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The GCMs differ substantially from each other in many aspects; to avoid dependences of results on individual GCMs, we evaluate the simple models of precipitation extremes using 20 GCM outputs in the CMIP5 achieve (Coupled Model Intercomparison Project Phase 5, Table S1 in the Electronic Supplementary Material, ESM). The outputs are daily data of the historical simulations between 1981 and 2000. The outputs of the 20 GCMs are interpolated to a 2.5° × 2.5° geographical grid so that they have the same horizontal resolution. The variables include pressure velocity (
$ \omega $ ), temperature ($ T $ ), specific humidity ($ q $ ) and relative humidity ($ r $ ) on vertical pressure ($ p $ ) levels, and surface precipitation. In GCMs, precipitation (and convection) is usually parameterized by several modules (e.g., convective precipitation produced by the convective parameterization of cumulus clouds, and grid-scale precipitation produced by the parameterization of stratus or layered clouds). This separation is an ad hoc treatment due to the insufficient resolution of GCMs. In this study, convection refers to clouds of all types.The precipitation extreme examined in this study is defined as the annual maximum daily precipitation (i.e., RX1day in the literature, Alexander et al., 2006, Pfahl et al., 2017; Nie et al., 2020). This definition is roughly equivalent to the 99.7th percentile of precipitation, close to the 99.9th percentile in some other previous studies (e.g., O’Gorman and Schneider, 2009a, b). As the threshold of precipitation extreme changes, the performances of the simple models vary, however, our conclusions are still valid (later see section 3.3). To obtain a better physical understanding of the full probability distribution of precipitation is important (e.g., Chen et al., 2019); however, it is beyond the scope of this study.
For the historical simulations, on each geographic grid we may find 20 extreme events (during the 20 years simulations) and their composites. We also extract the atmospheric variables conditioned on the extreme precipitation days, which are the inputs of the simple models. The precipitation extremes provided by the simple models are then compared with precipitation extremes from the direct outputs of GCMs. Their differences are treated as the errors of the simple models. The global mean relative error is the global sum of the absolute values of differences on each grid divided by the global sum of precipitation extremes. Unless otherwise specified, the results of the GCM outputs only show their multimodel means.
We use the high-resolution ERA-Interim reanalysis (Dee et al., 2011) as the observational basis to examine the sub-GCM-grid inhomogeneity of precipitation extremes. The ERA reanalysis provides daily data between 1979 and 2016, with a horizontal resolution of 0.25° × 0.25°. The ERA precipitation is from the short-range forecast, which shows reasonable agreement with those of the satellite- and rain gauge-based GPCP (Global Precipitation Climatology Project version 1.2; Huffman et al., 2001) precipitation (Dai and Nie, 2020). To match the resolution of the GCM outputs, we constructed a set of coarsened-resolution reanalyses (2.5° × 2.5°) based on the high-resolution (0.25° × 0.25°) reanalyses. Precipitation extremes are selected using the coarsened-resolution reanalyses, while the high-resolution reanalyses provide information on the sub-GCM-grid inhomogeneity.
