  While small, Neptune-like planets are among the most abundant exoplanets, our
understanding of their atmospheric structure and dynamics remains sparse. In
particular, many unknowns remain on the way moist convection works in these
atmospheres where condensable species are heavier than the non-condensable
background gas. While it has been predicted that moist convection could
shut-down above some threshold abundance of these condensable species, this
prediction is based on simple linear analysis and relies on strong assumptions
on the saturation of the atmosphere. To investigate this issue, we develop a 3D
cloud resolving model for H2 atmospheres with large amounts of condensable
species and apply this model to a prototypical temperate Neptune-like planet --
K2-18b. Our model confirms the shut-down of moist convection and the onset of a
stably stratified layer in the atmosphere, leading to much hotter deep
atmospheres and interiors. Our 3D simulations further provide quantitative
estimates of the turbulent mixing in this stable layer, which is a key driver
of the cycling of condensables in the atmosphere. This allows us to build a
very simple, yet realistic 1D model that captures the most salient features of
the structure of Neptune-like atmospheres. Our qualitative findings on the
behavior of moist convection in hydrogen atmospheres go beyond temperate
planets and should also apply to the regions where iron and silicates condense
in the deep interior of H2-dominated planets. Finally, we use our model to
investigate the likelihood of a liquid ocean beneath a H2 dominated atmosphere
on K2-18b. We find that the planet would need to have a very high albedo
(>0.5-0.6) to sustain a liquid ocean. However, due to the spectral type of the
star, the amount of aerosol scattering that would be needed to provide such a
high albedo is inconsistent with the latest observational data.
