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WARM STATE TRAPPING

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

INTRODUCTION:
An ocean temperature rise and/or a rise in the number of CO2 molecules in the ocean-atmosphere pool will cause an increase in the Earth's steady state atmospheric CO2 concentration. When the Earth's steady state atmospheric CO2 concentration exceeds about 433 ppmv the Earth's emission temperature T will spontaneously exceed 273.15 degrees K, the freezing point of water, and the Earth will become trapped in the "warm" state. In the "warm" state ocean heating adds yet more CO2 to the atmosphere causing the Earth to become more firmly trapped in the "warm" state.
 

ESCAPE FROM WARM STATE TRAPPING:
Once trapped in the "warm" state the atmosphere takes a very long time to return to the normal "cool" state because the return rate is limited by the rate at which photosynthesis removes CO2 from the ocean-atmosphere pool and forms fossil fuels and the rate at which exposed silicate rock converts to carbonate rock. Both of these processes are very slow.
 

OCEAN CO2 SOLUBILITY ISSUES:
There is a steady state balance between the number of CO2 molecules in the Earth's atmosphere and the number of CO2 molecules dissolved in the Earth's oceans. This steady state balance is ocean temperature dependent. The ocean temperature follows but lags far behind the average atmospheric temperature at the ocean surface.

If there is an increase in ocean temperature but no change in total number of CO2 molecules in the ocean-atmosphere pool there is a flow of CO2 from the ocean into the atmosphere which causes an increase in CO2 partial pressure in the atmosphere. This situation is made more complicated by the increase in emission temperature that occurs as a result of the increased CO2 concentration in the atmosphere.

If an increase in atmospheric CO2 concentration causes the emission temperature to exceed 273.15 degrees K there is a rapid drop in planetary albedo which causes a step increase in the Earth's emission temperature. After some time (many decades) the average ocean temperature follows the change in emission temperature.

The step change in emission temperature causes the existence of to two stable states, a normal "cool" state in which the emission temperature:
Tc < 273.15 K
and a "warm" state in which the emission temperature:
Tw > 273.15 K.

If while the Earth is in the normal "cool" state there is an injection of fossil CO2 into the Earth's atmosphere, which causes a transient increase in atmospheric CO2 concentration, most of the injected CO2 dissolves in the ocean with a time constant of about 41 years and a new steady state is established. If this CO2 injection raises the steady state atmospheric CO2 concentration from:
< 433 ppmv to > 433 ppmv
and hence causes emission temperature T to switch from:
T < 273.15 K
to
T > 273.15 K
the Earth will transition out of its normal "cool" state and into its "warm" state. In the "warm" state the ocean will warm up over decades further increasing the steady state atmospheric CO2 concentration and hence further increasing emission temperature T. A new steady state balance will be established with the Earth trapped in the "warm" state. Then the only way to achieve a "warm" state to normal "cool" state transition is to extract sufficient CO2 from the ocean-atmosphere pool by formation of fossil fuels to reduce the steady state atmospheric CO2 concentration to < 100 ppmv. This photosynthesis dependent CO2 extraction process can take hundreds of thousands of years.
 

TEMPERATURE DEPENDENT CO2 SOLUBILITY IN SEA WATER:
Sea water can be viewed as water in continuous contact with a very large supply of exposed marine CaCO3. Gaseous CO2 dissolves in sea water via the chemical reaction:
H2O + CaCO3 + CO2 = Ca(HCO3)^2 = Ca++ + 2 (HCO3)-
This chemical reaction is very temperature sensitive. As sea water temperature decreases the the reaction moves to the right. As sea water temperature increases the reaction moves to the left. In a steady state laboratory measurement with CO2 at a constant partial pressure above the sea water the amount of CO2 in solution at 0 degrees C is about 2.2 fold greater than at 24 degrees C and is about 4 fold greater than at 50 degrees C. Hence as the ocean temperature increases CO2 gas flows from the ocean into the atmosphere. As the ocean temperature decreases CO2 gas from the atmosphere is absorbed by the ocean. However, in the case of warming of the entire ocean the atmospheric CO2 partial pressure is not constant and hence changes in this partial pressure must be taken into account.
 

