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.

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	8.3	8.3 (?) 7.9

Performance for 20th Century sea level rise

For the 20th Century we can compute the “new” sea level rise due to 20th Century warming and the “old” rise due to earlier climate changes from Eq. 3. The results are shown in the table. From Grinsted et al, we show two versions fitted to different data sets, one only to “historical” data using the Jevrejeva et al. (2006) sea level from 1850, and one using the Moberg et al. (2006) temperature reconstruction with the extended Amsterdam sea level record starting in the year 1700.

First note that “old” and “new” rise are of similar magnitude for the 20th Century because of the small average warming of 0.25 ºC. But it is the a-term in Eq. (2) that matters for the future, since with future warming the temperature integral becomes many times larger. It is thus important to realise that the total 20th Century rise is not a useful data constraint on a, because one can get this right for any value of a as long as b is chosen accordingly. To constrain the value of a – which dominates the 21st Century projections — one needs to look at the “new rise”. How much has sea level rise accelerated over the 20th Century, in response to rising temperatures? That determines how much it will accelerate in future when warming continues.

The Rahmstorf model and the Grinsted “historical” case are by definition in excellent agreement with 20th Century data (and get similar values of a) since they have been tuned to those. The main difference arises from the differences between the two sea level data sets used: Church and White (2006) by Rahmstorf, Jevrejeva et al. (2006) by Grinsted et al. Since the “historical” case of Grinsted et al. finds a ~1200-year response time scale, these two models are almost fully equivalent on a century time scale (e^-100/1200=0.92) and give nearly the same results. The total model rise in the last column is just 1.5 percent less than that based on the linear Eq. 3 because of that finite response time scale.

For the Grinsted “Moberg” case the response time scale is only ~210 years, hence our linear approximation becomes bad already on a century time scale (e^-100/210=0.62, the total rise is 15% less than the linear estimate), which is why we give the linear estimates only in brackets for comparison here.

The rise predicted by Siddall et al is much lower. That is not surprising, since their parameters were fitted to the slow changes of the big ice sheets (time scale τ=2900 years) and don’t “see” the early response caused by thermal expansion and mountain glaciers, which makes up most of the 20th Century sea level rise. What is surprising, though, is that Siddall et al. in their paper claim that their parameter values reproduce 20th Century sea level rise. This appears to be a calculation error (4); this will be resolved in the peer-reviewed literature. Our values in the above table are computed correctly (in our understanding) using the same parameters as used by the authors in generating their Fig.3. Their model with the parameters fitted to glacial-interglacial data thus underestimates 20th Century sea level rise by a factor of two.

Future projections

It thus looks like R07 and Grinsted et al. both reproduce 20th Century sea level rise and both get similar projections for the 21st Century. Siddall et al. get much lower projections but also strongly under-estimate 20th Century sea level rise. We suspect this will hold more generally: it would seem hard to reproduce the 20th Century evolution (including acceleration) but then get very different results for the 21st Century, with the basic semi-empirical approach common to these three papers.

In fact, the lower part of their 7-82 cm range appears to be rather implausible. At the current rate, 7 cm of sea level rise since 2000 will be reached already in 2020 (see graph). And Eq. 1 guarantees one thing for any positive value of a: if the 21st Century is warmer than the 20th, then sea level must rise faster. In fact the ratio of new sea level rise in the 21st Century to new sea level rise in the 20th Century according to Eq. 2 is not dependent on a or b and is simply equal to the ratio of the century-mean temperatures, T₂₁/T₂₀ (both measured again relative to the 1900 baseline). For the “coldest” IPCC-scenario (1.1 ºC warming for 2000-2100) this ratio is 1.3 ºC / 0.25 ºC = 5.2. Thus even in the most optimistic IPCC case, the linear semi-empirical approach predicts about five times the “new” sea level rise found for the 20th Century, regardless of parameter uncertainty. In our view, when presenting numbers to the public scientists need to be equally cautious about erring on the low as they are on the high side. For society, after all, under-estimating global warming is likely the greater danger.

Does the world have to be linear?

How do we know that the relationship between temperature rise and sea level rate is linear, also for the several degrees to be expected, when the 20th century has only given us a foretaste of 0.7 degrees? The short answer is: we don’t.

A slightly longer answer is this. First we need to distinguish two things: linearity in temperature (at a given point in time, and all else being equal), and linearity as the system evolves over time. The two are conflated in the real world, because temperature is increasing over time.

