the Creative Commons Attribution 4.0 License.
the Creative Commons Attribution 4.0 License.
The multiscale structure of atmospheric energetic constraints on globally averaged precipitation
This study presents a multiscale analysis of crosscorrelations 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 subyearly 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 (${\mathit{\rho}}_{\mathrm{PWV},T}\approx \mathrm{0.9}$), (ii) surface temperature averaged over global land and over global ocean (sea surface temperature, SST) become strongly correlated (${\mathit{\rho}}_{\mathrm{Tland},\mathrm{SST}}\sim \mathrm{0.6}$); (iii) globally averaged precipitation variability is dominated by energetic constraints, specifically the surface downwelling longwave radiative flux (DLR) (${\mathit{\rho}}_{\mathrm{P},\mathrm{DLR}}\approx \mathrm{0.8}$) 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 clearsky DLR. At subyearly 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 2year resolution, relying on the emergent strong correlations (P vs. clearsky DLR). Such a simple linear sensitivity model was able to reproduce observed P anomaly time series with similar accuracy to an (uncoupled) atmospheric model (ERA20CM) and two climate reanalysis (ERA20C and 20CR). The linear sensitivity breaks down at subyearly timescales, whereby the underlying correlations become negligible. Finally, the relevance of the multiscale framework and its potential for stochastic downscaling applications are demonstrated by deriving accurate monthly P probability density functions (PDFs) from the reconstructed 2year P time series based on scaleinvariant arguments alone. The derived monthly PDFs outperform the statistics simulated by ERA20C, 20CR, and ERA20CM in reproducing observations.
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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 firstorder 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 longterm 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 multiscale 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 wellknown 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 nearsurface air temperature (ΔT) (e.g., Held and Soden, 2006; Schneider et al., 2010):
where ${\mathit{\alpha}}_{\mathrm{PWV},T}\approx \mathrm{0.07}$ K^{−1} at temperatures typical of the lower troposphere. Numerous studies have provided a robust confirmation for the Clausius–Clapeyron mechanism at multidecadal 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 subClausius–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 (L_{V}P, with P being the globally averaged precipitation flux and L_{V} the latent heat of vaporization) should be in balance with the net atmospheric radiative flux (R_{atm}) and the surface sensible flux (F_{SH}):
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 multiscale 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 satellitebased 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 subyearly 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 subyearly 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 multiscale 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 multiscale analysis framework; the results of multiscale 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 multiscale analysis is demonstrated by employing a simple linear model to reconstruct globally averaged precipitation time series from energetic constraints. At subyearly 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 scaleinvariant symmetries of precipitation. Finally, the main conclusions are summarized and discussed in Sect. 6.
2.1 Datasets
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 nearsurface 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 (CERESEBAF) edition 4.0 (Loeb et al., 2009), a monthly dataset covering the full globe at 1^{∘} resolution from March 2000 to June 2017.
Two stateoftheart 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 (NOAACIRES) twentiethcentury 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 ObservationBased Estimates of the Variability of SST and Marine Meteorological Variables version 2 (COBESST2). Here, globalmean time series of precipitation, precipitable water vapor, nearsurface 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 MediumRange Weather Forecasts (ECMWF) twentiethcentury reanalysis (ERA20C; 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 sealevel 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). Globalmean time series of precipitation, precipitable water vapor, nearsurface temperature, SST, and atmospheric radiative fluxes were obtained from ERA20C at daily resolution for the 1900–2010 period.
Finally, the uncoupled ECMWF twentiethcentury ensemble of 10 atmospheric model integrations (ERA20CM; Hersbach et al., 2015) was considered, which uses the same model, grid, initial conditions, and radiative and aerosol forcings as ERA20C. 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 10member ensemble of globalmean time series of precipitation, precipitable water vapor, nearsurface temperature, SST, and atmospheric radiative fluxes was obtained from ERA20CM at monthly resolution for the 1900–2010 period. Considering ERA20CM allowed for the testing of the sensitivity of the multiscale correlation structure derived from ERA20C to data assimilation, but different atmospheric evolutions associated with perturbations to the forcing fields (particularly to SST).
