We present significant differences in the simulated atmospheric flow for warm, tidally-locked small Neptunes and super Earths (based on a nominal GJ 1214b) when solving the simplified, and commonly used, primitive dynamical equations or the full Navier-Stokes equations. The dominant prograde, superrotating zonal jet is markedly different between the simulations which are performed using practically identical numerical setups, within the same model. The differences arise due to the breakdown of the so-called 'shallow-fluid' and traditional approximations, which worsens when rotation rates are slowed, and day-night temperature contrasts are increased. The changes in the zonal advection between simulations solving the full and simplified equations, give rise to significant differences in the atmospheric redistribution of heat, altering the position of the hottest part of the atmosphere and temperature contrast between the day and night sides. The implications for the atmospheric chemistry and, therefore, observations need to be studied with a model including a more detailed treatment of the radiative transfer and chemistry. Small Neptunes and super Earths are extremely abundant and important, potentially bridging the structural properties (mass, radius, composition) of terrestrial and gas giant planets. Our results indicate care is required when interpreting the output of models solving the primitive equations of motion for such planets.

We derive an analytical theory of the PDF of density fluctuations in supersonic turbulence in the presence of gravity in star-forming clouds. The theory is based on a rigorous derivation of a combination of the Navier–Stokes continuity equations for the fluid motions and the Poisson equation for the gravity. It extends previous approaches, first by including gravity and second by considering the PDF as a dynamical system, not a stationary one. We derive the transport equations of the density PDF, characterize its evolution, and determine the density threshold above which gravity strongly affects and eventually dominates the dynamics of turbulence. We demonstrate the occurrence of t wo power-law tails in the PDF, with two characteristic exponents, corresponding to two different stages in the balance between turbulence and gravity. Another important result of this study is to provide a procedure to relate the observed column density PDFs to the corresponding volume density PDFs. This allows us to infer, from the observation of column densities, various physical parameters characterizing molecular clouds, notably the virial parameter. Furthermore, the theory offers the possibility to date the clouds in units of t coll, the time since a statistically significant fraction of the cloud started to collapse. The theoretical results and diagnostics reproduce very well numerical simulations and observations of star-forming clouds. The theory provides a sound theoretical foundation and quantitative diagnostics to analyze observations or numerical simulations of star-forming regions and to characterize the evolution of the density PDF in various regions of molecular clouds.