Linearity in temperature is a very reasonable assumption often used by glaciologists. It is based on a heat flow argument: the global temperature anomaly represents a heat flow imbalance. Some of the excess heat will go into slowly warming the deep ocean, some will be used to melt land ice, a tiny little bit will hang around in the atmosphere to be picked up by the surface station network. If the anomaly is 2 ºC, the heat flow imbalance should be double that caused by a 1 ºC anomaly. That idea is supported by the fact that the warming pattern basically stays the same: a 4 ºC global warming scenario basically has the same spatial pattern as a 2 ºC global warming scenario, only the numbers are twice as big (cf. Figure SMP6 of the IPCC report). It’s the same for the heating requirement of your house: if the temperature difference to the outside is twice as big, it will lose twice the amount of heat and you need twice the heating power to keep it warm. It’s this “linearity in temperature” assumption that the Siddall et al. approach rejects.

Linearity over time is quite a different matter. There are many reasons why this cannot hold indefinitely, even though it seems to work well for the past 120 years at least. R07 already discusses this and mentions that glaciers will simply run out of ice after some time. Grinsted et al. took this into account by a finite time scale. We agree with this approach – we merely have some reservations about whether it can be done with a single time scale, and whether the data they used really allow to constrain this time scale. And there are arguments (e.g. by Jim Hansen) that over time the ice loss may be faster than the linear approach suggests, once the ice gets wet and soft and starts sliding. So ultimately we do not know how much longer the system will behave in an approximately linear fashion, and we do not know yet whether the real sea level rise will then be slower or faster than suggested by the linear approach of Eq. 1.

Getting soft? Meltwater lake and streams on the Greenland Ice Sheet near 68ºN at 1000 meters altitude. Photo by Ian Joughin.

Can paleoclimatic data help us?

Is there hope that, with a modified method, we may successfully constrain sea level rise in the 21st Century from paleoclimatic data? Let us spell out what the question is: How will sea level in the present climate state respond on a century time scale to a rapid global warming? We highlight three aspects here.

Present climate state. It is likely that a different climate state (e.g. the glacial with its huge northern ice sheets) has a very different sea level sensitivity than the present. Siddall et al. tried to account for that with their equilibrium sea level curve – but we think the final equilibrium state does not contain the required information about the initial transient sensitivity.

Century time scale. Sea level responds on various time scales – years for the ocean mixed layer thermal expansion, decades for mountain glaciers, centuries for deep ocean expansion, and millennia for big ice sheets. Tuning a model to data dominated by a particular time scale – e.g. the multi-century time scale of Grinsted et al. or the multi-millennia time scale of Siddall et al. – does not mean the results carry over to a shorter time scale of interest.

Global warming. We need to know how sea level – oceans, mountain glaciers, big ice sheets all taken together – responds to a globally near-uniform forcing (like greenhouse gas or solar activity changes). Glacial-interglacial climate changes are forced by big and highly regional and seasonal orbital insolation changes and do not provide this information. Siddall et al use a local temperature curve from Greenland and assume there is a constant conversion factor to global-mean temperature that applies across the ages and across different mechanisms of climate change. This problem is not discussed much in the paper; it is implicit in their non-dimensional temperature, which is normalised by the glacial-holocene temperature difference. Their best guess for this is 4.2 ºC (as an aside, our published best guess is 5.8 ºC, well outside the uncertainty range considered by Siddall et al). But is a 20-degree change in Greenland temperature simply equivalent to a 4.2-degree global change? And how does local temperature translate into a global temperature for Dansgaard-Oeschger events, which are generally assumed to be caused by ocean circulation changes and lead to a temperature seesaw effect between northern and southern hemisphere? What if we used their amplitude to normalise temperature – given their imprint on global mean temperature is approximately zero?

Overall, we find these problems extremely daunting. For a good constraint for the 21st Century, one would need sufficiently accurate paleoclimatic data that reflect a sea level rise (a drop would not do – ice melts much faster than it grows) on a century time scale in response to a global forcing, preferably from a climate state similar to ours – notably with a similar distribution of ice on the planet. If anyone is aware of suitable data, we’d be most interested to hear about them!

Update (8 Sept): We have now received the computer code of Siddall et al (thanks to Mark for sending it). It confirms our analysis above. The code effectively assumes that the warming over each century applies for the whole century. I.e., the time step for the 20^th Century assumes the whole century was 0.74 ºC warmer than 1900, rather than just an average of 0.25 ºC warmer as discussed above. When this is corrected, the 20^th Century rise reduces from 15 cm to 8 cm in the model (consistent with our linear estimate given above). The 21st Century projections ranging from 32-48 cm in their Table 1 (best estimates) reduce to 24-32 cm.

Martin Vermeer is a geodesist at the Helsinki University of Technology in Finland.

