Feynman explains how gravitational potential energy and kinetic energy convert to create the gravito-thermal greenhouse effect, without greenhouse gases

Excerpts from The Feynman Lectures, Chapter 40, The Principles of Statistical Mechanics, which prove the Maxwell/Clausius/Carnot gravito-thermal greenhouse effect is correct, and would occur in a pure atmosphere containing only the non-greenhouse gases N2 & O2 (99.94% of our atmosphere)(bolding and [additions] added):

If the temperature is the same at all heights, the problem is to discover by what law the atmosphere becomes tenuous as we go up. If N is the total number of molecules in a volume V of gas at pressure P, then we know PV=NkT, or P=nkT, where n=N/V is the number of molecules per unit volume. In other words, if we know the number of molecules per unit volume, we know the pressure, and vice versa: they are proportional to each other, since the temperature is constant in this problem. But the pressure is not constant, it must increase as the altitude is reduced, because it has to hold, so to speak, the weight of all the gas above it. That is the clue by which we may determine how the pressure changes with height. If we take a unit area at height h, then the vertical force from below, on this unit area, is the pressure P. The vertical force per unit area pushing down at a height h+dh would be the same, in the absence of gravity, but here it is not, because the force from below must exceed the force from above by the weight of gas in the section between h and h+dh. Now mg is the force of gravity on each molecule, where gis the acceleration due to gravity, and ndh is the total number of molecules in the unit section. So this gives us the differential equation Ph+dh−Ph= dP= −mgndh. Since P=nkT, and T is constant, we can eliminate either P or n, say P, and get

dndh=−mgkTn

for the differential equation, which tells us how the density goes down as we go up in energy.

We thus have an equation for the particle density n, which varies with height, but which has a derivative which is proportional to itself. Now a function which has a derivative proportional to itself is an exponential, and the solution of this differential equation is

n=n0e−mgh/kT.(40.1)

Here the constant of integration, n0, is obviously the density at h=0 (which can be chosen anywhere), and the density goes down exponentially with height.

Fig. 40–2.The normalized density as a function of height in the earth’s gravitational field for oxygen and for hydrogen, at constant temperature.

Note that if we have different kinds of molecules with different masses, they go down with different exponentials. The ones which were heavier would decrease with altitude faster than the light ones. Therefore we would expect that because oxygen is heavier than nitrogen, as we go higher and higher in an atmosphere with nitrogen and oxygen the proportion of nitrogen would increase. This does not really happen in our own atmosphere, at least at reasonable heights, because there is so much agitation which mixes the gases back together again. It is not an isothermal atmosphere. Nevertheless, there is a tendency for lighter materials, like hydrogen, to dominate at very great heights in the atmosphere, because the lowest masses continue to exist, while the other exponentials have all died out (Fig. 40–2).

40–2The Boltzmann law

Here we note the interesting fact that the numerator in the exponent of Eq. (40.1) is the [gravitational] potential energy of an atom. Therefore we can also state this particular law as: the density at any point is proportional to

e−the potential energy of each atom/kT.

That may be an accident, i.e., may be true only for this particular case of a uniform gravitational field. However, we can show that it is a more general proposition. Suppose that there were some kind of force other than gravity acting on the molecules in a gas. For example, the molecules may be charged electrically, and may be acted on by an electric field or another charge that attracts them. Or, because of the mutual attractions of the atoms for each other, or for the wall, or for a solid, or something, there is some force of attraction which varies with position and which acts on all the molecules. Now suppose, for simplicity, that the molecules are all the same, and that the force acts on each individual one, so that the total force on a piece of gas would be simply the number of molecules times the force on each one. To avoid unnecessary complication, let us choose a coordinate system with the x-axis in the direction of the force, F.

In the same manner as before, if we take two parallel planes in the gas, separated by a distance dx, then the force on each atom, times the n atoms per cm³ (the generalization of the previous nmg), times dx, must be balanced by the pressure change: Fndx=dP=kTdn. Or, to put this law in a form which will be useful to us later,

F=kTddx(lnn).(40.2)

For the present, observe that −Fdx is the work we would do in taking a molecule from x to x+dx, and if F comes from a potential, i.e., if the work done can be represented by a [gravitational] potential energy at all, then this would also be the difference in the [gravitational] potential energy (P.E.). The negative differential of [gravitational] potential energy is the work done, Fdx, and we find that d(lnn)=−d(P.E.)/kT, or, after integrating,

n=(constant)e−P.E./kT.(40.3)

Therefore what we noticed in a special case turns out to be true in general. (What if F does not come from a potential? Then (40.2) has no solution at all. Energy can be generated, or lost by the atoms running around in cyclic paths for which the work done is not zero, and no equilibrium can be maintained at all. Thermal equilibrium cannot exist if the external forces on the atoms are not conservative.) Equation (40.3), known as Boltzmann’s law, is another of the principles of statistical mechanics: that the probability of finding molecules in a given spatial arrangement varies exponentially with the negative of the potential energy of that arrangement, divided by kT.

This, then, could tell us the distribution of molecules: Suppose that we had a positive ion in a liquid, attracting negative ions around it, how many of them would be at different distances? If the potential energy is known as a function of distance, then the proportion of them at different distances is given by this law, and so on, through many applications...

40–4The distribution of molecular speeds

Now we go on to discuss the distribution of velocities, because sometimes it is interesting or useful to know how many of them are moving at different speeds. In order to do that, we may make use of the facts which we discovered with regard to the gas in the atmosphere. We take it to be a perfect gas, as we have already assumed in writing the potential energy, disregarding the energy of mutual attraction of the atoms. The only potential energy that we included in our first example was gravity. We would, of course, have something more complicated if there were forces between the atoms. Thus we assume that there are no forces between the atoms and, for a moment, disregard collisions also, returning later to the justification of this. Now we saw that there are fewer molecules at the height h than there are at the height 0; according to formula (40.1), they decrease exponentially with height. How can there be fewer at greater heights? After all, do not all the molecules which are moving up at height 0 arrive at h? No!, because some of those which are moving up at 0 are going too slowly, and cannot climb the potential hill to h. With that clue, we can calculate how many must be moving at various speeds, because from (40.1) we know how many are moving with less than enough speed to climb a given distance h. Those are just the ones that account for the fact that the density at h is lower than at 0...Since velocity and momentum are proportional, we may say that the distribution of momenta is also proportional to 

e−K.E./kT per unit momentum range. It turns out that this theorem is true in relativity too, if it is in terms of momentum, while if it is in velocity it is not, so it is best to learn it in momentum instead of in velocity:

f(p)dp=Ce−K.E./kTdp.(40.8)

So we find that the probabilities of different conditions of energy, kinetic and potential, are both given by e−energy/kT, a very easy thing to remember and a rather beautiful proposition.

