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THERMAL RUNAWAY

By Charles Rhodes, P. Eng., Ph.D.

INTRODUCTION:
Earth's atmosphere has two stable states, a "cool" state and a "warm" state. These two stable states exist because the physical properties of water go through major step changes at its freezing point. Thermal Runaway is the name given to uncontrolled spontaneous transition from the "cool" state to the "warm" state. This transition is triggered by an unusually high atmospheric CO2 concentration. Similarly a transition from the "warm" state to the "cool" state is triggered by a relatively low atmospheric CO2 concentration.

Thermal runaway is an immediate threat to the survival of mankind. If present trends continue thermal runaway is likely become out of control about the year 2035. The underlying problem is that present world governmental plans, even if fully implemented, will not result in a sufficient CO2 emission reduction to significantly reduce the atmospheric CO2 concentration. Achieving the required atmospheric CO2 concentration reduction requires an immediate 90% reduction in CO2 emissions by industrialized countries. As indicated by the 2015 Paris Agreement on climate change and by further investment in fossil fuel infrastructure most national governments, including the Canadian government, are simply unwilling to face this reality.

This web page addresses the physics of thermal runaway. Another web page addresses a related phenomena known as WARM STATE TRAPPING.

Non-linearity in the relevant radiant energy transfer equations causes Earth to have two locally stable temperature states, a "cool" state and a "warm" state. The non-linearity arises from:
a) A step change in the solar reflectance of water at 273.15 K where there is a solid-liquid phase transition;
b) Preferential infrared radiation emission by water as water goes through its liquid-solid phase transition;
c) Infrared power emission approximately proportional to the 4th power of absolute temperature;
d) Earth infrared emissivity dependence on upper atmosphere CO2 concentration;
e) Earth infrared emissivity dependence on atmospheric water vapor concentration.

On photos of Earth from deep outer space the "cool" regions and "warm" regions are readily apparent. The problem is that the area of the "cool" regions is shrinking and the area of the "warm" regions is expanding.


Earth From Space Apollo 17 Dec. 1972
 


Earth July 2015
 

Earth's atmospheric "cool" state is characterized by an infrared radiation emission temperature Tc, as seen from outer space, of less than 273.15 K.
In November 1996 the Mars Global Surveyor spacecraft measured Earth's average emission temperature as Tc = 270 K to 270.8 K. The actual infrared radiation was about 30% further reduced by CO2 and H2O absorption in Earth's upper atmosphere.

Earth's "warm" state is characterized by an infrared radiation emission temperature, as seen from outer space, of greater than 273.15 K.
At an emission temperature of 273.15 K the local Bond albedo drops from about 0.5 to as low as 0.035 over the ocean. There is also a sharp drop in infrared emissivity.

In 2016 the polar regions of Earth are still in a "cool" state but Earth's atmospheric CO2 concentration is close to the threshold of spontaneous thermal runaway.

Historically Earth's planetary Bond Albedo was about 0.30 and the Earth emission temperature was about Ta = 270 K. The above photos show that with complete ice melting Earth's planetary Bond albedo will drop from about 0.30 to about 0.10. Neglecting changes in Earth infrared emissivity the new average emission temperature Tb will be given by:
(Tb / Ta)^4 = (1.00 - 0.10) / (1.00 - 0.30)
or
Tb = Ta [0.90 / 0.70]^0.25
= 270 [1.064844317]

Hence:
(Tb - Ta) = 270 K [0.064844317]
= 17.5 K

The emission temperature rise due to a decrease in infrared emissivity is additional.

The problem is that elected politicians are unwilling to commit to immediate construction of sufficient nuclear power plants and related electricity transmission/distribution to prevent thermal runaway. These politicians fail to understand that due to its poor capacity factor wind power cannot displace a large enough fraction of the total fossil fuel consumption to prevent thermal runaway occurring.

There is much confusion around average Earth temperature measurements. There is average Earth surface temperature and average Earth emission temperature. Average Earth emission temperature is measured from outer space using a far infrared spectrometer. During thermal runaway the average Earth Bond Albedo is projected to drop from about 0.30 to 0.10 causing an increase in average Earth emission temperature of about 17.5 degrees C from about 270 degrees K as measured in 1996 to about 287.5 degrees K.

The average Earth surface temperature is presently about 18 degrees C above the average Earth emission temperature. Doubling Earth's atmospheric CO2 concentration will cause the average Earth emission temperature to rise by about 3 degrees C. There will be another 1 degree C emission temperature rise due to the corresponding increase in water vapor concentration in the upper atmosphere. The problem is that this 4 degree C rise in average Earth emission temperature will melt sufficient ice to significantly reduce Earth's planetary Bond Albedo, thus causing cause a large increase in average Earth emission temperature. The increase in average Earth emission temperature will melt more ice and hence further reduce Earth's planetary Bond Albedo. This feedback mechanism, known as thermal runaway, will run until there is no more ice left to melt.
 

CARBON STORAGE CYCLE:
In 2015 part of Earth was still in its "cool" state. However, natural accumulation over tens of millions of years of weakly bound carbon in fossil fuels has enabled a potential spontaneous transition from the "cool" state to the "warm" state.

When there is sufficient weakly bound carbon a "cool" state to "warm" state transition can be triggered by a geologically rare transient event such as a nearby passing star or by the emergence of a fossil carbon energy harvesting life form (humans) that via combustion of weakly bound carbon cause a further decrease in Earth's infrared emissivity.

In response to a "cool" state to "warm" state transition the ocean warms and releases more CO2 to the atmosphere which further reduces Earth's infrared emissivity trapping Earth in its "warm" state for several hundred thousand years.

Over several hundred thousand years in the "warm" state photosynthesis converts atmospheric CO2 back into weakly bound fossil carbon. The resulting decrease in atmospheric CO2 concentration gradually increases Earth's infrared emissivity and hence enables a future "warm" state to "cool" state transition.

Once a "warm" state to "cool" state transition is enabled it can be triggered by a geologically rare event such as a comet impact or a volcanic erruption that causes a transient increase in Earth's Bond albedo.

In response to a "warm" state to "cool" state transition ocean cooling absorbs further CO2 from the atmosphere which further increases Earth's infrared emissivity. As long as a substantial fraction of Earth is in the "cool" state large land animal life forms such as humans can obtain energy by converting a limited amount of weakly bound fossil carbon back to gaseous CO2. However, too much rapid conversion of weakly bound fossil carbon to gaseous CO2 will eventually enable another spontaneous "cool" state to "warm" state transition.

Once a spontaneous state transition is under way it cannot be stopped.

Thermal runaway occurred about 55 million years ago. As a result there was a global extinction of large land animals, the polar ice caps completely melted and Earth's ecosystem was disrupted for almost 500,000 years.
 

THERMAL RUNAWAY DANGER:
It is shown herein that today Earth's atmospheric CO2 concentration is close to the threshold for commencement of spontaneous thermal runaway. The thermal runaway state transition can only be avoided by an immediate and sustained reduction in Earth's atmospheric CO2 concentration. Otherwise thermal runaway will occur and over a few decades Earth's average surface temperature will further rise by as much as 17.5 degrees K.

In spite of the fact that science of thermal runaway is well understood by senior physicists and the remedies are well understood by senior engineers, the imminent danger of thermal runaway is not being seriously addressed by elected politicians. The atmospheric CO2 problem is global in nature and requires expensive solutions with international implementation and international enforcement. Elected politicians with authority to implement these solutions respond only to voters, who with few exceptions know nothing about the relevant science and engineering matters and are often unwilling to take personal responsibility for their own fossil CO2 emissions.

The simple reality is that absent strong leadership and effective action thermal runaway will occur during the 21st century. If present fossil fuel usage trends continue thermal runaway is certain and mankind is doomed. As a result of thermal runaway large land animals (including humans) will become extinct, Earth's polar ice caps will melt and the environmental conditions on Earth will be radically changed for hundreds of thousands of years.
 

THERMAL RUNAWAY AVOIDANCE:
The only way to avoid thermal runaway is for both voters and politicians to place their trust in senior scientists and senior engineers. The science of thermal runaway involves an advanced understanding of quantum mechanics, radiative energy exchange, non-linear differential equations and control system stability. Prevention of thermal runaway involves immediate widespread application of fast neutron nuclear power reactors, electricity generation/transmission/distribution systems and related technologies.

Presently mankind is emitting fossil CO2 to the atmosphere more than twice as fast as the ocean can absorb that CO2. Prevention of thermal runaway by reducing the atmospheric CO2 concentration requires as a minimum an immediate 75% reduction in total world wide fossil CO2 emissions. Fossil fuels must be left in the ground and must be immediately replaced by renewable and nuclear energy. At the present time elected politicians are unwilling to face these simple realities, so the future prospects for the human species are very poor.

The problem with renewable energy sources such as wind, solar and hydro is that in most jurisdictions they do not provide sufficient energy when and where required to support urban human populations. In most jurisdictions the combined costs of harvesting, transmitting, storing and retransmitting renewable energy to urban load centers are prohibitive. Hence mankind must rely heavily on nuclear energy.

At the heart of the CO2 / thermal runaway problem lies inadequate public education in the physical sciences. It should be possible to avoid thermal runaway through prompt widespread application of fast neutron reactors and related technologies for displacement of fossil fuels. However, the voting public has little or no understanding of these technologies and has irrational fears about nuclear energy. Moreover, voters are constantly misled by deceptive advertisements funded by the fossil fuel industry.

Hope for mankind lies primarily in rigorously enforced reduction of CO2 emissions combined with an immediate world wide expansion of nuclear electricity generation and related electricity transmission/distribution. Renewable energy, while helpful, can at best economically supply only about one third of the total energy required for displacement of fossil fuels.

