Linearity of Earth's outgoing longwave radiation with global mean surface temperature

In the study of Earth’s climate sensitivity, the response of longwave radiation to temperature anomalies is conventionally modeled as a linear damping mechanism. This assumption of linearity is robust across a temperature range of about 50 K. Previous studies have largely relied on single-column models with fixed relative humidity (RH) to explain this linearity, despite the diversity of surface conditions within the climate system. However, globally, there is a remarkably linear relationship between outgoing longwave radiation (OLR) and surface temperature ($T_s$).

We have derived conditions under which this linearity holds for the averages across the entire globe. We show that the analog for fixed RH of a single column is that the global RH histogram is invariant under warming.

We start with an expression of OLR as follows: \begin{equation} \mathrm{OLR}(T_s, \mathrm{RH}) = \alpha T_s +R(\mathrm{RH}), \end{equation} where $R$ represents a function of RH. This is equivalent to the expression: \begin{equation} \frac{\partial \mathrm{OLR}}{\partial T_s} = \alpha. \end{equation} Integrating local OLR values over the joint distribution of surface temperature ($T_s$) and RH yields the global average OLR: \begin{equation} \delta \overline{\mathrm{OLR}} = \alpha \delta \overline{T_s} + \int \mathrm{dRH}R(\mathrm{RH})\delta F(\mathrm{RH}), \end{equation} where $F$ denotes the global distribution of RH, particularly the column RH in the free troposphere. When the RH distribution remains invariant with warming, the relationship between the global average OLR and the global mean surface temperature reduces to \begin{equation} \frac{ \mathrm{d\overline{OLR}}}{\mathrm{d} \overline{T_s} }= \alpha. \end{equation} This reduction confirms the applicability of the single-column model to global averages. The assumption of invariant RH is generally supported by simulations of global warming in global climate models. This implies that the single-column results are broadly applicable to global climate projections. However, significant changes in the RH distribution could potentially change this linearity.

The linearity of global-mean outgoing longwave radiation (OLR) with surface temperature is a basic assumption in climate dynamics. This linearity manifests in global climate models, which robustly produce a global-mean longwave clear-sky (LWCS) feedback of 1.9 W/m2/K, consistent with idealized single-column models (Koll & Cronin, 2018). However, there is considerable spatial variability in the LWCS feedback, including negative values over tropical oceans (known as the “super-greenhouse effect”) which are compensated for by larger values in the subtropics/extratropics. Therefore, it is unclear how the idealized single-column results are relevant for the global-mean LWCS feedback in comprehensive climate models. Here we show with a simple analytical theory and model output that the compensation of this spatial variability to produce a robust global-mean feedback can be explained by two facts: (1) When conditioned upon free-tropospheric column relative humidity (RH), the LWCS feedback is independent of RH, and (2) the global histogram of free-tropospheric column RH is largely invariant under warming.

Linearity of Earth's outgoing longwave radiation with global mean surface temperature

References

2020