STEADY STATE MATHEMATICAL ANALYSIS:
Define:
N = total number of CO2 molecules in the Earth's ocean-atmosphere pool;
Na = [N|state "a"]
Nb = [N|state "b"]
Ps = steady state atmospheric CO2 partial pressure at sea level;
(Ps equivalent to 280 ppmv prior to the industrial revolution);
Psa = [Ps|state "a"]
Psb = [Ps|state "b"]
Ka Ps = number of CO2 molecules in the atmosphere;
Vw = volume of water in the oceans;
Ks = solubility of CO2 per unit volume in the ocean which is a function of ocean temperature;
Ps Ks Vw = number of CO2 molecules in solution in the oceans;
Then at steady state:
(N - Ka Ps) = Ps Ks Vw
or
Ps = N / (Ka + Ks Vw)
 

PETM ANALYSIS:
Define:
Too = average ocean water temperature;
Toa = [Too|state "a"]
Tob = [Too|state "b"]
t = time.

Now assume that due to sudden large scale combustion of biomatter and exposed fossil fuels N increases from Na to Nb. The consequent increase in average steady state ocean water temperature Too causes Ks to decrease from Ksa to Ksb. As a result the steady state atmospheric CO2 partial pressure Ps at sea level changes from Psa to Psb. Then:
(Na - Ka Psa) = Psa Ksa Vw
and
(Nb - Ka Psb) = Psb Ksb Vw

PETM isotope ratio measurements detailed on the PETM web page show that the ratio:
(Psb Ksb Vw) / (Psa Ksa Vw) = 1.1825

Hence at steady state during the PETM:
(Psb / Psa) = (Ksa / Ksb)(1.1825)

To a good approximation at steady state conditions:
(Ksa / Ksb) = (Tob / Toa)^Ko
where:
Ko = +ve constant to be determined.

Laboratory data relating to solubility of CO2 in sea water at a constant CO2 partial pressure shows that if condition "a" is 0 deg C = 273.15 K and condition "b" is 24 deg C = 297.15 K then:
(Ksa / Ksb) = 2.2.

(Ksa / Ksb) = (Tob / Toa)^Ko
or
(2.2) = (297.15 / 273.15)^Ko
or
Ln[2.2] = Ko Ln(297.15 / 273.15)
= Ko Ln(1.087863811)
or
Ko = Ln(2.2) / Ln(1.087863811)
= .7884573604 / 0.0842159669
= 9.362326284

In general: Ln(Ksa / Ksb) = 9.362326284 Ln(Tob / Toa)
or
(Ksa / Ksb) = Exp[9.362326284 Ln(Tob / Toa)]

For a 17 degree K ocean temperature rise from condition "a" at 4 degrees C (277.15 K) to condition "b" at 21 degrees C (294.15 K), such as occurred during the PETM:
(Tob / Toa) = (294.15 / 277.15) = 1.061338625
or
Ln(Tob / Toa) = Ln(1.061338625)
= (0.0595309652)

Hence:
(Ksa / Ksb) = Exp[9.362326284 Ln(Tob / Toa)]
= Exp[9.362326284 (0.0595309652)]
= Exp( 0.5573483199)
= 1.746036

PETM Data Analysis indicates that:
(Psb Ksb Vw) / (Psa Ksa Vw) = 1.14281

Hence:
(Psb / Psa) = 1.1825 (Ksa / Ksb)

Thus a 17 degree K ocean temperature rise during the PETM would cause:
(Psb / Psa) = (1.1825)[Ksa / Ksb]
= 1.1825 (1.7460)
= 2.0646

The minimum value of Psb necessary for thermal runaway with warm state trapping is 433 ppmv. Hence immediately prior to the PETM the steady state atmospheric CO2 concentration would have to have been less than 433 ppmv in order to prevent immediate thermal runaway and over:
433 ppmv / 2.0646 = 209.7 ppmv
in order to enable warm state trapping during the PETM.

