Why Greenhouse Gases Don't Affect the Greenhouse Equation or Lapse Rate

Some commenters on the greenhouse equation believe that greenhouse gas radiative forcing controls the adiabatic lapse rate, and claim without any mathematics or evidence, that that's allegedly how greenhouse gases control the Earth surface temperature. We'll now show the reasons why this claim is incorrect:

1. The adiabatic lapse rate equation is

dT = (-g/Cp)*dh

where

dT = change in temperature
dh = change in height
g = gravitational constant
Cp = heat capacity at constant pressure

Thus change in temperature from the lapse rate is dependent upon 3 variables that have no dependence whatsoever upon radiative forcing from greenhouse gases. None. 

Note temperature is inversely related to heat capacity (Cp), thus, water vapor increases heat capacity but this decreases temperature by up to 25C as we calculated in the first post of this series, due to the lapse rate changing from a dry to a wet adiabatic lapse rate. This proves water vapor acts as a negative-feedback cooling agent. 

N2, O2, and CO2 heat capacities (Cp) are less than half that of water vapor, but are all very similar. At only 0.04% of the atmosphere, an increase in CO2 would cause a trivial increase in Cp, which since the lapse rate equation says Cp and temperature are inversely related, would cause a slight cooling. 

2. Secondly, even if the atmosphere was 100% nitrogen N2, with no greenhouse gases, the pressure change with altitude and thus convection and the lapse rate would be almost the same as the current atmosphere. 

We can prove this using the well-known barometric formulae, which are based on the ideal gas law and kinetics of gases (a "Boltzmann distribution"), are completely independent of greenhouse gas radiative forcing, and demonstrate a 100% greenhouse-free atmosphere of the same mass would have a nearly identical lapse rate, nearly identical height at the center of mass, and the surface would be just as warm or warmer than the current atmosphere. 

As the lecture notes below show, to answer the question, 

"What is the ratio of atmospheric pressure in Denver at 1 mile to that at sea level (assume the atmosphere is 100% Nitrogen N2)?

The solution uses the molecular weight of N2, the Boltzmann gas constant, Avogadro's number (constant) and well known standard barometric formula to determine the pressure ratio between sea level and one mile is 0.822, which is the same ratio (0.823) calculated by the US Standard Atmosphere calculator for our current atmosphere. Therefore the presence of greenhouse gases in our atmosphere insignificantly affect the barometric formulae, ideal gas law, and convection which are the basis of the greenhouse equation. 

We show using the same barometric formula for our atmosphere that the center of mass and equilibrium temperature with the Sun occur at ~5100 meter height:

Solving for the center of mass of our atmosphere, which is located at 0.50 atmospheres, and which sets the location of the ERL at 255K and near mid-point of the lapse rate

Compared to a pure Nitrogen N2 atmosphere in which the center of mass is only slightly higher altitude and at the equilibrium temperature with the Sun at ~5300 meter height:

Solving for the center of mass of a pure Nitrogen atmosphere, which is also is located at 0.5 atmospheres, and which sets the location of the ERL at 255K and near mid-point of the lapse rate

Thus, a 100% Nitrogen atmosphere without any greenhouse gases would have an atmosphere with essentially the same dry adiabatic lapse rate (10C/km) as Earth's, and the average adiabatic lapse rate in a N2 atmosphere would be higher than Earth's average 6.5C/km lapse rate due to the lack of water vapor, which increases Cp to decrease the lapse rate and cause cooling. 

Additional slides from the lecture notes above on why the composition of gases don't significantly affect the barometric formulae or ideal gas law:

Quick and dirty explanation of the Greenhouse Equation and theory

Here's a quick and dirty explanation of the greenhouse equation, which I hope will be helpful to understand the theory behind it. 

In the first post of this series we derived the following equation from the 1st law of thermodynamics and ideal gas law to calculate the temperature at any height in the troposphere:

T = Te + (lapse rate)*(h - he)   (1)

where

T = calculated T at height (h), or (s) used in equation below
Te = equilibrium temperature with the Sun (a constant)
lapse rate = -g/Cp = -gravity/heat capacity at constant pressure
he = height at the average "effective radiating level" or ERL, where T= equilibrium Te with the Sun 

Since we are calculating the gravitational greenhouse effect on the mass of the atmosphere, in order to conserve energy, one-half of the gravitational potential energy of the atmosphere has to be above the center of mass and one-half below. This point is at 1/2 of the surface pressure after a logarithmic adjustment for pressure and density with altitude. Since the surface pressure in atmosphere units is by definition = 1 atmosphere, in the greenhouse equation log(P/2) = log(1/2) below.

