Atmospheric Carbon Dioxide Lags Temperature: the Proof

Written by Dr Pierre Latour PE

Man-made global warming promoters claim the high correlation between carbon dioxide (CO2) and atmospheric temperature (T) in the 420,000 year ice core record proves CO2 causes T to change. Herein is demonstrated how the evidence conflicts with that belief.

CO2 fraud

Basics. First, correlation alone only proves correlation, not cause and effect. Physics is required to describe and prove cause and effect. Second if increasing CO2 did cause T to increase, there must be some physical lag or delay in the response of T to CO2; average T of whole atmosphere, oceans and land masses cannot respond instantaneously to CO2, no matter how strong the cause.

In fact many researchers claim CO2 actually lags T, proving CO2 cannot cause T changes at all. Rather T causes CO2.

What could cause CO2 to lag warming? Its solubility in water? Yes, that explains the data well. Simply put, when oceans warm due to greater solar energy absorption, they outgas dissolved CO2 just like soda water does because CO2 is less soluble in warm water than cold. When oceans are chilled, they absorb CO2 gas and hold it because CO2 is more soluble. Tropical seas hold less CO2/m3 than polar seas do.

The lag is measured to be about 800 years and confirmed by theory.

Data analysis. After studying that mechanism in 2009 and Al Gore’s 420,000 years of T and CO2 data in his “Inconvenient Truth” movie and National Geographic June 2007 Big Thaw article and insert, their data confirm it was a lag, not a lead.

Houston’s University of St Thomas, Environmental Sciences Department also confirmed the 800 year lag at their April 21, 2009 Conference with Jill Hasling, Weather Research Center, Houston, as have many others.

The measurement is done by inputting T data to a lag model with an assumed lag time constant, τ, and comparing lagged T data output with raw CO2 data. This rigorously accounts for different frequencies. If a lag time can be found that provides a close match between lagged T and CO2, the lag time assumption would be verified by measurement.

The lag model is yi = f*xi + (1-f)yi-1, a discrete form of low-pass filter or electrical RC circuit.

xi is the series of T data for i = 1, 2, 3, …….. and yi is series of lagged or filtered T data.

Filter factor f = del t/(τ + del t) < 1, where del t is the sampling time interval between xi data points, probably about 10 years.

Computer programs easily find the value for f and τ that best fits the 420,000 year data sequence of lagged yi to CO2i. The result is τ = about 800 years. Of course it may vary by a 100 years or so. The point is the lag τ > 0.

Role of Science. This correlation becomes a proven cause and effect relation when physics explains and predicts what is observed. This value is predicted by ocean circulation rates and mass transfer rate of CO2 across oceans’ gas-liquid surface interface.

The science is this. Solubility of CO2 in water decreases with T. Just like in beer, soda and Champagne.

Air is well mixed with only minor variation in CO2 content with latitude. Ocean is saturated with CO2 in contact with CO2 in air. Ocean water circulates up and down, north and south, east and west, all around. As warm water flows north and cools by convective and radiant heat transfer, it absorbs CO2 from air. As cold water flows south and warms by convective and radiant heat transfer, it releases, outgasses, CO2 to air. If global average Tw increases, water outgasses and CO2 in air increases. If global average Tw decreases, water absorbs and CO2 in air decreases. Oceans are a CO2 reservoir, a sink and source, depending on T changes, average about 14.9C and solar incidence absorbed/emitted changes, average about 161 w/m2 of surface.

Chemical engineering. Engineers know the rate of mass transfer of any component, x, across any gas-liquid interface is proportional to the difference between partial pressure of x in liquid and its partial pressure in vapor. Partial pressure = mol fraction* total pressure = x*Pt.  x = 0.000400

PPa = Partial pressure CO2 in air is 400 ppmv * 1 atm = 0.0004 atm

PPw = Partial pressure CO2 in liquid = f*X*Tw, where f = fugacity, X = CO2 concentration in water and Tw = water T. When Tw increases, solubility decreases and PPw increases.

Transfer rate of CO2 from air to water is = TR = r*A*(PPa – PPw) > 0, where r is the interfacial film coefficient, a mass transfer rate constant, and A = interface surface area of transfer. If TR < 0, transfer is in other direction, from water to air.

TR, mol/hr = r*A(x*Pa – f*X*Tw) = r*A*(0.0004 – f*X*Tw).