During the three year period between 1942 and 1945, with the aid of World War II priorities, my grand parents generation took nuclear reactor technology from a scientific concept to breeding enough plutonium for two functional atomic bombs. If mankind is to avoid thermal runaway a similar priority must be assigned to construction of new liquid sodium cooled fast neutron nuclear power reactors and related electricity generation/transmission/distribution. To meet the time constraints the existing pre-construction licensing processes and other legal and administrative obstacles to rapid construction of nuclear power reactors and electricity transmission lines must be set aside.
 

PREVIOUS THERMAL RUNAWAY:
The geologic record shows that about 55 million years ago, during a 200,000 year period known as the PETM (Paleocene Eocene Thermal Maximum), analogous atmospheric CO2 triggered thermal runaway caused complete melting of the polar ice caps and caused a global extinction of all large land animals. A further 300,000 years passed before the Earth returned to normal. Isotopic analysis of PETM sediments confirms that Earth has at least two locally stable emission temperature states, a normal "cool" state and a "warm" state.
 

BOND ALBEDO:
The bond albedo of Earth is the fraction of solar energy incident on Earth that is reflected into space. The Bond albedo increases with the number of water molecules in Earth's atmosphere that are in the solid state and hence are clumped together as air borne ice micro-crystals. This number is the product of the total number of water molecules in Earth's atmosphere and the fraction of these water molecules that are in the solid state. This fraction is highly dependent on the infrared emission temperature. In November 1996 Earth's "cool" state infrared emission temperature Tc was measured as:
Tc = 270.0 K.

As the Earth's cloud temperature rises through 273.15 K, the freezing point of water, the number of air borne water molecules in the solid state will rapidly decrease and hence Earth's Bond albedo will rapidly decrease from an average of about 0.30 in the "cool" state to an average of about 0.10 in the "warm" state. The dependance of local Bond albedo on atmospheric temperature is readily apparent on solar illumination photos of Earth taken from deep space. The average local Bond albedo near Earth's equator is about 0.10 as compared to a local bond albedo of about 0.50 near Earth's poles. During the late 1990s the average Bond Albedo of the whole Earth was measured to be about 0.297.
 

NON-LINEARITY:
The rapid change in solar bond albedo as Earth's emission temperature T rises through the freezing point of water at:
To = 273.15 K
in combination with infrared radiative power emission proportional to T^4 leads to Earth's energy balance equations having two separate locally stable steady state solutions for emission temperature T. These solutions are referred to herein as Tc for the "cool" state solution and Tw for the "warm" state solution. These separate solutions indicate the existence of a spontaneous thermal runaway transition between the "cool" state and the "warm" state.
 

THERMAL RUNAWAY THRESHOLD DEFINITION:
The best available experimental data indicates that the atmospheric CO2 concentration threhold that triggers a "cool" state to "warm" state transition is about 433 ppmv. However, this calculated threshold is quite sensitive to small errors in the experimental measurements of the "cool" state emission temperature Tc and "cool" state bond albedo Fr.

It would be prudent for the National Research Council of Canada to implement or supervise a program for ongoing monitoring of Earth's bond albedo to an absolute accuracy of three significant figures and to report the results annually to the Canadian public because the precise value of Earth's bond albedo will heavily impact the Canadian fossil fuel industry. The bond albedo can be determined by measurement of the intensity of Earthshine reflected from the moon.
 

THERMAL RUNAWAY:
Thermal runaway occurs when the conditions for stability in Earth's "cool" state are no longer met. In these circumstances Earth will spontaneously drift toward the locally stable "warm" state. The commencement of thermal runaway is imperceptible. However, as thermal runaway progresses Earth's infrared emission temperature will rise faster and faster until the Canadian winter becomes like the present Canadian summer and the Canadian summer becomes like the present equatorial climate.

The theoretical average Earth emission temperature rise associated with thermal runaway is 17.5 K. The near term temperature rise is somewhat less due to the long thermal response time of the oceans.

When thermal runaway occurs Canada will face uncontrollable immigration from the USA, Mexico and Central America. Parts of Canada's major sea port cities, such as Vancouver's Fraser River Delta and surrounding communities, will quickly be submerged by the rising sea level.
 

IPCC FAILURES:
The IPCC (International Panel on Climate Change) has totally failed to address the issue of thermal runaway. The IPCC has ignored reliable astrophysical data, possibly because relatively few people understand it. The IPCC treats the Earth energy balance as a linear system, which it is not, and assumes local temperature stability without any mathematical or physical justification for that assumption. The IPCC methodology of "Radiation Forcing" is only valid for very small changes in climate system parameters. In reality if:
T = infrared emission temperature
and
To = 273.15 K = the freezing point of water
the infrared radiation emitted by Earth varies in proportion to:
(T / To)^4
and the solar radiation absorbed by Earth has a 20% component that varies in proportion to the near step function:
[(T / To)^Kf / (1 + (T / To)^Kf)]
where:
Kf ~ 600

This high degree of non-linearity is completely ignored by the IPCC and leads to major errors in the IPCC's future projections.

One of the sources of the IPCC's problems is inadequate understanding of the dominant mechanism by which the Earth emits thermal infrared photons. Spectral analysis shows that in the "cool" state these photons are emitted during the liquid-solid phase transition of water droplets. The resulting ice micro-crystals contribute to Earth's Bond albedo. In the "warm" state Earth's infrared emission occurs without formation of ice micro-crystals.

The IPCC reports do not address future changes in Earth's Bond albedo.

The IPCC's reports do not address the Earth's measured thermal infrared emission spectrum.

The IPCC reports fail to quantitatively link Earth's infrared emission temperature T to Earth's infrared emissivity Ft and Earth's planetary Bond albedo Fr.

The IPCC reports also do not address the control system parameter relationships required to provide Earth long term temperature stability.
 

TWO STABLE TEMPERATURE STATES:
The Earth continuously absorbs a fraction of incident solar radiation and continuously emits infrared radiation. At steady state conditions the absorbed solar power equals the emitted infrared power so that the Earth's net energy change is zero. However, today we are not at steady state conditions. There is a non-equilibrium concentration of CO2 in the Earth's atmosphere that is causing net energy absorption by the oceans. This energy absorption in combination with a decreasing infrared emissivity due to an increasing atmospheric CO2 concentration is causing Earth to gradually move toward the thermal runaway threshold.

Today in 2015 Earth is close to the threshold of a spontaneous "cool" state to "warm" state transition. If the present high atmospheric CO2 concentration persists cumulative net energy absorption by the oceans will trigger thermal runaway. Escape from thermal runaway is only possible in the very near term and then only via an immediate major reduction in the Earth's atmospheric CO2 concentration and / or via an immediate major increase in the Earth's bond albedo. Neither of these two physical objectives is easy to achieve.

The Earth's thermal infrared radiation emission spectrum exhibits a well defined emission temperature. An emission temperature of:
T = 270.0 degrees K
was measured in November 1996 from deep space by a far infrared spectometer (Thermal Emission Spectrometer) on board the Mars Global Surveyor spacecraft.

The infrared radiation that presently cools Earth is primarily emitted by water micro-droplets in the atmosphere during liquid-solid phase transitions.

In November 1996 the Earth's atmospheric CO2 concentration, as measured at Mona Loa, Hawaii, was about 360.76 ppmv.
 

STATE COMPARISON:
In the "cool" state part of the Earth is snow or ice covered and brilliant white clouds, formed from ice micro-crystals, cover about half of the Earth's sun facing surface. In the "warm" state the entire Earth is free of snow and ice and dark grey clouds formed from liquid water micro-droplets cover a portion of the Earth's sun facing surface.

In the "cool" state ice micro-crystals in clouds and surface ice reflect incident solar radiation photons back into space much more effectively than do "warm" state liquid water micro-droplets in clouds and the open ocean. Hence the solar power absorbed by the Earth in the "warm" state is ~ 28.5% larger than in the "cool" state. At steady state conditions in the "warm" state the increase in solar power absorption is balanced by an increase in infrared radiant energy emission due to an increase in the Earth's steady state emission temperature T from its "cool" state value Tc to its "warm" state value Tw
 

EFFECT OF CO2 INJECTION:
Combustion of fossil fuels causes a transient increase in atmospheric CO2 concentration. At present ocean temperatures the transient CO2 in the atmosphere has a half life of about 28 years.(Ref:CARBON DIOXIDE)

The increased concentration of CO2 in the Earth's upper atmosphere reduces Earth's far infrared emissivity. This effect causes global warming which raises the "cool" state emission temperature Tc.

Define:
Tr = emission temperature at commencement of Thermal Runaway.
Fr = Earth bond albedo
= fraction of incident solar radiation that is reflected into space

For T > Tr the formation rate of solar light reflecting ice micro-crystals rapidly diminishes. Hence, as the atmospheric CO2 concentration increases Fr decreases and the Earth transitions from its "cool" state into its "warm" state.

In the "warm" state ocean warming causes an increase in the steady state atmospheric CO2 concentration. If there is sufficient ocean warming the Earth will be trapped in its "warm" state long after the initial transient fossil CO2 concentration decays. The Earth will remain trapped in its "warm" state for hundreds of thousands of years until photosynthesis transfers enough CO2 from the ocean-atmosphere pool to the weakly bound carbon pool to enable a "warm" state to "cool" state transition.

A "cool" state to "warm" state transition is accompanied by an increase in Earth's infrared emission temperature T of about 17.4 degrees C and is accompanied by a decrease in the Earth's bond albedo Fr from ~ 0.30 to ~ 0.10.
 