Again to conserve energy, the equation has to balance the local vertical equilibrium (since gravity is a vertical forcing vector) at every given local height the gravitational potential energy with the opposing thermodynamic energy of convection. Thus, the center of mass where log(P/2) must be the balance point where the two opposing forces balance, and this same point must also be at the equilibrium temperature with the Sun and near the mid-point of the adiabatic lapse rate. 

We previously calculated he to be located at the h where P=log(Ps/2) and the same point where Te = 255K = equilibrium temperature with the Sun.

The gravitational acceleration constant (g) appears twice in the equation, since to calculate the gravitational forcing we used Newton's second law of motion F=ma, which applied to the atmosphere is F=mg. The second use of the gravitational constant is from the dry adiabatic lapse rate.

Here's the greenhouse equation and my quick and dirty notes on how the components I just discussed enter into equation (1) above to calculate the temperature T at any height including the surface, as well as the entire 33C greenhouse effect, and without ever once using any radiative forcing whatsoever from greenhouse gases:

The Greenhouse Equation predicts 1% change in cloud cover changes global temperature by 1°C

The albedo terms in the greenhouse equation can be used to determine the effect of an increase or decrease of cloud cover upon the global temperature, and finds a 1% change up or down in cloud cover/albedo produces a 1°C temperature change at the surface. This indicates the Earth temperature is quite sensitive to swings in global cloud cover or other albedo sources, such as the well-known global dimming that produced the ice-age scare of the 1970's, which was followed by global brightening since, and which is likely responsible for much of the warming since the 1970's.

Climate models are unable to skillfully model clouds and thus albedo, and use a rough assumption of a 30% global average albedo. If we decrease albedo by 1% from 0.30 to 0.29, the greenhouse equation predicts an increase in surface temperature from 288.433°K to 289.457°K, an increase of 1.024°C warming, mathematically verifying what Dr. Roy Spencer writes in his book, 

"The most obvious way for warming to be caused naturally is for small, natural fluctuations in the circulation patterns of the atmosphere and ocean to result in a 1% or 2% decrease in global cloud cover. Clouds are the Earth’s sunshade, and if cloud cover changes for any reason, you have global warming — or global cooling."

The 2 albedo terms in the equation are notated

Why the atmosphere is in horizontal thermodynamic equilibrium but not vertical equilibrium

Willis Eschenbach and others claim to have refuted the greenhouse equation and gravitational greenhouse theory on the basis that the atmosphere is not in equilibrium, and others have claimed the theory predicts an atmosphere without greenhouse gases would be isothermal and in equilibrium. 

Both of these "proofs" and strawman arguments fail because the troposphere is in horizontal equilibrium due to gravity and center of mass at a particular latitude, but is in complete vertical disequilibrium due to convection and the lapse rate between the surface vertically rising to the tropopause. Further, if an atmosphere was 100% non-greenhouse gases, it would still be subject to gravity and convection calculated by the greenhouse equation, and thus absolutely not isothermal. A container of pure nitrogen with a heat source at the bottom would definitely convect, and convection is what controls the lapse rate and vertical temperature profile in the troposphere. 

Basically, the greenhouse equation determines the unique solution for temperature at any height given the opposing vertical disequilibrium from convection and the opposing horizontal gravitational equilibrium at that same height. 

Thus, the apparent misunderstanding of these "refutations" has arisen from a false assumption that fails to differentiate vertical and horizontal equilibrium. 

We have previously shown the greenhouse equation precisely determines the unique temperature at any vertical height from the surface up to ~11,000 meter top of the tropopause, based upon the local horizontal equilibrium at that particular height. 

Satellite observations indeed show a remarkable horizontal equilibrium at a given center of mass of the overlying vertical atmospheric mass:

Annotated:

Newsflash: Heat rises

and the vast majority of that heat rise is vertical due to the vertical, not horizontal, disequilibrium due to the vertical vector of gravitational force with zero horizontal gravitational forcing. Deal with it, and I hope that clears it up. 

h/t to ren at Tallbloke's talkshop for posting link to top graphic which was perfect to illustrate this point and said 

"It is clear that the troposphere is heated uniformly from the surface of the Earth. Thus, the center of mass of the troposphere is logical."

couldn't agree more. 