This quantifies the rate of CO2 from air to water increases with its 400 ppmv content in air, a stabilizing effect, and also when Tw decreases. As Tw increases, absorption rate decreases and can turn to outgassing.

This is the part AGW promoters miss.

Control systems engineering. The instantaneous CO2 mass balance differential equation assuming ocean is well mixed is mCp dX/dt = TR + other inputs – other outputs, where m = mass of ocean and Cp is heat capacity of water. This is the form of a low-pass filter.

The theoretical time lag is Tlag = mCp/r*A*f*Tw.

This says big oceans, large m, with high heat capacity, Cp, have big lag. Thin oceans with large A have small lag. If mass transfer rate r is large, lag is small. If Tw is high, outgassing rate is large and lag is small.

m = 1.3*10**9 km3.

Cp = 75.327 J/mol-K or or 4.187 kJ/kg-K

Ocean A = 3.61*10**8 km2. A more realistic value accounting for waves is 2x = 7.2*10**8 km2. and

Tw = 288K, approximately, and r and f are available from chemistry.

Using the known values of these physical parameters predicts Tlag = 800 years.

This explains why mixing or residence time of different species between atmosphere and oceans varies so greatly, from 200 years for iron to 100 million years for chloride; they have widely different mass transfer film coefficients and fugacity’s.

Since ocean circuits complete in much less than 800 years, the well-mixed assumption is good and the circuit rates are irrelevant over 800 year periods.

Dynamics. I must add there is another rate step in the ocean, the consumption of CO2 by phytoplankton and production of CO2 from dissolving inorganic carbonates. I shall assume they are not rate limiting, and interfacial mass transfer at the surface is. If not, there will be a significant second-order lag.

The response of Earth’s global radiating temperature to changes in solar intensity has second-order dynamics, a small thermal lag of atmosphere and a much larger thermal lag of oceans and continents. You can think of these time lags as system residence times; volume/volumetric throughput rate or mass/mass throughput rate. These dynamic properties are neglected by AGW modelers that simply use instantaneous radiation physics, even get that wrong (I proved it would constitute a perpetual motion machine to create energy in violation of two laws of thermodynamics), resort to empirical correlations, and get that wrong too. So they can’t forecast anything. So they can do more research, to scare people into paying higher taxes.

Conclusions. Chemists and chemical engineers know about this, AGW promoters don’t, even if they are radiation astrophysicists.

So Al Gore’s 420,000 years of data confirms CO2 lags T and chemical engineering mass transfer theory proves it. That is a Truth. Whether it is Convenient or Inconvenient depends on your agenda.

Further, any attempt to use raw data rather than lag synchronized data to develop empirical statistical regression correlation models to forecast weather, global warming and climate change will lead to widely different models and wildly divergent forecasts, as proven by University of East Anglia, UK and UN IPCC research since 2000.

I have not seen any better explanation for the lag. I have never seen any credible explanation for or evidence of lead, CO2 causing T in either direction.

This does not prove that CO2 does not cause any warming whatsoever; only that 420,000 years of data doesn’t prove it does. It is not the AGW skeptics’ job to prove the GHGT, but to disprove it. For society’s sake.

Feel free to inform your elected officials.

Confirmation References.

Jo Nova, “The 800 year lag – graphed”, 28Oct2008.

Joe Martino, “420,000 years of Data Suggests Global Warming is Not Man-Made”, 8Feb2013.

Jeff Severinghaus, “What does the lag of CO2 behind temperature tell us about global warming?”, 3Dec04.

Forum, “CO2 lags temperature increase by 800 years”, undated.

Anthony Watts, “New research in Antarctica shows CO2 follows temperature by ‘a few hundred years’ at most”, 23Jul12.

Rebuttal Attempts References.

Unconvincing, hypothesis not seen in data.

Catherine Brahic and Michael Le Page, “Climate myths: Ice cores show CO2 increases lag behind temperature rises, disproving the link to global warming”, 16May07.


Thomas Schueneman, “Temperature leads CO2. The 800-year lag?”, 14Dec12.

The AGW promoter rebuttal attempts are unconvincing because they postulate a “positive feedback” effect which is not explained or verified because they do not exist in nature.

Dana, “CO2 lags temperature – what does it mean?”, 19Apr14.


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