Rahmstorf vs. Rahmstorf

March 5, 2012

Oh, what a tangled web we create, when first we practice to exaggerate.

with apologies to Sir Walter Scott

Intrepid mathematician Stefan Rahmstorf has calculated the global temperature increase rate for the last 31 years.  (Global temperature evolution 1979–2010, Foster and Rahmstorf, Environ. Res. Lett. 6, 2011) For the fun of it, lets take him at his word.  The problem is that when his temperatures from this new paper are inserted into his sea level rise rate formula from one of his earlier papers (Global sea level linked to global temperature, Vermeer and Rahmstorf, PNAS, 2009), the calculated sea level rise rate isn’t anywhere close to reality.

These papers can’t both be correct.  My guess is that neither of them are. 

In the 2011 paper he starts with five different global temperature records and adds his version of corrections for volcanoes, el Nino and solar variations.  He then calculates the temperature rate of change per decade for each of the five temperature records.  The five ranged between 0.141 °C/decade to 0.175 ° C/decade, but the average was 0.163 °C/decade as shown in figure 1,  below.

He also calculated the temperature rise rate acceleration, and found none.  In his own words

“To look for changes in the warming rates over time, we computed the rate in adjusted data sets for different time intervals, for all start years from 1979 to 2005 and ending with the present. The results show no sign of a change in the warming rate during the period of common coverage.”

Figure 1 Rahmstorf's version of global temperature for 1979 to 2010. This is figure 4 and table 1 from Foster and Rahmstorf. Trendline, based on the average of table 1, added by ClimateSanity

You know what higher temperatures mean: higher sea level rise rates.  Nobody knows this better than Herr Rahmstorf, who has spent the better part of his career making the point.  He has even provided a formula in his 2009 paper to translate the global temperature to the sea level rise rate.

Some easy math

Assuming his calculated temperature increase rates for the last three decades are correct, what does his sea level rise rate formula tell us?  In Rahmstorf’s parlance H is the sea level and dH/dt is the sea level rise rate.  His formula, from which sprang the famous 1.8 meter sea level rise for the 21st century meme, looks like this…

From Rahmstorf’s graph of global temperature from 1979 to 2010 (figure 1, above), we see that his temperature and the rate of temperature change are given by …


Substituting equations II & III into equation I and gathering terms reveals

While equation IV won’t tell us the exact sea level rise rate for a particular year, it will tell use how much the sea level rise rate changes between two years.  That is

Let’s say that Rahmstorf’s temperature data from the his 2011 Environmental Research Letters paper is correct and his formula relating sea level rise rate from his 2009 PNAS paper is correct.  And let’s say that we wanted to know how  much the sea level rise rate had increased between (oh, I don’t know – how about) 1993 and 2010. Then equation V would tell us that the sea level rise rate should have increased by 1.55 mm/year (0.09128 mm/year X (2010-1993)). 

Comparing to reality

Lucky for us, we have measured sea level data to compare the calculated value to.  As figure 2, below makes abundantly clear, the sea leve rise rate has been about 3.1 mm/year over this time period.   The combination of Rahmstorf’s 2009 PNAS paper and 2011 Environmental Research Letters paper indicate that it should have increased by 1.55 mm/year (an additional 50%).
How can this discrepancy be explained?  Oh yeah, I almost forgot, we already know the Rahmstorf’s formula relating sea level rise rate to global temperature is totally bogus.



  1. The Second Law can be illustrated with a hose used as a siphon to empty a swimming pool, for example. It works if the other end of the hose goes down a slope and is significantly below the bottom of the pool.

    The water flows and entropy increase because we have a single process. The SLoT requires a single process, as is obvious in everyday life.

    If you cut the hose at the highest point you now have two processes, and the water no longer goes upwards from the pool.

    Any heat flow from a cooler atmosphere to a warmer surface is a single completed process. The energy is not constrained to return by radiation or to do anything in particular. It could be conducted elsewhere in the surface for example.

    Because it is a single process from atmosphere to surface, there is no justification for saying that any subsequent process can create a net effect and thus excuse the violation of the Second Law. It would be like water flowing uphill to the town’s water tank on the basis that it would subsequently flow further downhill through pipes into houses. But there is no constraint enforcing this, as there was with the siphon before the hose was cut. After all, the tank might leak.

    Hence, thermal energy cannot transfer spontaneously from a cooler atmosphere to a warmer surface. Fullstop.

    See my publication Radiated Energy and the Second Law of Thermodynamics

    Click to access jo120314.pdf

    • OK,

      I have let Doug Cotton have his say. I have read his stuff and find it uninformed.


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