Thursday, November 27, 2014

Derivation of the effective radiating height & entire 33°C greenhouse effect without radiative forcing from greenhouse gases

The purpose of the recent series of physical proofs is to demonstrate that the greenhouse effect theory is entirely explained by the force of gravity, i.e. "gravity forcing" upon the mass of the atmosphere, rather than "radiative forcing" from greenhouse gases. This alternative "gravity forcing theory" of the greenhouse effect will be demonstrated to be completely independent of greenhouse gas radiative forcing, and compatible with all physical laws and millions of observations, as opposed to the radiative forcing theory.

We will use the ideal gas law, 1st law of thermodynamics, Newton's second law of motion (F = ma), and well-known barometric formulae in this derivation to very accurately determine Earth's surface temperature, the height in the atmosphere at which the effective equilibrium temperature of Earth with the Sun is located, and show that this height is located as expected at the center of mass of the atmosphere on Earth and Titan.

We will show that the mass/pressure greenhouse effect theory can also be used to accurately determine the temperatures at any height in the troposphere from the surface to the tropopause, and compute the mass/gravity/pressure greenhouse effect to be 33.15C, the same as determined from radiative climate models and the conventional radiative greenhouse effect theory. 

1. Conservation of energy and the ideal gas law

We will start once again with the ideal gas law 

PV = nRT (1)

an equation of state that relates the pressure P, volume V, temperature T, number of moles n of gas and the gas law constant R = 8.3144621 J/(mol K)

The properties of gases fall into two categories: 

1. Extensive variables are proportional to the size of the system: volume, mass, energy
2. Intensive variables do not depend on the size of the system: pressure, temperature, density

To conserve energy (and to ensure that no radiative imbalances from greenhouse gases are affecting this derivation) of the mass/gravity/pressure greenhouse effect, we assume

Energy incoming from the Sun (Ein) = Energy out (Eout) from Earth to space

Observations indeed show Ein = Eout = 240 W/m2 (2)

which by the Stefan-Boltzmann law equates to a blackbody radiating at 255 K or -18C, which we will call the effective or equilibrium temperature (Te) between the Sun and Earth. As seen by satellites, the Earth radiates at the equilibrium temperature 255K from an average height referred to as the "effective radiating level" or ERL or "effective radiating height."

2. Determine the "gravity forcing" upon the atmosphere

Returning to the ideal gas law above, pressure is expressed using a variety of measurement units including atmospheres, bars, and Pascals, and for this derivation we will use units in atmospheres, which is defined as the pressure at mean sea level at the latitude of Paris, France in terms of Newtons per square meter [N/m2]

Newtons per square meter corresponds to the force per unit area [or "gravity forcing" upon the atmospheric mass per unit area of the Earth surface]. 

Now let's determine the mass of the atmosphere above one square meter at the Earth surface:

By Newton's 2nd law of motion equation, force (F) is 

F = ma  (3)   where m = mass and a = acceleration

As we noted above, the atmospheric pressure is a force or forcing per unit area. The force in this case is the weight or mass of the atmosphere times the gravitational acceleration, therefore

F = mg  (4) where g is the gravitational constant 9.8 m/s2, i.e. the acceleration due to gravity in meters per second (s)

If we assume that g is a constant for the entire column of the atmosphere above the 1 meter2 area (A) we obtain

m = PA/g = (1.0325 x 10^5 N/m2 )(1 m2 )/(9.8 m/s2 ) = 1.05 x 10^4 kg

thus, the weight of the atmosphere over 1 square meter of the surface is 10,500 kilograms, quite a remarkable gravitational forcing upon the atmosphere.

If m is the mass of the atmosphere and g is the gravitational acceleration, the gravitational force is thus

F = mg

The density (p) is the mass (m) per unit Volume (V), thus,

p = m/V

SI units of pressure refer to N/m2 as the Pascal (Pa). There are 1.0325 x 10^5 Pa per atmosphere (unit). 

Starting again with equation (3) above

F = ma  (3)

F = mg  (4)

F = (PA/g)g = PA  (5)

P = F/A = mg/A = phAg/A = phg (6) 

where 

h=height along either a gas or liquid column under pressure or gravity field
g = gravitational constant
p = density = mass/volume

3. Determine the atmospheric pressures from gravitational forcing, and the height of the effective equilibrium temperature (ERL)

Now we will determine the atmospheric pressures in a gravitational field using (6) above

First let's determine the pressure at the ERL since the temperature must equal the equilibrium temperature of 255K at the ERL.