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For precipitation extremes within an area of a typical GCM grid, previous models may be roughly divided into two categories. Models in the first category (named model 1, e.g., Emori and Brown, 2005) are based on the column moisture budget. Since in heavy precipitation events the moisture sink due to precipitation is mainly balanced by vertical moisture advection, model 1 approximates precipitation extremes (
$ P $ ) aswhere the overline denotes GCM-grid-mean variables, and {} denotes the vertical integral from the surface level to the tropopause (here defined as the layer where the pressure level below 50 hPa has a lapse rate of 2 K km−1). The subscript in
$ {P}_{1} $ denotes the model number (the same applies to model 2 and model 3). The variables in the simple models are conditioned on the extreme precipitation day. In model 1, the budget terms of moisture storage, horizontal moisture advection, surface evaporation, and moisture flux at the tropopause are neglected.The second-category model (model 2, O’Gorman and Schneider, 2009a, b) suggests that during heavy rainfall, the air column is close to saturation. Thus, precipitation is the excess of water vapor of saturated rising air following moist adiabatic processes, which has the formula of
where
$ {d\bar{{q}^{*}}}/{dp}{|}_{{\theta }_{\mathrm{e}}^{\mathrm{*}}} $ is the vertical material derivative of the saturation specific humidity at a constant saturation equivalent potential temperature ($ {\theta }_{\mathrm{e}}^{\mathrm{*}} $ ). Recent studies (e.g., Pfahl et al., 2017; Nie et al., 2020) prefer using Eq. (2) to Eq. (1) due to its better performance. However, its key assumption that the whole air column is horizontally homogeneous and saturated may be oversimplified.Now, we evaluate the performances of model 1 and model 2 by comparing the precipitation given by Eq. (1) and Eq. (2) and precipitation from the direct GCM outputs (denoted as
$ {P}_{0} $ , Fig. 1a). Both models reasonably reproduce the general geographic patterns of$ {P}_{0} $ ; however, there are sizeable errors both globally and regionally [Figs. 1b–c for errors and Fig. S1 in the electronic supplementary material (ESM) for relative errors]. Eq. (1) underestimates precipitation extremes in most regions; especially in middle and high latitudes, the relative error is close to 50%. Eq. (2) also leads to a general underestimation, although not as badly as Eq. (1). In addition, Eq. (2) shows large overestimations over dry zones such as the Sahara and western Australia. Since the dynamic components ($ \bar{\omega } $ ) in the two equations are the same, the differences between them come from the thermodynamic components. The global-mean profiles of the thermodynamic components of the two models are compared in Fig. S2. The amplitudes of$ \partial \bar{q}/\partial p $ and$ {({\rm{d}}\bar{{q}^{*}}/{\rm{d}}p)|}_{{\theta }_{e}^{*}} $ are similar.$ \partial \bar{q}/\partial p $ decreases monotonically with height since water vapor is mostly confined near the surface. On the other hand,$ {({\rm{d}}\bar{{q}^{*}}/{\rm{d}}p)|}_{{\theta }_{e}^{*}} $ peaks in the middle troposphere. Since$ \bar{\omega } $ peaks in the middle to upper troposphere during precipitation events, precipitation estimated by Eq. (2) is greater than that estimated by Eq. (1). The global-mean relative errors of the two models are 27.2% and 10.6% (Table 1), respectively.Figure 1. (a) Multimodel-mean climatology of precipitation extremes from the direct GCM outputs (
$ {P}_{0} $ ) in the CMIP5 historical simulations. (b) and (c) show the errors of model 1 and model 2 in reproducing precipitation extremes, respectively.Historical simulations RCP8.5 simulations Model 1 27.2% 24.3% Model 2 10.6% 11.5% Model 3 5.5% 5.4% Table 1. The global-mean relative errors of the simple models. The time period for the RCP8.5 simulations is between 2081 and 2100. Note the global mean value of precipitation extreme is 22.8 mm d−1 for the CMIP5 historical simulations and 27.9 mm d−1 for the RCP8.5 simulations.
The above evaluation shows that model 1 and model 2 both have sizeable errors. Model 2 has better performance than model 1 has; however, it still has large errors in many regions. Over a GCM-grid-size column, saturated convective updrafts only occupy a fraction of area; saturation throughout the whole column is very rare even during heavy precipitation. Figure. S3 shows composites of relative humidity during precipitation extremes at several representative latitudes. Relative humidity during precipitation extremes can only reach up to approximately 70%–90% in the troposphere. Actually, many GCMs set an upper limit on the grid’s relative humidity by including a large-scale condensation parameterization. In the following, we propose a two-plume convective model for precipitation extremes that takes the sub-GCM-grid inhomogeneity into account and shows its improved performance.