WARM STATE TRAPPING:
Warm State Trapping is the name given to the ocean warming and CO2 release process that can cause the Earth to be trapped in the "warm" state for hundreds of thousands of years.
 

WARM STATE TO COOL STATE TRANSITION:
After ocean temperature stabilization in the "warm" state over hundreds of thousands of years photosynthesis by plants converts CO2 in the ocean-atmosphere pool into fossil carbon. Eventually photosynthesis in the "warm" state will cause a sufficient cumulative drop in the amount of CO2 in the ocean-atmosphere pool (and hence in the atmospheric CO2 concentration) to enable the Earth to transition from its "warm" state back to its "cool" state.
 

MECHANICAL ANALOG:
The concept of multiple locally stable temperature states can be illustrated with a two state mechanical analog. Consider a coin. The coin has two locally stable positions, heads up or heads down. If the coin is slightly mechanically disturbed it will return to its initial locally stable position. However, if the coin is more heavily mechanically disturbed it can change to its other locally stable position.

Thus a coin has two locally stable position states.
 

MATHEMATICAL ANALOG:
A component of high school mathematics is the study of quadratic equations of the form:
A X^2 + B X + C = 0
where A, B, C are constants and X is an unknown quantity. There are two solutions for X of the form:
X = {- B + [B^2 - 4 A C]^0.5} / 2 A
and
X = {- B - [B^2 - 4 A C]^0.5} / 2 A

If:
B^2 > 4 A C
then there are two distinct real solutions for X.

Similarly the non-linear equations that establish the Earth's infrared emission temperature have two distinct real solutions, one below the freezing point of water known as the "cool" state at:
T = Tc
and the other above the freezing point of water known as the "warm" state at:
T = Tw.
 

ENERGY BALANCE:
In order to understand why thermal runaway occurs it is necessary to understand the natural processes that regulate the Earth's thermal infrared emission temperature T.

The Earth constantly receives solar energy from the sun. Fraction Fr of the incident solar energy, is reflected into space. The average thermal power per unit area produced via combustion of fossil fuels, via human initiated nuclear reactions and via volcanic activity is Pn. The Earth achieves long term energy balance via infrared radiative energy emission. From quantum mechanics the infrared power emission P per unit area is approximately given by the formula:
P = Ft Cb T^4
where:
P = average infrared radiant power emission per unit area.
and
T = absolute emission temperature
and
Cb = 5.6697 X 10^-8 W m^-2 K^-4
= the Stefan-Boltzmann constant.
and
Ft = Earth's far infrared emissivity
where:
0 < Ft < 1

The emissivity Ft is a function of the Earth's atmospheric gas mixture and is typically in the range 0.7 to 0.9. The Earth's emissivity is dependent on the upper atmospheric concentrations of H2O, CO2, O3 and CH4.

Define:
Ho = Solar irradiance
= (total radiant power emitted by the sun) / [4 Pi (Earth orbit radius)^2]
= 1367 W / m^2,
which calculation is valid because the Earth's orbit around the sun is nearly circular. The solar irradiance has been accurately measured by numerous artificial satellites over a 50 year period and is quite stable.

Define:
Po = average radiant solar power per unit of Earth surface area incident upon the Earth.
Re = Earth radius

Due to the Earth's nearly spherical shape:
Po = Ho Pi Re^2 / (4 Pi Re^2)
= Ho / 4
= (1367 W / m^2) / 4
= 341.75 W / m^2

The average solar power per unit area absorbed by the Earth is:
Po (1 - Fr)
where:
Fr = fraction of incident solar energy reflected off the Earth
= Bond Albedo for the entire Earth under solar illumination

Published astronomical data, believed to be reliable, indicates that the bond albedo of the Earth, as measured by comparing the reflection of Earthshine off the dark side of the moon to the reflection of direct solar illumination off the moon, has decreased from 0.31 in the mid 1990s, to 0.297 in 1998-1999 to 0.290 today in 2015.

The law of conservation of energy gives the differential equation for the Earth's change in total thermal energy E as a function of time t as:
(dE / dt) / (4 Pi Re^2)
= (average solar radiative absorbed power per unit area) + (average internally generated thermal power per unit area) - (average infrared radiated emitted power per unit area)
= Po (1 - Fr) + Pn - Ft Cb T^4

At steady state the Earth is neither net absorbing nor net emitting energy, giving:
dE / dt = 0

Numerical evaluation shows that Pn is four orders of magnitude smaller than the other terms. Thus:
Pn << Po (1 - Fr)
and
Pn << Ft Cb T^4

Hence for the purposes of this web page, at steady state:
Po (1 - Fr) - Ft Cb T^4 = 0
or
[(1 - Fr) / Ft] = (Cb / Po) T^4

In this equation Fr is a function of T and Ft is a function of atmospheric CO2 concentration. This web page is primarily concerned with finding solutions to this equation. If the ratio [(1 - Fr) / Ft] was constant independent of temperature T there would be a unique solution for T and there would be no thermal runaway. However, in reality due to the strong temperature dependence of Fr near:
T = To
= 273.15 K
= the freezing point of water
,
there are two real locally stable solutions for T and one real unstable solution for T.

At high temperatures [T > (To + 10 K)]:
Fr ~ 0.1
giving:
{d[(1 - Fr) / Ft] / dT} < 4 (Cb / Po) T^3

At low temperatures [T < To - 20 K]:
Fr ~ 0.5
giving:
{d[(1 - Fr) / Ft] / dT} < 4 (Cb / Po) T^3

If at moderate temperatures:
{d[(1 - Fr) / Ft] / dT} > 4 (Cb / Po) T^3
then T has three real values, two of which are locally stable and are known as the "cool" state temperature Tc and the "warm" state temperature Tw. The unstable solution at T = Tr lies between the two stable solutions and is referred to herein as the thermal runaway temperature.
 

NON-STEADY STATE:
If:
(Cb / Po) T^4 > [(1 - Fr) / Ft]
the emitted infrared radiation is greater than the absorbed solar radiation, the Earth is loosing net energy and the emission temperature T is gradually decreasing over time at a rate proportional to the power per unit area difference:
[Po (1 - Fr) - Ft Cb T^4] and the effective heat capacity per unit area.

If:
(Cb / Po) T^4 < [(1 - Fr) / Ft]
the emitted infrared radiation is less than the absorbed solar radiation, the Earth is gaining net energy and the emission temperature T is gradually increasing over time at a rate proportional to the power per unit area difference:
[Po (1 - Fr) - Ft Cb T^4] and the effective heat capacity per unit area.
 

STEADY STATE:
If:
(Cb / Po) T^4 = [(1 - Fr) / Ft]
the Earth is neither losing nor gaining net energy and the emission temperature T is constant. This is the steady state condition.
 

EARTH CLIMATE STATUS:
At any instant in time the Earth has a characteristic value of infrared emission temperature T, which at steady state defines the present climate. At any instant in time the Earth has a characteristic value of (dT / dt), where t = time, which allows us to make limited projections regarding the climate in the future.

At steady state conditions (dT / dt) = 0 so we can reliably conclude that the future climate will be very similar to the present climate.

In circumstances in which there is known ongoing climate change an issue of great importance in public policy is projecting the future emission temperature at which the climate change will stop. Such projections can be made by taking advantage of known mathematical stability criteria. These same stability criteria can be used to warn us when the Earth is approaching a dangerously unstable situation, such as triggered the PETM.
 

GRAPHICAL PORTRAYAL OF THE EARTH'S CLIMATE AND CLIMATE CHANGE STATUS:
A useful technique for portraying the climate and climate change status of the Earth at a particular time is to plot:
(Cb / Po) T^4 versus T with a red line
and on the same graph to plot:
[(1 - Fr) / Ft] versus T with a blue line.

At any instant in time the Earth will be at a point on the red line defined by the emission temperature T at that time. However, the rate of movement left or right along the red line is proportional to the thermal power difference:
[Po (1 - Fr) - (Ft Cb T^4)]
which is simply (Po Ft) times the amplitude difference between the blue line and the red line. Points where the blue and red lines cross are solutions for T that should be individually examined for conformance with stability criteria.
 

STABILITY CRITERIA:
At a locally stable state:
[(1 -Fr) / Ft] = (Cb / Po) T^4
and
{d[(1 -Fr) / Ft] / dT} < 4 (Cb / Po) T^3

At a locally stable state if the temperature T is too high then:
(Cb / Po)T^4 > [(1 -Fr) / Ft]
and the Earth spontaneously loses energy to return to the locally stable temperature.

Similarly at a locally stable state if the temperature T is too low then:
(Cb / Po)T^4 < [(1 -Fr) / Ft]
and the Earth spontaneously gains energy to return to the locally stable temperature.

At an unstable state if:
(Cb / Po) T^4 = [(1 - Fr) / Ft]
but
{d[(1 - Fr) / Ft] / dT } > 4 (Cb / Po) T^3
then temperature T is unstable enabling a spontaneous state change.

At the locally unstable state if the temperature T drifts too high then due to:
[(1 -Fr) / Ft] > (Cb / Po) T^4
the Earth gains more energy causing T to drift even further away from the point of instability Tr.

Similarly at the locally unstable state if the temperature T drifts too low then due to:
[(1 -Fr) / Ft] < (Cb / Po)T^4
the Earth loses more energy causing T to drift even further from the point of instability. Thus at the unstable temperature Tr the Earth switches from one locally stable state to the other locally stable state.
 