The pressure is a function of height 

P(h) = ρgh 

and the change in pressure dP is related to the change in height dh by 

dP = -ρg dh 

The minus sign arises from the fact that pressure decreases with height, subject to an adjustment for density which changes with height. We will determine this adjustment from the ideal gas law. The density is 

ρ = nM/V 

where n is the number of moles, M is the molar mass, and V is the volume. We can obtain n/V from the ideal gas law: 

n/V = P/RT 

thus 

ρ = MP/RT

We can now substitute the density into the pressure vs. height formula:

dP = -(MPg/RT)dh

 dP/P = -(Mg/RT) dh  (the first integral is from 1 to P, second from 0 to h)

ln(P) = -(Mgh/RT)

P = exp^-((Mgh/(RT))

We will now determine the height (h) at the ERL where the temperature = the effective equilibrium temperature = 255K, and without use of radiative forcing from greenhouse gases.

Plugging in numbers of M = 29 grams/mole (0.029 kg/mole) as average molar mass for atmosphere, g = 9.8 m/s^2, Pressure = 0.50 atmospheres at the approximate center of mass of the atmosphere, R=8.31, and T=Te=255K effective equilibrium temperature we obtain:

0.50 atmosphere P at the ERL= e^-((.029*9.8*5100)/(8.31*255))

So the height of the ERL set by gravity forcing is located at 5100 meters and is where T=Te=255K and pressure = 0.5 atmospheres, right at the center of mass of the atmosphere as we predicted from our gravity forcing hypothesis. 

4. Determine the temperatures at any location in the troposphere, and the magnitude of the mass/pressure greenhouse effect

Now that we have solved for the location of the ERL at 5100 meters, we can use the adiabatic lapse rate equation to determine all troposphere temperatures from the surface up to the ERL at 255K and then up to the top of the troposphere. The derivation of the lapse rate equation from the ideal gas law and 1st law of thermodynamics is described in this post, thus will not be repeated here, except to mention that the derivation of the lapse rate 

dT/dh = -g/Cp where Cp = heat capacity of the atmosphere at constant pressure

is also completely independent of any radiative forcing from greenhouse gases, greenhouse gas concentrations, emission/absorption spectra from greenhouse gases, etc., and is solely a function of gravity and heat capacity of the atmosphere. 

Plugging the average 6.5C/km lapse rate and 5100 meter or 5.1 km height of the ERL we determine above into our derived lapse rate equation (#6 from prior post) gives

T = -18C - (6.5C/km × (h - 5.1km)) 

Using this equation we can perfectly reproduce the temperature at any height in the troposphere as shown in Fig 1. At the surface, h = 0, thus temperature at the surface Ts is calculated as

Ts = -18 - (6.5 × (0 - 5.1)) 

Ts = -18 + 33.15C (gravity forced greenhouse effect)

Ts = 15.15°C or 288.3°K at the surface

which is exactly the same as determined by satellite observations and is 33.15C above the equilibrium temperature -18C or 255K with the Sun as expected.

Thus, we have determined the entire 33.15C greenhouse effect, the surface temperature, and the temperature of the troposphere at any height, and the height at which the equilibrium temperature with the Sun occurs at the ERL entirely on the basis of the Newton's 2nd law of motion, the 1st law of thermodynamics, and the ideal gas law, without use of radiative forcing from greenhouse gases, nor the concentrations of greenhouse gases, nor the emission/absorption spectra of greenhouse gases at any point in this derivation, demonstrating that the entire 33C greenhouse effect is dependent upon atmospheric mass/pressure/gravity, rather than radiative forcing from greenhouse gases. Also note, it is absolutely impossible for the conventional radiative theory of the greenhouse effect to also be correct, since if that was the case, the Earths temperature would be at least twice the present temperature. 

In essence, the radiative theory of the greenhouse effect confuses cause and effect. As we have shown, temperature is a function of pressure, and absorption/emission of IR from greenhouse gases is a function of temperature. The radiative theory tries to turn that around to claim IR emission from greenhouse gases controls the temperature, the heights of the ERL and tropopause, and thus the lapse rate, pressure, gravity, and heat capacity of the atmosphere, which is absurd and clearly disproven by basic thermodynamics and observations. The radiative greenhouse theory also makes the absurd assumption a cold body can make a hot body hotter,disproven by Pictet's experiment 214 years ago, the 1st and 2nd laws of thermodynamics, the principle of maximum entropy production, Planck's law, the Pauli exclusion principle, and quantum mechanics. There is one and only one greenhouse effect theory compatible with all of these basic physical laws and millions of observations. Can you guess which one it is?

