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Using the local one-dimensional atmospheric boundary layer (ABL) model of vertical turbulent diffusion, we performed simulations of the moderately stable and strongly stable atmospheric boundary layer using scenarios previously used for numerical experiments with eddy-resolving turbulence models (Beare et al., 2006; Glazunov et al., 2016; van der Linden et al., 2019). We tested a locally one-dimensional ABL models with first-order turbulent closures while substituting various stability functions proposed in the literature and used in climate and weather prediction models. Based on these tests, it has been shown that the best fit with eddy-resolving simulation data is given by models with stability functions that allow for turbulence maintenance at large Richardson gradient numbers (Zilitinkevich et al., 2013; Esau and Byrkjedal 2007) and calibrated on eddy-resolving and direct numerical simulation data (Mortikov et al., 2019; Zilitinkevich et al., 2019). It is shown that these turbulent diffusion models are significantly superior to the parameterization based on (Louis 1979) and currently used in the INMCM5 climate model. The turbulent diffusion models that have performed best in independent tests have been selected for testing within the climate system model. The parameterizations of vertical mixing in the ABL selected on the basis of the described independent tests were included in the INMCM5 model. Test simulations with the climate model were carried out to investigate the influence of the new parameterization on the reproduction of the modern climate. The basic version of the INMCM5 model was used. In order to obtain climatic averages and to be able to compare with ERA-Interim reanalysis data (Dee et al. 2009), simulations were carried out from 1996 to 2014. For both the annual and seasonal averages, the largest changes in the model-reproduced climate were found in the polar regions in surface temperature and sensible heat flux. This is consistent with occurance and lifetime of stable boundary layers in high latitudes. The improvement of the parameterization of the stable atmospheric boundary layer reduced the average error in the model reproduction of the surface temperature from -0.45°C to -0.25°C, and the root-mean-square error by 8% compared to the reanalysis. These improvements are mainly due to the reproduction of a warmer Arctic and colder Antarctic continent compared to those observed in simulations with the baseline version of INMCM5 (Volodin et al. 2017), similar improvements are observed in other meteorological parameters.