Note that the clearsky 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. Clearsky estimates from ERA20C and ERA20CM cover the full globe area and not just the cloudfree regions at each time instant. However, they are available for net radiative fluxes, but not for downwelling or upwelling radiation fluxes.
2.2 Multiscale 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 longterm average (over all the years in the record) of each calendar day (or month, depending on temporal resolution):
where y_{ds} is the deseasonalized anomalies time series.
Traditionally, fluctuations are defined by the difference $\mathrm{\Delta}y\left(\mathrm{\Delta}t\right)=y(t+\mathrm{\Delta}t)y\left(t\right)$. 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 lowfrequency 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 $t+\mathrm{\Delta}t/\mathrm{2}$ to $t+\mathrm{\Delta}t/\mathrm{2}$ and from t to $t+\mathrm{\Delta}t/\mathrm{2}$, i.e.,
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 Haarfluctuationbased correlation methodology used here (e.g., Lovejoy et al., 2017). Additionally, the analogous method of detrended crosscorrelation 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 crosscorrelation analysis. Since the multiscale crosscorrelation structure obtained with Haar fluctuations is identical to the results using detrended crosscorrelation 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 crosscorrelation 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 powerlaw 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.
3.1 Multiscale 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 nearsurface 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 subyearly timescales. This result suggested that the Clausius–Clapeyron relationship (see Eq. 1) holds to a very good approximation at multiyear timescales, but not at subyearly timescales. Interestingly, Lovejoy et al. (2017) computed the Haar fluctuation correlations for GISSTEMP surface temperatures and found a similar transition in the multiscale 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 nearsurface 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 ERA20C, ERA20CM, 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.
Nogueira (2019) also reported a transition in the multiscale 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 subyearly 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 multiscale 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 ERA20C and ERA20CM, ∼0.6 in 20CR, and ∼0.4 in observations). Furthermore, considering different time lags in computing the crosscorrelations between precipitation and surface temperature did not reveal the presence of significant lagged correlations over the daily to decadal timescale range.
3.2 Multiscale 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 multiscale correlation coefficients between the deseasonalized anomaly time series for precipitation and net atmospheric radiative fluxes were strongly (negatively) correlated at multiyear timescales ($\left\mathit{\rho}\right=\mathrm{0.8}$ in ERA20C, ERA20CM, and observations), in agreement with the balance in Eq. (2). In contrast, at subyearly timescales the correlation magnitude decreased rapidly, changed sign around monthly timescales, and reached values ∼0.4 at timescales below about 10 days.
Considering the combined effect of the net atmospheric radiative fluxes and sensible heat flux in Eq. (2) slightly increased the (positive) correlations at submonthly 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 submonthly timescales, but essentially <0.4 at multiyear timescales (Fig. 2a, dotdashed lines). Given the results in Fig. 2a, the following linear relation was hypothesized: ${L}_{\mathrm{V}}\mathrm{\Delta}P\approx {c}_{\mathrm{1}}\times (\mathrm{\Delta}{R}_{\mathrm{atm}})+{c}_{\mathrm{2}}$, where c_{1} and c_{2} are arbitrary constants, and Δ represents fluctuations taken as deseasonalized anomalies at multiyear resolutions. At subyearly 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 subyearly timescales, most likely due to nonnegligible 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., ${R}_{\mathrm{atm}}={R}_{\mathrm{LW},\mathrm{net}}+{R}_{\mathrm{SW},\mathrm{net}}$. 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., ${\mathit{\rho}}_{\mathrm{P},{\mathrm{R}}_{\mathrm{atm}}}\approx {\mathit{\rho}}_{\mathrm{P},{\mathrm{R}}_{\mathrm{LW},\mathrm{net}}}$) over the full range of timescales considered (Fig. 2b). On the other hand, ${\mathit{\rho}}_{\mathrm{P},{\mathrm{R}}_{\mathrm{SW},\mathrm{net}}}$ also displayed significant values (∼0.6) at multiyear timescales, but the latter magnitude was nearly 0.2 lower when compared to ${\mathit{\rho}}_{\mathrm{P},{\mathrm{R}}_{\mathrm{atm}}}$ and ${\mathit{\rho}}_{\mathrm{P},{\mathrm{R}}_{\mathrm{LW},\mathrm{net}}}$ (Fig. 2b). Consequently, the above linear relationship for multiscale P anomalies was further refined as ${L}_{\mathrm{V}}\mathrm{\Delta}P\approx {c}_{\mathrm{1}}\times (\mathrm{\Delta}{R}_{\mathrm{atm}})+{c}_{\mathrm{2}}\approx {c}_{\mathrm{3}}\times (\mathrm{\Delta}{R}_{\mathrm{LW},\mathrm{net}})+{c}_{\mathrm{4}}$, where c_{3} and c_{4} are arbitrary constants.