Note the gravity forcing greenhouse theory also perfectly predicts the height of the ERL and surface temperature of Titan, the closest Earth analog in our solar system, and the only planet other than Earth with an atmosphere comprised of mostly non-greenhouse gases. The theory would not apply to any planets with thin atmospheres such as Mars which is unable to sustain significant convection. In the odd case of Venus, which I will pursue next, the atmospheric temperatures will likely be as determined by the mass/pressure theory plus additional warming from conduction downward from the thick opaque cloud top of the atmosphere, but this work is in progress.

New paper finds strong evidence the Sun has controlled climate over the past 11,000 years, not CO2

A paper published today in Journal of Atmospheric and Solar-Terrestrial Physics finds a "strong and stable correlation" between the millennial variations in sunspots and the temperature in Antarctica over the past 11,000 years. In stark contrast, the authors find no strong or stable correlation between temperature and CO2 over that same period. 

The authors correlated reconstructed CO2 levels, sunspots, and temperatures from ice-core data from Vostok Antarctica and find
"We find that the variations of SSN [sunspot number] and T [temperature] have some common periodicities, such as the 208 year (yr), 521 yr, and ~1000 yr cycles. The correlations between SSN and T are strong for some intermittent periodicities. However, the wavelet analysis demonstrates that the relative phase relations between them usually do not hold stable except for the millennium-cycle component. The millennial variation of SSN leads that of T by 30–40 years, and the anti-phase relation between them keeps stable nearly over the whole 11,000 years of the past. As a contrast, the correlations between CO2 and T are neither strong nor stable."
Thus, the well known ~1000 year climate cycle responsible for the Holocene Climate Optimum 6000 to 4000 years ago, the Egyptian warm period ~4000 years ago, the Minoan warm period ~3000 years ago, the Roman warm period ~2000 years ago, the Medieval warm period ~1000 years ago, and the current warm period at present all roughly fall in this same 1000 year sequence of increased solar activity associated with warm periods. 


a) sunspots, b) temperature, c) CO2, d-i show the amplitudes of the strongest cycle lengths (period in years) shown in the data for sunspots, temperature, and CO2


Wavelet analysis in graph a shows the most prominent solar periods in red and graph b for temperature. The most stable period for both is at ~1024 years, shown by the horizontal region in red/yellow/light blue.
The authors find a lag of 30-40 years between changes in solar activity driving temperature, likely due to the huge thermal capacity and inertia of the oceans. Lead time shown in bottom graph of 40 years shows the temperature response following an increase or decrease of solar activity lags by about 40 years. Top graph shows "the anti-phase relation between [solar activity and temperature] keeps them stable nearly over the whole 11,000 years of the past."

The authors find temperature changes lag solar activity changes by ~40 years, which is
 likely due to the huge heat capacity and inertia of the oceans. Warming proponents attempt to dismiss the Sun's role in climate change by claiming 20th century solar activity peaked at around 1960 and somewhat declined from 1960 levels to the end of the 20th century (and have continued to decline in the 21st century right along with the 18+ year "pause" of global warming). 


Firstly, the assumption that solar activity peaked in 1960 and declined since is false, since it is necessary to determine the accumulated solar energy over multiple solar cycles, which is the accumulated departure from the average number of sunspots over the entire period, which I call the "sunspot integral." The sunspot integral is plotted in blue and shows remarkable correction with global temperatures plotted in red below. Correlating sunspot and temperature data with and without CO2, we find the sunspot integral explains 95% of temperature change over the past 400 years, and that CO2 had no significant influence (also here).

Source

Secondly, this paper finds strong evidence of a 30-40 year lag between solar activity and temperature response. So what happened ~40 years after the 1960 peak in sunspot activity? Why that just so happens to be when satellite measurements of global temperature peaked with the 1998 El Nino [which is also driven by solar activity], followed by the "pause" and cooling since. 

We have thus shown
  • Strong correlation between solar activity and climate over the past 11,000 years of the Holocene
  • Strong lack of correlation between CO2 and climate over the past 11,000 years of the Holocene
  • Solar activity explains all 6 well-known warming periods that have occurred during the Holocene, including the current warm period
  • The 20th century peak in sunspot activity is associated with a 40 year lag in the peak global temperature
What more proof do you need that it's the Sun!

