Open Mind

Sea Level Rise

August 31, 2009 · 21 Comments

Stefan Rahmstorf and Martin Vermeer have an interesting post on RC about very simple models of sea level rise. The simple model is

dS/dt = aT + b,

where S is sea level and t is time, so dS/dt is the rate of sea level rise. T is temperature difference from the average temperature around 1900 (averaged over many years to reduce the noise in that figure), and a,~b are constants defining the model. Rahmstorf & Vermeer discuss various attempts by various authors to estimate the constants. They quote the following results for the constants a,~b (the numbers in the RC post use cm as the unit of distance; I’ve converted to mm):

Study a (mm/deg.C/yr) b (mm/yr)
Rahmstorg 3.4 0.77
Grinsted et al. “historical” 3.0 1.41
Grinsted et al. “Moberg” 6.3 0.85
Siddall et al. 1.7 0.4

One can directly estimate the constants a,~b if one has time series data for T and dS/dt. For the former I’ll use GISS data; for the latter I’ll combine sea level data from Church & White (2006, Geophysical Research Letters, 33, L01602, doi:10.1029/2005GL024826) with data from satellite estimates. Those two data sets are on a different zero point, so I reset the zero point of the satellite data so that its average is that same as the Church & White data during their period of overlap. Here’s the sea level data:


I then estimated dS/dt by applying a modified lowess smooth and using it to estimate the time derivative of sea level. Finally we can construct a scatter diagram of the rate of change of sea level vs global temperature:


There is clearly a correlation, although just as clearly it’s far from perfect. Nonetheless we can use these data to estimate the constants by linear regression; this gives estimates a=2.8 mm/deg.C/yr and b=0.96 mm/yr. The estimate of a is closest to the Grinsted et al. (historical) value, although it’s also not too far from the value estimated by Rahmstorf. The estimate of b is closest to the Grinsted et al. (Moberg) value, but again not too different from the value estimated by Rahmstorf.

We can also directly compare the estimated rate of sea level rise from the combined Church & White/satellite data to what it would be according to the model using GISS temperature data:


What does a model like this portend for the next century? Let’s take a conservative estimate of temperature evolution until the year 2100: T = 0.55 + 0.02 (t-2010). In my opinion this will give us a lower bound for the impact of temperature on sea level rise this century. Using this estimate, the simple model, and the constants as I’ve estimated them gives a sea level rise by the year 2100 of 515 mm, about 20 inches. A more extreme picture of temperature evolution is T = 0.55 + 0.04(t-2010). Using the simple model and estimated constants now yields expected sea level rise of 1.2 meters, about 47 inches.

20 inches of sea level rise is definitely trouble for coastal regions. 47 inches (nearly four feet) is big trouble. But of course it’s possible sea level rise could be more; the simple model doesn’t allow for nonlinear behavior of the system, in particular it doesn’t anticipate accelerating disintegration of large ice sheets. As the RC article notes, “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.” I think it extremely unlikely that sea level rise will be less than estimated by these simple models.

Categories: Global Warming

21 responses so far ↓

  • David B. Benson // August 31, 2009 at 10:32 pm | Reply

    The estimates being used by coastal engineers in Britian, The Netherlands and Northern California are closer to the 1.2 m estimate for century end. As I understand it, each group did their own estimate and all arrived at 1+ m.

  • S2 // August 31, 2009 at 10:54 pm | Reply

    The model implies that even if we did magically manage to cut emissions enough to hold the temperature at today’s value, sea level would continue to increase at about today’s value (at least for a while).

    Thanks for this, it has helped me to see why the rate of increase in sea level has been increasing as fast as it has (as a function of temperature, rather than as a function of temperature change).

    (PS. small typo in your table – it should be Rahmstorf)

  • naught101 // September 1, 2009 at 1:11 am | Reply

    Don’t you really need some kind of logistic model for this? If global temps rise a certain amount, and then stop, this model would have the sea level rise be constant and positive indefinitely – no equilibrium.

    I realise it’s a simple model intended for use while temperature rise is relatively constant. Perhaps a logistic model would just make the constants that much harder to estimate?

    [Response: As discussed in the RC article, this model is only applicable because the time constant for equilibration of sea level response to temperature change is much longer than the time spans under consideration. Of course sea level will not rise indefinitely due to a temperature change.

    Estimating the time constant appears to be much less accurate, but all indications are that it is on the order of thousands of years.]

  • DrC // September 1, 2009 at 5:41 am | Reply

    Maybe a quick comment on the way that conceptual, mathematical, and process models intersect here (a la your great recent post on climate models Seems to me to be another great example of a simple concept (sea levels and temperature) being reinforced by statistical, empirical analysis and detailed mechanistic simulations. Yet another rallying cry for the role of common sense from Tamino!