In the intervening 55 million years some of the CO2 in the ocean-atmosphere pool converted into carbonate rock and fossil fuels so that the steady state value of Psa decreased to 280 ppmv by the commencement of the industrial revolution. However, due to fossil carbon injection the steady state value of Psa increased during the 20th century so that now it is now over 300 ppmv.

Hence an ocean temperature rise of ~ 17 degrees C, caused by:
dFr = -0.200
would trap the Earth in its "warm" state.

In summary, a large transient injection of CO2 can switch the Earth from its normal "cool" state to its "warm" state. If the CO2 injection is sufficiently large that with consequent ocean warming the steady state atmospheric CO2 concentration exceeds 433 ppmv the Earth will be trapped in the "warm" state. In that event the Earth can only extricate itself from the "warm" state via photosynthesis fossil fuel formation and carbonate rock formation processes that gradually extract CO2 from the ocean-atmosphere pool.
 

TRANSIENT INJECTED CO2 LIFETIME Tt:
Prior to the industrial revolution the sun constantly evaporated water from the ocean which contained CO2 in solution. Evaporation of this water constantly released CO2 gas to the atmosphere at a rate given by:
Ks Ps Fs (volumetric evaporation rate)
where the fraction Fs of dissolved CO2 in the evaporated water that enters the atmosphere is in the range:
0 < Fs < 1.

This CO2 gas flux is reabsorbed by precipitation and the ocean surface. At steady state conditions the net CO2 absorption rate is equal to the solar driven CO2 release rate. Injection of transient CO2 into the Earths atmosphere increases the CO2 partial pressure from its steady state value Ps to a new transient value Pt. However, at the commencement of a CO2 injection transient the amount of CO2 in the ocean is:
Ks Psa Vw.
The rate of emission of CO2 from the ocean remains at:
(Evaporation rate) Ks Psa Fs.
Hence the net flow of CO2 into the ocean becomes:
(Ks Fs) (Pt - Psa)(evaporation rate)

The rate of accumulation of CO2 in the atmosphere is:
Ka (dPt / dt).

The number of fossil CO2 molecules injected into the atmosphere per unit time is:
dN / dt.

Hence conservation of CO2 molecules gives:
dN / dt = Ka (dPt / dt) + Ks Fs (Pt - Psa)(evaporation rate)

If (dN / dt) = 0
and if
Psa = constant
then:
0 = Ka (d(Pt - Ps) / dt) + Ks Fs (Pt - Psa)(evaporation rate)
or
d(Pt - Psa) / dt = - (Ks Fs / Ka)(evaporation rate)(Pt - Psa)
which has solution:
(Pt - Psa) = [(Pt - Psa)|t=0] Exp[-t / Tt]
where:
(1 / Tt) = {(Ks Fs / Ka)(evaporation rate)}
or
Tt = Ka / [Ks Fs (evaporation rate)]
which allows calculation of the minimum lifetime of CO2 injected into the Earth's atmosphere by combusion of fossil fiuels.

Typically:
Ks Ps Vw = 33.35 Ka Ps
giving:
Ka / Ks = Vw / 33.35
Hence:
Tt = Ka / [Ks Fs (evaporation rate)]
= Vw / [33.35 Fs (evaporation rate)]

Numerical evaluation:
From the web page CARBON DIOXIDE
Vw = water volume of the oceans
= 1339.67 X 10^15 m^3
and
(water evaporation rate) = 0.981 X 10^15 m^3 / year
giving:
Tt = Vw / [33.35 Fs (evaporation rate)]
= 1339.67 X 10^15 m^3 / [33.35 Fs (.981 X 10^15 m^3 / year)]
= (40.948 / Fs) years

Recall that:
d(Pt - Ps) / dt = - (Ks Fs / Ka)(evaporation rate)(Pt - Ps)
= - (Pt - Ps) / To
= - (Pt - Ps) Fs / (40.94 years)

Rearranging gives:
Fs = [- d(Pt - Ps) / dt] [40.94 years / (Pt - Ps)

Numerical substitution of current experimental data gives:
Fs ~ (2.66 ppmv / year) 40.948 years / (400 ppmv - 280 ppmv)
= 0.908

These results for To and Fs compare well with experimental measurements set out on the web page titled: CARBON DIOXIDE.