But wait, there's more. Please see the two previous posts demonstrating that the alternate 33C greenhouse effect is due to atmospheric mass/gravity/pressure, not CO2 or water vapor, physical proof & observations that water vapor is a strong negative-feedback cooling agent, and physical proof that CO2 cannot cause any significant global warming. All of the above also strongly suggests the increase in CO2 levels is primarily due to ocean outgassing from warming oceans from the Sun, not from CO2 radiative forcing warming the oceans, and not primarily from man-made CO2 emissions.

SSN [Sunspot Number] and Vostok temperature (T) had common periodicities in past 11,000 years.
The millennial variations of SSN and T had a strong and stable correlation.
The millennial variation of SSN led that of T by 30–40 years.
Correlations between CO2 and T were neither strong nor stable.

Abstract

The solar impact on the Earth's climate change is a long topic with intense debates. Based on the reconstructed data of solar sunspot number (SSN), the local temperature in Vostok (T), and the atmospheric CO2 concentration data of Dome Concordia, we investigate the periodicities of solar activity, the atmospheric CO2 and local temperature in the inland Antarctica as well as their correlations during the past 11,000 years before AD 1895. We find that the variations of SSN and T have some common periodicities, such as the 208 year (yr), 521 yr, and ~1000 yr cycles. The correlations between SSN and T are strong for some intermittent periodicities. However, the wavelet analysis demonstrates that the relative phase relations between them usually do not hold stable except for the millennium-cycle component. The millennial variation of SSN leads that of T by 30–40 years, and the anti-phase relation between them keeps stable nearly over the whole 11,000 years of the past. As a contrast, the correlations between CO2 and T are neither strong nor stable. These results indicate that solar activity might have potential influences on the long-term change of Vostok's local climate during the past 11,000 years before modern industry.

Wednesday, November 26, 2014

Why greenhouse gas radiative forcing doesn't explain Earth's energy budget

We have previously demonstrated that the atmospheric mass/gravity/pressure theory of the 33C greenhouse effect explains Earth's surface temperature and the temperatures throughout the troposphere, rather than radiative forcing from greenhouse gases.

We have also demonstrated why the atmospheric mass/gravity/pressure theory of the greenhouse effect also perfectly explains the observed greenhouse effect on Titan, the closest Earth analog in our solar system, and the only planet other than Earth with an atmosphere comprised of mostly non-greenhouse gases (Titan: 98.4% Nitrogen, 0.1% hydrogen, and only 1.5% greenhouse gas methane compared to Earth's 78.09% nitrogen, 20.95% oxygen, 0.93% argon, 0.04% carbon dioxide).


We now address three additional reasons why the conventional anthropogenic CO2 warming theory is flawed due to incorrect assumptions regarding the energy budget. In contrast, the mass/gravity/pressure alternative greenhouse theory is entirely 
compatible with Earth's energy budget, physical laws, and observations. 

The radiative greenhouse theory is commonly represented by the Earth energy budget devised by Kiehl and Trenberth as shown in this diagram from their 2008 publication:


Trenberth, Fasullo, Kiehl 2008 Earth energy budget shows "atmospheric window" transmitting only 40 W/m2 from the surface directly to space
which shows an "atmospheric window" where the greenhouse gases don't significantly absorb/emit infrared radiation, and thus most of this radiation travels directly from the surface to space unimpeded by greenhouse gases, or any radiative forcing as a result. 



The "atmospheric window" of radiation direct to space falls between 8-12 micron wavelengths, as shown in the figures above and below, which includes the peak emission wavelengths (~8-11 microns) from Earth, thus ~80% of these peak emissions from Earth pass directly to space without being absorbed or emitted by greenhouse gases, thus don't contribute to radiative forcing. 


Blue curve shows the Planck curve assuming Earth radiated as a true blackbody, with peak emissions located at ~10 microns, which is at the middle of the 8-12 micron "atmospheric window" that is not absorbed/emitted by greenhouse gases.
However, according to the Trenberth et al Earth energy budget above, only 40 W/m2 passes through the "atmospheric window," i.e. only 10% of the 396 W/m2 radiative surface emissions in his energy budget.

The Planck–Einstein relation is a formula integral to quantum mechanics, and states that the energy of a photon (E) is proportional to its frequency (ν). The constant of proportionality, h, is known as the Planck constant:



and since frequency v is inversely related to wavelength λ by

v = c/
λ where c = speed of light

therefore

E = hc/λ

thus, the shorter the wavelength/higher the frequency of a photon, the higher the energy it contains.

Therefore, the highest energy photons at the short 8-12 micron wavelengths emitted from Earth's surface fall within the direct atmospheric window to space without any interaction with greenhouse gases.