So far we have, of course, only the distribution of the velocities “vertically.” We might want to ask, what is the probability that a molecule is moving in another direction? Of course these distributions are connected, and one can obtain the complete distribution from the one we have, because the complete distribution depends only on the square of the magnitude of the velocity, not upon the z-component. It must be something that is independent of direction, and there is only one function involved, the probability of different magnitudes. We have the distribution of the z-component, and therefore we can get the distribution of the other components from it. The result is that the probability is still proportional to e−K.E./kT, but now the kinetic energy involves three parts, mv2x/2, mv2y/2, and mv2z/2, summed in the exponent. Or we can write it as a product:

f(vx,vy,vz)dvxdvydvz∝e−mv2x/2kT⋅e−mv2y/2kT⋅e−mv2z/2kTdvxdvydvz.(40.9)

You can see that this formula must be right because, first, it is a function only of v2, as required, and second, the probabilities of various values of vz obtained by integrating over all vx and vy is just (40.7). But this one function (40.9) can do both those things!

40–5The specific heats of gases

Now we shall look at some ways to test the theory, and to see how successful is the classical theory of gases. We saw earlier that if U is the internal energy of N molecules, then PV= NkT= (γ−1)U holds, sometimes, for some gases, maybe. If it is a monatomic gas, we know this is also equal to 

23 of the kinetic energy of the center-of-mass motion of the atoms. If it is a monatomic gas, then the kinetic energy is equal to the internal energy, and therefore γ−1=23. But suppose it is, say, a more complicated molecule, that can spin and vibrate, and let us suppose (it turns out to be true according to classical mechanics) that the energies of the internal motions are also proportional to kT. Then at a given temperature, in addition to kinetic energy 32kT, it has internal vibrational and rotational energies. So the total U includes not just the kinetic energy, but also the rotational and vibrational energies, and we get a different value of γ. Technically, the best way to measure γ is by measuring the specific heat, which is the change in energy with temperature. We will return to that approach later. For our present purposes, we may suppose γ is found experimentally from the PVγ curve for adiabatic compression...

40–6The failure of classical physics

So, all in all, we might say that we have some difficulty. We might try some force law other than a spring, but it turns out that anything else will only make γ higher. If we include more forms of energy, γ approaches unity more closely, contradicting the facts. All the classical theoretical things that one can think of will only make it worse. The fact is that there are electrons in each atom, and we know from their spectra that there are internal motions; each of the electrons should have at least 12kT of kinetic energy, and something for the potential energy, so when these are added in, γ gets still smaller. It is ridiculous. It is wrong.

The first great paper on the dynamical theory of gases was by Maxwell in 1859. On the basis of ideas we have been discussing, he was able accurately to explain a great many known relations, such as Boyle’s law, the diffusion theory, the viscosity of gases, and things we shall talk about in the next chapter. He listed all these great successes in a final summary, and at the end he said, “Finally, by establishing a necessary relation between the motions of translation and rotation (he is talking about the 12kT theorem) of all particles not spherical, we proved that a system of such particles could not possibly satisfy the known relation between the two specific heats.” He is referring to γ (which we shall see later is related to two ways of measuring specific heat), and he says we know we cannot get the right answer...

Also see Feynman's lecture 42 in which he states,

"It is like the atmosphere in equilibrium under gravity, where the gas at the bottom is denser than that at the top because of the work mghneeded to lift the gas molecules to the height h."

."

n=(constant)e−P.E./kT

Physicist Richard Feynman proved the Maxwell gravito-thermal greenhouse theory is correct & does not depend upon greenhouse gas concentrations

The great physicist Richard Feynman adds to three other giants of physics, Maxwell, Clausius, and Carnot, who have explained the "greenhouse effect" is solely a consequence of gravity, atmospheric mass, pressure, density, and heat capacities, and is not due to "trapped radiation" from IR-active or 'greenhouse' gas concentrations. 

Only one 33C greenhouse theory can be correct, either the 33C Arrhenius radiative greenhouse theory (the basis of CAGW alarm and climate models) or the 33C Maxwell/Clausius/Carnot/Feynman gravito-thermal greenhouse effect, since if both were true, the surface temperature would be an additional 33C warmer than the present. As we have previously shown, the Arrhenius greenhouse theory confuses the cause (gravito-thermal) with the effect (radiation from greenhouse gases).

In addition, the US Standard Atmosphere, the International Standard Atmosphere, the HS 'greenhouse equation,' Chilingar, et al derive the observed atmospheric temperature profile without use of a single greenhouse gas radiative transfer equation or calculation, and using the same basic atmospheric physics discussed by Feynman in his lecture below. Feynman does not make a single mention of radiation, radiative transfer, greenhouse gases, CO2, nor does he derive any radiative transfer equations to derive the atmospheric temperature profile, and instead utilizes the barometric and statistical mechanics formulas necessary to describe the gravito-thermal greenhouse effect of Maxwell et al (who Feynman quotes extensively below). Feynman demonstrates that an atmosphere comprised solely of the non-greenhouse gases N2 & O2 (99.94% of our atmosphere, but 100% in Feynman's demonstrations) would establish the temperature gradient/"greenhouse effect" observed in the troposphere.

Feynman demonstrates that the conservative force of gravity does indeed do continuous thermodynamic Work upon the atmosphere (a common false argument by those who do not accept the gravito-thermal GHE theory is that gravity allegedly can't do Work upon the atmosphere), and describes gravitational potential energy (PE) accumulated as air parcels rise/expand/cool, which is then exchanged for kinetic energy (KE) as the air parcel descends/compresses/warms, creating the temperature gradient & greenhouse effect. 

Another online version here with larger print

'Greenhouse Gases Warmed the Earth Somewhat, but Additions Now Cool the Earth'

Reblog of a new post by materials physicist Dr. Charles Anderson, which discusses the recent paper by Chilingar et al posted here. Dr. Anderson improves upon some aspects of the paper, but in general comes to the same basic conclusion that additions to the present levels of the greenhouse gases CO2, H2O, and methane will cause cause cooling, not warming, of the Earth surface & atmosphere. 