The Greenhouse Equation predicts temperatures within 0.28°C throughout entire troposphere without radiative forcing from greenhouse gases

In this continuing series of posts on the greenhouse equation, we will now calculate and plot the tropospheric temperature profile as a function only of the balance of radiative forcing from the Sun and gravitational forcing upon atmospheric mass, and without any contribution from greenhouse gas radiative forcing. We will see that the greenhouse equation reproduces observations and the 1976 standard atmosphere database within 0.28°C at every height from the surface all the way to the ~11,000 meter average height at the top of the troposphere. 

The greenhouse equation in essence determines the temperature based upon the balance of gravitational potential energy and radiative forcing from the only source of energy, the Sun, and is unique for each height from the surface to top of the troposphere. The greenhouse equation maintains the balance of solar/radiative and gravitational/convective equilibrium, and there is one and only one temperature that maintains this balance for each height in the troposphere. 

Note the only radiative forcing considered by the greenhouse equation is from the Sun, and nothing from greenhouse gases. The solar forcing warms the surface, which causes all gases to convect, rise, and expand, and which keeps the atmosphere inflated against the force of gravity, and these forces balance at every local height to maintain a horizontal local equilibrium at that particular height. Note the overall atmosphere is not in equilibrium, which is what drives this whole process of convection/adiabatic lapse rate/and the tropospheric temperature gradient.   

Thus, the opposing forces of convection and gravitation at each local height are in local equilibrium. The solar radiative forcing causes all gases at the surface to warm, rise, and expand against the gravitational force which is constantly pulling all gases back to the surface. Gravitational potential energy increases the higher a gas packet rises and expands, until that gas packet reaches a height where it's temperature is the same as the surrounding air, at which the gravitational potential energy starts to exceed its kinetic energy, then the gravitational potential energy accumulated takes over to make that gas packet subside/fall/compress and warm due to the ideal gas law. Once that gas packet becomes warm enough from compression to once again overcome gravitational potential energy, it starts to rise once again and that whole process continues over and over again ad infinitum. 

This rising/falling and expansion/compression of gas packets is thus driven by two opposing forces:

• Solar radiative forcing to make gases warm, rise, and expand until they cool to the same temperature as the surrounding air, followed by 
• Gravitational forcing pulling the now cooled gases back to the Earth

The balance between these two opposing forces is what generates the entire 33C greenhouse effect. Radiation spectra from greenhouse gases are the result, not the cause, of the temperature gradient in the troposphere that these two dominating, very strong, and opposing forces create. (As an example, in an earlier post, we calculated that the effect of gravity on the atmosphere is creating 10,500 kilograms of gravitational forcing per square meter at the surface.)

The greenhouse equation was derived in the recent series of posts: 

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.

and I'll now show the unique numeric solutions perfectly reproduce within 0.28C the temperature at every height in the 1976 Standard Atmosphere database from the surface all the way to ~11,000 meters height, after which the atmosphere at < ~0.223 atmospheres (units) becomes too thin to sustain convection and the equation no longer applies. This is near the transition between the troposphere and tropopause, which is where Robinson & Catling, Nature 2014, demonstrated the transition begins due to loss of convection, not greenhouse gas radiative forcing. Above the tropopause transition level (which is isothermal for several km), increased greenhouse gases cause increased cooling. 

The greenhouse equation solves for the single unique temperature at every height necessary to balance the vertical equilibrium of the atmosphere (since gravitational forcing is vertical only). The atmosphere is in horizontal equilibrium at every height relative to the center of mass at that coordinate. The equation applies as long as the troposphere is capable of sustaining convection, up to ~12,000km in the tropics, but beyond that the atmosphere is too thin to sustain convection and the adiabatic lapse rate, thus the equation is not applicable above this point around 11-12 km average:

We find the greenhouse equation perfectly reproduces (within 0.28C) the standard tropospheric temperature profile (black=Standard Atmosphere, red=the greenhouse equation). Here's the data calculated by Wolfram Alpha for the greenhouse equation, and the 1976 Standard Atmosphere data:

height (km)	Greenhouse Equation T in K	Std Atmos 1976 T in K
0	288.433	288.15
1	281.933	281.65
2	275.433	275.15
3	268.933	268.65
4	262.433	262.15
5	255.933	255.65
6	249.433	249.15
7	242.933	242.65
8	236.433	236.15
9	229.933	229.65
10	223.433	223.15
11	216.933	216.65
12	210.433	216.65
13	203.933	216.65