We can get a sense for the huge effect just a few microns change in wavelength has on the corresponding blackbody emission temperature by plotting the peak blackbody emission wavelength in microns vs. the peak emission temperature of a blackbody determined by Wien's Displacement Law, which shows the emission temperature at the beginning of the atmospheric window at 8 microns is 89C, dropping all the way down to -31.7C at the end of the atmospheric window at 12 microns, a temperature change of 120.7C.


The catastrophic greenhouse gas CO2, however, absorbs and emits line spectra centered around 15 microns, so what does Wien's displacement law calculate for a blackbody (which CO2 is not) emitting at 15 microns? Whoa, a toasty peak emission temperature of minus 80C:



Thus, the radiative greenhouse theory makes the absurd assumption that CO2 radiating at a blackbody temperature of -80C contributes 20% of the radiative greenhouse effect and heats the Earth by 6.6C from the equilibrium temperature with the Sun of -18C to -11.4C. How can a cold body radiating at -80C cause a hotter body at -18C to warm by 6.6C or at all? Pictet's experiment 214 years ago, the 1st and 2nd laws of thermodynamics, the principle of maximum entropy production, Planck's law, the Pauli exclusion principle, and quantum mechanics all prove it cannot. CO2 radiating at an equivalent blackbody temperature of  -80C cannot possibly account for 20% of the ~33C greenhouse effect to cause 20% of the warming from -18C to 15C.

Further, even within the CO2 equivalent -80C blackbody emission peak at 15 microns [line spectra range ~13.5 - 17 microns], the CO2 and water vapor absorption/emission spectra significantly overlap such that almost 70% is independently due to water vapor and thus would be completely unaffected by increased CO2. Thus, despite overwhelming evidence that CO2 cannot significantly warm the planet, if you still believe the radiative greenhouse theory and reject the alternate mass/pressure greenhouse theory, the overlapping spectra of CO2 with water vapor prove CO2 is a bit player at most:

Shaded area shows difference between CO2 and water vapor absorption/emission spectra

Blowup of above figure. Area with asterisk is absorbed by CO2 only, remainder below overlaps with absorption spectra of water vapor.

Another huge flaw in Trenberth's energy budget is the false assumption that infrared radiation from greenhouse gases can heat the oceans, and just as effectively as the land surface. This is disproven by both theory and observations, since IR can only penetrate the ocean surface a few microns to cause evaporative cooling of the ocean skin surface, not warming. Since the oceans cover 70% of the Earth's surface, this problem alone suggests the radiative greenhouse theory is exaggerated by at least 70% (not even considering the cooling effect of evaporation and that evaporation results in clouds and further cooling).




These are only three of many fatal flaws of conventional radiative greenhouse theory. There is one and only one explanation for the entire ~33C greenhouse effect that satisfies all physical laws and is in accordance with millions of weather balloon and satellite observations, the atmospheric mass/gravity/pressure greenhouse theory. 

Sunday, November 23, 2014

Derivation of the entire 33°C greenhouse effect without radiative forcing from greenhouse gases

We will derive the entire 33°C greenhouse effect using the 1st law of thermodynamics and ideal gas law without use of radiative forcing from greenhouse gases, nor the concentrations of greenhouse gases, nor the emission/absorption spectra of greenhouse gases at any point in this derivation, thus demonstrating that the entire 33C greenhouse effect is dependent upon atmospheric mass/pressure/gravity, rather than radiative forcing from greenhouse gases. Secondly, we will show why multiple observations perfectly confirm the mass/gravity/pressure theory of the greenhouse effect, and disprove the radiative forcing theory of the greenhouse effect.

Note, this physical derivation is absolutely not suggesting the ~33C greenhouse effect doesn't exist. On the contrary, the physical derivation and observations demonstrate the 33C greenhouse effect does exist, but is explained by a different mechanism not dependent on radiative forcing from greenhouse gases. Also note, it is impossible for both explanations of the greenhouse effect to be true, since the global temperature would have to increase by an additional 33C (at least) above the present. You cannot have it both ways. We will show how the 
mass/gravity/pressure theory causes the temperature gradient and that the emission spectra of greenhouse gases seen from space are a consequence rather than the cause of that temperature gradient. 