27 July 2015

Greenhouse Gases Warmed the Earth Somewhat, but Additions Now Cool the Earth

By Charles R. Anderson, PhD, physics

Let us examine the net effect of infra-red active (so-called greenhouse) gases on the Earth’s surface temperature under present conditions and then the effect of a perturbation of that condition.   First, the net effect of the greenhouse gases presently on the surface temperature is usually found as the presently measured surface temperature minus the temperature predicted by a simple black body radiation calculation.  The average power flux of energy from solar insolation at the top of the atmosphere on the Earth system is usually given as

S (1-A)/ 4,

where S is the total solar insolation or radiation, A is the albedo or the fraction of the solar radiation reflected without absorption by the Earth system, and the factor of 4 is the average reduction of solar flux due to the projection of a rotating sphere onto a disk in the daily cycle.  However, the Earth has a precession angle of ψ = 23.44° or 0.4094 radians relative to its rotational axis.  According to Sorokhtin, Chilingar, Khilyuk and Gorfunkel in Evolution of the Earth’s Global Climate, Energy Sources, Part A, (2007), 1-19 and Sorokhtin, Chilingar and Khilyuk, Global Warming and Global Cooling: Evolution of Climate on Earth, Elsevier, Amsterdam (2007), p.313, the correction factor for the precession effect, ø, replaces the factor 4 in the divisor above with 4ø, where ø is

[π/2 – ψ]/π/2 + (ψ/π/2) [1/(1+cos ψ)] = 0.8754 for the Earth

So 4ø = 3.5016 for the Earth.

However, to calculate surface temperature of the Earth without any infra-red active gases such as water vapor, carbon dioxide, or methane, one has to delete the losses of reflected solar insolation due to reflections from clouds.  If there is no water vapor, there are no clouds.  Let us examine the 2013 NASA Earth Energy Budget of Fig. 1 or a means to estimate the fraction of the solar insolation incident upon the surface, the only location where absorption occurs, which is reflected.  The albedo A of an Earth without infra-red active gases is 0.127 from this NASA Earth Energy Budget, rather than the 0.3 value for our present Earth with infra-red active gases.  The Earth’s surface temperature without infra-red active gases, T_S, is then

T_S = [S(1-A)/(3.5016)σ]^0.25 = [1367 W/m² (1 - 0.127)/ (3.5016)(5.6697 x 10^-8 W/m² K⁴)]^0.25

T_S = 278.4 K

So if the present average temperature of the Earth is taken to be 288.2 K, the net warming effect of all of the present infra-red active gases is 9.8 K.  This is a far cry from the 33 K warming effect which is often claimed as the result of the so-called greenhouse gas effect.  But it is true that without the so-called greenhouse gases, the Earth’s surface would be cooler than it is now because the surface itself would be in radiative equilibrium with space instead of a combination of the surface, a more heavily weighted altitude at the top of the troposphere, and a much lighter weighting of the stratosphere.

Fig. 1.  The NASA Earth Energy Budget of 2013 is shown.  There is a great deal of nonsense in this energy budget, but the one thing we are taking from it is the fraction of solar radiation incident upon the surface which is reflected, which is 7% / (48% + 7%) = 0.127.

There are many effects that are caused by the infra-red active gases.  The first molecules of these gases added to the atmosphere were able to absorb energy that would otherwise have been radiated directly from the surface straight out into space.  That absorbed energy was then most often transferred to non-active infra-red molecules of nitrogen, oxygen, and argon gas which then mostly transported the energy upward by convection processes until the energy was deposited in the atmosphere where the molecular collision rate was lower and the mean free path for infra-red energy absorption was longer.  This absorption effect is large at first, but becomes rapidly smaller as the number of infra-red molecules becomes larger.  Other effects do not shrink as rapidly or at all as the number of infra-active molecules increases, however.  For instance, water vapor and CO₂ also absorb incoming solar insolation in the atmosphere and that absorption is less saturated at the present concentrations of water vapor and CO₂ in the atmosphere.  This is a surface cooling effect in that the radiation never arrives at the surface to warm it.  The differential effects of water vapor and CO₂ compared to N₂ and O₂ on the heat transported by convection scale linearly with the increase in water vapor and CO₂, so they do not diminish as their concentrations are increased.  Water vapor condensation in the atmosphere also increases linearly with the amount of water vapor.

So, it is not a foregone conclusion that adding CO₂ to the present mix of gases in the Earth atmosphere will cause further warming, just because the additions of the first molecules did cause warming.  We do not immediately know whether the so-called greenhouse effect is increasing or decreasing with further additions of greenhouse gases.  This is a question I have been discussing for years on this blog and since I wrote a book chapter called "Do IR-Absorbing Gases Warm or Cool the Earth’s Surface?, in Slaying the Sky Dragon -- Death of the Greenhouse Gas Theory, Stairway Press, published in January 2011.  Of course, the presence of water on the Earth’s surface and water vapor in the atmosphere causes the Earth’s surface to be warmer than it would be without water, but unlike the common assumption, this does not tell us that further additions of the so-called greenhouse gases will cause further warming.  I have many times explained why the physics commonly and vaguely offered as the reason why such gases would continue to warm the Earth’s surface is wrong.

The recent paper by G.V. Chilingar, O.G. Sorokhtin, L.F. Khilyuk, and M. Liu entitled Do Increasing Contents of Methane and Carbon Dioxide in the Atmosphere Cause Global Warming?, Atmospheric and Climate Sciences, Vol.04 No.05 (2014), Article ID:51443 addresses this question.  They note that the adiabatic temperature distribution with pressure p, gas heat capacity at constant pressure of c_P and heat capacity at constant volume of c_V, is given by

T^γ p^1-γ = constant, where γ = c_P/c_V, or

T = (constant) p^α, with α = (γ – 1)/γ

They note that for atmospheres with a pressure greater than 0.2 atm,

T_h = b^α [S(1-A)/(4 ø σ)]^0.25 (p_h / p₀)^α,

Where T_h is the temperature in K at altitude h, p_h is the pressure at altitude h, σ is the Stefan-Boltzmann constant, and b is a constant.  For Earth, S = 1367 W/m², the albedo A = 0.3, and 4 ø = 3.5016.  Taking the surface temperature T_S = 288.2 K, one can calculate the value of b^α to be 1.094.  For the Earth’s present atmosphere, α = 0.1905.

