# Science Story: the Making of a Sea Level Study

Guest commentary by Martin Vermeer

On December 7, 2009 the embargo expired, and my and Stefan’s joint paper ‘Global sea level linked to global temperature’ appeared in the Proceedings of the U.S. National Academy of Sciences. It had been a long time coming! But this post is not so much about the science as about the process, and about how a geodesist from Helsinki and an oceanographer from Potsdam, who to this day have never even met, came to write, to the surprise of both of us, a joint paper on sea level rise.

My own entry into climatology happened only a few years ago. A significant trigger was RealClimate, which I had learned to appreciate as one of the rare reliable Internet sources amidst the junk. Contributing to the oft-slandered science is my small ‘thank you’ and revenge as a scientist.

As I remember, it was the commenter calling himself Rod B. who enquired, sometime August 2008, what the story really was with Rahmstorf (2007). Trying to answer, I ended up reading the paper and getting interested. What seduced me was the simplicity of this, so-called semi-empirical approach: linear regression of sea level rise dH/dt against temperature T, yielding two unknown parameters: a regression coefficient a, and an intercept, or ‘equilibrium temperature’, T0. See our Ups and downs of sea level projections for a more detailed explanation.

The curve of temperature as a function of time over the 20th century has three parts: a steep rise in the beginning, a flat middle part commonly attributed to aerosols, and a very steep upswing at the end. Physically one would expect for the curve of the sea level rise rate dH/dt as a function of time to look rather similar, as indeed it does: this justifies the Rahmstorf (2007) approach of regressing the one against the other. Looking more carefully however one sees that the dH/dt curve has slightly more of an S-like shape, turning downward in the middle, before swinging up again at the end.

This suggested to me that, in addition to a proportionality to temperature T, sea level rise would also contain a term proportional to the time derivative of temperature, dT/dt. In other words, global sea level would be a good global thermometer, but with a ‘quirk’. I could even think of a physical mechanism for such behaviour.

I contacted Dr. Rahmstorf, proposing the idea: one would expect the ocean surface to warm up rapidly to completion, contrary to the deep ocean and the continental ice sheets. This would argue for a term, in addition to the secular a (TT0) term, of form b dT/dt. Stefan’s response was cautious; not surprising, as being something of a media figure in Germany surely means that he has to contend with his share of cranks. But he suggested I look myself into the idea, which I subsequently did: in for a penny, in for a pound.

I downloaded Stefan’s script, modified it, did the first computations with the same real tide gauge and temperature data Stefan had used — surprise: negative b. Hmmm, strange. That was for real data from the real Earth; what would happen if I applied the extended relationship to simulated data from the same general circulation model (actually, an Earth system model) for the period 1900-2100 that Stefan had used in his paper for testing his relationship? This model was in one essential way very much simpler than reality: it completely lacked the contribution of land ice melting to sea level.

Stefan helpfully sent me Matlab snippets and model output, and indeed I got it all working. What was more, the disagreement found by Stefan for the late 21st Century — between sea level rise as predicted directly by the model, and indirectly through the semi-empirical relationship between temperature and sea level rise — went almost completely away when using the new, extended relationship. With a positive value for b, just as expected from theory for an ocean surface water response.

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