IMPORTANT EXPERIMENTAL DATA:

Note that in the normal "cool" state the experimentally measured value of Fr early in the 1990s was 0.31 and during 1998 - 1999 was:
Fr = 0.297 +/- .005
Reference: Earthshine Observations of the Earth's Reflectance

More recent astronomical data indicates that currently Fr = 0.29. Reference:Hyperphysics

The problem with direct astronomical measurements of Fr is that the error bars slightly overlap, which introduces substantial uncertainty into calculations of the rate of change of Fr with time. There are small variations in solar output, hourly and seasonal changes in Earth reflectance and hour by hour variations in Earth atmospheric attenuation, all of which must be eliminated by averaging over sufficient time. The problem is further complicated by the necessity to ensure that the angles of incidence and viewing of the solar illumination on the moon are close to the angles of incidence and viewing of the Earthshine. Typically an accurate measurement of Fr takes about two years to perform.

Satellites can effectively measure local albedo but satellites are too close to the Earth to directly and accurately measure bond albedo Fr. Values of Fr derived from satellite data involve so many assumptions that the results have little useful value. The bond albedo of the Earth could potentially be directly measured by an interplanetary space vehicle. However, the cost of such a space vehicle measurement to three significant figures would be several billion US dollars and would need to be repeated annually to be useful.

The normal "cool" state emission temperature Tc experimentally measured in November 1996 by the Mars Global Surveyor space craft was Tc = 270.0 K.

Reference: Thermal Emission Spectrometer

Reference: Initial Data from the Mars Global Surveyor Thermal Emission spectrometer Experiment: Observations of the Earth

Reference: Initial Data from the Mars Global Surveyor thermal emission spectrometer experiment-Observations of the Earth by Philip R. Christensen and John C. Pearl.

This emission temperature appears to be uniform over the Earth's surface, indicating that the dominant source of the emitted infrared radiation is from the liquid-solid phase transition of water in the Earth's atmosphere which occurs at a uniform temperature, independent of latitude and almost independent of altitude.

The triple point of water is at:
273.16 degrees K at a ambient pressure of 6 millibars.

The freezing point of water is at:
273.15 degrees K = 0.00 degrees C at an ambient pressure of one atmosphere (1 bar).

Hence at steady state the temperature at which dominant clouds change from being composed of microscopic ice crystals to being composed of water droplets is in the narrow range 273.15 K to 273.16 K and is almost independent of ambient pressure and hence of cloud altitude and cloud latitude. This issue is illustrated on the phase diagram shown in Figure 1.


Figure 1
 


 

The temperature difference (273.15 K - 270.0 K) = 3.15 K drives the infrared emission reaction forward at a rate sufficient to balance the absorbed solar radiation power.
 

PHOTON ENERGY LIBERATED BY LATENT HEAT OF FUSION OF WATER:
The latent heat of fusion of water is:
334 J / gm

One mole of water has a mass of 18 gm and contains 6.023 X 10^23 molecules.

Hence the energy per molecule liberated by the liquid-solid phase transition is:
334 J / gm X 18 gm / mole X 1 mole / 6.023 X 10^23 molecules = 998.17367 X 10^-23 J/ molecule

Assume that release of the latent heat of fusion on average liberates one photon per H2O molecule.
Then:
998.17367 X 10^-23 J = h F
where:
h = Planck Constant
= 6.62607004 10-34 m2 kg / s
and
F = radiation frequency
and
C = speed of light
= 3 X 10^8 m / s.

Then:
Wave Number = F / C
= 998.17367 X 10^-23 J / (h C)
= 998.17367 X 10^-23 J / (6.62607004 10-34 m2 kg / s X 3 X 10^8 m / s)
= 50.21 X 10^3 J s^2/ m^3 kg
= 50.21 X 10^3 kg m^2 s^2/ s^2 m^3 kg
= 50.21 X 10^3 / m
= 502.1 / cm

As shown on Figure 2, the experimenatlly recorded graph of infrared power emission versus wave number, 502.1 / cm is close to the center wave number of the far infrared emission from the Earth recorded by the thermal emission spectrometer carried by the Mars Global Surveyor space craft. Hence the dominant source of thermal infrared radiation emitted by the Earth in November 19996 was the liquid-solid phase change (freezing) of water micro-droplets in the Earth's atmosphere. This conclusion is consistent with the results of numerous other investigations of the interaction of water with infrared radiation.

References: Infrared Absorption By Water #1 and
Infrared Absorption By Water #2
 


Figure 2


Reference: Initial Data from the Mars Global Surveyor Thermal Emission spectrometer Experiment: Observations of the Earth
 

CALCULATE Ft:
Recall that:
[(1 - Fr)) / Ft] = (Cb / Po) T^4
or
Ft = [(1 - Fr) Po / (Cb T^4)]
giving the value of Ft in November 1996 as:
Ft = [(1 - Fr) Po / (Cb T^4)]
= [(1 - 0.297) (341.75 W / m^2)] / [(5.6697 X 10^-8 W / m^2-K^4 (270.0 K)^4)]
= 0.7973494737
 

CRITERIA FOR THE EXISTENCE OF TWO LOCALLY STABLE EMISSION TEMPERATURE STATES:
Recall that the criteria for the potential existence of two locally stable temperature states is:
{d[(1 - Fr) / Ft] / dT} > 4 (Cb / Po) T^3

The freezing point of water is at 273.15 K.

Numerical evaluation of the right hand side of this inequality at 273.15 K gives:
4 (Cb / Po) T^3 = 4 [(5.6697 X 10^-8 W /m^2-K^4)/(341.74 W / m^2)] [273.15 K]^3
= 0.01352431 / K

If Fr changes by - 0.2 over the temperature range:
2 (To - Tc)
then:
{d[(1 - Fr) / Ft] / dT} = (.2 / 0.7973494737) /( 2 (273.15 K - 270.0 K))
= 0.03981 / K

Hence the inequality:
{d[(1 - Fr) / Ft] / dT} > 4 (Cb / Po) T^3
is satisfied even if the change in albedo is only - 0.1 instead of - 0.2.

At the freezing point of open water the albedo decreases from about 0.5 for ice to as low as .035 for open sea water. For clouds the albedo change is less but clouds still satisfy the criteria:
{d[(1 - Fr) / Ft] / dT|T = Tr} > 4 (Cb / Po) Tr^3
which enables two separate locally stable temperature states.

Note that the rapid change in albedo with temperature occurs because the emission temperature of the Earth is nearly constant over a wide range of latitudes. The emission temperature is nearly constant because the infrared emission originates at points where liquid water freezes to form ice. These points form a uniform temperature shell around much of the Earth.

As the emission temperature T decreases below the "cool" state temperature Tc the ocean surface progressively freezes. Today the average ocean surface temperature is about 15 degrees C which is about 18 degrees K above the "cool" state emission temperature. Hence if the emission temperature fell more than 18 degrees K below Tc the ocean surface would freeze and the planetary albedo of the Earth would increase to about 0.5.

Hence as the emission temperature T rises from 230 K to 270 K the Bond albedo decreases from about 0.50 to about 0.30. At T = 273.15 K the bond albedo rapidly falls from 0.30 to 0.10 and then remains almost constant at higher temperatures.

Thus the change in bond albedo has two main step components. One step is associated with a solid-liquid phase change of melting ice on the ocean surface. The other step is associated with the solid-liquid phase change of ice micro-crystals in clouds. Typically the average temperature on the Earth's surface is about 18 degrees K higher than the Earth emission temperature. Hence, much of the ocean surface will be frozen at:
T = To - 18 K
= 252 K.
At T = 270 K only near the poles does floating ice remain on the Earth's surface. The ice micro-crystals in clouds change to liquid water at:
T = 273.15 K.

Note that Fr is an average of many individual local albedo Fri values. The local albedo Fri varies widely with position and time. If there is dense white cloud:
Fri ~ 0.5
whereas when there is no cloud:
over the ocean:
Fri = .035
and over dry land the average value of Fri is about:
Fri = 0.2576
 

GRAPHICAL SOLUTION FOR T:
The practical way to find solutions for T is to superimpose two graphs. Using a red line plot (Cb / Po) T^4 vs T. Using a blue line plot [(1 - Fr) / Ft] vs T for:
[dFr|T = 273.15 K] = - 0.200
and for
[dFr|T = 252 K] = - 0.200

Define:
To = 273.15 K
= freezing point of water

Toc = 270.0 K - 18.0 K
= 252.0 K

For T < Toc the blue line is given by:
[(1 - Fr) / Ft] = [(1 - .497) / 0.7973494737]
= 0.6308400728

For Toc < T < To the blue line is given by:
[(1 - Fr) / Ft] = [(1 - .297) / 0.7973494737]
= 0.8816711156

For T > To the blue line is given by:
[(1 - Fr) / Ft] = [(1 - .097) / 0.7973494737]
= 1.132502158

At the points where the red and blue graph lines intersect are steady state points at which the Earth's net energy change per unit time is zero. These points are at (248.5 K, 0.63084), (270.0 K, 0.88167) and (287.4, 1.13250). Thus:
Tc = 270.0 K
and
Tw = 284.4 K

The emission temperature rise due to thermal runaway is:
(Tw - Tc) = 284.4 K - 270.0 K
= 14.4 K

The choice of dFr = - 0.200 is based on a measured "cool" state value of:
[Fr|T = Tc] = 0.297
a known "frozen" state value of:
[Fr|T =Tf] = [Fr|T = Tc] + 0.2
= 0.497

and a known "warm" state value of:
[Fr|T = Tw] = [Fr|T = Tc] - 0.2
= 0.097

These Fr values come from measurements of planetary albedo, measurements of local albedo and solar illumination photographs of the Earth from space.

