Here we will discuss articles on the "Hockey Schtick" climate change blog which are similar also to articles on Tallbloke's Talkshop. Each blog site recognises that gravity plays a part, but they misunderstand the convective heat transfer process.

Before considering the main articles on the gravito-thermal effect, there is an article about the Second Law of Thermodynamics (January 2nd, 2015) which is incorrect.

The Second Law applies to an isolated system which, in physics, is different from a closed system. But, more importantly, it applies to each individual INDEPENDENT process. If applied to more than one process then those processes must be DEPENDENT upon each other and, in effect, be components of one combined process. One way radiation into the surface from the atmosphere is an independent process. If it transferred thermal energy from cold to hot it would violate the Second Law of Thermodynamics.  

The argument in their article implies that we only consider the whole Earth plus atmosphere system when considering the need for entropy to increase. If that were all we needed to do then we could "prove" (using the Second Law) that water could flow up a hillside provided that it subsequently flowed further down on the other side. Indeed, that could happen in a siphon, but the processes on each side of a siphon are dependent processes. 

The Hockey Schtick blog ran this series of articles on the gravito-thermal effect (late 2014) but they failed to explain the actual thermal energy transfers correctly. 

They assume that atmospheric circulation (that is, wind in all its forms) somehow forms the temperature gradient supposedly by convection.  

But what physicists call "convective heat transfer" is not wind, and the temperature gradient results from the process described in statements of the Second Law of Thermodynamics.

Now, because the Second Law applies only to an independent  process within an isolated system, if we consider an imaginary bounded column of air, the density and temperature gradients form in calm conditions when there is no energy being added to that column of air.  Wind entering the column adds energy and so we do not get the calculated temperature gradient. This disproves their hypothesis.
Errors in the Hockey Schtick blog
Centrifugal force in a vortex
cooling tube creates a temperature 
gradient which evolves at the molecular
level. Gravity does likewise in the troposphere. 
Ranque-Hilsch vortex tube confirms the existence of
 a gravitationally-induced
temperature gradient