1976 Standard Atmosphere:

(Note the greenhouse equation does correct for the change in density with pressure, and the natural logarithmic decay of pressure with altitude)

We annotate the plot which shows the height of the "effective radiating level" or ERL, the height at which the local horizontal equilibrium at that particular height must by definition be where T = equilibrium temperature with the Sun = 255K, at essentially exactly at the center of mass of the atmosphere, which has to coincide with the location where half of the mass is above and half below, where the pressure is one-half the surface pressure after altitude and density correction - both of which are done by the greenhouse equation (and is the reason for the natural logarithms in the equation):

There is nothing in the greenhouse equation which is dependent upon radiative forcing from greenhouse gases, nor greenhouse gas concentrations, nor greenhouse gas absorption/emission spectra, and yet we can perfectly reproduce the temperature within 0.28C anywhere in the troposphere and at the surface. Thus, the absorption and emissions of IR from greenhouse gases are the consequence, and not the cause of the real 33C greenhouse effect. 

Here are some of the numerical solutions from Wolfram Alpha which did the for the one and only unique value T that satisfies the greenhouse equation horizontal equilibrium at a particular height (s) in kilometers above the surface, which is the one and only variable that I changed for each of the these numeric solutions, as you can see from the input and Wolfram Alpha solutions:

Try it yourself: Here's the code for the greenhouse equation to determine the T at height = 0 (i.e. at the surface). I have bolded that 0 in the code below so you can find it. Simply change that variable from 0 to whatever height in kilometers from the surface to calculate the one and only unique local equilibrium at that particular height solution for the greenhouse equation temperature gradient:

T = [1367 (1 - 0.3) / (4*1* (5.6704*10^-8))]^1/4 +((-6.5)*(0-[-[8.31*{[1367 (1 - 0.3) / (4*1* (5.6704*10^-8))]^(1/4)}*log(1/2)]/[9.8*0.029*log(e)]/1000]))

The Greenhouse Equation

A recent series of Hockey Schtick posts

• Derivation of the entire 33°C greenhouse effect without radiative forcing from greenhouse gases
• Derivation of the effective radiating height & entire 33°C greenhouse effect without radiative forcing from greenhouse gases
• Why greenhouse gas radiative forcing doesn't explain Earth's energy budget

have derived the entire ~33°C greenhouse effect as a consequence of gravitational forcing rather than radiative forcing from greenhouse gases, and entirely independent of radiative forcing from greenhouse gases. We have also determined the effective radiating height (average) or ERL in the troposphere (where T = the equilibrium temperature of Earth with the Sun), and found the ERL to be located as expected at the center of mass of the atmosphere if the ERL height and temperature are a function of mass/gravity/pressure rather than radiative forcing from greenhouse gases.

We now join the gravitational greenhouse effect to the only source of energy that the Earth receives, the Sun, and show that solar shortwave radiative forcing plus gravitational forcing calculates the Earth's surface temperature, ERL height and temperature, and the entire tropospheric temperature profile perfectly, without any contribution from greenhouse gas forcing, nor dependence on greenhouse gas concentrations, nor dependence upon emission/absorption spectra from greenhouse gases. 

We show that the entire 33°C greenhouse effect that raises Earth's equilibrium temperature with the Sun of -18C or 255K up to +15C or 288K at the surface, and the temperature at any height in the atmosphere from the surface to top of the troposphere (above which the atmosphere is too thin to sustain convection), can be fully explained by the following equation, which I'm calling "the greenhouse equation":

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. 

which solves T as a function of mass/pressure/gravity for which none of the variables are dependent upon radiative forcing from greenhouse gases, and for which the only radiative forcing we require to reproduce the entire tropospheric temperature profile is that from the Sun. Note none of the constants and variables on the right side of the greenhouse equation are related to GHG radiative forcing, and temperature does not appear on the right side of the equation and thus it can't be a tautology of temperature.