Subsequently, the net atmospheric longwave radiative flux was further decomposed into the topofatmosphere (TOA) and surface net longwave fluxes, i.e., ${R}_{\mathrm{LW},\mathrm{net}}={R}_{\mathrm{LW},\mathrm{TOA}}+{R}_{\mathrm{LW},\mathrm{SFC}}$. At multiyear timescales, ${\mathit{\rho}}_{\mathrm{P},{\mathrm{R}}_{\mathrm{atm}}}\approx {\mathit{\rho}}_{\mathrm{P},{\mathrm{R}}_{\mathrm{LW},\mathrm{SFC}}}$ (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 $\sim \mathrm{0.6}$ in ERA20C and ERA20CM datasets, but much lower in 20CR in which the magnitude of the correlation was <0.4 at multiyear timescales. Nonetheless, the former correlations (in ERA20C and ERA20CM) 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 ${\mathit{\rho}}_{\mathrm{P},{\mathrm{R}}_{\mathrm{LW},\mathrm{TOA}}}$ at multiyear timescales was nearly 0.2 lower than for ${\mathit{\rho}}_{\mathrm{P},{\mathrm{R}}_{\mathrm{LW},\mathrm{SFC}}}$. Consequently, a further approximation was considered on the linear model for precipitation fluctuations at multiyear timescales: ${L}_{\mathrm{V}}\mathrm{\Delta}P\approx {c}_{\mathrm{1}}\times (\mathrm{\Delta}{R}_{\mathrm{atm}})+{c}_{\mathrm{2}}\approx {c}_{\mathrm{3}}\times (\mathrm{\Delta}{R}_{\mathrm{LW},\mathrm{net}})+{c}_{\mathrm{4}}\approx {c}_{\mathrm{5}}\times (\mathrm{\Delta}{R}_{\mathrm{LW},\mathrm{SFC}})+{c}_{\mathrm{6}}$.
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 (ρ_{P,DLR}) or against net atmospheric radiative fluxes (i.e., ${\mathit{\rho}}_{\mathrm{P},{\mathrm{R}}_{\mathrm{atm}}}$) were within 0.1 in observations, ERA20C, and ERA20CM (R_{atm} 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: ${L}_{\mathrm{V}}\mathrm{\Delta}P\approx {c}_{\mathrm{7}}\times (\mathrm{\Delta}\mathrm{DLR})+{c}_{\mathrm{8}}$. 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 ERA20CM and ERA20C. Thus, a similar linear relationship between ΔP and $\mathrm{\Delta}{R}_{\mathrm{LW},\mathrm{SFC},\mathrm{UP}}$ 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 clearsky net radiative atmospheric heating (i.e., ${\mathit{\rho}}_{\mathrm{P},{\mathrm{R}}_{\mathrm{atm},\mathrm{cs}}}$) was nearly identical to ${\mathit{\rho}}_{\mathrm{P},{\mathrm{R}}_{\mathrm{atm}}}$ 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 ${\mathit{\rho}}_{\mathrm{P},{\mathrm{R}}_{\mathrm{atm},\mathrm{cs}}}$ and ${\mathit{\rho}}_{\mathrm{P},{\mathrm{R}}_{\mathrm{atm}}}$. The clearsky 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 clearsky 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 clearsky approximation provided support for this hypothesis.
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 multiscale 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 subyearly 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 nearsurface 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 crosscorrelation 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 multiscale correlation obtained from Haar fluctuations recently introduced by Lovejoy et al. (2017).