The adiabatic exponent α is known to be

α = R / µ (c_P + c_W + c_R),

where R is the gas constant or 1.987 cal/K mole, µ is the air molecular weight, c_W is the heat capacity per gram due to water vapor, c_R is the additional specific heat capacity per gram due to infra-red radiation, and µ c_P is the partial pressure weighted average of the c_P per gram of each gas molecule given as

µ c_P = [µ_N2 p_N2 c_P (N₂) + µ_O2 p_O2 c_P (O₂) + µ_CO2 p_CO2 c_P (CO₂) + µ_Ar p_Ar c_P (Ar)]/p,

which is not the way this is expressed in the paper.  Note that µ_N2 c_P (N₂) is the heat capacity per mole of nitrogen gas and each atmospheric gas component should be handled similarly.  c_W + c_R is the effective heat capacity of the sum of the water condensation processes and the absorption by infra-red active gases of the incoming solar insolation in the atmosphere.  A decrease in the value of α will cause a temperature decrease at any given altitude in the troposphere and a temperature decrease at the surface.

The value of µ should also be adjusted for additions with a weighted average based on component gas partial pressures as I showed above, though the paper does not present the issue in this way.  Additions of carbon dioxide with a mass of 44 amu increase the overall air µ since N₂ has a mass of 28 amu and O₂ has a mass of 32 amu, with normal air being about 28.96 amu on average.  So additions of carbon dioxide will decrease α by increasing the average molecular mass.  On the other hand, additions of water vapor (18 amu) or methane (16 amu), both reduce the average air molecular weight, which acts to increase α.  To find the overall effect of a gas component in convection, however, one needs to examine the heat capacity of each gas in terms of its µ c_P or its constant pressure heat capacity per mole.

Unfortunately, the paper incorrectly equates specific heat with heat capacity in the discussion.  Specific heats are given in relation to that of water.  While they misuse the term, the results are handled correctly.

Because the infra-red active gases have internal modes of vibration which are excited and hence carry energy in addition to the translational kinetic energy of these molecules, they have larger heat capacities per mole than do the non-infra-red active gases such as N₂ and O₂.  For instance, at atmospheric pressure N₂ has a heat capacity at constant pressure of 6.96 cal/K mol, while H₂O vapor has a heat capacity of 8.02 cal/K mol, CO₂ has a heat capacity of 8.87 cal/K mol, and methane, CH₄, has a heat capacity of 8.44 cal/K mol.  The constant pressure heat capacities per mole of water vapor, carbon dioxide, and methane are all greater than those of nitrogen gas, so they reduce the value of α by increasing the convective heat capacity in the denominator of α.  A reduced α means a reduced temperature.  The paper confuses this issue in the discussion because it gives the heat capacities for each molecule as the heat capacity per gram, which is lower for CO₂ than it is for N₂ and O₂ due to its substantially greater molecular weight.  They state the right conclusion, but the reasoning is hard to follow.

More water vapor increases both c_W and c_R, while an increase in carbon dioxide or methane increases c_R.  So α and the temperature are still further reduced by the increased net heat capacity.

The effective temperature of radiative equilibrium with space, T_e, is not defined in the paper, but is this:

T_e = [S(1-A)/(3.5016)σ]^0.25 = [1367 W/m² (1 - 0.3)/ (3.5016)(5.6697 x 10^-8 W/m² K⁴)]^0.25

T_e = 263.5 K

In addition, the heat in the atmosphere per gram, Q is given as

 Q = c_R T_e

But we also have 

Q = (c_P + c_W) (T_S – T_e)

Consequently,

C_R = (c_P + c_W) (T_S – T_e)/ T_e

Note that Equation 5 in the paper is in error, though 5’, which is derived from equation 5, is correct.  Using the fact that α = R / µ (c_P + c_W + c_R), we find that 

c_R = (R/µα) (T_S – T_e)/ T_S

Also, 

C_W = (R/µα) (T_e/T_S) - c_P

Calculating these values for Earth with α = 0.1905, µ = 29, the dry air heat capacity c_P = 0.2394 cal/g K, T_S = 288 K, T_e = 263.5 K, one finds that 

c_R = 0.306 cal/g K

c_W = 0.0897 cal/g K

The heat energy transport by convection, water condensation, and radiation of infra-red active gases is proportional to the c_P, c_W, and c_R values.  Convection is responsible for 66.56% of the heat transfer, water condensation for 24.94%, and radiation by infra-red active gases accounts for 8.51% of the energy transport in the troposphere.

The paper uses this methodology to show an excellent match with the surface temperatures and the lower atmosphere temperature gradients for both Earth and Venus.  It points out that an all methane Earth atmosphere would have almost exactly the same surface temperature, while an all CO₂ Earth atmosphere would have a surface temperature of about 281K, instead of 288K.  These are under the assumption that the total weight of the atmosphere is preserved in the comparisons.

So, as I have often said, the net warming of the Earth’s surface by infra-red gases is much less than it is claimed to be.  It is about 9.8 K, not about 33 K.  Also, as I have said by other empirical approaches, the effect of adding water vapor to the atmosphere is now a cooling effect, though water vapor is responsible for most of the prior warming due to its role in preventing a direct radiative equilibrium between the surface and space for most of the heat at the surface.  I have also said that adding CO₂ has a very small effect on the surface temperature, which is borne out by this paper where CO₂ is only responsible for a small portion of the small c_R effect and a very small increase of c_P.  I have long said that it was not clear that adding CO₂ would not decrease the temperature a wee bit.  It now appears clear that just as adding water vapor now decreases the surface temperature, so too does adding either CO₂ or methane gas.  This paper I have just discussed shows why additions of the infra-red active (greenhouse) gases now have a net cooling effect upon our troposphere and upon surface temperatures.

There is a warming of the surface by infra-red active gases, the so-called greenhouse gases, but that effect was maximized at lower concentrations of those gases than we now have.  Increases in those gases now cause small decreases in surface and general tropospheric temperatures.  This is because the mean free length for infra-red absorption by these gases is now too short for them to move the upper troposphere radiative equilibrium altitude to higher altitudes in the dense troposphere.  With that space radiation shell at the top of the troposphere relatively stabilized, the increased role of the gases in transporting heat energy upward from the surface means they are stronger coolants than they are “greenhouse” heaters.