This derivation uses very well-known physical principles and barometric formulae possibly first described by the great physicist Maxwell in 1872, who demonstrated that the atmospheric temperature gradient and greenhouse effect are due to pressure from Earth's gravitational field, not radiative forcing. Maxwell makes no mention of any influence of radiation as the cause of the temperature gradient of the atmosphere, but rather relates temperature at a given height to pressure. He discusses the convective (dominated) equilibrium of the atmosphere in his book Theory of Heat, pp. 330-331:

"...In the convective equilibrium of temperature, the absolute temperature is proportional to the pressure raised to the power (γ-1)/γ, or 0,29..."
Twenty four years later, Arrhenius devised his radiative forcing theory of the greenhouse effect, which unfortunately makes a huge false assumption that convection doesn't dominate over radiative-convective equilibrium in the lower atmosphere, and thus Arrhenius completely ignored the dominant negative-feedback of convection over radiative forcing in his temperature derivations. Johns Hopkins physicist RW Wood completely demolished Arrhennius' theory in 1909, as did other published papers in 1963, 1966, 1973, (and others below), but it still refuses to die given its convenience to climate alarm.

We now know from Robinson & Catling's paper in Nature 2014 (and others) that radiative-convective equilibrium on all planets with thick atmospheres in our solar system (including Earth of course) is dominated by convection/pressure/lapse rate in the troposphere up to where the tropopause begins at pressure = 0.1bar. When P < 0.1 bar, the atmosphere is too thin to sustain convection and radiation from greenhouse gases takes over to cause cooling of the stratosphere and above. 

Since Maxwell's book was published in 1872, many others have confirmed that the greenhouse effect is due to atmospheric mass/pressure/gravity, rather than radiative forcing from greenhouse gases, including Hans JelbringConnolly & ConnollyNikolov & ZellerMario Berberan-Santos et alClaes Johnson and hereVelasco et alGiovanni Vladilo et alHeinz ThiemeJacques HenryStephen WildeAlberto MiatelloGerhard Gerlich and Ralf D. TscheuschnerVerity JonesWilliam C. Gilbert & hereRichard C. TolmanLorenz & McKayPeter Morecombe, Robinson and Catling, and many others, so this concept is not new and preceded the Arrhenius theory. 


Step 1: Derivation of the dry adiabatic lapse rate from the 1st Law of Thermodynamics and ideal gas law:

First, the basic assumption can be adopted that the atmosphere, in hydrostatic terms, is a self-gravitating system in constant hydrostatic equilibrium due to the balance of the two opposing forces of gravity and the atmospheric pressure gradient, according to the equation:

 dP/dz = - ρ × g (1)

where ρ is the density (mass per volume) and g the acceleration due to gravity. This equation, from a mathematical point of view, can be derived by considering the hydrostatic equilibrium function as a system of partial derivatives depending on P and ρ and considering all three spatial dimensions:

 ∂P/∂x = ρ × X, ∂P/∂y = ρ × Y, ∂P/∂z = ρ × Z (2)

As, within a fluid mass in equilibrium, pressure and density does not vary along the horizontal axes (X and Y), the related partial derivatives equal zero. But, in the remaining vertical dimension, the partial derivative is non-zero, with density and pressure varying inversely as a function of fluid height (density and pressure decrease with increasing height relative to the bottom) and, considering gravitational force as a constant connected to the measure of density, thus equation (2) can be derived.

For a precise calculation involving the valid parameters, the 1st Law of Thermodynamics can be used:

 Δ U = Q – W (3)

where U is the total internal energy of the system, Q its heat energy, and W the mechanical work the system is undergoing. Applying this relationship to Earth's atmosphere, yields:

 U = C(p)T + gh (4)

where U is the total energy of atmospheric system in hydrostatic equilibrium and equal to the sum of the thermal energy (kinetic plus dissipative and vibro-rotational), the specific heat C(p) multiplied by the temperature T plus the gravitational potential energy, with gravitational force g at height h of the gas. In this case, because the force of gravity has a negative sign as the system is undergoing work, the potential energy ( -g × h) can be equated to the mechanical work (-W) that the system undergoes in the 1st Law of Thermodynamics.

Based on this equation, the atmosphere's "average" temperature change can be found for any point with the system in equilibrium; for now and for simplicity, weather phenomena and disturbances at specific locations are not considered because, with the system in overall hydrostatic and macroscopic equilibrium, any local internal, microclimatic perturbation by definition triggers a rebalancing reaction. In fact, to calculate the energy change of the system in equilibrium (here U is constant) as a function of temperature and height change, differentiation yields: 

dU = 0 = C(p)dT + gdh,

which becomes: 

dT/dh = -g/C(p), or dT = (-g/C(p))dh.  [Dry adiabatic lapse rate equation]

This is a splendid equation, describing precisely the temperatures’ distribution of a gas (as the air of Earth’s atmosphere) in hydrostatic equilibrium between the 2 forces of the lapse-rate (preventing the collapse of the atmosphere at the Earth’s surface) and gravity (preventing the escape of the atmosphere in the void of space). 

