Physics Rate Equations
W. Happer and W. A. van Wijngaarden June 16, 2020 Summary We conclude this review of Ed Berry’s paper with a summary of our views. 1. Unlike many others, Ed is very clear about his mathematical assumptions. He writes down crisp, linear rate equations for the flow of carbon between four reservoirs, the land, the atmosphere, the shallow ocean and the deep ocean. 2. IPCC’s original Bern model was probably based on rate equations similar to Ed’s. But a natural question is whether it is possible, in principle, to model the carbon cycle with linear rate equations. Carbon exchanges between the atmosphere, land and oceans are very complicated and involve a great deal of biology. The equations of fluid flow for the atmosphere and oceans (e.g. Navier-Stokes) are also famously nonlinear. 3. Ed’s rate equations have an equilibrium distribution of carbon between the reservoirs. Without human emissions, the equilibrium distribution does not change with time. In Ed’s equilibrium, about 1.45% of carbon is airborne, 90.70% is in the deep ocean, 2.21% is in the shallow ocean and 5.64% is on the land. 4. The existence of an unperturbed equilibrium distribution of carbon is consistent with the CO2 fractions measured in air bubbles trapped in ice cores taken from the Law Dome in Antarctica [6], shown in Fig. 14. The atmospheric fraction has remained close to 280 ppm from about the year 1000 to 1850. 5. Ed’s rate equations conserve carbon. If there are no human emissions, carbon lost from one reservoir flows to the other three so the total carbon content of all reservoirs remains constant. If there are human emissions, they equal the total increase of carbon in all four reservoirs. 6. As we have pointed out earlier in the review, normal relaxation modes are a more efficient way to solve Ed’s rate equations than the numerical integration he used. But Ed does not make mistakes, in numerical integration. 7. The IPCC is very vague about what the numbers it cites really mean. It will be difficult to write Ed’s paper in a way that demonstrates that the IPCC is mistaken in its estimates of carbon inventories and flows. IPCC can simply say that whatever Ed (or anyone else) writes down is a straw-man argument, criticizing equations that IPCC never wrote down. But the IPCC’s “Bern Model” looks like it started from rate equations similar to Ed’s. 8. The IPCC can say that the transfer rates that Ed assumed between the reservoirs are much too big. As we showed in Section 9, one can fix most of the problems that Ed identified by assuming that the real flow rates from the atmosphere to the land and 1
ocean are ten times smaller than those inferred from IPCC cartoons. With ten times smaller flow “admittances,” about half of emissions would remain in the atmosphere, and the time for removing a pulse of atmospheric carbon would be reasonably close to what was observed for atmospheric tests of nuclear weapons. 9. The treatment of the all-important oceans in Ed’s paper is probably too sketchy. Ed does not discuss the biological pump that can carry carbon quickly through the shallow ocean to the deep ocean. Airborne carbon can be transferred directly into the deep ocean through the formation of cold, salty, “carbonated” water near the poles. The release of carbon into the atmosphere involves upwelling of deep ocean water that was formed centuries ago. 10. If Ed’s rate equations are correct some other source of airborne carbon is needed. Ed is not very clear about what this other source could be, but the two largest reservoirs are the deep oean and the land. It is hard to understand why, after some 800 years of apparent equilibrium (as implied by Fig. 14), the deep ocean might start to outgas CO2 more rapidly around the year 1850. 11. An increase of land emissions due to human activities sounds more reasonable than an increase of ocean emissions. Carbon from respiration of living things on the land would be depleted in the heavy isotope 13C, compared to the dominant light isotope 12C. So land emissions would be consistent with the observed “lightening” [7] of the airborne fraction of CO2. But it is not clear that more carbon is being released from the land. In the US, east of the Mississippi river, the land has almost certainly been a major sink for carbon because of the regrowth of forest on land that can no longer be profitably farmed. References [1] E. X. Berry, Preprint, March 7, 2020, The Core Issues of the Human Carbon Cycle. [2] Historical Emissions of Carbon Dioxide, https://cdiac.ess-dive.lbl.gov/trends/ emis/glo_2010.html [3] IPCC, Carbon and Other Biogeochemical Cycles,https://www.ipcc.ch/site/assets/ uploads/2018/02/WG1AR5_Chapter06_FINAL.pdf [4] E. X. Berry, Preprint, March 9, 2020, The IPCC Bern Model. [5] SCRIPPS O2, CO2 AND APO, https://climatedataguide.ucar.edu/ climate-data/scripps-o2-co2-and-apo [6] D. M. Etheridge et al., Historical Records from the Law Dome DE08, DE08-2 and DSS Ice Cores, https://cdiac.ess-dive.lbl.gov/trends/co2/lawdome.html [7] 13C/12C isotope ratios, https://www.esrl.noaa.gov/gmd/outreach/isotopes/ mixing.html 2