New paper finds greenhouse gases causing radiative cooling, not warming, at current Earth surface temperatures

A new paper published in the Quarterly Journal of the Royal Meteorological Society finds radiation from greenhouse gases only begins to cause a positive-feedback warming effect at Earth temperatures 7C warmer than the present (& significantly higher than IPCC projections for the next century), and that at the current Earth temperature of 288K or 15C, greenhouse gas IR radiation has a negative-feedback cooling effect upon surface temperatures. Thus, addition of greenhouse gases at the present surface temperature of 288K (and up to 7C warmer or 295K) would have a negative-feedback cooling effect, not warming effect as claimed by IPCC theory and models.

These findings are incompatible with conventional Arrhenius radiative greenhouse/IPCC theory, which postulates radiative forcing from greenhouse gases has always caused a positive-feedback 'greenhouse' warming effect at all historical Earth surface temperatures and greenhouse gas concentrations. 

However, the findings of this paper demonstrating that greenhouse gases have a negligible or cooling effect at present Earth surface temperatures are compatible with the Maxwell/Clausius/Carnot gravito-thermal greenhouse theory, the HS 'greenhouse equation,' Chilingar et al, Kimoto, Wilde, and others.

The authors find (excerpts),

"shortwave [solar] radiation is a strong positive feedback at low surface temperatures but weakens at higher temperatures, and longwave radiation [from greenhouse gases] is a negative feedback at low temperatures [295K or 15C], but becomes a positive feedback for temperatures greater than 295–300 K [current Earth temperature is 288K or 15C]. 

It has been recently suggested that some modes of convective organization may result from an instability of the background state of radiative-convective equilibrium, which results in separation of the atmosphere into moist regions with ascent and dry regions with subsidence (Emanuel et al. 2014). If such an instability indeed exists in the real [not modelled] atmosphere, it would reshape our understanding of tropical circulations, and could help to explain the growth and life cycle of large-scale organized convective systems such as tropical cyclones and the Madden-Julian Oscillation (e.g., Bretherton et al. 2005; Sobel and Maloney 2012). If this instability is temperature-dependent, as suggested by numerical modeling studies (Khairoutdinov and Emanuel 2010; Wing and Emanuel 2014; Emanuel et al. 2014), then the increasing tendency of convection to organize with warming could also alter the climate sensitivity significantly (Khairoutdinov and Emanuel 2010); it is unclear whether current global climate models capture this process adequately. [Described in a recent post by Stephen Wilde]

Is the phenomenon of self-aggregation in a 200 km × 200 km domain in a model with explicit convection and clouds possibly the same as that in a 20000 km × 20000 km domain in a model with parameterized [fudge-factored] convection and clouds? This question has largely gone unaddressed, but it is essential to answer if we want to understand the robustness of self-aggregation across our modeling hierarchy and its relevance to the real atmosphere. 

In all simulations, the troposphere warms and dries relative to the initial condition (Figure 2(a)), though the stratosphere cools in simulations where Ts is lower than 300 K (Figure 2(b); Table 1). Tropospheric warming overall, and the increase in tropospheric warming with Ts, are consistent with the finding by Singh and O’Gorman (2013) that the lapse rate in RCE depends on entrainment and free-tropospheric relative humidity. In our simulations, aggregation decreases the free-tropospheric relative humidity in the domain mean, but increases the freetropospheric relative humidity in convectively active regions, plausibly reducing the influence of entrainment on the lapse rate and driving the thermal structure of the troposphere closer to a moist adiabat. Warming of the troposphere with aggregation can also be explained as a consequence of convective cores in moist regions drawing air with higher moist static energy from deeper within the boundary layer (Held et al. 1993).

The mean outgoing longwave radiation increases over the course of each simulation as a consequence of this drying, by an amount that increases with TS, ranging from ∼11 W m−2 at 280 K to ∼24 W m−2 at 310 K (Table 1).

The frozen moist static energy (hereafter referred to as h) is conserved in dry and moist adiabatic displacements, as well as freezing and melting of precipitation; h is given by the sum of the internal energy, cpT, the gravitational energy, gz, and the latent energy, Lvq − Lf qc,i (cp is the specific heat of dry air at constant pressure and g is the gravitational acceleration). In the latent energy term, Lv is the latent heat of vaporization, q is the water vapor mixing ratio, Lf is the latent heat of fusion, and qc,i is the condensed ice water mixing ratio: 

h = cpT + gz + Lvq − Lf qc,i. 

Atmospheric heating and cooling lead, respectively, to moistening and drying, because the weak temperature gradient approximation implies that anomalous heating is largely balanced by ascent, converging moisture into the column, while anomalous cooling is largely balanced by descent, diverging moisture out of the column.

In the four coldest simulations, (TS = 280K, 285K, 290K, 295K), the longwave radiation is at first a negative feedback, but in the warmer simulations [>295K], it is an important positive feedback. The magnitude of the shortwave feedback decreases by nearly a factor of 10 as the surface temperature increases from 280 K to 310 K, and the shortwave feedback also becomes much less important relative to the other feedbacks.

Although the behavior of the longwave radiation feedback term in our channel simulations appears to be consistent with the temperature-dependence suggested by Emanuel et al. (2014), cloud effects rather than clear-sky radiative transfer lead to our negative longwave feedback at low Ts (Figure 5). As predicted by Emanuel et al. (2014), the clear sky longwave feedback is weaker in the colder simulations – near zero or slightly negative – but this contributes only a small amount to the total longwave feedback. Aggregation occurs in spite of an initially negative longwave feedback at Ts <= 295 K, because this negative feedback is overridden by the combination of a positive surface flux and shortwave feedbacks; recall that the increasing strength of the shortwave feedback with decreasing temperature is largely due to clouds. 

A negative longwave cloud feedback implies that the atmosphere itself is cooling more in the moist regions and cooling less in the dry regions, due to the presence of clouds. We speculate that this occurs because a low-temperature atmosphere is optically thin, so the addition of clouds can increase the atmospheric longwave cooling by increasing its emissivity. An increase in longwave cooling due to greater cloud fraction in moist regions (where bh 0 > 0) is then a negative feedback on aggregation.

A key result is that the behavior of the radiative feedbacks varies with temperature, primarily due to the contribution of clouds. The longwave radiative feedback at the beginning of the simulation becomes negative as Ts is decreased, which is compensated for by an increase in the magnitude of the shortwave radiative feedback."

