Here we will discuss an article on the most viewed climate blog which is operated by Anthony Watts.

Refutation of Stable Thermal Equilibrium Lapse Rates  (January 24, 2012)

Robert Brown cites an author who correctly claims that the temperature gradient exists as the state of thermodynamic equilibrium. He then twists the discussion to thermal equilibrium (which is different) and he claims internal conductivity is ignored.  It is not.  It is the very mechanism whereby the gradient evolves. 

He then postulates a wire connecting the top and bottom of the cylinder under consideration.  He ignores the fact that the wire will also develop a temperature gradient.  He also ignores the fact that the new combined system will now develop a new temperature gradient based on the weighted mean specific heat of the gases and solids involved, including the walls of the container.  As we saw on the Home page, there can be convective heat transfers up a temperature gradient, as would happen on one side of the combined system, whether it be two cylinders with different gases connected thermally at each end, or a single cylinder with a wire. 

So, even though there may be cyclic movement of thermal energy, it would lead to a state of hydrostatic equilibrium in which the temperature at each point remained constant.  No work could be done by the system without extracting at least the energy required to do such work, and thus cooling the whole system whilst maintaining the temperature gradients.

Robert Brown shows a complete misunderstanding of thermodynamic equilibrium, which he does not even name correctly, and he fails to understand that there will always be a propensity towards establishing a temperature gradient in a gravitational field.  ​You should note this final comment on that thread written by a "BigWaveDave" and pointing out how the gravito-thermal effect has been "forgotten" by many PhD's.  It is not a matter for acceptable disagreement.  Also see the comment below that was posted on wattsupwiththat.

Now people will tell you that Maxwell made a similar suggestion in a weak attempt to refute Josef Loschmidt's gravito-thermal effect.  But Maxwell did not have centrifuges in those days to settle the issue, whereas we do, and the easiest centrifuge to use for this purpose is the Ranque Hilsch vortex tube illustrated at the foot of the 'Evidence' page. As air spirals down the tube in a helical motion it experiences a huge centrifugal force acting along each radius of the cylinder.  As expected, this force field, like gravity, forms a temperature gradient along each such radius, enabling separation of the air into streams that are hotter than the original air on the outside and colder along the center of the tube.
Errors in WattsUpWithThat blog
vortex cooler
Centrifugal force forms a temperature gradient in
a vortex tube, just as gravity does in air.
Oh what a wicked web we weave
When in our head we do believe
That we can sway another's mind
And, with some hoaxsters, lead the blind
To fear and tremble at the warning
That CO2 does all the warming
By sending down its radiation,
Fooling leaders of the Nation,
That all from flooded houses sailing
Will join the weeping and the wailing ...
While Mother Nature calmly ruling
Turns that warming into cooling.

by Doug Cotton, 8 August 2015

to prove me wrong here.