As shown on Figure 3, these superimposed plots intersect at up to five emission temperatures. The lowest of these emission temperatures is the locally stable frozen state temperature Tf. The middle emission temperature is the locally stable "cool" state temperature Tc. The highest of these emission temperatures is the locally stable "warm" state temperature Tw. The emission temperatures Tf, Tc and Tw are indicated on the graph by bold blue dots. In between the locally stable temperature states are unstable temperature states at 252 K and at 273.15 K which are indicated by vertical dotted blue lines. In this simple mathematical model the Earth emission temperature at commencement of thermal runaway is 273.15 K.
 


Figure 3
Note that if the blue lines rise with respect to the red line, as will happen with an increased atmospheric CO2 concentration that reduces Ft, then the only stable state is the "warm" state and the Earth reaches the "warm" state via thermal runaway.


The change in local albedo with temperature, the PETM O-18 / O-16 ratio data, the PETM C-13 / C-12 ratio data, and fossil data all confirm that the existence of two locally stable temperature states and hence thermal runaway.

We currently live in the "cool" state. The practical issues that we face are accurate determination of the atmospheric CO2 concentration at which thermal runaway will occur and the magnitude of the temperature rise (Tw - Tc) associated with thermal runaway.
 

DEFINITIONS:
The equation:
[(1 - Fr) / Ft] = (Cb / Po) T^4
has stable solutions Tc and Tw where:
Tc < Tr < Tw
where:
Tc = locally stable "cool" state emission temperature that in November 1996 was 270.0 K
Tr = thermal runaway emission temperature
Tw = locally stable "warm" state emission temperature
 

PARAMETER VALUES:

To = 273.15 degrees K

Tc|1996 = 270.0 degrees K

Cb = 5.6697 X 10^-8 W / m^2-K^4

Po = 341.75 W / m^2
 

ANALYTIC REPRESENTATION OF Fr:
In reality there is some scatter in local T values near To. Hence in a real situation the value of the slope:
{d[(1 - Fr) / Ft] dT|T = To}
is not infinite. The finite slope leads to:
(1 - Fr)
having a functional representation in the temperature range:
265 K < T < 295 K
of the form:
(1 - Fr) = [{([(T/To)^Kf] - 1) / (10([(T/To)^Kf] + 1))} + 0.803]
where:
To = 273.15 K
at which temperature the solar spectrum optical properties of water change rapidly with temperature;
and
Kf ~ 600
where Kf is constrained by [(1 - Fr) / Ft] function slope requirements at:
T = Tr,
and at:
T = Tc
and at:
T = Tw.

Note that Fr varies over the range Tc < T < Tw and takes nearly constant values for T < Tc and for T > Tw.

From a mathematical perspective, if:
[(1 - Fr) / Ft] = (Cb / Po) T^4
and if
{(d[(1 - Fr) / Ft] / dT)|T = Tr} > {(Cb / Po)[d(T^4) / dT]|T = Tr}
then T potentially has two locally stable real solutions. The temperature separation between these two solutions depends on the values of Fr at saturation. Saturation occurs because when the Earth is hot Fr will not fall below about:
Fr = 0.100
and when the Earth is cool Fr will not exceed about:
Fr = 0.30
until Earth is so cold that the ocean surface freezes.

Recall that:
(1 - Fr) = [{([(T/To)^Kf] - 1) / (10([(T/To)^Kf] + 1))} + 0.803]

If:
(T / To)^Kf << 1
then:
(1- Fr) = (-.100 + .803)
= 0.703
If Kf is sufficiently large to meet the boundary conditions on (dFr / dT) at T = Tc, then:
(1 - Fr) = 0.703

If:
(T / To) = 1
then:
(1- Fr) = (0.000 + .803)
= 0.803

If:
(T / To)^Kf >> 1
then:
(1 - Fr) = (.100 + .803)
= 0.903
 

DETERMINATION OF Ft:
At steady state conditions in the "cool: state:

{[(1 - Fr) / Ft]|T = Tc} = (Cb / Po) Tc^4
or
[{([(Tc / To)^Kf] - 1) / (10([(Tc / To)^Kf] + 1))} + 0.803] / Ft = (Cb / Po) Tc^4

At T = Tc the quantity:
[(Tc / To)^Kf] << 1
giving:
0.703 / Ft = (Cb / Po) Tc^4

However:
(Cb / Po) Tc^4 = [(5.6697 X 10^-8 W / m^2-K^4) / (341.75 W / m^2)][270.0 K]^4
= 0.8816711156

Thus at the stable "cool" state temperature Tc we have the equation:
[0.703 / Ft]|Tc = 0.8816711156
or
Ft = 0.703 / 0.8816711156
= 0.7973494737

Note that Ft is dependent on the atmospheric CO2 concentration. Changing the atmospheric CO2 concentration changes Ft which can cause a change in state.

Absent ocean warming the ratio [(1 - Fr) / Ft] varies from:
.703 / .7973494737 = 0.8816711156 in the "cool" state to:
0.903 / .7973494737 = 1.132502158 in the "warm" state.

The corresponding November 1996 graphs of [(1 - Fr) / Ft] versus T and [(Cb / Po) T^4] versus T are shown on Figure 4 for Ft = 0.7973494737. The "cool" state temperature was 270.0 K. The potential "warm" state temperature was 287.4 K. The atmospheric CO2 concentration was 360.76 ppmv.

NOVEMBER 1996

Figure 4
Note that in November 1996 the Earth was still in a safe stable state.
 

DETERMINATION OF Tw:
The warm state temperature Tw lies on the steady state line. Hence:
[{([(Tw / To)^Kf] - 1) / (10([(Tw / To)^Kf] + 1))} + 0.803] / Ft = (Cb / Po) Tw^4

However, in the warm state:
[(Tw / To)^Kf] >> 1
giving:
0.903 / Ft = (Cb / Po) Tw^4
or
0.903 / 0.7973494737 = (Cb / Po) Tw^4
which can be solved for Tw. Note that the "warm" state temperature Tw varies slightly with variations in Ft.
 

PRE-INDUSTRIAL, PRESENT AND FUTURE GRAPHS:
The corresponding pre-industrial, present and future graphical solutions can be found by using the relevant atmospheric CO2 concentrations to find the relevant Ft values.

Define:
Ta = steady state emission temperature at recovery from thermal runaway at the atmospheric CO2 concentration Pa;
Tb = steady state emission temperature at pre-industrial atmospheric CO2 concentration of Pb = 280 ppmv;
Tc = 270.0 K = measured emission temperature at the November 1996 atmospheric CO2 concentration of Pc = 360.76 ppmv;
Td = emission temperature at the 2014 atmospheric CO2 concentration of Pd = 400 ppmv;
Tr = 271.50 K = thermal runaway temperature at the atmospheric CO2 concentration of Pr;
To = 273.15 K = freezing point of water;
Te = (Tc + 4.015 K)
= (270.0 K + 4.015 K)
= 274.015 K
= theoretical emission temperature at an atmospheric CO2 concentration of 721.52 ppmv if there is no change in planetary albedo Fr.
Reference: GLOBAL WARMING
P = atmospheric CO2 concentration at steady state temperature T
Pa = atmospheric CO2 concentration at recovery from thermal runaway T
Pr = atmospheric CO2 concentration at thermal runaway trip point
Pc = atmospheric CO2 concentration at temperature Tc in the "cool" state in November 1996.
Pe = 2 Pc
Ft = emissivity
Fta = emissivity at atmospheric CO2 concentration Pa = 280 ppmv;
Ftc = emissivity at atmospheric CO2 concentration Pc = 360.76 ppmv;
Ftd = emissivity at atmospheric CO2 concentration Pd = 400 ppmv;
Ftr = emissivity at atmospheric CO2 concentration Pr;
Fte = emissivity at atmospheric CO2 concentration Pe = 2 Pc = 721.52 ppmv;

Note that:
Ta < Tb < Tc < Td < Tr < To < Te
 

CO2 PRESSURE DEPENDENCE OF Ft:
Note that for T in the range:
Ta <= T <= Tc
Fr is constant.

At steady state conditions in circumstances of constant Fr:
Ftc Cb Tc^4 = Ft Cb T^4

Hence:
Ftc / Ft = (T / Tc)^4

We know that for:
Tc = 270.0 K, Pc = 360.76 ppmv
and for:
Pe = 2 Pc
= 2 (360.76 ppmv)
= 721.52 ppmv

Te = (270.0 K + 4.015 K) = 274.015 K

Note that if the Earth's atmospheric CO2 concentration doubles there is about a 6.0% increase in Ft which causes an emission temperature increase is about 4.015 K. Reference: GLOBAL WARMING.

Hence:
(Te / Tc) = (274.015 / 270.0)
= 1.01487037
= (Pe / Pc)^(Ki)

Thus:
(Pe / Pc) = (Te / Tc)^(1 / Ki)
or
(Pe / Pc) = [(Ftc / Fte)^0.25]^(1 / Ki)

Ln(Pe / Pc) = Ln[(Ftc / Fte)^(1 /4 Ki)]
= (1 / 4 Ki) Ln[Ftc / Fte]
= (1 / 4 Ki) Ln[(Te / Tc)^4]
= (1 / Ki) Ln[(Te / Tc)]

Thus:
Ln(2) = (Ln[(274.015 /270.0)] / Ki)
or
Ki =[1 / (Ln(2))]Ln[(274.015 /270.0)]
= [1 / 0.6931471806][.0147608901]
= 0.0212954629

In general:
Ln(P / Pc) = (1 / 4 Ki) Ln[Ftc / Ft]
or (P / Pc) = Exp{(1 / 4 Ki) Ln[Ftc / Ft]}
= Exp{[1 / (4 X 0.0212954629)] Ln[Ftc / Ft]}
= Exp{11.73958984 Ln[Ftc / Ft]}

Hence:
P = Pc Exp{11.73958984 Ln[Ftc / Ft]}
which equation relates CO2 partial pressure P to emissivity Ft under circumstances of constant bond albedo Fr.

