at monthly resolution
showed better accuracy in reproducing observed statistics than atmospheric-model-based products, as measured by the PDF matching S score computed over
decadal and 30-year periods. These results highlight the relevance and
potential applications of multi-scale frameworks for atmospheric sciences.
Data availability
ERA-20C (Poli et al., 2015) and ERA-20CM (Hersbach et al., 2015)
were provided by ECMWF and are available through the website http://apps.ecmwf.int/datasets.
20CR reanalysis (Compo et al., 2011), GISSTEMP (Hansen et al., 2010), and GPCP
(Adler et al., 2003) precipitation products were provided by NOAA/OAR/ESRL PD,
Boulder, Colorado, USA, from their website at http://www.esrl.noaa.gov/psd.
The CERES-EBAF data (Loeb et al., 2009) were obtained from the NASA Langley Research Center
Atmospheric Science Data Center from their website at https://eosweb.larc.nasa.gov/project/ceres/ebaf_surface_table.
The supplement related to this article is available online at: https://doi.org/10.5194/esd-10-219-2019-supplement.
Competing interests
The author declares no conflict of interest.
Acknowledgements
The author would like to thank Shaun Lovejoy for his detailed comments and
suggestions and for making available the codes for computing the Haar
fluctuations. The author also thanks the anonymous reviewer for comments
and suggestions, which helped to improve the paper. This study was
funded by the Portuguese Science Foundation (F.C.T.) under project CONTROL
(PTDC/CTA-MET/28946/2017). The author was funded by the Portuguese Science
Foundation (F.C.T.) under grant UID/GEO/50019/2013.
Review statement
This paper was edited by Rui A. P. Perdigão and reviewed
by Shaun Lovejoy and one anonymous referee.
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