Self-aggregation of convection in long channel geometry

Allison A. Wing1,* and Timothy W. Cronin2

Abstract: Cloud cover and relative humidity in the tropics are strongly influenced by organized atmospheric convection, which occurs across a range of spatial and temporal scales. One mode of organization that is found in idealized numerical modeling simulations is self-aggregation, a spontaneous transition from randomly distributed convection to organized convection despite homogeneous boundary conditions. We explore the influence of domain geometry on the mechanisms, growth rates, and length scales of self-aggregation of tropical convection. We simulate radiative-convective equilibrium with the System for Atmospheric Modeling (SAM), in a non-rotating, highly-elongated 3D channel domain of length > 10⁴ km, with interactive radiation and surface fluxes and fixed sea-surface temperature varying from 280 K to 310 K. Convection self-aggregates into multiple moist and dry bands across this full range of temperatures. As convection aggregates, we find a decrease in upper-tropospheric cloud fraction, but an increase in lower-tropospheric cloud fraction; this sensitivity of clouds to aggregation agrees with observations in the upper troposphere, but not in the lower troposphere. An advantage of the channel geometry is that a separation distance between convectively active regions can be defined; we present a theory for this distance based on boundary layer remoistening. We find that surface fluxes and radiative heating act as positive feedbacks, favoring self-aggregation, but advection of moist static energy acts as a negative feedback, opposing self-aggregation, for nearly all temperatures and times. Early in the process of self-aggregation, surface fluxes are a positive feedback at all temperatures, shortwave [solar] radiation is a strong positive feedback at low surface temperatures but weakens at higher temperatures, and longwave radiation [from greenhouse gases] is a negative feedback at low temperatures but becomes a positive feedback for temperatures greater than 295–300 K [current Earth temperature is 288K]. Clouds contribute strongly to the radiative feedbacks, especially at low temperatures [ < 295 K].

PDF here

Collapse of the AGW theory of the IPCC; 'Most influential climate paper of all time' contains multiple false assumptions

Introduction

Kyoji Kimoto, a Japanese chemist, scientist, and fuel-cell computer modeler & inventor, has submitted his latest work as a guest post to The Hockey Schtick, and which refutes multiple false physical assumptions which underlie the alleged "first physically sound climate model" described in "the most influential climate change paper of all time." These same erroneous physical assumptions also continue to serve as the fundamental basis of James Hansen's NASA/GISS climate model, many other models including the 'state-of-the-art' IPCC climate models, and form the basis of the wide range of modeled CO2 climate sensitivity estimates.

In Kimoto's new work below (and in his prior published paper also below), he addresses the multiple unphysical assumptions made by Manabe & Wetherald, Hansen/GISS, and IPCC modelers et al, a few of which include:

1. An artificially fixed tropospheric lapse rate of 6.5K/km, which does not adjust to perturbations in the atmosphere. This false assumption artificially limits negative lapse rate feedback convection. Using physically correct assumptions, Kimoto finds the climate sensitivity to doubled CO2 to be a negligible 0.1-0.2C.

2. Mathematical error in the calculation of the Planck response parameter, due to a false assumption of fixed emissivity, an error which continues to be promulgated by the IPCC

3. Positive feedback from water vapor (whereas millions of radiosonde & satellite observations demonstrate water vapor has a net negative-feedback cooling effect)

4. Fixed relative humidity (contradicted by observations showing a decline of mid-troposphere relative and specific humidity) (A new paper also finds specific humidity is the most non-linear and non-Gaussian variable in weather models, also implying relative humidity is non-linear, and borne out by observations)

5. Neglect of the < 15 micron ocean penetration depth of GHG IR radiation, which greatly limits potential greenhouse gas warming of the top ocean layer. 

[An upcoming HS post will discuss additional other unphysical assumptions of Manabe et al. including improper application of blackbody assumptions & the Stefan-Boltmann Law, gross failure to calculate maximum emitting temperatures of greenhouse gases, and absolutely false assumption that CO2 can absorb/emit at an equivalent blackbody temperature of 300K (but is limited to 193K maximum by basic physical chemistry & quantum theory). Also note, the 1976 US Standard Atmosphere was published 9 years after Manabe et al, did not reference Manabe et al, and did not use one single radiative transfer equation or calculation to determine the entire atmospheric temperature profile 0-100km, including the stratosphere which is grossly inaccurate in Manabe et al] 

According to a recent 'consensus' by The Carbon Brief of 36 IPCC authors, "one paper clearly takes the top spot" as "the most influential climate change paper of all time:" Manabe & Wetherald's 1967 paper entitled, "Thermal Equilibrium of the Atmosphere with a Given Distribution of Relative Humidity"

According to The Carbon Brief article,

'the work was the first to represent the fundamental elements of the Earth's climate in a computer model, and to explore what doubling carbon dioxide (CO2) would do to global temperature." 

Manabe & Wetherald (1967), Journal of the Atmospheric Sciences 

The Manabe & Wetherald paper is considered by many as a pioneering effort in the field of climate modelling, one that effectively opened the door to projecting future climate change. And the value of climate sensitivity is something climate scientists are still grappling with today. 

Prof Piers Forster, a physical climate scientist at Leeds University and lead author of the chapter on clouds and aerosols in working group one of the last IPCC report, tells Carbon Brief: 

"This was really the first physically sound climate model allowing accurate predictions of climate change." 

The paper's findings have stood the test of time amazingly well, Forster says.
"Its results are still valid today. Often when I've think I've done a new bit of work, I found that it had already been included in this paper." 

Prof Steve Sherwood, expert in atmospheric climate dynamics at the University of New South Wales and another lead author on the clouds and aerosols chapter, says

"[The paper was] the first proper computation of global warming and stratospheric cooling from enhanced greenhouse gas concentrations, including atmospheric emission and water-vapour feedback."

All of the above claims regarding Manabe & Wetherald, et al, are refuted in Kimoto's new work below:

Collapse of the Anthropogenic Warming Theory of the IPCC

By Kyoji Kimoto

 
PDF file

A prior peer-reviewed, published paper by the same author, which is referenced in his new work above is available here and reproduced below:

Erasing AGW: How Convection Responds To Greenhouse Gases To Maintain The Hydrostatic Equilibrium Of The Atmosphere

How Convection Responds To Greenhouse Gases So As To Maintain The Hydrostatic Equilibrium Of An Atmosphere

Guest post by Stephen Wilde, who has been a member of the Royal Meteorological Society since 1968.

Introduction:

This article is complementary to Stephen Wilde’s earlier works at:

http://www.newclimatemodel.com/

and is consistent with the cause of the so called greenhouse effect being atmospheric mass held off a surface within a gravitational field and subjected to insolation.