Footnotes

(1) Siddall et al. use two steps. First they determine an equilibrium sea level for each temperature (their Eq 1, and shown in their Fig. 1). Second, they assume an exponential approach of sea level to this equilibrium value in their Eq. 2, which (slightly simplified, for the case of rising sea level) reads:

dS/dt = (S_e(T) – S(t)) / τ.

Here S is the current sea level (a function of time t), S_e the equilibrium sea level (a function of temperature T), and τ the time scale over which this equilibrium is approached (which they find to be 2900 years).
Now imagine the temperature rises. Then S_e(T) increases, causing a rise in sea level dS/dt. If you only look at short time scales like 100 years (a tiny fraction of those 2900 years response time), S(t) can be considered constant, so the equation simplifies to

dS/dt = S_e(T)/ τ + constant.

Now S_e(T) is a non-linear function, but for small temperature changes (like 1 ºC) this can be approximated well by a linear dependence S_e(T) = s * T + constant. Which gives us

dS/dt = s/τ * T + constant, i.e. Eq (1) in the main post above.

R07 on the other hand used:
dS/dt = a * (T – T0), which is also Eq. (1) above.
Note that a = s/τ and b = –a*T0 in our notation.

(2) Here is a very basic matlab/octave script that computes a sea level curve from a given temperature curve according to Eq. 2 above. The full matlab script used in R07, including the data files, is available as supporting online material from Science

% Semi-empirical sea level model - very basic version
T1900=mean(tempg(11:30)); T=tempg-T1900;

a=0.34; % sea level sensitivity parameter [cm/degree/year]
b=0.077; % note this value depends on a and on the temperature
% baseline, here the mean 1890-1909

% rate of rise - here you need to put in an annual temperature time series T
% with same baseline as chosen for fitting b!
dSdt = a*T + b;

% integrate this to get sea level over the period covered by the temperature series
S = cumsum(dSdt); plot(S);

(3) Here is a matlab/octave script to compute the equilibrium sea level curve of Siddall et al. Note the parameters differ in some cases from those given in the paper – we obtained the correct ones from Mark Siddall.

% Siddall et al equilibrium sea level curve, their Fig. 1, NGRIP scenario
A = 15.436083479092469;
b = 0.012630000000000;
c = 0.760400212014386;
d = -73.952809369848552;

Tdash=[-1.5:.05:2];
% Equilibrium sea level curve
Se=A*asinh((Tdash+c)/b) + d;
% Tangent at current temperature
dSe=A/sqrt(1+((0+c)/b)^2)/b;
Se0= A*asinh((0+c)/b) + d;
Te=dSe*Tdash + Se0;
plot(Tdash, Se, 'b', Tdash, Te, 'c', Tdash, 0.0*Se, 'k', [0 0], [-150 40], 'k')
xlabel('Dimensionless temperature')
ylabel('Equilibrium sea level (m)')
fprintf(1, 'Slope: %f m/K, Sensitivity: %f cm/K/year, zero offset: %f m\n\n', dSe/4.2, 100*dSe/4.2/2900, Se0);

(4) We did not yet receive the code at the time of writing, but based on correspondence with the authors conclude that for their values in Fig. 3 and table 1, Siddall et al. integrated sea level with 100-year time steps with a highly inaccurate numerical method, thus greatly overestimating the a-term. In their supporting online information they show a different calculation for the 20th Century with annual time steps (their Fig. 5SI). This is numerically correct, giving an a-term of about 4 cm, but uses a different value of b close to 0.12 cm/year to obtain the correct total 20th Century rise.

References

Church, J. A. & White, N. J. A 20th century acceleration in global sea-level rise. Geophysical Research Letters 33, L01602 (2006).

Grinsted, A., Moore, J. C. & Jevrejeva, S. Reconstructing sea level from paleo and projected temperatures 200 to 2100 ad. Climate Dynamics (2009).

Jevrejeva, S., Grinsted, A., Moore, J. C. & Holgate, S. Nonlinear trends and multiyear cycles in sea level records. Journal of Geophysical Research 111 (2006).

Moberg, A., Sonechkin, D. M., Holmgren, K., Datsenko, N. M. & Karlen, W. Highly variably Northern Hemisphere temperatures reconstructed from low- and high-resolution proxy data. Nature 433, 613-617 (2005).

Rahmstorf, S. A semi-empirical approach to projecting future sea-level rise. Science 315, 368-370 (2007).

Rahmstorf, S. Response to comments on “A semi-empirical approach to projecting future sea-level rise”. Science 317 (2007).

Siddall, M., Stocker, T. F. & Clark, P. U. Constraints on future sea-level rise from past sea-level change. Nature Geoscience (advance online publication, 26 July 2009).