In other words, temperature variation (dT) is a function of altitude variation (dh), whose solution at any point of height (h°) and for any temperature (T°), can be found by integrating as follows:

∫dT = -g/C(p) × ∫dh (5)

and whose solution is:

 T - T° = -g/C(p) × (h - h°)   (6)

where:

T – T° = ∆ T (or dT) = Interval of temperatures
g = Newton’s gravitational constant = 6.67 × 10^-11 N (m/Kg)^2
h – h° = ∆ h (or dh) = Space interval (vertical) in the atmosphere
Cp = heat capacity at constant pressure

Step 2: Determine the height at the center of mass of the atmosphere

We are determining the temperature gradient within the mass of the atmosphere using a linear function of atmospheric mass (the lapse rate), therefore the equilibrium temperature is located at the center of mass. The "effective radiating level" or ERL of planetary atmospheres is located at the approximate center of mass of the atmosphere where the temperature is equal to the equilibrium temperature with the Sun. The equilibrium temperature of Earth with the Sun is commonly assumed to be 255K or -18C as calculated here. As a rough approximation, this height is where the pressure is ~50% of the surface pressure. It is also located at the approximate half-point of the troposphere temperature profile set by the linear adiabatic lapse rate, since to conserve energy in the troposphere, the increase in temperature from the ERL to the surface is offset by the temperature decrease from the ERL to the tropopause.


Fig 1. From Robinson & Catling, Nature, 2014 with added notations in red showing at the center of mass of Earth's atmosphere at ~0.5 bar the temperature is ~255K, which is equal to the equilibrium temperature with the Sun. Robinson & Catling also demonstrated that the height of the tropopause is at 0.1 bar for all the planets in our solar system with thick atmospheres, as also shown by this figure, and that convection dominates over radiative-convective equilibrium in the troposphere to produce the troposphere lapse rates of each of these planets as shown above. R&C also show the lapse rates of each of these planets are remarkably similar despite very large differences in greenhouse gas composition and equilibrium temperatures with the Sun, once again proving pressure, not radiative forcing from greenhouse gases, determines tropospheric temperatures. 

Step 3: Determine the surface temperature

For Earth, surface pressure is 1 bar, so the ERL is located where the pressure ~0.5 bar, which is near the middle of the ~10 km high troposphere at ~5km. The average lapse rate on Earth is 6.5C/km, intermediate between the 10C/km dry adiabatic lapse rate and the 5C/km wet adiabatic lapse rate, since the atmosphere on average is intermediate between dry and saturated with water vapor. 

Plugging the average 6.5C/km lapse rate and 5km height of the ERL into our equation (6) above gives

T = -18 - (6.5 × (h - 5)) 

Using this equation we can perfectly reproduce the temperature at any height in the troposphere as shown in Fig 1. At the surface, h = 0, thus temperature at the surface Ts is calculated as

Ts = -18 - (6.5 × (0 - 5)) 

Ts = -18 + 32.5  

Ts = 14.5°C or 288°K

which is the same as determined by satellite observations and is ~33C above the equilibrium temperature with the Sun.

Thus, we have determined the entire 33C greenhouse effect, the surface temperature, and the temperature of the troposphere at any height, entirely on the basis of the 1st law of thermodynamics and ideal gas law, without use of radiative forcing from greenhouse gases, nor the concentrations of greenhouse gases, nor the emission/absorption spectra of greenhouse gases at any point in this derivation, demonstrating that the entire 33C greenhouse effect is dependent upon atmospheric mass/pressure/gravity, rather than radiative forcing from greenhouse gases.

The greenhouse gas water vapor does have a very large negative-feedback cooling effect on the surface and atmospheric temperature by reducing the lapse rate by half from the 10C/km dry rate to the 5C/km wet rate. Increased water vapor increases the heat capacity of the atmosphere Cp, which is inversely related to temperature by the lapse rate equation above:

dT/dh = -g/Cp

Plugging these lapse rates into our formula for Ts above:

Ts = -18 - (10 × (0 - 5)) = 32C using dry adiabatic lapse rate

Ts = -18 - (5 × (0 - 5)) = 7C using wet adiabatic lapse rate [fully saturated]

showing a cooling effect of up to 25C just from changes in the lapse rate from water vapor. Water vapor also cools the planet via evaporation and clouds, and which is confirmed by observations. Water vapor is thus proven by observations and theory to be a strong negative-feedback cooling agent, not a positive-feedback warming agent as assumed by the overheated climate models to amplify warming projections by a factor of 3-5 times. 