Convective overturning within any horizontal layer of gases around a planet and held off the surface against the force of gravity revolves around the point where the upward pressure gradient force within the atmosphere achieves hydrostatic balance with the downward force of gravity. At that point kinetic energy (KE or heat) matches potential energy (PE which is not heat) in any molecules present.

http://www.public.asu.edu/~hhuang38/mae578_lecture_06.pdf

It is Earth’s surface temperature enhancement of 33K above the temperature predicted by radiative physics that provides the kinetic energy required at the surface to maintain the upward pressure gradient force. That kinetic energy is locked into constant convective overturning and cannot be radiated to space without the mass of the atmosphere falling to the surface.

Background:

If tropopause height is raised by upward convection by the force of rising air from below then it must descend at some other location where the force of uplift is absent or less strong. Thus tropopause height will always be irregular and fluctuating up and down from place to place. Uneven surface heating causing density variations in the horizontal plane makes convective overturning unavoidable with or without GHGs.

Rising air has enough kinetic energy (KE) to overcome the downward force of gravity which seeks to both pull down and compress gases. That downward force can conveniently be represented by the quantity of potential energy (PE) held by atmospheric molecules suspended off the surface.

It follows that descending air does not have sufficient KE to overcome the downward force of gravity. PE can then be said to exceed KE.

At the surface, molecules contain only KE and as height is gained PE takes over from KE because as molecules rise into regions of lower density they move further apart and vibrate less. Kinetic energy (heat) is transformed into potential energy which is not heat and does not radiate.

For Earth’s troposphere, molecules involved in convection can only rise as far as the tropopause because at that level there is an inversion layer containing warmer, lighter air.  Colder, denser air, being heavier, will always remain at a lower height than warmer, lighter air.

This diagram shows the essential features:

Narrative:

The coldest air is found at the top of section A having been cooled by adiabatic ascent and held at that height by warmer lighter air pushing up from below. If convection is enhanced by GHGs absorbing IR from the ground then that will tend to push convection higher than if there are no GHGs and produce more upward distortion of tropopause height above ascending columns. Section A becomes larger as does section B. Section A is then storing more PE than before and, as long as non-condensing GHGs remain present in an atmosphere, that greater energy storage facility in section A is permanent. More GHGs leads to more energy storage in section A. 

PE is not heat and does not radiate.

Since warmer, lighter air continues to move upwards beneath it, that cold air at the top of section A is forced to displace sideways but being colder, denser and heavier air than the air in section B (and beneath section B) it will be forced downwards as shown above . Due to the Coriolis force it circles around in high level winds from the top of a low pressure cell containing ascending cooling air towards the nearest high pressure cell containing descending warming air, descending and warming adiabatically as it does so.

The important point to note is that due to the temperature and pressure gradient within the troposphere the surroundings will warm and become denser at the same rate as the descending cold air from the top of section A warms and compresses.

The consequence is that the density and temperature difference between the descending air and its surroundings will be maintained all the way to the surface or until an inversion layer is encountered. The descending column is therefore colder than it would have been if descending from a lower height in an atmosphere with no GHGs. The lapse rate slope in the descending column is distorted to the colder side of the DALR.

Air being a poor conductor we can ignore conduction in and out of the descending parcel of air for present purposes.

So, just as KE was greater than PE throughout the column of ascending air then PE is greater than KE throughout the column of descending air.

Let us look at the consequences for the lapse rate structure of the tropopause in that situation:

This is a simplified diagram that should be considered in conjunction with the top diagram that shows how the tropopause behaves when rising air starts to descend.

We can see that, above the height of hydrostatic balance, radiation to space from GHGs steadily increases the KE deficit in the descending column so that the lapse rate slope diverges further and further from the DALR until the point of hydrostatic balance is reached.

Below that point KE starts being added via radiative absorption from the ground by the GHGs in the descending column which slowly warms to the surface temperature it would have attained with no GHGs present.

The CO2 emissions gap at the top of the atmosphere as viewed from space is caused by the absorption of IR from the surface in both ascending and descending columns below the height of hydrostatic balance.

Due to the additional storage of KE as PE in the expanded section A at the top of the atmosphere the surface temperature need not rise. The distortion of the lapse rate slope to the warm side in the ascending column is enough to provoke enhanced uplift without the surface needing to be warmer.

AGW theory mistakenly relies on the lapse rate not being distorted so that the surface temperature needs to be higher to enhance convection.

At the bottom of the descending column, surface temperature rises above the temperature predicted by radiative physics via the so called S-B equation with or without GHGs because the descending air is inhibiting upward convection, just like a greenhouse. That is why sunny deserts beneath semi-permanent descending high pressure cells get hot in the daytime. It then takes time for surface winds to transfer the excess surface energy to the nearest region of convective uplift and it is that delay averaged globally that raises surface temperature 33K above S-B whether GHGs are present or not.

That process is a consequence of atmospheric mass absorbing insolation from the heated surface by conduction and convection and not downward radiation from greenhouse gases as proposed by AGW theory.

Any downward radiation towards the surface alters the local hydrostatic pressure at height between the surface and the absorbing GHG molecule which conducts KE to adjoining non GHG molecules putting the local pressure gradient force out of balance with the gravitational force so that the warmed molecules rise spontaneously due to their lower density. In the process of rising they cool by decompression but whilst rising they remain warmer than they should be for their position along the DALR because their surroundings cool at the same rate as they ascend adiabatically. That is where the DALR distortion to the warm side comes from and that is what enhances upward convection rather than any surface warming.

Exactly the opposite occurs in the descending column (all the molecules are cooler than they should be for their position along the DALR because the GHGs have caused the ascending column to rise to a higher colder location and KE has been taken out and stored permanently in section A so that it is no longer available for recovery at the surface in convective descent.

In the above diagram Section A represents the sum total of the tops of all ascending columns around the globe. As some dissipate they are replaced by others so the average global volume of all sections A remains the same. 

The net thermal effect from the GHGs is zero when the two columns are averaged which is why the diagram shows the lines all converging at the same local temperature at both surface and tropopause.

GHGs cause section A in the above diagram to be higher and colder (with more PE) than it otherwise would be. The only KE which is recovered from PE during the subsequent descent and then radiated out from the surface is that which is not stored permanently in the expanded Sections A i.e. the original quantity for a GHG free atmosphere.

