| Summary: | The Earth’s global radiation budget depends critically on the relationship between outgoing longwave radiation (OLR) and surface temperature ( $T_\mathrm{s}$ ). Using the fifth generation of European ReAnalysis dataset, we find that although OLR appears to be linearly dependent on $T_\mathrm{s}$ over a wide range, there are significant deviations from the linearity in the OLR– $T_\mathrm{s}$ relationship for regions warmer than 270 K $T_\mathrm{s}$ , which covers 89% of the surface of Earth. While the AMIP runs of CMIP6 models largely capture the overall OLR– $T_\mathrm{s}$ relationship, considerable discrepancies are found in clear-sky OLR at given $T_\mathrm{s}$ ranges. In this study, we investigate physical mechanisms that control the clear-sky OLR– $T_\mathrm{s}$ relationship seen in reanalysis and CMIP6 models by using accurate radiative transfer calculations. Our study identifies three key mechanisms to explain both the linearity and departure from linearity of the clear-sky OLR– $T_\mathrm{s}$ relationship. The first is a surface contribution, controlled by the thermal emission of the surface and the infrared opacity of the atmosphere, accounting for 60% of the observed clear-sky OLR– $T_\mathrm{s}$ linear slope. The second is changes in atmospheric emission induced by a foreign pressure effect on water vapor and other greenhouse gases, which accounts for 30% of the linear slope in a clear-sky condition. The third is changes in atmospheric emission induced by variations in relative humidity (RH), particularly in the mid-troposphere (250 to 750 hPa), which determines the non-linearity in the clear-sky OLR– $T_\mathrm{s}$ relationship and adds to the remaining 10% of the slope. The inter-model spread in mid-tropospheric RH explains a large fraction of the differences in clear-sky OLR across CMIP6 models at given surface temperatures. Furthermore, the three key mechanisms outlined here apply to the OLR– $T_\mathrm{s}$ relationship in all-sky conditions: clouds disguise the surface contribution but increase the atmospheric contribution, retaining a similar linear slope to the clear-sky condition while amplifying the non-linear curvature.
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