Spectrally dependent emission by the surface (i.e., surface spectral emissivity) is commonly ignored by current climate models. Surface spectral emissivity matters more in cold and dry environments than in hot and humid environments. Recent modeling studies confirmed that, for current climate simulations, this process affects the polar climate more than the extra-polar climate. As for the Snowball Earth, a period characterized by global polar-like conditions of extreme cold and low humidity, including surface spectral emissivity could alter the simulated global radiation budget. This, in turn, could affect the simulated climate of the Snowball Earth. Here, we use anaqua-planet slab-ocean simulation of Snowball Earth by the ICON model to perform offline radiative transfer calculations to quantify such impact on the outgoing longwave radiation (OLR). The offline radiative transfer model is used to compute the clear-sky OLR for two surfaces that would be present in the extremely cold simulation: ice and snow. Compared to the results with assumed blackbody surface, the globalmean OLR decreases by 3.4 and 1.1 W m-2 for ice and snow surfaces, respectively. The impact of surface spectral emissivity on the OLR is strongest at the equator and weakens towards the poles, presenting a noticeable meridional gradient. The seasonal variation of the impact is also obvious. The radiative effects of this often-neglected process warrant further scrutiny for the Snowball Earth simulations, particularly the Jormungand mechanism, as well as simulations of other cold and dry climates.

Earth has experienced several pan-glaciations in its history, often interpreted as hard Snowball Earth periods with complete global ice cover. Alternatively, waterbelt states with a narrow equatorial strip of ice-free ocean provide a compelling explanation for the survival of life during these extreme glaciations. In this study, we establish a framework to quantify three atmospheric factors that influence waterbelt states: the spatial extent of bare dark sea ice, cloud masking of the ice-albedo feedback, and cloud shortwave feedback. We first explore these factors in the Budyko-Sellers energy balance model, and then investigate them in simulations with three versions of the three-dimensional climate model ICON. This allows us to relate differences in the waterbelt states between climate model versions to differences in the three factors.   A broader Hadley circulation shifts the boundary between snow-covered and bare sea ice poleward, leading to waterbelt states with an ice line at higher latitudes. Clouds work in favor of stable waterbelt states by weakening the ice-albedo feedback through cloud masking. A positive shortwave cloud feedback due to increasing cloud condensate over the open ocean destabilizes waterbelt states. While our study does not answer which of the model versions is most realistic, it provides a quantitative framework for understanding the atmospheric mechanisms that govern the existence and hysteresis of waterbelt states.