To infer surface temperatures from the radiative forcing and remove the effects of clouds, we start with a simple physical formulation of the surface energy balance. The surface of the Earth is heated by solar absorption and downwelling long-wave radiation. This heat is released back into the atmosphere through surface emission of long-wave radiation and exchange of turbulent fluxes of sensible and latent heat. This balance between the incoming and outgoing energy fluxes at the Earth's surface is described by Eq. (1): 1Rs+Rl,down=Rl,up+J, where Rs is the surface net solar absorption, Rld is the downwelling long-wave radiation, Rl,up is the upwelling long-wave radiation emitted from the surface and J is turbulent flux exchange between the surface and the atmosphere (comprising of sensible and latent heat). We neglect the ground heat flux as it is generally small when averaged over a few days or longer. While Rs and Rl,down can be obtained using radiation datasets for different sky conditions, the partitioning between Rl,up and J is poorly constrained by surface energy balance alone. To have these additional constraints on J, we used a thermodynamic systems approach to view the Earth system. Similar approaches had also been used (Kleidon and Renner, 2013; Kleidon et al., 2014; Dhara et al., 2016) and were found to capture the observed surface temperatures, energy partitioning and climate sensitivities very well.

To do this, we conceptualise the surface atmosphere system as a heat engine, with warm Earth surface as the heat source and cooler atmosphere being the sink (Fig. 1). Heat and mass are transported within this engine by the exchange of turbulent fluxes (J) between the surface and the atmosphere. The differential radiative heating and cooling between the surface and the atmosphere maintains the temperature difference and drives the vertical convective motion. The power (G) associated with the work done by the heat engine, required to sustain convective motion in the form of vertical mixing and exchange of turbulent fluxes, can be derived by simply using the first and second law of thermodynamics and can be represented by the well-established Carnot limit as 2G=J1-TaTs. Detailed derivation about this can be found in Kleidon and Renner (2013) and Kleidon et al. (2014). Here, Ta and Ts are the representative temperatures of cold atmosphere and the hot surface, respectively.

Both temperatures are inferred from their respective energy balances. The atmospheric temperature (Ta) is assumed to be equal to the radiative temperature (Tr) of atmosphere and is estimated using the outgoing long-wave radiation at top of atmosphere (Rl,toa) 3Ta=Rl,toaσ1/4, where σ is the Stefan–Boltzmann constant (σ= 5.67×10-8 Wm-2 K-4). A correction of 15 K was applied to the radiative temperature to account for the assumption of black atmosphere and effective height of convection (Dhara et al., 2016). We consider the atmosphere as opaque to terrestrial radiation and hence it is assumed that all outgoing long-wave radiation emitted into space originates from the atmosphere.

The heat engine source temperature, i.e. surface temperature (Ts) can be expressed from the emitted long-wave radiation from the surface (Rl,up) as 4Ts=Rl,upσ1/4. Using the surface energy balance (Eq. 1), we can then express the surface temperature in terms of net solar absorption, downwelling long-wave radiation and turbulent fluxes (J) as 5Ts=Rs+Rl,down-Jσ1/4. The differential radiative heating and cooling between the surface and the atmosphere maintains the temperature difference and drives the vertical convective motion. Thermodynamics sets a limit to this conversion and thus constrains the amount of turbulent flux exchange. Less turbulent fluxes result in a hotter surface (Eq. 5), which will increase the temperature difference between the surface and atmosphere. This will subsequently increase the efficiency term in the generation rate, i.e. the second term on the right-hand side of Eq. (2). On the other hand, an increase in turbulent fluxes (J) increases the first term in the generation rate of Eq. (2), but it will, in turn, reduce the surface temperature and temperature difference between surface and atmosphere (Eq. 5). Thus, a trade-off exists that sets the limit for the power to maintain vertical energy and mass exchange between the surface and the atmosphere. This limit is termed as the maximum power limit and provides an additional constraint to partitioning of the surface energy balance that we used here to infer surface temperatures.