Thus the exponential decay time constant Tt for excess CO2 in the Earth's atmosphere at the present ocean temperature is about:
to = 40.948 years.
The corresponding half life of excess transient CO2 in the Earth's atmosphere is:
40.948 years Ln(2.0)
= 40.948 years (0.693147)
= 28.383 years

Thus during the PETM the initial transient CO2 partial pressure would largely dissipate in the first century after injection.

Recall that:
Tt = Ka / (Ks Fs (evaporation rate))

Due to the ocean temperature dependence of CO2 solubility Ks in ocean water the decay time constant To and the half life of excess CO2 in the atmosphere will increase as the average ocean temperature rises. This issue is important in enabling warm state trapping from a transient fossil CO2 injection into the atmosphere.
 

ICE AGES:
As the atmospheric CO2 concentration rises slightly increases global warming increases causing further ocean warming and hence more emission of CO2 from the ocean to the atmosphere. Hence:
dT / dt > 0
and there is mild positive feedback.

As the atmospheric CO2 concentration decreases global warming decreases, which allows cooling of the ocean and hence allows the ocean to reabsorb CO2. Hence:
dT / dt < 0
and there is mild positive feedback.

This positive feedback mechanism also contributes to slowly oscillating ocean temperatures and accompanying glaciations known as ice ages. The frequency of these oscillations is about one complete thermal cycle per 11,000 years due to the large thermal mass of the ocean and the comparatively small net heating or net cooling rate. These natural oscillations are typically accompanied by +/-20 ppm changes in steady state atmospheric CO2 concentration. Today the transient atmospheric CO2 concentration is about 400 ppmv which is about 120 ppmv above the steady state atmospheric CO2 concentration of 280 ppmv.
 

NORMAL OSCILLATIONS:
Normally the amplitude of the atmospheric CO2 concentration changes associated with ice ages is limited by the changing Arctic ocean ice cover. Increasing Arctic ocean floating ice cover reduces the atmospheric CO2 absorption rate by the ocean while CO2 transfer to the atmosphere via evaporation continues unchanged at low latitudes. Hence at large Arctic Ocean floating ice cover the net CO2 absorption rate changes sign and becomes negative which reverses the direction of (dT / dt).

Decreasing the Arctic Ocean floating ice cover increases the atmospheric CO2 absorption rate by the ocean while while CO2 transfer to the atmosphere via evaporation continues unchanged at low latitudes. Hence at low Arctic Ocean floating ice cover the net CO2 absorption rate changes sign and becomes positive which reverses the direction of (dT / dt).

For over 2.5 million years the atmospheric CO2 concentration and the ocean temperature have oscillated over relatively narrow ranges.
 

DAILY CYCLE:
There is also a daily cycle in which solar energy evaporates sea spray droplets, which emit CO2 to the atmosphere during the evaporation process. The rate of CO2 emission by this evaporation process is proportional to the absorbed solar radiation and the density of (HCO3)- ions in the evaporated sea water. The resulting water vapor rises in the atmosphere, emits its latent heat of vaporization via infrared photons and falls back into the ocean as precipitation at about the same average temperature as it was at during the original evaporation. While on this path it absorbs CO2 from the atmosphere.

The CO2 concentration in the atmosphere is enlarged by combustion of fossil fuels. The ocean absorbs CO2 at a rate proportional to the open ocean area, and the differential pressure between the CO2 partial pressure in the atmosphere and the effective CO2 partial pressure in the ocean.

A transient change in atmospheric CO2 concentration, which does not have time to affect the ocean temperature, causes a corresponding change in the CO2 absorption rate by the ocean, which corrects the atmospheric CO2 concentration by stable negative feedback control. Hence over short periods the atmospheric CO2 concentration is stable.
 