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 clearsky DLR at multiyear timescales is tested. Additionally, whether the more robust correlation between globally averaged precipitation and clearsky 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 clearsky DLR can be derived, to a good approximation, from the globally averaged nearsurface 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 clearsky 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 clearsky 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,
where a_{1}=59.38 W m^{−2}, a_{2}=113.7 W m^{−2}, and a_{3}=96.96 W m^{−2} are the model parameters, and PWV_{c}=22.5 kg m^{−2} is the climatological value for precipitable water vapor. The subscript “2 y” (e.g., DLR_{2 y}) indicates a fluctuation for Δt=2 years. Note that DLR${}_{\mathrm{c},\mathrm{DO}}={a}_{\mathrm{1}}+{a}_{\mathrm{2}}+{a}_{\mathrm{3}}$ is obtained by replacing the climatological values of PWV and SST in Eq. (8).
The Prata model for ΔDLR_{2 y,Pr} is based on the Stefan–Boltzmann equation:
where ${\mathit{\sigma}}_{\mathrm{SB}}=\mathrm{5.67}\times {\mathrm{10}}^{\mathrm{8}}$ W m^{−2} K^{−4} is the Stefan–Boltzmann constant and
The anomaly time series is computed from ΔDLR${}_{\mathrm{2}\phantom{\rule{0.125em}{0ex}}\mathrm{y},\mathrm{Pr}}={\mathrm{DLR}}_{\mathrm{2}\phantom{\rule{0.125em}{0ex}}\mathrm{y},\mathrm{Pr}}{\mathrm{DLR}}_{\mathrm{c},\mathrm{Pr}}$, where DLR_{c,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 PWV${}_{\mathrm{2}\phantom{\rule{0.125em}{0ex}}\mathrm{y}}\approx {\mathit{\alpha}}_{\mathrm{PWV},\mathrm{SST}}\mathrm{\Delta}{\mathrm{SST}}_{\mathrm{2}\phantom{\rule{0.125em}{0ex}}\mathrm{y}}+{\mathrm{PWV}}_{\mathrm{c}}$. The forcing ΔSST_{2 y} time series were obtained by coarsegraining the deseasonalized (using Eq. 3) globally averaged SST obtained from the GISSTEMP dataset. The sensitivity coefficient, ${\mathit{\alpha}}_{\mathrm{W},\mathrm{SST}}\approx \mathrm{0.08}$ K^{−1}, was estimated by leastsquares regression of ΔPWV_{2 y}∕PWV_{c} against ΔSST_{2 y}, pooling together all datasets (ERA20C, ERA20CM, and 20CR). The α_{PWV,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.
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 P_{c}≈2.7 mm d^{−1} was estimated from the GPCP dataset. The sensitivity coefficient ${\mathit{\alpha}}_{\mathrm{P},\mathrm{DLR}}\approx \mathrm{0.004}$ (W m^{2})^{−1} was estimated by leastsquares regression of ΔP_{2 y}∕P_{c} against ΔDLR_{2 y}, pooling together all available datasets (ERA20C, ERA20CM, 20CR, and GPCP against CERESEBAF). Note that, in estimating α_{P,DLR}, clearsky DLR time series were used where available (i.e., for ERA20C and ERA20CM), but they were replaced by (fullsky) DLR otherwise (i.e., for 20CR and CERESEBAF). The α_{P,DLR} estimates are summarized in Table 2, including values obtained from each dataset (no estimate was made for GPCP against CERESEBAF due to the limited duration of the latter), ranging between 0.003 (W m^{2})^{−1} and 0.005 (W m${}^{\mathrm{2}}{)}^{\mathrm{1}}$.
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., ${P}_{\mathrm{2}\phantom{\rule{0.125em}{0ex}}\mathrm{y},\mathrm{SST}}\approx {\mathit{\alpha}}_{\mathrm{P},\mathrm{SST}}\mathrm{\Delta}{\mathrm{SST}}_{\mathrm{2}\phantom{\rule{0.125em}{0ex}}\mathrm{y}}{P}_{\mathrm{c}}+{P}_{\mathrm{c}}$. Again, ΔSST_{2 y} was obtained from the GISSTEMP dataset. The sensitivity coefficient, ${\mathit{\alpha}}_{\mathrm{P},\mathrm{SST}}\approx \mathrm{0.02}$ K^{−1}, was estimated by leastsquares regression of ΔP_{2 y}∕P_{c} against ΔSST_{2 y}, pooling together all datasets (ERA20C, ERA20CM, 20CR, and GPCP against GISSTEMP). The α_{P,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).