This equation can be rearranged to give:
Ln(P / Pc) = {11.73958984 Ln[Ftc / Ft]}
or
Ln[Ftc / Ft] = [Ln(P / Pc)] / 11.73958984
or
Ftc / Ft = Exp{[Ln(P / Pc)] / 11.73958984}
or
Ft = Ftc / Exp{[Ln(P / Pc)] / 11.73958984}
 

FIND Ftb = PRE-INDUSTRIAL VALUE OF Ft:
For Ftc = 0.7973494737, Pc = 360.76 ppmv, Pb = 280 ppmv

Ftb = Ftc / Exp{[Ln(Pb / Pc)] / 11.73958984}
= 0.7973494737 / Exp{[Ln(280 / 360.76)] / 11.73958984}
= 0.7973494737 / 0.978644266
= 0.8147490374

For a the pre-industrial atmospheric CO2 concentration of 280 ppmv:
Ft = 0.81475
provides a "cool" state temperature:
Tc = 268.7 K
and for a 0.20 reduction in Fr with increasing temperature, provides a corresponding locally stable "warm" state temperature:
Tw = 286.0 K.

The corresponding graphs of [(1 - Fr) / Ft] versus T and [(Cb / Po) T^4] versus T are shown on Figure 5.

PRE-INDUSTRIAL

Figure 5
Until the 20th century the Earth was thermally stable at a "cool" state infrared emission temperature of 268.7 K.
 

FIND Ftd - 2014 VALUE OF Ft:
For Ftc = 0.7973494737, Pc = 360.76 ppmv, Pd = 400 ppmv
Ftd = Ftc / Exp{[Ln(Pd / Pc)] / 11.73958984}
= 0.7973494737 / Exp{[Ln(400 / 360.76)] / 11.73958984}
= 0.7973494737 / 1.008833956
= 0.7903674028

The Earth reached this Ft value in 2014.
The corresponding Tc and Tw values are:
Tc = 270.66 K
and
Tw = 288.16 K

As shown in Figure 6 in 2014 the atmospheric CO2 concentration was just below the threshold for commencement of thermal runaway.
 


 

2014

Figure 6
 

FIND THE ATMOSPHERIC CO2 CONCENTRATION Pr AT COMMENCEMENT OF THERMAL RUNAWAY:
For Ftc = 0.7973494737 and Ftr = 0.7850

Recall that:
Pr = Pc Exp{11.73958984 Ln[Ftc / Ftr]}
= 360.76 ppmv Exp{11.73958984 Ln[.7973494737 / .7850]}
= 433.312 ppmv

As shown on Figure 7 at Ftr = 0.7850 there is a spontaneous transition from the "cool" state to the "warm" state known as thermal runaway. The only stable temperature is in the "warm" state at T = 288.7 K.

THERMAL RUNAWAY

Figure 7
The Earth is projected to reach the calculated thermal runaway atmospheric CO2 concentration by the year 2027.

Whether or not thermal runaway is avoided will depend on atmospheric CO2 concentration reductions actually achieved in the very near future and on possible small errors in the experimental measurements of Tc and Fr.
 

DETERMINATION OF Tr:
The instability temperature at T = Tr lies on the steady state line. Hence:
{[(1 - Fr) / Ft]|T = Tr} = (Cb / Po) Tr^4
or
[{([(Tr / To)^Kf] - 1) / (10([(Tr / To)^Kf] + 1))} + 0.803] / Ft = (Cb / Po) Tr^4

The practical way to solve this equation for Tr is graphically. The above graphs indicate a thermal runaway temperature of 271.5 K.

Note that under an accurate calculation the emission temperature:
Tr ~ 271.5 K
at which thermal runaway commences is significantly less than:
To = 273.15 K.

Hence the CO2 partial pressure at which thermal runaway commences is substantially less than is indicated by a simple calculation based on the assumption that thermal runaway commences at emission temperature:
T = To
= 273.15 K
.

Examination of the above graphs shows that any further reduction in the value of Ft caused by increasing the atmospheric CO2 concentration will trigger thermal runaway.
 

FIND ATMOSPHERIC CO2 CONCENTRATION Pa REQUIRED TO RECOVER FROM THERMAL RUNAWAY:
Increasing the value of Ft to about 0.95 would enable a "warm" state to "cool" state transition.

For Ftc = 0.7973494737 and Fta = 0.95

Recall that:
Pa = Pc Exp{11.73958984 Ln[Ftc / Fta]}
= 360.76 ppmv Exp{11.73958984 Ln[.7973494737 / .95]}
= 46.15 ppmv

This is the low atmospheric CO2 concentration that must be achieved for the Earth to recover from the "warm" state.

As shown on Figure 8 at Ft = 0.95 the only stable temperature is at Tc = 258.7 K and there is a spontaneous transition from the "warm" state to the "cool" state.
 

RECOVERY FROM THERMAL RUNAWAY

Figure 8


Note that at commencement of thermal runaway Ft = 0.785 and at recovery from thermal runaway Ft = 0.95. Achieving this Ft change requires about a (Pr / Pa) ratio of about:
(Pr / Pa) = (433 / 46.15)
= 9.4 fold
change in atmospheric CO2 concentration. The recovery process is aided by a companion reduction in atmospheric water vapor concentration which also affects Ft. Even so, recovery from thermal runaway will require hundreds of thousands of years for photosynthesis to convert CO2 that has accumulated in the ocean-atmosphere pool back into fossil carbon.

It is theoretically possible that the change in Fr is only - 0.10 instead of - 0.20. That reduced value of dFr would cause proportionately more closely spaced Ft extremes and would proportionately reduce the temperature difference (Tw - Tc). The corresponding change in atmospheric CO2 concentration required to recover from thermal runaway would be about 3 fold.
 

DERIVATIVE PROPERTIES:

Note that:
Tc < Tr < To < Tw

{d[(1 - Fr) / Ft] / dT|T = Tc} < 4 (Cb / Po) Tc^3
This inequality sets a minimum value on Kf

{d[(1 - Fr) / Ft] / dT|T = Tr} > 4 (Cb / Po) Tr^3
This inequality further constrains Kf

{d[(1 - Fr) / Ft] / dT|T = Tw} < 4 (Cb / Po) Tw^3
This inequality sets another minimum value on Kf

If Kf is sufficiently large to meet the derivative constraints on (dFr / dT) then:
(Tc / To)^Kf << 1
and at T = Tc:
(1 - Fr) = 0.703

At T = Tc and at T = Tw the large value of Kf established by the derivative constraints forces:
(dFr / dT) = 0
 

FIND d[(1 - Fr) / Ft] / dT
Recall that:
[(1 - Fr) / Ft] = [{([(T / To)^Kf] - 1) / (10([(T / To)^Kf] + 1))} + 0.803] / Ft

d[(1 - Fr) / Ft] / dT
= (1 / Ft){(10([(T / To)^Kf] + 1)) Kf [(T / To)^(Kf-1)](1 / To) - ([(T / To)^Kf] - 1) 10 Kf [(T / To)^(Kf-1)](1 / To)}
/ (10([(T / To)^Kf] + 1))^2
 
= (1 / Ft){(10([(T / To)^Kf] + 1)) Kf [(T / To)^Kf](1 / T) - ([(T / To)^Kf] - 1) 10 Kf [(T / To)^Kf](1 / T)}
/ (10([(T / To)^Kf] + 1))^2
 
= (1 / Ft){ Kf [(T / To)^Kf](1 / T) + 10 Kf [(T / To)^Kf](1 / T)}
/ (10([(T / To)^Kf] + 1))^2
 
= (1 / Ft){11 [(T / To)^Kf](Kf / T)} / (10([(T / To)^Kf] + 1))^2
 

FIND {d[(1 - Fr) / Ft] / dT|T = To}
{d[(1 - Fr) / Ft] / dT|T=To} = (1 / Ft){11 [(T / To)^Kf](Kf / T)} / (10([(T / To)^Kf] + 1))^2
= (1 / Ft){11 [1](Kf / To)} / (10([1] + 1))^2
= [(11 Kf) / (400 To Ft)]
 

FIND {d[(1 - Fr) / Ft] / dT|T = Tc}
{d[(1 - Fr) / Ft] / dT|T=Tc} = (1 / Ft){11 [(T / To)^Kf](Kf / T)} / (10([(T / To)^Kf] + 1))^2
= (1 / Ft){11 [(Tc / To)^Kf](Kf / Tc)} / (10([(Tc / To)^Kf] + 1))^2
= (1 / Ft){11 [0](Kf / Tc)} / (10([0] + 1))^2
= 0
 

FIND {d[(1 - Fr) / Ft] / dT|T = Tw}
{d[(1 - Fr) / Ft] / dT|T=Tw} = (1 / Ft){11 [(T / To)^Kf](Kf / T)} / (10([(T / To)^Kf] + 1))^2
= (1 / Ft){11 [(Tw / To)^Kf](Kf / Tw)} / (10([(Tw / To)^Kf] + 1))^2
= (1 / Ft){11 (Kf / Tw)} / (100 [(Tw / To)^Kf]])
= 0
 

FIND {d[(1 - Fr) / Ft] / dT|T = Tr}
{d[(1 - Fr) / Ft] / dT|T=Tr} = (1 / Ft){11 [(T / To)^Kf](Kf / T)} / (10([(T / To)^Kf] + 1))^2
= (1 / Ft){11 [(Tr / To)^Kf](Kf / Tr)} / (10([(Tr / To)^Kf] + 1))^2

Recall that:
{d[(1 - Fr) / Ft] / dT|T = Tr} > 4 (Cb / Po) Tr^3
giving:
(1 / Ft){11 [(Tr / To)^Kf](Kf / Tr)} / (10([(Tr / To)^Kf] + 1))^2 > 4 (Cb / Po) Tr^3
or
{(11 Kf / Ft){[(Tr / To)^Kf]} / (10([(Tr / To)^Kf] + 1))^2} > 4 (Cb / Po) Tr^4
which is an important constraint on the choice of Kf.
 