Using Eqs. (2), (3) and (5), the rate of work done (power) produced by the heat engine can be expressed as a function of turbulent fluxes (J) as 6G=J1-TaRs+Rl,down-Jσ-1/4. Note that power G= 0 when J= 0 or when J=Rs+Rl,down-Rl,toa. Hence, there is a maximum Gmax=G (Jmaxpower) for a value between 0<Jmaxpower<Rs+Rl,down-Rl,toa. The optimum J that maximises power was calculated numerically. This flux was then used to determine the surface temperatures: 7Ts,maxpower=Rs+Rl,down-Jmaxpowerσ1/4. Surface temperatures were estimated using Eq. (7) for all-sky and clear-sky radiative conditions using radiative forcing from the NASA-CERES datasets. We then refer to these two temperatures derived using Eq. (7) as all-sky and clear-sky temperatures.

Radiative fluxes of short- and long-wave radiation at surface and top of atmosphere (TOA) were obtained from the NASA-CERES (EBAF 4.1) dataset (Loeb et al., 2018; Kato et al., 2018) and the NASA-CERES Syn1deg dataset (Doelling et al., 2013, 2016). These datasets are available for both all-sky and clear-sky conditions at monthly and daily scales, respectively with a 1∘ latitude × 1∘ longitude spatial grid resolution. They were used as a forcing in our energy balance model. We evaluated our model using observation-derived gridded temperature data from the Indian Meteorological Department (IMD; Rajeevan et al., 2008). To estimate the precipitation–temperature scaling, we used daily gridded precipitation and temperature datasets with a spatial resolution of 1∘ latitude × 1∘ longitude from IMD (Rajeevan et al., 2008) and 3-hourly gridded rainfall data from NASA-TRMM_3B42 with a spatial resolution of 0.25∘ × 0.25∘. We repeated the analysis using daily gridded precipitation and temperature data from the Asian Precipitation Highly-Resolved Observational Data Integration towards Evaluation of water resources (APHRODITE) dataset, available at a spatial resolution of 0.25∘ × 0.25∘ (Yatagai et al., 2012). To further ensure the robustness of our results, we also used daily precipitation–temperature observations based on three stations in India (Mumbai Airport, Bangalore Airport and Chennai Airport) from the global surface summary of the day (GSOD) data provided by the National Oceanic and Atmospheric Administration (NOAA). Daily dew point temperatures were obtained from the ERA-5 reanalysis. Based on the availability of all datasets, the period of analysis is 2003–2015.

Extreme precipitation events were scaled with observed, all-sky and clear-sky temperatures using two widely adopted scaling approaches: the binning method (Lenderink and Van Meijgaard, 2008) and quantile regression (Wasko and Sharma, 2014). For the binning method, we defined extreme precipitation events using a threshold of the 99th percentile precipitation contained at each grid cell. Precipitation–temperature pairs were then divided into the increasing order of non-overlapping bins of 2 K width. Only those bins with at least 150 data points have been considered for the analysis (Utsumi et al., 2011). The median value of each bin was then used to examine the variation of precipitation extremes with temperature. Bins with temperatures less than 3 ∘C were discarded to remove the effects of freezing, thawing and snowfall. To ensure that our results are not biased with the number of data points in each bin and bin sizes (which may affect the nature of the scaling relationship), we further used the quantile regression method to estimate the scaling rates.

Quantile regression estimates the conditional quantile of the dependent variable (in our case, precipitation) over the given values of the independent variable (temperature). We first fitted a quantile regression model between the logarithmic precipitation and temperature values at the target quantile of 99 %: 8Log(Pi)=βo99+β199(Ti),

where Pi denotes the mean daily precipitation intensity, Ti is the daily mean temperature and βo99 and β199 are the regression coefficients for the 99th quantile of precipitation. The slope coefficient β199 is then exponentially transformed to estimate the scaling rate (α1): 9α1=100⋅eβ199-1. The aforementioned methodology had been widely adopted to estimate the extreme precipitation–temperature scaling in previous studies (Lenderink and Van Meijgaard, 2008, 2010; Utsumi et al., 2011; Wasko and Sharma, 2014; Schroeer et al., 2018).