When compared against ΔP_{2 y} directly derived from GPCP for the 1979 to 2010 period, the errors in the proposed linear ΔP_{2 y} reconstructions were generally close to those for atmosphericmodelbased products (Fig. 4). ΔP_{2 y,Pr} displays the highest mean bias, somewhat higher than for atmosphericmodelbased datasets, but also higher than the mean bias for ΔP_{2 y,DO} and ΔP_{2 y,SST} (Fig. 4a). Note that all atmosphericmodelbased 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 atmosphericmodelbased and linearmodelbased ΔP_{2 y} time series. Figure 4c shows that the normalized standard deviation (${\mathit{\sigma}}_{\mathrm{n}}={\mathit{\sigma}}_{\mathrm{2}\phantom{\rule{0.125em}{0ex}}\mathrm{y},\mathrm{model}}/{\mathit{\sigma}}_{\mathrm{2}\phantom{\rule{0.125em}{0ex}}\mathrm{y},\mathrm{obs}}$) estimated from ΔP_{2 y,DO} (∼0.4) and, particularly, from ΔP_{2 y,SST} (∼0.3) was lower than the values estimated from atmosphericmodelbased products (∼0.5–0.9). In contrast, σ_{n} estimated from ΔP_{2 y,Pr} was nearly 0.8, which was higher than 20CR and most members in the ERA20CM ensemble, and was only outperformed by the ERA20C dataset. The root mean square error after bias correction (RMSE_{bc}) estimated from ΔP_{2 y,Pr} and ΔP_{2 y,DO} was well within the range of values obtained from atmosphericmodelbased products (Fig. 4b), with the Prata model slightly overperforming the Dilley–O'Brien model. RMSE_{bc} estimated from ΔP_{2 y,SST} was on the high end of the atmosphericmodelbased range of values and somewhat worse than for the DLRbased linear models. Finally, the Pearson correlation coefficient between models and observations (Fig. 4d) was similar amongst all linearbased models and well within the range of values estimated from the atmosphericmodelbased products. The latter result was expected since all linear models were forced by the same SST time series.
Overall, these results suggested that ΔP_{2 y,Pr} (after bias correction) reproduced the observations with similar accuracy to atmosphericmodelbased products, including similar RMSE_{bc}, variability amplitude, and phase of the signal. ΔP_{2 y,DO} displayed a similar performance for RMSE_{bc} and for the phase, but not for the variability amplitude. Finally, ΔP_{2 y,SST} had the worst performance concerning RMSE_{bc}, 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 ΔP_{2 y,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.
At subyearly 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 nonnegligible (Fig. 3). Consequently, the linear relationships underlying the above simple linear reconstructions of globally averaged precipitation at 2year resolution are no longer appropriate at subyearly timescales. Previous investigations have suggested that this transition should be related to a fundamental transition in the stochastic scaleinvariant behavior of climate variables, which separates a highfrequency 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 lowfrequency 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 multiscale 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 multiscale structure of atmospheric variables and its remarkable potential for stochastic downscaling applications.
Building on the strong scaleinvariant 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 subyearly 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 quasiGaussian and quasinonintermittent 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 scaleinvariant properties, fractional Gaussian noise was used here to generate multiple realizations of downscaled ΔP at monthly resolution from each member of each ΔP_{2 y} time series:
where fGn_{1 m} is a fractional Gaussian noise, which was computed by first generating a random Gaussian noise (g), then taking its Fourier transform ($\stackrel{\mathrm{\u0303}}{g}$), multiplying by ${k}^{\mathit{\beta}/\mathrm{2}}$, and finally taking the inverse transform to obtain fGn_{1 m}. The mean of fGn_{1 m} was forced to be equal to the number of data points of ΔP_{2 y}. Then fGn_{2 y} was obtained by coarsegraining fGn_{1 m} using 24point (i.e., 2 years) length boxes. In this way, ΔP_{1 m,DO}, ΔP_{1 m,Pr}, and ΔP_{1 m,SST} ensembles are respectively generated from the biascorrected ΔP_{2 y,DO}, ΔP_{2 y,Pr}, and ΔP_{2 y,SST} time series. A total of 100 plausible realizations are generated for each ensemble, corresponding to 100 different realizations of fGn_{1 m}. 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 2year resolution time series were assumed to have a constant value for every month inside each 2year 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 nonnegligible and the quasiGaussian approximation to the statistics is no longer robust.
