Mathematical & observational proof that CO2 has no significant effect on climate

A recent comment at WUWT sets out the mathematical and observational proof that the effect of CO2 on climate is effectively nil. 

The comment is a succinct summary of many recent posts here on the Hockey Schtick. 

Bart says:

September 28, 2013 at 11:09 am

‘As Monk would say, “Here’s what happened”.’

That Monk is a smart guy. It’s been pretty obvious, actually, for a long time that the climate modelers had conflated the natural cyclical upswing with a sudden anthropogenic rise. But, the fact that the rise from approximately 1970-2000 was almost precisely the same as the rise from 1910-1940 gave the game away.

“This analysis shows that the real AGW effect is benign and much more likely to be less than 1 °C/century than the 3+ °C/century given as the IPCC’s best guess for the business-as-usual scenario.”

It’s actually pretty obvious from other data that it must be even less than that, and effectively zero. If we look at the relationship between CO2 and temperatures, it is apparent that to a very high degree of fidelity that

dCO2/dt = k*(T – Teq)

CO2 = atmospheric concentration
k = sensitivity factor
T = global temperature anomaly
Teq = equilibrium temperature

k and Teq are parameters for a 1st order fit. They may change over time, but are well represented by constants for the modern era since 1958 when precise measurements of CO2 became available.

This is a positive gain system – an increase in temperatures produces an increase in CO2 concentration. If we now presume that there is a positive feedback between CO2 and temperature, we get a positive feedback loop, which would be unstable.

There are other negative feedbacks, e.g., the T^4 radiation of heat. But, to maintain stability, these would have to be dominant, in which case the overall effect of CO2 on temperature would be negligible anyway. All roads lead to Rome –whatever the overall system response is, it must be such that the effect of CO2 on temperatures is effectively nil.

Now, a note on how the relationship above comes about. Atmospheric CO2 obeys a partial differential diffusion equation. The interface with the oceans sets boundary conditions. The boundary condition can be considered to obey something akin to Henry’s law (buffering processes complicate the actual relationship)

CO2(boundary) = Kh*CO2_Oceans(boundary)

The derivative of this is

dCO2(boundary)/dt = dKh/dt*CO2_Oceans(boundary) + Kh*dCO2_Oceans(boundary)/dt

Kh is a function of temperature, and thus can be expanded to first order as

Kh = Kh_eq + Kh_partial*(T – Teq)

where Kh_partial is the partial derivative of Kh to temperature. The oceans have been a net source of CO2 to the atmosphere. Assuming these are dominant, then

dCO2(boundary)/dt := (Kh_partial*dCO2_Oceans(boundary)/dt) * (T – Teq)

which is the form of the equation above with

k = Kh_partial*dCO2_Oceans(boundary)

In words, the influx of CO2 from the oceans produces a temperature dependent pumping action into the atmosphere.

The full dynamics are an atmospheric diffusion equation, with ocean boundary conditions as above, as well as a boundary condition with the land, which establishes a flow from the atmosphere into the miinerals and biota of the land, and an outflow from anthropogenic release of latent CO2. This is vastly simplified, of course, as the oceans contain their own biota and other CO2 absorbing processes. So, rather than division strictly into oceans and land, there is some overlap between the two reservoirs. In any case, though I have not yet worked out the details, it is clear where all this is heading. A very simplified ODE system model is

dCO2/dt = (CO2eq – CO2)/tau + H
dCO2_eq/dt = k*(T – Teq)

CO2 = atmospheric CO2
CO2eq = equilibrium CO2 established by the oceanic boundary condition
H = human inputs
tau = a time “constant”

The equilibrium CO2 is established by the interface with the oceans, and is relentlessly driven upward by temperatures above the equilibrium level. These feed into the atmospheric diffusion equation, which is being driven by human inputs, but is also being depleted by natural sinks which react in proportion to the CO2 level above equilibrium.

If “tau” is short, then H will be dramatically attenuated, and have little overall effect, and CO2 will track CO2eq. The actual dynamics are undoubtedly much more complicated, and “tau” would be more precisely modeled as an operator theoretic value which smooths the CO2 differential, leading to a “long tail” response, though not too long in the most significant components, as the data show that human inputs are being fairly rapidly sequestered.

But, this is effectively what the data show is happening. There really is no doubt about it. And, because of the positive feedback effect noted above, CO2 concentration cannot have a significant effect on temperature, because otherwise, we already would have maxxed out at some enormous level of CO2 and exceedingly high temperatures eons ago.