This material is not easy to understand. But we can demonstrate the concept of entropy quickly in the context of application of the First Law of Thermodynamics to an Ideal Gas. Guidance is provided by Arnold Sommerfeld, Thermodynamics and Statistical Mechanics, Academic Press, 1964. The First Law may be written
       dq = du + p dV
Here d means a small amount, q is heat, u is internal energy, p is pressure and V is volume per unit mass. A little bit of heating causes internal energy u (and thus temperature T) to increase, and work to be done by the action of pressure p accompanying an expansion of volume dV.

     Specific heat C introduced to relate internal energy to temperature T with du = C dT. If this change is at constant volume, the specific heat C = Cv; if at constant pressure, C = Cp. The internal energy u is a function solely of T, and du = Cv dT.

     The Ideal Gas Law reads
       pV = RT
where R is the appropriate gas constant. (For dry air R = 287.058 Joules/kg/K). This is a delightful little law, very easy to remember; it's not exactly true but the approximation is brilliant! Call it the International Gas Law. Let's see what it looks like when differentiated:
       p dV + V dp = R dT

     Return to the First Law and apply it to a reversible change to unit mass of a perfect gas, substituting for du and for p. We obtain
       dq = Cv dT + (RT/V) dV
Now divide through by T:
       (dq/T) = Cv (dT/T) + R (dV/V)
With Cv and R constant, both terms on the right hand side can be integrated. The result of integrating over a change can be expressed solely in terms of the initial and final states. The integral of (dq/T) is perfect. We can replace (dq/T) with dS. The function of state, entropy S, is defined - a result of stunning brilliance!
[Who's TOtally Brilliant Ydea was that? Probably Rudolf Clausius in the 1850's. A useful discussion is provided by Ian Stewart in his book 17 Equations that Changed the World, Profile Books, 2012]

     Specific volume V is an extensive variable of state which is related to work. It is coupled with the intensive variable pressure p. A small quantity p dV has dimensions of energy.

     Entropy S is an extensive variable of state which is related to thermal energy. It is coupled with the intensive variable temperature T. A small quantity T dS has dimensions of energy. The kinetic energy of winds is generally a very small term.

       In making calculations about the atmosphere, it would be unthinkable to be sloppy about specific volume (1/density); but meteorologists have been disgracefully sloppy about entropy. In 1997 Donald Johnson pointed out that this was probably the cause of the "general coldness of climate models" (J. Clim., 10, 2826-46).

       In their paper "Reversible and irreversible processes of radiation entropy" (Quart. J. Roy. Meteorol. Soc., Jan 1996 Part B, 122, 483-94) Richard Goody and Wedad Abdou wrote: "So far as the fluid is concerned, these events take place outside the fluid system and are of no direct relevance to meteorology or climate theory, .." I protested about this work. Having at length made friends with Professor Goody, I got on to him about the work of Donald Johnson, of which he had not heard. Then Professor Goody wrote his classic paper about "Sources and Sinks of Climate Entropy" (QJRMS, July 2000 Part A, 126, 1953-70), largely in response to Donald Johnson. All may be vanity, but occasionally there's a twinkle.

       With great stupidity I protested against an excellent paper by G.L. Stephens and D. M. O'Brien, Energy and Climate. I: ERBE Observations of the entropy production of the earth (QJRMS, Jan 93 A, 121-152). This onslaught was treated with immaculate courtesy. I have not heard from Graeme Stephens. A useful reference to the work of Stephens and collaborators in Nature Geosci. (2012) is given by Axel Kleidon. His book Thermodynamic Foundations of the Earth System (Cambridge University Press, 2016) gives an ample summary of this very difficult subject; it reads easily but the subject is very difficult to comprehend as he does. I recommend also Non-Equilibrium Thermodynamics and the Production of Entropy (Springer, 2005, Axel Kleidon and R. D. Lorenz, Eds.); this book contains several short essays.

       After a long struggle to clear up my confusion with the difficulties, I published a paper on "Entropy sources in equilibrium conditions over a tropical ocean" in J. Atmos. Sci., 62, 1588-1600 (2005). I surveyed the sources of entropy arising at various levels from various different processes within the atmosphere. Greatest contributions are from evaporation of precipitation and mechanical work done by falling precipitation. Mixing of air of varying temperature and humidity makes a large contribution, difficult to quantify. Mechanical efficiency of conversion to kinetic energy, interesting in the context of the atmosphere as a heat engine, is only a few percent. Intense conversions take place at very small scales, while gentle processes occur through the whole extent of the atmosphere, so a comprehensive grasp of overall entropy production within the atmosphere is elusive. I found entropy production amounting to 53 mW/m²/K, corresponding to a flux of free energy, or rate of dissipation within the Earth system, of 14.9 W/m² at a representative temperature of 281 K. My results corresponded with those of Vyacheslav Zakharov and his collaborators. He told me that this was an excellent paper. He has not been able to pursue his important innovation owing to pressure to study consequences of melting permafrost at high latitudes.

       This paper of mine was ignored in a fine survey of work by Olivier Pauluis and Juliana Dias published in Science (Vol. 335, 24 Feb 2012). A perspective on this paper called "Frictional Dissipation - Blame It on the Rain" was written by Dargan M. W. Frierson. I was sad to see that along with many other people, he did not know of my work.

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       There is only one 1974 reference to the great Russian M. I. Budyko. We have also M. Budyko, A. Ronov and A. Yanshin (1987), History of the Earth's Atmosphere (Springer Verlag, 1987). This is very interesting in its presentation of contents in the Earth's atmosphere of oxygen and carbon dioxide over the last 500 Ma, showing cycles. Are these recently confirmed?

       Axel has told me that he was unaware of my paper of May 2005 on entropy sources in the atmosphere. It was overlooked, and I would draw attention to it as a useful contribution.

       Scattering of sunlight is an important and under-appreciated business. In Axel's section 6.3.2 he covered the Nature Geosciences review by Graeme Stephens and co-authors (2012), but we could use more attention on this matter.

       The work of Vyacheslav Zakharov should receive attention. V. I. Zakharov, R. Imasu and K. G. Gribanov, "Regarding Free Energy Net of the Earth and its Monitoring from Space Concept" (SPIE Proc., 2005, vol. 5655, pp 540-547) made clear the respective meanings of thermal energy, entropy and free energy in the context of radiation. (The SPIE in California hosts Conferences for photographic instrumentation engineers.)

       Zakharov et al. gave just a few results from their analyses of budgets of free energy. They assumed an albedo of 0.3 and 50% cloudiness to get a net free energy flux of 19 W/m², indicating useful work inside the climate system of 19 W/m². For a tropical latitude of 15° they found 52 W/m² at 20° latitude they found 20 W/m² and at 80° they found -8 W/m². Near the poles, therefore, a net import of free energy from lower latitudes was determined.

       The 2005 conference paper was followed in 2008 by V.I. Zakharov, R. Imasu, K. G. Gribanov and S. V. Zakharov, "Free energy balance at the upper boundary of the atmosphere", Atmospheric and Oceanic Optics, Vol. 21, 2008, No. 3, 211-218. In this paper they contrived brilliantly to separate mathematically thermal energy and free energy, and worked directly with fluxes of free energy. They found that the annual mean flux of free energy imported by the Earth is about 59 W/m². This finding should be exhaustively re-examined.