All-sky temperatures were estimated using the daily observed radiative fluxes from CERES in conjunction with surface energy partitioning constrained by maximum power (see Eq. 7). We found an extremely good agreement between these estimated temperatures when compared to surface temperature observations over India with R2 > 0.9 and RMSE < 1.5 K over most regions (Fig. 2). This signifies that our formulation strongly captures the surface temperature variation over India and thus validates our approach. We then extend this for clear-sky conditions by forcing our model with clear-sky radiative fluxes from CERES and estimating clear-sky temperatures. It worth noting that clear-sky temperatures are reconstructed temperatures estimated by removing the effect of clouds from radiative transfer.

We used the difference between the all-sky and clear-sky temperatures as a measure to quantify the effect of cloud-driven cooling during rainfall events. This cooling strongly increases with precipitation across regions, resulting in a stronger reduction in surface temperature with greater precipitation (Fig. 3a); it is predominantly caused by the substantial reduction in absorbed solar radiation at the surface for all-sky conditions compared to clear-sky conditions (Fig. 3b). On the other hand, changes in long-wave radiation are comparatively small and remain largely insensitive to precipitation.

To examine the spatial consistency in precipitation variability and associated cooling, we isolated extreme daily precipitation days over each grid. Figure 4a shows the mean magnitude of daily extreme precipitation events over India. The pattern was consistent with the cloud-cover map from NASA-CERES (shown in Appendix C). Figure 4b shows the cooling of clouds associated with these days. This cloud-cooling effect and precipitation shows a clear, systematic variation across India. The cooling effect is greater where precipitation rates are high. In contrast, in the more arid regions of northwestern India, the cooling effect almost disappears with low precipitation rates. In the northernmost Himalayan region, the difference in clear-sky and all-sky temperatures is negative. These high-altitude regions are more sensitive to changes in long-wave radiations. As a result, there is a significant increase in long-wave radiation with an increase in cloud cover that compensates for the cooling due to reduction in short wave over those grids. Figure 4c further shows the mean all-sky temperature during these days. We find that the heaviest events occur at a relatively lower temperature as a result of stronger cooling. Figure 4d shows the mean number of rainfall days per year. More rainy days implies more cloudy conditions and thus a stronger cloud radiative cooling over that region. Having quantified this effect of cloud radiative cooling and its systematic variation across regions, we then estimate its impact on the precipitation–temperature scaling.

We performed a binning analysis (Lenderink and Van Meijgaard, 2008) to understand the scaling of precipitation extremes with temperature using observed temperatures as well as our estimated clear-sky and all-sky temperatures. Precipitation events were isolated and binned into P–T pairs and the resulting scaling relationships are shown in Fig. 5. The scaling relationship using observed and all-sky temperatures showed similar scaling behaviour (yellow and red lines in Fig. 5a). Extreme precipitation increases close to the CC rate up to a threshold of around 23–24 ∘C, above which the scaling becomes negative. This breakdown in scaling behaviour with observed temperatures is consistent with the findings of previous studies (Hardwick Jones et al., 2010; Ghausi and Ghosh, 2020) and is commonly referred to in literature as “hook” or “peak structure” (Wang et al., 2017; Gao et al., 2018). However, when precipitation extremes are scaled with clear-sky temperatures that exclude the cloud-cooling effect, the resulting scaling relationship does not show a breakdown and increases consistently, close to the CC rate over the whole temperature range (blue line in Fig. 5a). Similar results were obtained when the scaling curves were reproduced for station-based observations (see Appendix A).