ICE AGE PREVENTION:
Now assume that there is a transient increase in the atmospheric CO2 concentration due to combustion of biomass and fossil fuels. Assume that this transient increase in atmospheric CO2 concentration causes an increase in the CO2 absorption rate by the ocean sufficient to overwhelm any decrease in CO2 absorption rate due to increasing Arctic Ocean floating ice cover. Hence the whole system is driven toward the no ice cover condition. In this state (dT / dt) is postive and cannot change sign. There will be an gradual increase in atmospheric CO2 concentration which causes additional ocean heat absorption. This additional heat absorption will cause additonal CO2 emission by the ocean. The ocean temperature will rise to a new steady state condition instead of oscillating.
 

PETM THERMAL RUNAWAY SEQUENCE INITIATION:
We know from mass spectrometer isotopic analysis of PETM ocean sedimentary data that during the PETM there was an injection of organic C-13 depleted CO2 that caused a 0.1825 fractional increase in steady state ocean dissolved CO2 concentration. However, before this concentration increase occurred this carbon was injected into the atmosphere over the ocean and caused transient ocean heating. It is instructive to calculate the amount of the CO2 injection and the initial transient increase in temperature.

Recall that:
(Na - Ka Psa) = Psa Ksa Vw
and
(Nb - Ka Psb) = Psb Ksb Vw
and from PETM DATA ANALYSIS:
Psb Ksb / Psa Ksa = 1.1825

Hence: (Nb - Ka Psb) = Psb Ksb Vw
= 1.1825 Psa Ksa Vw
= 1.185 (Na - Ka Pa)

However, we know from present day measurements that for Pa = 280 ppmv:
Ka Pa = Na (1 / 33.76)

Hence: (Nb - Ka Psb) = 1.1825 (Na - Ka Pa)
= 1.1825 Na [1 - (1 / 33.76)]
= 1.1825 Na [32.76 / 33.76]

Hence:
Nb - Na = Na{1.1825[32.76 / 33.76] - 1}
or
[(Nb - Na) / Na] = {1.1825[32.76 / 33.76] - 1}
= 0.147473341

This amount of this initial organic CO2 injection indicates that shortly before the PETM there were likely extensive tar pools on the surface of the Earth.

It appears that 56 million years ago there was an initial trigger event which caused widespread combustion of biomatter and exposed fossil fuels, causing about an 8 fold transient increase in the atmospheric CO2 concentration. This CO2 injection caused a transition from the normal "cool" state to the "warm" state. Then a century of ocean warming trapped the Earth in its "warm" state.

It is likely that in the case of the PETM the initial step increase in atmospheric CO2 concentration was initiated by transient nearby passage of another star which for a few months increased the solar irradiance sufficiently to prevent formation of cloud ice microcrystals. The consequent reduction in albedo led to wide spread combustion of biomass and exposed fossil fuels, which added sufficient CO2 to the atmosphere to cause thermal runaway.
 

WARNING:
A signal indicating that global warming is occurring today is melting of the Arctic Ocean floating ice. The accompanying increase in ocean CO2 absorption due to increased exposed open water area is not enough to counteract the increase in CO2 emission due to fossil fuel combustion. Another signal is the experimentally observed increase in the atmospheric CH4 concentration. Experimental data shows that the atmospheric CH4 concentration has increased from 680 ppbv in pre-industrial times to about 1800 ppbv (1.8 ppmv) today. Hence there is ongoing net conversion of biomatter in soils into atmospheric CH4.
 

EARTH EVOLUTION:
Large scale combustion of fossil fuels since WWII has caused a major increase in the atmospheric CO2 concentration which is causing ongoing net absorption of heat by the ocean. Thus far the rate of average ocean temperature rise has been constrained by melting of floating polar ice. However, the floating polar ice pack is now close to being exhausted. Within a few years the ongoing heat absorption by the oceans will cause the average ocean temperature to increase. An increase in ocean temperature will cause an increase in steady state atmospheric CO2 concentration, placing the Earth at increased risk for both thermal runaway and "warm" state trapping.

