# Ups and downs of sea level projections

*By Stefan Rahmstorf and Martin Vermeer*

The scientific sea level discussion has moved a long way since the last IPCC report was published in 2007 (see our post back then). The Copenhagen Synthesis Report recently concluded that “The updated estimates of the future global mean sea level rise are about double the IPCC projections from 2007″. *New Scientist* last month ran a nice article on the state of the science, very much in the same vein. But now Mark Siddall, Thomas Stocker and Peter Clark have countered this trend in an article in *Nature Geoscience*, projecting a global rise of only 7 to 82 cm from 2000 to the end of this century.

Coastal erosion: Like the Dominican Republic, many island nations are

particularly vulnerable to sea level rise. (Photo: S.R.)

**Semi-empirical sea level models**

Siddall et al. use a semi-empirical approach similar to the one Stefan proposed in *Science *in 2007 (let’s call that R07) and to Grinsted et al. (2009), which we discussed here. What are the similarities and where do the differences come from?

For short time scales and small temperature changes everything becomes linear and the two new approaches are mathematically equivalent to R07 (see footnote 1). They can all be described by the simple equation:

dS/dt = a ΔT(t) + b *(Eq 1)*

dS/dt is the rate of change of sea level S, ΔT is the warming above some baseline temperature, and *a *and *b *are constants. The baseline temperature can be chosen arbitrarily since any constant temperature offset can be absorbed into *b*. This becomes clear with an example: Assume you want to compute sea level rise from 1900-2000, using as input a temperature time series like the global GISS data. A clever choice of baseline temperature would then be the temperature around 1900 (averaged over 20 years or so, we’re not interested in weather variability here). Then you can integrate the equation from 1900 to 2000 to get sea level relative to 1900:

S(t) = a ∫ΔT(t’) dt’ + b t *(Eq 2)*

There are two contributions to 20th C sea level rise: one from the warming in the 20th Century (let’s call this the “new rise”), and a sea level rise that results from any climate changes prior to 1900, at a rate *b* that was already present in 1900 (let’s call this the “old rise”). This rate is constant for 1900-2000 since the response time scale of sea level is implicitly assumed to be very long in Eq. 1. A simple matlab/octave code is provided below (2).

If you’re only interested in the total rise for 1900-2000, the temperature integral over the GISS data set is 25 ºC years, which is just another way of saying that the mean temperature of the 20th Century was 0.25 ºC above the 1900 baseline. The sea level rise over the 20th Century is thus:

S(1900-2000) = 25 a + 100 b *(Eq. 3)*

Compared to Eq. 1, both new studies introduce an element of non-linearity. In the approach of Grinsted et al, sea level rise may flatten off (as compared to what Eq 1 gives) already on time scales of a century, since they look at a single equilibration time scale τ for sea level with estimates ranging from 200 years to 1200 years. It is a valid idea that part of sea level rise responds on such time scales, but this is unlikely to be the full story given the long response time of big ice sheets.

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