Previous studies (Hardwick Jones et al., 2010; Chan et al., 2015; Wang et al., 2017) have attributed the breakdown in precipitation–temperature scaling to a lack of moisture availability as relative humidity tends to decrease at high temperatures. To account for this effect of moisture limitation, some studies used dew point temperature, a measure of atmospheric humidity, as an alternative scaling variable (Wasko et al., 2018; Barbero et al., 2018). They showed that the breakdown and negative scaling disappear when scaled with dew point temperatures (Zhang et al., 2019; Ali et al., 2021). To evaluate this interpretation and compare it to ours, we used the dew point temperature from the ERA-5 reanalysis. We derived the extreme precipitation scaling using this temperature (Fig. 5b) and compared it to our all-sky and clear-sky temperatures (Fig. 5c).

At first sight, the scaling relationship using dew point temperatures looks very similar to our clear-sky relationship (compare Fig. 5a and b, but note the difference in temperature scale). Yet, its interpretation differs because the use of dew point temperatures merely implies that the intensity of extreme precipitation events scales with the moisture content of the air, with moister air resulting in higher intensity events. Dew point scaling thus carries less insight about the response of extreme precipitation to global warming (Bao et al., 2018). To infer the precipitation sensitivity with temperature from dew point scaling, one then needs to see how dew point temperatures change with actual temperatures (dTdew / dT) (Fig. 5c). This is further demonstrated using Eq. (10): 10dPdT=dPdTdew×dTdewdT. If relative humidity remains unchanged, we would expect the dew point temperature to increase continuously with surface temperature, representing a moisture increase of 7 % K-1. However, when dew point temperatures are compared to all-sky temperatures (red line, Fig. 5c), we note that a breakdown occurs in this scaling as well. Dew point temperatures increase with all-sky temperatures for colder temperatures more strongly than what would be expected from an unchanged relative humidity when air gets warmer. However, at temperatures above 23–25 ∘C, dew point temperatures fall, reflecting a decrease in relative humidity that is typical for warm, arid regions. Thus, one does not see a breakdown in precipitation–dew point scaling because the information on the breakdown is contained in how dew point temperatures change with surface air temperatures (second term in Eq. 10). Similar findings were also reported in Roderick et al. (2019).

The scaling of dew point temperatures with clear-sky temperatures is much more uniform and consistent across the whole temperature range; it does not show a breakdown or a super CC scaling in the relationship. This is because the clear-sky temperatures reflect the radiative conditions and not the effects of atmospheric humidity or clouds. In contrast, observed temperatures and all-sky temperatures covary with cloud effects, which in turn are linked to precipitation and humidity, thus resulting in less clear scaling relationships that are not as straightforward to interpret. This further implies that moisture loading of the atmosphere primarily occurs during the non-precipitating periods that are more representative of clear-sky radiative conditions.

The breakdown in scaling effect can thus be explained by the cooler temperatures associated with precipitation events. This cooling shifts the precipitation extremes to lower-temperature bins while the high-temperature bins correspond to more arid regions or to the drier pre-monsoon season temperatures with lower values of precipitation extremes. We refer to this as a “bin-shifting” effect. The cooling effect is proportional to the amount of precipitation (Fig. 3a) and hence, the heavier the precipitation, the stronger the cooling and bin shifting becomes. When the cloud-cooling effect is removed, as in the case of clear-sky temperatures, extreme precipitation then shows a scaling that is consistent with the CC rate. This bin-shifting effect arising due to the presence of clouds also causes a decrease in relative humidity at higher temperatures. This effect can be seen by the stronger increase in dew point temperatures below 25 ∘C and the decline above this temperature (Fig. 5c). The breakdown in scaling is thus not directly related to changes in aridity or moisture availability, but rather to the radiative effect of clouds on surface temperature.