The proposed downscaling methodology corresponds to treating the subyearly frequencies as random “weather noise”, which is characterized, to a good approximation, by scaleinvariant behavior with quasiGaussian statistics (Vallis, 2009; Lovejoy et al., 2015). A similar downscaling methodology has been previously demonstrated to reproduce the spatial subgridscale 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 ΔP_{1 m,DO}, ΔP_{1 m,Pr}, and ΔP_{1 m,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 atmosphericmodelbased products. This improved PDF accuracy was quantified using the Perkins skill score, S score (Perkins et al., 2007), defined as:
where f_{mod}(i) and f_{obs}(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 %.
The linearbased models showed S score values around 95 %, which were significantly higher than the ∼80 % found for the atmosphericmodelbased 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 linearbased models were consistently better than the S scores obtained from atmosphericmodelbased products (∼80 %). Despite some differences between PDFs obtained from ΔP_{1 m,DO}, ΔP_{1 m,Pr}, and ΔP_{1 m,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 coarseresolution forcing.
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 multiscale structure, which is still ill understood and responsible for significant uncertainties in climate models. Here a multiscale analysis framework was employed, aiming to disentangle the complex structure of globally averaged precipitation variability.
The multiscale correlation structure obtained from Haar fluctuations suggested that globalmean 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 multiscale 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 ($\left\mathit{\rho}\right\mathrm{0.8}$), 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 nearsurface 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 subyearly 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 subyearly 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 topofatmosphere 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 (subyearly) timescales, but the dominant effect of topofatmosphere 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 multiscale 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 2year 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. clearsky DLR correlation) with an empirical algorithm for clearsky DLR estimation and the Clausius–Clapeyron relationship. After correcting for their systematic mean bias, the highly idealized model for ΔP_{2 y} based on clearsky DLR estimated from the Prata algorithm displayed similar accuracy in reproducing observations as atmosphericmodelbased products, as measured by RMSE_{bc}, 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 subyearly timescales because all the correlations upon which they rely become negligible. This abrupt transition in the multiscale correlation structure implies that at subyearly 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 multiscale analysis framework provides another path for the reconstruction of precipitation statistics at subyearly resolution. A stochastic downscaling algorithm based on scaleinvariant symmetries of precipitation was applied to ΔP_{2 y} 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 atmosphericmodelbased products, as measured by the PDF matching S score computed over decadal and 30year periods. These results highlight the relevance and potential applications of multiscale frameworks for atmospheric sciences.
ERA20C (Poli et al., 2015) and ERA20CM (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 CERESEBAF 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/esd102192019supplement.
The author declares no conflict of interest.
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/CTAMET/28946/2017). The author was funded by the Portuguese Science Foundation (F.C.T.) under grant UID/GEO/50019/2013.
This paper was edited by Rui A. P. Perdigão and reviewed by Shaun Lovejoy and one anonymous referee.
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 Abstract
 Introduction
 Data and methodology
 Analysis of the mechanisms governing P variability across timescales
 Evaluation of the multiyear linear relationships between globally averaged precipitation, clearsky DLR, and surface temperature
 Exploring scale invariance for stochastic downscaling of precipitation to monthly resolution
 Discussion and conclusions
 Data availability
 Competing interests
 Acknowledgements
 Review statement
 References
 Supplement
 Abstract
 Introduction
 Data and methodology
 Analysis of the mechanisms governing P variability across timescales
 Evaluation of the multiyear linear relationships between globally averaged precipitation, clearsky DLR, and surface temperature
 Exploring scale invariance for stochastic downscaling of precipitation to monthly resolution
 Discussion and conclusions
 Data availability
 Competing interests
 Acknowledgements
 Review statement
 References
 Supplement