NUMERICAL EVALUATION OF Kf:
Recall that:
{d[(1 - Fr) / Ft] / dT|T=To} = [(11 Kf) / (400 To Ft)]

Recall that:
{d[(1 - Fr) / Ft] / dT|T=To} > 4 (Cb / Po) To^3 = 0.013 / K

Hence:
[(11 Kf) / (400 To Ft)] > 0.013 / K
or
Kf > (0.013 / K)(400 To Ft) / 11
or
Kf > 129.1

In practice to simultaneously meet all the boundary conditions Kf ~ 600.
 

GRAPH DISCUSSION REVIEW:
1) At any instant in time the emission temperature of the Earth is represented by a point on the red line.

2) The red line is defined by the equation:
[(1 - Fr) / Ft] = (Cb / Po) T^4

3) Everywhere on the red line the slope of the red line is given by:
4 (Cb / Po) Ta^3

4) In order for a stable "cool" state to exist at T = Tc:
{[(1 - Fr) / Ft]|T = Tc} = (Cb / Po) Tc^4
and
{(d[(1 - Fr) / Ft] / dT)|T = Tc} < 4 (Cb / Po) Tc^3

5) In order for the point of instability to exist at T = Tr:
{[(1 - Fr) / Ft]|T = Tr} = (Cb / Po) Tr^4
and
{(d[(1 - Fr) / Ft] / dT)|T = Tr} > 4 (Cb / Po) Tr^3

6) In order for a stable warm state to exist at T = Tw:
{[(1 - Fr) / Ft]|T = Tw} = (Cb / Po) Tw^4
and
{(d[(1 - Fr) / Ft] / dT)|T = Tw} < 4 (Cb / Po) Tw^3

7) In order for Tc and Tw to be separate:
{[d[(1 - Fr) / Ft] / dT]| Tr} > 4 (Cb / Po) Tr^3

8) The steep slope of [(1 - Fr) / Ft] at To, the freezing point of water, together with the low slopes of [(1 - Fr) / Ft] at temperatures Tc and Tw that are respectively below and above the freezing point of water causes formation of a stable "cool" state at:
T = Tc
below the freezing point of water and a "warm" state at:
T = Tw
above the freezing point of water.

9) At temperatures far from the freezing point of water Fr is nearly constant so that [(1 - Fr) / Ft] is nearly constant.

10) As the ocean temperature increases the ocean emits CO2 causing Ft to gradually decrease. Thus at steady state at temperatures far from the freezing point of water as T increases the ratio:
[(1 - Fr) / Ft]
gradually increases. However, close to the freezing point of water:
[(1 - Fr) / Ft]
rapidly increases.

11) The intersection point between the blue line and the red line at Tc = 270.0 K was measured by the Mars Global Surveyor spacecraft in November 1996. The corresponding value of planetary albedo Fr that was measured during the period 1998 to 2000 was:
Fr = 0.297.

Applying the formula:
(Cb / Po) T^4 = [(1-Fr) / Ft]
gives:
[Ft|Tc = 270.0 K] = (1 - Fr) Po / (Cb Tc^4)
= [(1 - 0.297) (341.75 W / m^2) / ((5.6697 X 10^-8 W / m^2-K^4) X (270.0 K^4))]
= 0.7973494737

12) The corresponding value of [(1 - Fr) / Ft] at Tc = 270.0 K is given by:
[(1 - Fr) / Ft] = [(1 - .297) / 0.7973494737]
= 0.8816711156

13) The freezing point of water at 273.15 degrees K is accurately known and is almost independent of pressure and hence is independent of the altitude of the emitting liquid-solid transitioning water.

14) At steady state the formula:
(Cb / Po) T^4 = [(1 - Fr) / Ft]
implies that at steady state temperature T as compared to steady state temperature Tc:
(T^4 / Tc^4) = {[(1 - Fr) / Ft]|T} / {[(1 - Fr) / Ft]|Tc}
or
{[(1 - Fr) / Ft]|T} = {[(1 - Fr) / Ft]|Tc}(T^4 / Tc^4)

Thus for T = Tr and Tc = 270.0 K:
{[(1 - Fr) / Ft]|Tr = {[(1 - Fr) / Ft]|Tc}(Tr^4 / Tc^4)
= 0.8816711156 (Tr / 270.0)^4

15)For a decrease in Fr of 0.200 between T = Tc and T = Tw:
{[(1 - Fr) / Ft]|Tw} = {[(1 - Fr) / Ft]|Tc}(Tw^4 / Tc^4)
or
{[(1 - Fr) / Ft]|T = Tw} = [(1 - .297) / 0.7973494737] [Tw / 270.0 K]^4
= 1.132502158

Hence:
Tw^4 = [270.0 K]^4 [0.7973494737 / (1 - .297)] [1.132502158]
or
Tw = [270.0 K] {[0.7973494737 / (1 - .297)] [1.132502158]}^0.25
= 287.44 K
 

Some of the aforementioned topics are detailed below.
 

INFRARED RADIATION:
The Earth, as viewed from outer space on November 24, 1996, appears to be a uniform 270.0 degree K black body covered by an upper atmosphere with prominent far infrared emission "absorption bands" due to atmospheric H2O, CO2 and O3. Along with these prominent "absorption bands" are lesser "absorption bands" due to other gases such as CH4 and N2O.

The thermal infrared radiation that is emitted by the Earth into outer space primarily originates at altitudes:
A ~ Af
where Af is the altitude at which the liquid-solid phase change of water occurs. In spite of the wide range of Earth surface temperatures, for radiation frequencies outside of the upper atmosphere "absorption bands" the emission temperature of the infrared radiation emitted by the Earth is almost constant at 270.0 K.

Note that the altitude of the 270.0 K infrared emission is generally higher at low latitudes than at high latitudes due to variations in Earth surface temperature.

It should be noted that the upper atmosphere "absorption bands" are really frequency bands in which radially propagating far infrared radiation is scattered in random directions. However, from the perspective of a distant observer in outer space these bands appear to be "absorption bands".
 

PHYSICAL EXPLANATION:
Consider a vertical column of air at a particular latitude. At sea level, due to solar energy absorption by the ocean, water evaporates. The latent heat of vaporization is expressed by water molecules as linear and rotational kinetic energy. As a water vapor molecule diffuses up the air column this kinetic energy is gradually transferred to N2 and O2 molecules in the air via molecular collisions. The resulting nearly stationary water molecules clump together (condense) to form liquid water micro-droplets.

In that column of air energy is lost out the top of the column via emission of infrared radiation into space so the air temperature in the column decreases with increasing altitude A. At altitude:
A = Af
in the air column the local air temperature is:
273.15 degrees K = To,
= the freezing point of water.

At:
A > Af
liquid water micro-droplets in the air column form ice micro-crystals by emission of infrared radiation photons which carry into outer space the latent heat of fusion of the H2O. This photon emission occurs because the infrared emissivity of H2O near 273.15 K (its freezing point) is very high. Near this temperature there is strong coupling between H2O molecular vibrations and infrared radiation.

After some time the resulting ice micro-crystals fall and at:
A < Af
the ice micro-crystals acquire sufficient kinetic energy from N2 and O2 molecules in the air column to become liquid water micro-droplets.

These micro-droplets may again diffuse up the air column and emit infrared photons into space. This infrared energy emission process continues as long as there is heat available to drive it.

The number of H2O molecules per unit volume in the air column decreases exponentially with increasing altitude. Hence as altitude Af increases due to an increasing Earth emission temperature the number of H2O molecules in the column of air that exist as ice micro-crystals decreases.

Ice micro-crystals reflect solar radiation back into space much more efficiently than do liquid water micro-droplets. Thus as the Earth's emission temperature increases the number of ice micro-crystals reflecting solar radiation back into space decreases and hence the Earth's bond albedo decreases.

Visible light photographs of the Earth taken from deep space clearly show that the local albedo of the Earth is much less in the tropics where the atmosphere is warm than near the poles where the atmosphere is cooler. These photographs show that the bond albedo of the whole Earth could potentially decrease from about 0.30 to about 0.10 due to an increasing Earth emission temperature. If the infrared emissivity of the Earth sufficiently decreases due to an increase in Earth's atmospheric CO2 concentration then all the requirements for thermal runaway are met.
 

MEASUREMENTS MADE OF NOVEMBER 24, 1996:
The emission temperature Tc of the Earth in its near normal "cool" state was obtained from a far infrared spectrum of the Earth recorded on November 24, 1996 by the Mars Global Surveyor spacecraft while on its way to Mars. The original data is shown on Figure 9.