What about CO2? At only 0.04% of the atmosphere, CO2 contributes negligibly to atmospheric mass and only slightly increases the heat capacity Cp of the atmosphere, which as we have shown above, is inversely related to temperature. CO2 would thus act as a cooling agent by slightly increasing troposphere heat capacity. Increased CO2 also increases the radiative surface area of the atmosphere to enhance outgoing radiation to space, analogous to putting a larger heat sink on your microprocessor which increases radiative surface area and convection to cause cooling. 

It is well-known that CO2 and ozone are the primary cooling agents of the stratosphere up to the thermosphere, but even the warmist proponents are unable to agree on a coherent explanation why CO2 would assume the opposite role of a warming agent in the troposphere. As the mass/gravity/pressure greenhouse theory shows, and just like water vapor, CO2 also acts to cool the troposphere, and the rest of the atmosphere by increasing radiative surface loss and outgoing radiation to space. 

Millions of weather balloon observations confirm that there is no greenhouse gas-induced "hot spot" in the mid-upper troposphere, which is the alleged "fingerprint of AGW." The 2nd law of thermodynamics principle of maximum entropy production also explains why such a "hot spot" will not form. However, observations do show a cooling of the stratosphere over the satellite era, which would be consistent with increased CO2 increasing outgoing radiation to space. Observations also show an increase of outgoing longwave radiation to space over the past 62 years, which is entirely consistent with increased outgoing radiation from greenhouse gases and a decrease of "heat trapping", the opposite of AGW theory. 

In essence, the radiative theory of the greenhouse effect confuses cause and effect. As we have shown, temperature is a function of pressure, and absorption/emission of IR from greenhouse gases is a function of temperature. The radiative theory tries to turn that around to claim IR emission from greenhouse gases controls the temperature and thus pressure and heat capacity of the atmosphere, which is absurd and clearly disproven by basic thermodynamics and observations. The radiative greenhouse theory also makes the absurd assumption a cold body can make a hot body hotter, disproven by Pictet's experiment 214 years ago, the 1st and 2nd laws of thermodynamics, the principle of maximum entropy production, Planck's law, the Pauli exclusion principle, and quantum mechanics. There is one and only one greenhouse effect theory compatible with all of these basic physical laws and millions of observations. Can you guess which one it is?

Update: The atmospheric center of mass assumption in step 2 above also appears to be applicable to Titan, the closest Earth analog with a thick atmosphere in our solar system. For Titan, the surface temperature is 94K, equilibrium temperature with the Sun is 82K, and surface pressure is 1.47 bar. 

Thus, the center of mass of the atmosphere is located at ~1.47/2 = ~0.74 bar, which observations show is where Titan's atmospheric temperature is ~82K, the same as the equilibrium temperature with the Sun. I have added the notations in red to Robinson and Catling's graph below:




Update 2: Some still claim the ERL is set by radiative forcing, but this is false because:

1) A purely radiative model of the atmosphere without convection sets the ERL too high, at ~7km instead of ~5 km where observations show it is located. At 7km altitude, observations show the temperature to be 242K instead of the equilibrium temperature of 255K.

Solely radiative "greybody" model of the atmosphere without convection shown in blue. I have added in red the actual temperatures from the US Standard Atmosphere calculator. Note how the purely radiative model is up to 20K hotter, e.g. at the top of the troposphere, than the observations show. This is because convection dominates and "short-circuits" radiative forcing in the troposphere to cause cooling.


Open symbols show no relationship between tropopause height and troposphere temperatures
That's because, as Robinson and Catling have shown, the height of the troposphere and the adiabatic lapse rate that extends from the 0.1 bar tropopause all the way to the surface is controlled by pressure, not temperature nor radiative forcing from greenhouse gases:
"At higher pressures [P > 0.1 bar], atmospheres become opaque to thermal radiation, causing temperatures to increase with depth and convection to ensue. A common dependence of infrared opacity on pressure, arising from the shared physics of molecular absorption, sets the 0.1 bar tropopause"

3. We have already shown that temperature is a function of pressure, and radiance and emission spectra from greenhouse gases are in turn a function of temperature, not the other way around.