A GHG such as CO2 blocks certain wavelengths from escaping the atmosphere to space but since the convective adjustment returns kinetic energy back towards the surface the blocked wavelengths are simply reconverted to the whole range of wavelengths again at the surface.

The wavelength changes when that KE returns to the surface for another chance at upward emission to space ensure that the blocking effect of the GHGs is overcome by more radiation escaping to space directly from the surface otherwise the sections A would keep expanding indefinitely.

Note that here we are discussing only two radiatively active ‘pipes’ for the exit of radiative energy to space, namely GHGs within the atmosphere and the surface.

For a more complex atmosphere with other types of radiatively active material such as other GHGs, water vapour condensate and inorganic or organic particulates then the other ‘pipes’ will also be active and take over more of the compensating radiation to space.

That is why, for Earth, the surface only radiates a little more to space when GHGs increase. The other ‘pipes’ are helping it out. If those other ‘pipes were not present it would all happen from the surface. If there were no GHGs water vapour or particulates at all then the lapse rate structure would look like this:

This scenario would involve ALL radiation escaping directly to space from the surface when there is no radiative capability in the atmosphere. There is inevitably still convective overturning and no isothermal atmosphere even in that situation.

Every atmosphere has a decline in density and temperature with height as long as it is held off the ground against the force of gravity in hydrostatic balance.

 All mass has some radiative capability, even oxygen and nitrogen and there will always be aerosols and particulates so the above diagram is an idealised one. In reality there would always be some distortion of the DALR in both ascent and descent.

For a condensing GHG such as water vapour and no other ‘pipes’ other than the surface, the lapse rate structure looks like this:

There will be condensate in the form of clouds at various heights in this case and such condensate radiates readily to space depending on height and density hence the faster decline in temperature with height above the point of hydrostatic balance. Because of the wide variability of cloud types, heights and densities the height of the point of hydrostatic balance varies constantly within rising columns of air containing water vapour.

So, whatever is the atmospheric composition the only factors that influence surface temperature are the strength of the gravitational field, the mass of the atmosphere and the strength of incoming insolation.

The infinitely variable lapse rate structure available in three dimensions around a rotating sphere stabilises any thermal effects from everything else by causing variable convection which leads to variations in tropopause height so as to deposit surplus KE into the atmospheric PE reservoir or draw it out as necessary to maintain stability. In the process there are changes in the balance of radiation escaping to space from each of any available ‘pipes’ and if there are none other than the surface then the surface alone it will be.

New paper finds increased CO2 or methane will have 'essentially no effect' upon global temperature or climate

A new paper by USC Professor Emeritus of Geology, Dr. George Chilingar (with three co-authors), finds that increasing levels of the greenhouse gases CO2 & methane will have "essentially no effect" upon global temperatures or climate. 

The authors utilize a one-dimensional adiabatic model of climate to demonstrate that the entire tropospheric temperature profile of the atmosphere on both Earth and Venus may be mathematically derived solely on the basis of atmospheric pressure/mass and solar activity, confirmed by observations on both planets, despite vast differences in atmospheric composition and mass/pressure on Earth and Venus. The paper corroborates the 33C Maxwell/Clausius/Carnot greenhouse theory and thereby excludes the alternative 33C Arrhenius radiative greenhouse theory.

Excerpts:

"The writers investigated the greenhouse effect using their adiabatic model, which relates the global temperature of troposphere to the atmospheric pressure and solar radiation. This model allows one to analyze the global temperature changes due to variations in mass and chemical composition of the atmosphere. Even significant releases of anthropogenic carbon dioxide and methane into the atmosphere do not change average parameters of the Earth’s heat regime and have no essential effect on the Earth’s climate warming. Moreover, based on the adiabatic model of heat transfer, the writers showed that additional releases of CO2 and CH4 lead to cooling (and not to warming as the proponents of the conventional theory of global warming state) of the Earth’s atmosphere. The additional methane releases possess a double cooling effect: First, they intensify convection in the lower layers of troposphere; Second, the methane together with associated water vapor intercept part of the infrared solar irradiation reaching the Earth. Thus, petroleum production and other anthropogenic activities resulting in accumulation of additional amounts of methane and carbon dioxide in the atmosphere have practically no effect on the Earth’s climate."

Physically, an explanation of the cooling effect of the atmosphere with the high content of “greenhouse gases” is the high efficiency of the convective heat transfer from the planet’s surface to the lower stratosphere, from which this heat is rapidly dissipating into the outer space through radiation. As the greenhouse gases absorb the Earth’s heat radiation in the lower layers of troposphere, its energy transforms into the heat oscillations of the gas molecules. This, in turn, leads to expansion of the gas mixture and its rapid ascent to the stratosphere where the heat excess is lost through radiation into the outer space.  

To replace these volumes of the warm air, the already cooled air descends from the upper troposphere. As a result, the global average atmospheric temperature slightly decreases. One particular consequence of it is that with an increase in the carbon dioxide and methane contents in troposphere the convective mass exchange of the atmospheric gases must substantially accelerate. Thus, it is not out of the question that the intensification of synoptic processes in Earth troposphere (but not temperature increase) may be a result of the carbon dioxide and other “greenhouse gases” accumulation."

The primary equation of the paper [2] is similar to the 'greenhouse equation' described in a recent series of posts on the 33C Maxwell/Clausius/Carnot greenhouse theory. 

The "Greenhouse Equation" calculates temperature (T) at any location from the surface to the top of the troposphere as a function of atmospheric mass/gravity/pressure and radiative forcing from the Sun only, and without any radiative forcing from greenhouse gases. Note the pressure (P) divided by 2 in the greenhouse equation is the pressure at the center of mass of the atmosphere (after density correction), where the temperature and height are equal to the equilibrium temperature with the Sun and ERL respectively.

The primary differences between Chilingar et al equation [2] and the 'greenhouse equation' are:

1. Chilingar et al introduce a correction for solar insolation based on the Earth's precession angle of 23.44 degrees 

2. Chilingar et al assume an Earth surface temperature of 288K or 15C, whereas the HS 'greenhouse equation' only assumes the equilibrium temperature of the Earth with the Sun (255K or -18C) & atmospheric mass/pressure to derive the surface temperature, as well as that of the entire troposphere, replicating the 1976 US Standard Atmosphere. 

An upcoming post will join the mathematics of these two equations to explain the entire temperature profile of the atmosphere from the surface to the edge of space at 100+ km geopotential altitude, without incorporating 'radiative forcing' from CO2.