Ups and downs of sea level projections

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.

Siddall et al. in contrast find a time scale of 2900 years, but introduce a non-linearity in the equilibrium response of sea level to temperature (see their curve in Fig. 1 and footnote 3 below): it flattens off strongly for warm temperatures. The reason for both the long time scale and the shape of their equilibrium curve is that this curve is dominated by ice volume changes. The flattening at the warm end is because sea level has little scope to rise much further once the Earth has run out of ice. However, their model is constructed so that this equilibrium curve determines the rate of sea level rise right from the beginning of melting, when the shortage of ice arising later should not play a role yet. Hence, we consider this nonlinearity, which is partly responsible for the lower future projections compared to R07, physically unrealistic. In contrast, there are some good reasons for the assumption of linearity (see below).

Comparison of model parameters

But back to the linear case and Eq. 1: how do the parameter choices compare? a is a (more or less) universal constant linking sea level to temperature changes, one could call it the sea level sensitivity. b is more situation-specific in that it depends both on the chosen temperature baseline and the time history of previous climate changes, so one has to be very careful when comparing b between different models.

For R07, and referenced to a baseline temperature for the year 1900, we get a = 0.34 cm/ºC/year and b = 0.077 cm/year. Corresponding values of Grinsted et al. are shown in the table (thanks to Aslak for giving those to us!).

For Siddall et al, a = s/τ where s is the slope of their sea level curve, which near present temperatures is 4.8 meters per ºC and τ is the response the time scale. Thus a = 0.17 cm/ºC/year and b = 0.04 cm /year (see table). The latter can be concluded from the fact that their 19th Century sea level rise, with flat temperatures (ΔT(t) = 0) is 4 cm. Thus, in the model of Siddall et al, sea level (near the present climate) is only half as sensitive to warming as in R07. This is a second reason why their projection is lower than R07.

Model

a [cm/ºC/year]

b [cm /year]

“new rise” [cm] (25a)

“old rise” [cm] (100b)

25a+100b [cm]

total model rise [cm]

Rahmstorf

0.34

0.077

8.5

7.7

16.2

16.2

Grinsted et al “historical”

0.30

0.141

7.5

14.1

21.6

21.3

Grinsted et al “Moberg”

0.63

0.085

(15.8)

(8.5)

(24.3)

20.6

Siddall et al

0.17

0.04

4.3

4

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