# Fat-Tail Impulse Response of CO2

This is a combination of the last two posts. The first post called Explaining “The Missing Carbon” reviewed the fat-tail response of atmospheric CO2 adjustment time to a carbon forcing function, and detailed how the Fokker-Planck (FP) diffusion models sequestering of CO2 into deep stores. The first step is to approximate the impulse response, R(t):

$$\large [CO_2] = f(t) \otimes R(t) = f(t) \otimes \frac{1}{1+0.15\sqrt{t}}$$

Next we convolve this impulse response with the fossil fuel estimates, f(t), from the last 260 years (data from the Carbon Dioxide Information Analysis Center at Oak Ridge National Labs)

Next we convert the carbon emissions in metric tons to a CO2 concentration in ppm, and plot it in comparison to the historically measured CO2 concentrations at Mauna Loa with the NASA GISS global temperature anomaly alongside. We choose a baseline of 290 ppm because that fits better than 280 ppm (and this is in agreement with a previous estimate made).

Now we zero in on the CO2 sensitivity so that we can compare it to the last post “The sensitivity of global temperature to CO2”. The BLUE curve below shows the yearly deviations as differential CO2, short-handed as d[CO2], from the convolved impulse response, calculated since 1960. The RED curve below is yearly d[CO2] data from NOAA. The GREEN curve is data directly computed from the fossil fuel emissions, that is no impulse response convolution.

This data removes all the seasonal adjustments and though there is likely a weak correlation with measured d[CO2], most of the CO2 variation is associated with multi-year temperature fluctuations.This link http://www.skepticalscience.com/print.php?r=145 points to work by (Bacastow and Keeling 1981) and Section 7.3.2.4 of the IPCC AR4 Working Group 1 report. The latter has all the details:

[IPCC] — The atmospheric CO2 growth rate exhibits large interannual variations (see Figure 3.3, the TAR and http://lgmacweb.env.uea.ac.uk/lequere/co2/carbon_budget). The variability of fossil fuel emissions and the estimated variability in net ocean uptake are too small to account for this signal, which must be caused by year-to-year fluctuations in land-atmosphere fluxes. Over the past two decades, higher than decadal-mean CO2 growth rates occurred in 1983, 1987, 1994 to 1995, 1997 to 1998 and 2002 to 2003. During such episodes, the net uptake of anthropogenic CO2 (sum of land and ocean sinks) is temporarily weakened. Conversely, small growth rates occurred in 1981, 1992 to 1993 and 1996 to 1997, associated with enhanced uptake.

Those years do indeed match, just odd that no one ever thought to actually plot the numbers and show the cross-correlation. This seems like such obvious scientific book-keeping that I am dumb-founded by the lack of a published plot.

[IPCC] — Since the TAR, many studies have confirmed that the variability of CO2 fluxes is mostly due to land fluxes, and that tropical lands contribute strongly to this signal (Figure 7.9). A predominantly terrestrial origin of the growth rate variability can be inferred from (1) atmospheric inversions assimilating time series of CO2 concentrations from different stations (Bousquet et al., 2000; Rödenbeck et al., 2003b; Baker et al., 2006), (2) consistent relationships between δ13C and CO2 (Rayner et al., 1999), (3) ocean model simulations (e.g., Le Quéré et al., 2003; McKinley et al., 2004a) and (4) terrestrial carbon cycle and coupled model simulations (e.g., C. Jones et al., 2001; McGuire et al., 2001; Peylin et al., 2005; Zeng et al., 2005). Currently, there is no evidence for basin-scale interannual variability of the air-sea CO2 flux exceeding ±0.4 GtC yr–1, but there are large ocean regions, such as the Southern Ocean, where interannual variability has not been well observed.

Take a look at figure 7.9 in particular and one can see how the trends change quite a bit for CO2 measurements over ocean versus land. Curiously Mauna Loa is in the ocean yet it shows the strong correlation of the land stations.

In any case, the long-term atmospheric CO2 concentration is clearly explainable by a fossil-fuel forcing function. The short-term deviations in CO2 are explained by a combination of fluctuating carbon emissions along with a strong dependence on natural short-term temperature fluctuations (shown below).
with a strong cross-correlation between Temperature and the Proportional-Derivative d[CO2] model: