tisdagen den 11:e juni 2013

The Dog and the Tail: Global Temperature vs CO2

Prof. Murry Salby's presentation in Hamburg in April is a showcase of effective scientific communication based on mathematics. Salby gives strong evidence based on observation that the offset of concentration C(t) of atmospheric CO2 as a function of time t is determined by the offset of global temperature T(t) by an equation of the form
  • dC/dt  = T   for all t > 0, C(0) = 0,
after suitable scaling of C(t). In other words, C(t) is the integral of T(t), so that if T(t) = cos(t) then C(t) = sin(t) with a time lag of a quarter of a period.  

The fact that in the equation dC/dt = T the concentration C(t) is determined by T(t), comes out as an aspect of stability (or wellposedness): Integration is a stable or well posed mathematical operation in the sense that small variations in the integrand T(t) gives small variations in the integral C(t). 

On the other hand, differentiation is a an unstable or ill posed mathematical operation: small variations dC(t) in C(t) can give rise to large variations in dC(t)/dt as a result of division by a small dt. This means that viewing T(t) in the relation dC/dt = T to be determined by C(t) corresponds to an unstable mathematical operation. 

To make a connection from cause to effect in physics, requires stability and thus in the observed relation dC/dt = T, it is C(t) which is determined by T(t) as the cause and not the other way around. Another way of expressing this fact is to say that C(t) lags T(t) with a quarter of a period, so that variations in the cause T(t) precedes the effect as variations C(t). 

This is the observation from ice core proxies showing that temperature changes before CO2 and thus temperature is the dog and CO2 the tail with the dog wagging the tail, and not the other way around as the basic postulate of CO2 alarmism:

   

4 kommentarer:

  1. I prefer the summary analysis presented at appinsys.com. And if you look at the temperature record (the first graph at the appinsys link), you will see what everyone should know, that the temperature decreased from 1880 to 1910, increased from 1910 to 1940, decreased again from 1940 to 1975, and increased again from 1975 to about 2000. Now the CO2 record from 1957 onward shows dC/dt has increased monotonically over that period, i.e., d/dt(dC/dt)>0 from at least 1957 onward. But your CO2 vs. T relation above requires d/dt(dC/dt)=dT/dt, and dT/dt was negative from 1940 to 1975 (including the period 1957 to 1975). So the posited relation dC/dt=T is empirically wrong. And physically, that relation flies in the face of what I (a general physicist, not a climate scientist) have heard repeatedly as common knowledge: That a warmer ocean cannot hold as much CO2 dissolved in it, and releases more CO2 into the atmosphere the warmer it gets (in line with the skeptic's claim, "atmospheric CO2 follows temperature, not vice-versa as the consensus claims"). So the posited CO2 vs. T relation is also falsified according to (the general understanding of) the basic physics of CO2 sequestration by the ocean as a function of T. So, again, I prefer the appinsys summary of the situation. I commented on Dr. Salby's presentation at hockeyschtick also.

    SvaraRadera
  2. The model is simple but yet seems to capture an essential aspect. As long as the temperature anomaly or offset is positive the concentration of CO2 increases, which is seen during the recovery from the little Ice Age.

    SvaraRadera
  3. But the relation should be more like C ~ T, due to ocean CO2 solvency, not dC/dt ~ T.

    SvaraRadera
  4. Claes, I thought you knew enough mathematics to call Salby's bluff about the natural contribution to rising CO2. But I suppose you are no real skeptic.

    He is looking at correlations with "surface conditions" which are insensitive to the almost linear anthropogenic increase. See, the derivative of a linear function gives a constant. Correlating a constant with something gives zero.

    Which means the correlations say nothing about what causes the rise even if they are equal to 1.

    SvaraRadera