The "warm" state will melt all ice on the surface of the Earth and will thermally expand the ocean volume, causing a sea level rise of about 80 m. The related increase in average Earth surface temperature will drive all large land animals, including humans, into extinction.

After the oceans reach steady state at the Earth's new higher atmospheric CO2 concentration photosynthesis will gradually cause reformation of fossil fuels. Over a period of about 500,000 years this reformation of fossil fuels will reduce the carbon content of the ocean-atmosphere pool, reducing both the atmospheric CO2 concentration and the ocean (HCO3)- ion concentration. The accompanying reduction in Earth surface temperature will cause the oceans to reabsorb much of the excess atmospheric CO2 and the Earth will gradually return to its normal steady state atmospheric CO2 concentration of about 280 ppm. However, mankind will be extinct.
 

THERMAL RUNAWAY SEQUENCE PREVENTION:
The existence of the PETM fossils and sedimntary layers with their characteristic isotopic concentration ratios confirm that this sequence of thermal runaway and warm state trapping events occurred in the past and will reoccur again in the near future unless sufficien immediate remedial action is taken.

The only way for mankind to prevent thermal runaway and warm state trapping is to cease combustion of fossil fuels. The day may soon come when there is a world war, the objective of which is to force unwilling parties to close fossil fuel production. In this respect it is of paramount importance to use nuclear energy to form synthetic hydrocarbons for displacement of fossil fuels in both heating and transportation applications. The single most important action for governments to take is to remove all production based financial incentives from the fossil fuel industry and to transfer these financial incentives to the nuclear power industry. Simultaneously the nuclear power cycle must be made sustainable via adoption of fast neutron reactor technology. Highly dynamic electricity rates must be introduced to encourage high efficiency farm and forestry based methanol production for efficient capture of carbon from the atmosphere.

An issue of some concern is that mankind has already pushed the Earth far outside its normal range of stable ice age oscillations. To switch from an ocean warming half cycle to an ocean cooling half cycle there must be a major decrease in the atmospheric CO2 concentration. At this moment, other than via a nuclear war, there does not appear to be any practical way of achieving that objective.
 

OCEAN SWITCHING RATE:
In thermal runaway with dFr = - 0.200 the increased absorbed solar energy flux available to drive the ocean warming is:
0.2 X Po = 0.2 X 341.75 W / m^2
= 68.35 W / m^2
= 68.35 J / s-m^2

The amount of heat required to raise a 3 km tall column of ocean water (1 m X 1 m X 3 km) up 17 degrees K is:
3000 m^3 X 1000 kg / m^3 X 1000 cal / kg deg K X 17 deg K X 4.18 J / cal
= 213.18 X 10^9 J

Hence the switching time for a 3 km tall column of water is:
(213.18 X 10^9 J / 68.35 J / s) X 1 hour / 3600 s X 1 year / 8766 hour
= 98.8 years.

This time will be extended by supplying heat of fusion to the polar ice caps.

Note that for "warm" state trapping to occur the high transient CO2 concentraion must exist until the ocean temperature follows the emission temperature. In the case of the PETM that required an initial atmospheric CO2 concentration that was about 2000 ppmv.

However, very soon after "cool" state to "warm" state switching commences the high atmospheric temperatures will be intolerable for large animal life forms.
 

RECOVERY FROM WARM STATE TRAPPING:
Once the Earth is trapped in its "warm" state the only way for the Earth to switch back to the locally stable "cool" state is via extremely slow photosynthesis and silicate rock weathering processes which extract CO2 from the atmosphere and oceans and sequester the carbon in carbohydrates, fossil fuels and carbonate rock. These processes will gradually raise the Earth's Ft value which will cause gradual ocean cooling and CO2 reabsorption by the ocean until T falls below 273.15 K causing the Earth to transition back to its normal "cool" state. This recovery process may take over 200,000 to 500,000years.

While the Earth is in the "warm" state ocean ice cannot form, so ice ages cannot occur.
 

This web page last updated November 5, 2015.

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