Figure 9

Reference: Initial Data from the Mars Global Surveyor Thermal Emission spectrometer Experiment: Observations of the Earth


The red line on the above graph shows a theoretical black body (Ft = 1.0) emission curve for an emission temperature of 270.0 degrees K. The blue line on the above graph shows a corresponding theoretical black body emission curve for an emission temperature of 215.0 degrees K and is the effective temperature of the upper atmosphere in the radiation frequency band where CO2 most greatly attenuates infrared radiation emission originating in the lower atmosphere. The wiggling black line is the experimentally recorded infrared thermal emission spectrum of the whole Earth as viewed from deep space over Hawaii on a November 24, 1996. The infrared absorption bands due to H2O, CO2 and O3 are marked.

The frequency dependent emissivity Ft(w) is the ratio of:
(amplitude of black line) / (amplitude of red line red line).

The above graph shows that Ft(w) is strongly affected by the concentrations of H2O, CO2 and O3 in the Earth's atmosphere. Note that the main infrared emission is due to liquid water giving up its latent heat of fusion. This emission spectrum is further filtered by CO2, H2O and O3 molecules at higher altitudes.

The above graph can be numerically analyzed to find the dependence of Ft on the atmospheric CO2 concentration and on the atmospheric H2O concentration.

An atmospheric CO2 concentration measurement at Mona Loa, Hawaii, on November 24, 1996 indicated an atmospheric CO2 concentration of 360.76 ppmv.

The corresponding atmospheric temperature at the Earth's surface in Honolulu, Hawaii on November 24, 1996 was measured to be about:
76.9 F = 24.94 C
= 298.09 K
.
 

FEEDBACK:
As the atmospheric CO2 concentration increases Ft decreases which causes the steady state value of T to increase. At T = 273.15 degrees K microscopic ice crystals in clouds become liquid water which causes a large decrease in Fr which in turn causes a step increase in the steady state value of T. The increase in T tends to increase the atmospheric H2O concentration which further decreases Ft. With increasing ocean water temperature the partial pressure of CO2 over the ocean increases due to the temperature dependence of the chemical reaction:
Ca ++ + 2(HCO3)- = CaCO3 + H2O + CO2

The combination of these effects causes a further increase in the ratio:
[(1 - Fr) / Ft]
with increasing temperature at T = 273.15 deg K.
 

BOND ALBEDO Fr:
When the Earth is in its normal "cool" state the dominant clouds are mostly composed of microscopic ice crystals and part of the Earth's surface is ice covered. When the Earth is in its "warm" state the dominant clouds are mostly composed of microscopic water droplets and most of the Earth's surface is snow and ice free. In between these two extremes the Earth is in an unstable transition region. As the Earth's average emission temperature rises from below 273.15 K to above 273.15 K the dominant clouds change from being brilliant white to dark grey and surface snow and ice melts. The change in local albedo with temperature is readily apparent on visible light photos of the Earth from deep space, as shown on Figure 10 and Figure 11.


Figure 10



Figure 11

Note that the local albedo of the Earth is lower at low latitudes than at high latitudes.
 

In the normal "cool" state:
Fr = Frc.
In the "warm" state:
Fr = Frw.
Experimental measurements of the Earth's "cool" state planetary albedo Frc gave:
Frc = 0.31 during the early 1990s
and
Frc ~ 0.297 during 1998 and 1999.
Any measurement of Frc is of necessity a time averaged value. Due to cloud movement and seasonal changes in folliage reflectivity the instantaneous value of Frc varies from day to day and season to season.

At any instant in time the planetary albedo is an average of the local albedos over the Earth's sun facing surface. The local albedo at any particular geographical position varies with time due to foliage, surface water, snow, ice, cloud formation and cloud movement.

Note that the bond albedo Fr is an average of time varying local bond albedo values. Thus while Fr contributes to average ocean temperature it does not directly result in warm state trapping. Warm State trapping occurs when the ocean temperature is high enough to keep the equilibrium atmospheric CO2 concentration above the minimum level that must be achieved to execute a "warm" state to "cool" state transition.
 

ORIGIN OF Frc:
Frc = Fr|"cool" state
Data assumptions:
Fraction of Earth's sun facing surface covered by cloud = 0.50
Local albedo of normal "cool" state cloud = 0.50

Fraction of Earth's sun facing surface not covered by cloud = (1.0 - 0.5) = 0.5
Local albedo of ocean with no cloud cover = 0.035
Fraction of Earth's sun facing surface covered by ocean = 0.708
Average local albedo of land with no cloud cover = 0.2576
Average local albedo of Earth's sun facing surface not covered by cloud:BR> = 0.2576 (1 - .708) + .708 (.035) = 0.10

"Cool" state bond albedo Frc is given by:
Frc = 0.5 (0.5) + 0.5 (0.1)
= 0.25 + 0.05
= 0.30
(The measured value is 0.297)
 

ORIGIN OF Frw for dFr = -0.200:
Frw = Fr|"warm" state Data assumptions:
Fraction of Earth's sun facing surface covered by cloud = x
Local albedo of "warm" state cloud = 0.10

Fraction of Earth's sun facing surface not covered by cloud = (1.0 - x)
Local albedo of ocean with no cloud cover = 0.035
Fraction of Earth's sun facing surface covered by ocean = 0.708
Average local albedo of land with no cloud cover = 0.2576
Average local albedo of Earth's sun facing surface not covered by cloud:BR> = 0.2576 (1 - .708) + .708 (.035) = 0.10

"Warm" state bond albedo Frw is given by:
Frw = x (0.1) + (1 - x) (0.1)
= 0.10
The corresponding value of dFr is:
dFr = (Frw - Frc)
(0.1 - 0.3) = - 0.2
 

"WARM" STATE DATA:
Much of the experimental data relating to the "warm" state comes from mass spectrometer analysis of the world wide PETM sedimentary layer and contemporary fossils, both in oceans and on land.
 

THE TREND:
Combustion of fossil fuels injects transient CO2 into the Earth's atmosphere which reduces the Earth's thermal infrared emissivity Ft causing net energy absorption and eventually an increase in emission temperature. Increasing the emission temperature raises the saturation water vapor pressure which further decreases the average emissivity Ft and hence further increases the emission temperature. The increase in emission temperature further decreases the Earth's planetary albedo and hence causes more solar energy absorption.

During the 20th century mankind injected so much transient CO2 into the Earth's atmosphere via combustion of fossil fuels that the Earth's emissivity Ft significantly decreased causing net heat absorption which melted most of the floating polar ice. The transient atmospheric CO2 concentration is continuing to increase. Over time the ocean temperature will gradually rise causing an increase in the steady state atmospheric CO2 concentration. When a similar increase in atmospheric CO2 concentration occurred during the Paleocene-Eocene Thermal Maximum (PETM) the Earth was trapped in its "warm" state for over 200,000 years.

If we let thermal runaway commence we are condemning our children to thermal extinction. The ground level temperature rise corresponding to the thermal runaway emission temperature rise is larger than the adaption capability of large land animals..
 

THERMAL RUNAWAY:
As CO2 is added to the atmosphere the local emission temperature increases due to a decrease in local infrared emissivity Ft. However this temperature increase also causes a decrease in Bond albedo Fr. There is an atmospheric CO2 concentration of about 433.33 ppmv where this positive feedback effect triggers a spontaneous change of atmospheric state from the "cool" state to the "warm" state.

During thermal runaway emission temperature T will spontaneously rise due to the positive feedback driven changes in Fr. In order to keep the Earth out of the "warm" state fossil CO2 emissions must be stopped now. The biggest threat to mankind as a species is failure of major governments to recognize the scope of the thermal runaway danger and /or failure to act collectively now to prevent thermal runaway happening.

The atmospheric CO2 concentration required to trigger thermal runaway can be reached either by transient CO2 injection or by steady state CO2 accumulation in the ocean-atmosphere pool. In present circumstances the Earth is in accute danger of being trapped in the "warm" state.

The projected average emission temperature increase, if it is allowed to occur, will cause massive changes in the Earth's natural environment.

It is reasonable to assume that over many decades the 17.5 K increase in emission temperature will result in about a 17 degree K increase in average ocean temperature. Hence a 17.0 K increase in ocean temperature is assumed on the web page titled WARM STATE TRAPPING. This temperature increase quantitatively explains the behavior of the Earth during the PETM.

In thermal runaway the Bond albedo will spontaneously decrease and the Earth will switch from its locally stable "cool" state to its locally stable "warm" state. Once thermal runaway is seriously under way there is no practical way to prevent continuation of the state switching process. The "warm" state warms the ocean which emits CO2 and H2O and hence further reduces Ft, which tends to trap the Earth in the "warm" state.

If the quantity of CO2 in the ocean-atmosphere pool is sufficiently large, ocean warming will raise the steady state atmospheric CO2 concentration above the level which will trap the Earth in the "warm" state, even if combustion of fossil fuels is totally stopped. The ocean warming will continue in the "warm" state until stopped by a new steady state balance between solar energy absorption and infrared emission in the "warm" state.
 

CONSEQUENCE OF WARM STATE ON GROUND LEVEL TEMPERATURE:
Note that Tc = 270.0 K is the Earth's emission temperature as seen by an infrared spectrometer mounted in a distant space craft on November 24, 1996. The corresponding mean Earth surface temperature measured in Honalulu, Hawaii on November 24, 1996 was 76.9 F = 24.94 C = 298.09 K. A more typical average Earth surface temperature is 15 C = 288.15 K. Hence the average Earth surface temperature is about 18 K higher than the average emission temperature.

The average emission temperature of:
270.0 + 17.5 = 287.5 K
which likely occurred during the initial PETM transient, caused a daily average Earth surface temperature of:
287.5 K + 18 K = 305.5 K
= 32.35 C
This average Earth surface temperature is too hot for sustaining large animal life. This average temperature may have been raised a few more degrees C by an increased water vapor concentration in the upper atmosphere.
 

This web page last updated January 18, 2017.

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