T = temperature at height (s) meters above the surface, thus at the surface s = 0

s = height in meters above the surface to calculate the temperature T, thus at the surface s=0
S = the solar constant = 1367 W/m2, derivation here
ε = emissivity = 1 assuming Sun and Earth are blackbodies
σ = the Stefan-Boltzmann constant = 5.6704 x 10-8 W m-2 K-4
g = gravitational acceleration = 9.8 m/s^2

m = average molar mass of the atmosphere = 29g/mole = 0.029kg/mole
α = albedo = 0.3 for earth

C = Cp = the heat capacity of the atmosphere at constant pressure, ~ 1.5077 average for Earth

P = surface pressure in the unit atmospheres, defined as = 1 atmosphere for latitude of Paris
R = universal gas constant = 8.3145 J/mol K
e = the base of the natural logarithm, approximately equal to 2.71828

As shown by the prior posts listed above, all of the components of this entire gravitational "greenhouse equation" were first derived from the ideal gas law, the First Law of Thermodynamics, the Stefan-Boltzmann equation, Newton's second law of motion (F = ma), and well-known barometric formulae, without ever once introducing any variables dependent upon radiative forcing from greenhouse gases. 

In a prior post we determined surface pressure by the relation:

P = e^-((Mgh/(RT))  (15) where T = T at height (s) (or (h) in the prior post)

for which we substitute for P in the greenhouse equation above (note s = h from the prior post) to yield:

After plugging in the numerical values, Wolfram Alpha solves the greenhouse equation to find Earth's surface (where the height (s)=0) the temperature T is equal to 288.433K or 15.28C, which is the same as determined from satellite measurements:

Note the 1000 in the numeric solution is the conversion factor between meters and kilometers

We can now use the greenhouse equation for many other purposes including determining the effect of a change in solar activity on the expected Earth surface temperature. If we increase the solar constant in the above numerical solution by 1 W/m2 from 1367 to 1368 W/m2, we find an increase in surface temperature from T=288.433K to 288.486K, an increase of 0.056C (this includes the division by 4 to convert solar insolation from a flat disk to a sphere):

Note Wolfram Alpha is solving for T, providing a unique solution of T as a function of the mass/gravity/pressure forcing at each height (s) in meters above the surface (s = 0 at the surface). Each solution for T as a function of height (s) perfectly reproduces the observed tropospheric temperature profile at each height (s), from the surface to the ERL located at the atmospheric center of mass and then on up to the top of the troposphere. 

Wolfram Alpha solves for the unique temperature (T) that satisfies the greenhouse equation for a given height (s)  with P substituted by e^-((Mgh/(RT)) (15) above, showing there is a unique solution of T for every value of height (s), with no T on the right-hand side of the solution for T, proving the greenhouse equation is not a tautology:

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).

Note this equation would not be expected to hold for planets with thick, opaque cloud tops, such as Venus, which in addition to heating from the bottom up due to gravity/pressure, is also heated from top down by absorption of sunlight at the TOA and probable conduction of heat downward from the opaque cloud tops. It also cannot be applied to planets with thin atmospheres such as Mars, which has a surface pressure of only ∼0.006 bar. In addition, the ideal gas law and barometric equations are only true if the heat capacity C in the greenhouse equation stays constant while the temperature changes, which is true for N2 and O2 (more than 99% of our atmosphere and 98.4% of Titan's atmosphere), but CO2 does not. Since CO2 comprises 96.5% of the Venus atmosphere, the heat capacity C (Cp) in the greenhouse equation would have to be adjusted as a function of temperature.

I welcome all suggestions and refutations of the "greenhouse equation." There can be only one valid theory of the 33C greenhouse effect, since if both the gravitational and greenhouse gas radiative forcing theories had merit, the Earth would be at least 33C warmer than the present. 

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/s^2, i.e. the acceleration due to gravity in meters per second squared.

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 (4)

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 (7)

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

dP = -ρg dh (8)

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  (9)

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 (10)

thus 

ρ = MP/RT  (11)

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

dP = -(MPg/RT)dh  (12)

 ∫dP/P = -(Mg/RT) ∫dh  (13) (the first integral is from 1 to P, second from 0 to h)  

ln(P) = -(Mgh/RT)  (14)

P = e^-((Mgh/(RT))  (15)

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 Earth's greenhouse effect would be at least double (66C+ rather than 33C). 

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.

Correlation between solar activity and the local temperature of Antarctica during the past 11,000 years

• X.H. Zhao^, , 
• X.S. Feng^, 

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SSN [Sunspot Number] and Vostok temperature (T) had common periodicities in past 11,000 years.

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The millennial variations of SSN and T had a strong and stable correlation.

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The millennial variation of SSN led that of T by 30–40 years.

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Correlations between CO₂ and T were neither strong nor stable.

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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.