To demonstrate the implications of our interpretation for precipitation scaling across regions, we estimated regression slopes of the 99th percentile precipitation events for both sub-daily (TRMM) and daily (IMD & APHRODITE) precipitation with the different temperatures using the quantile regression method (Wasko and Sharma, 2014). We found that extreme precipitation scaling was negative for both observed and all-sky temperatures over most regions (Fig. 6), except for the Himalayan foothills to the north of India. The scaling rates for sub-daily extremes were slightly higher than those estimated for daily extremes, yet they remain negative over most grids. When the cooling effect of clouds is removed by using clear-sky temperatures, extreme precipitation scaling then shows a diametric change and scaling estimates come close to CC rates over most of the regions. A similar diametric change in the scaling was also obtained with the APHRODITE precipitation dataset (Appendix B). The highest positive sensitivities were found over the region of central India where a widespread increase in rainfall extremes has already been reported (Roxy et al., 2017). There seems to be a minor difference between the clear-sky scaling in IMD and TRMM in the foothills of the Himalayas to the north of India, which is likely because of the underestimation of rainfall by TRMM over this region (Sharma et al., 2020; Shukla et al., 2019).

We also note that negative scaling was found over a few regions of south-central and south-east India with clear-sky temperatures at both daily and sub-daily scales (Fig. 6c, f). To our understanding, this negative scaling primarily arises for two reasons. Firstly, these grids receive rainfall contributions during both the summer and winter monsoon, however, a relatively higher proportion of the rain occurs during the winter monsoon (Fig. C1). The is due to the fact that this region lies over the leeward side of the Western Ghats for the incoming south-west monsoon winds during the summer monsoon. Contrastingly, during the winter monsoon, north-east winds blow over the Bay of Bengal leading to large moisture advection and more rain over this region. As a result of this seasonality effect, more extreme precipitation is sampled during the winter season over this region while moisture supply may limit these extremes to increase during the summer season. This may lead to a negative scaling when a single quantile regression slope is fitted over the whole temperature range. Another reason could be the development of a low-pressure system in the Bay of Bengal during winter months which causes cyclones over the eastern coast of India. These cyclonic systems cause very high rainfall at very low temperatures which can lead to negative scaling (Traxl et al., 2021). It remains an important area for future research and more work is necessary to resolve these systems in conventional scaling approaches.

The effect of seasonality on precipitation scaling was also evaluated by producing the scaling curves for different seasonal subsets (summer and winter monsoon). We find a change in scaling during summer season after removing the cloud effects as the drop disappears (see Appendix C). The winter season, on the other hand, is associated with reduced rainfall amounts (less than 20 %) and less clouds over most regions resulting in a similar scaling for both all-sky and clear-sky temperatures.

While some differences exist, the cloud-cooling effect largely explains the negative scaling across most of the grid points over India. Extreme precipitation increases monotonically with temperature when the cloud-cooling effect is removed. This implies that the peak structures obtained with observed scaling will not constrain the rise in extremes with anthropogenic warming. Moreover, the confounding effect between precipitation and temperature on observed scaling relationships, also termed “apparent scaling”, had been argued by some recent studies (Bao et al. 2017; Visser et al., 2020). Our results agree with these studies in that the observed scaling relationships also reflect the impact of synoptic conditions and cooling associated with precipitation events on temperature. However, we suggest that this confounding effect is largely associated with cloud radiative effect, which is removed by our use of clear-sky temperatures as a scaling variable. Additionally, we address the arguments raised to resolve apparent scaling using dew point temperature (Barbero et al., 2018). Our results confirm that precipitation extremes scale well with dew point temperatures as a measure for atmospheric moisture, but that the breakdown in scaling actually originates from the scaling of dew point temperatures with observed temperatures. This response of dew point temperature to warming is further affected by the presence of clouds and associated radiative cooling. Clear-sky temperatures are independent of the covariations arising from cloud effects and are thus a better, more independent measure and scaling variable for understanding the precipitation response to global warming.