  • David Horton // September 1, 2009 at 6:03 am | Reply

    Yes, and then there is this suggesting a much faster melting Greenland ice cap than any previous estimate, and therefore an even faster potential sea level rise.

  • suricat // September 2, 2009 at 12:25 am | Reply


    With all due respect, I have a problem with this basic model.

    The model can only be applied to the first kilometre of ocean depth (at most) and as sea level rises the bottom kilometre also rises for the ocean thermocline. See this representation for an example:
    Where the lower depths are warmed only by UV solar insolation until the standard graph line of natural thermal transfer can be seen from below ~1,000m.

    More to this, the model can’t account for lowland topology. As sea levels increase they encroach upon land and this means that the global ocean area increases as the land area decreases. This can only result in the reduction of sea level increase due to the greater volume of water needed to cover the encroached land area.

    I know that you only comment on another article, but is that article really worth a comment?

    Well, perhaps it is. If only to attract attention to an oddity.

    Best regards, suricat.

    [Response: My intuition is that the ratio of open-ocean area to coastal-region are is so microscopically small that the impact of lowland geometry is truly negligible.

    In my opinion yes the article is worth a comment, if only to show (as the final graph does) that there's at least a rough correspondence over the last century between temperature and sea level rise rate.]

  • Hank Roberts // September 2, 2009 at 2:58 am | Reply

    > As sea levels increase they encroach upon land
    And they’re encroaching on land that’s had time to warm to a considerable depth since the last ice age, adding a little bit more heat to the oceans.

    Not to mention the buried oil and gasoline tanks, and waste dumps, and all the other civilized miscellania that will be taken by the sea.

  • Simon D // September 2, 2009 at 8:43 pm | Reply

    Statistically, the simple historical model is sensible. I think it’s a mistake to use it for even a back-of-the-envelope future prediction. There’s no reason to expect the relationship between dS/dt and dT/t will remain constant. Most of the sea level rise to date is from thermal expansion. The contribution from ice sheet melt may now be increasingly rapidly, changing the dS/dt and dT/t relationship.

    become a larger contributor.

  • Andrew Dodds // September 4, 2009 at 1:23 pm | Reply

    Suricat -

    If the amount of land transgressed by the ocean due to rising sea levels actually becomes a significant element in the calculations, then I’d suggest correcting the calculations would be the least of our worries.

  • Hank Roberts // November 18, 2009 at 8:59 pm | Reply

    someone’s calling for attention:

  • Rattus Norvegicus // November 19, 2009 at 3:08 am | Reply

    Oh dear, oh dear, oh dear, oh dear. Rahmstorf and Vermeer use a butt stupid model to estimate future sea level rise and come up with a number in line with the IPCC. Stop the presses, this must be nonsense!

    Come on, somebody here man up and leave a comment on NN’s stern rebuttal of R09, a blog post. It think the comments were on the original RV07 article in Science seemed (since NN wasn’t polite enough to provide a link) were pointing out that bog simple models like R07 don’t provide an appreciation of the non linear effects which are likely to exist in the system. I don’t think this was ever disputed, Rahmstorf was just attempting to reset the lower bound with a bog simple model which had been in existence for a while.

    I would note that Nicholas Nirenberg is a businessman. I just don’t get what it is with businessmen that makes them think they understand everything about the world (see a certain Steve McIntyre). I would note that while Unify when somewhere, it eventually imploded. Actuate — umm never really heard of them and I lived the life in the gulch, if any of you know what I mean (Mashey knows, I worked for SGI in it’s heydey — he was just a lot better at his job than I was…).

    Come on. Somebody man up and comment at NN’s site! He needs the comments since he only has two spam comments from someone trying to sell hookahs.

  • Paul Middents // November 24, 2009 at 8:34 pm | Reply

    NN is now call for attention on RC

    He just might get some.

  • Paul Middents // November 25, 2009 at 11:32 pm | Reply

    nn finally gets some love from Martin V.

    I don’t think this is enough to justify his characterization of Rahmstorf’s work as “nonsense” over on Stoat. See the Oh Dear^5 thread.

  • Pterosaur // December 16, 2009 at 1:20 am | Reply

    Sorry for being a bit OT, but I have recently encountered the statement that :

    “Sea level rises must be constant. It is impossible for seas to rise more in one part of the globe than in another.”

    I’m pretty sure that it’s not true, but haven’t been able to find a direct refutation – any pointers ?

    • Sceptical Guy // December 16, 2009 at 2:03 am | Reply

      Pterosaur – My understanding is that the water moving from ice in one place to water in another causes subtle changes in the earth’s gravity – which causes a small but detectable change in where the water collects.

      I can’t find any papers (someone else probably can) but this article may help:

      NASA’s GRACE experiment may also be worth a look.

      Also, and this is me (an engineer) talking and therefore very UN-scientific, I would expect the moon to have an effect. Sea level is never exactly the same everywhere ‘cos of the tides.

  • Ray Ladbury // December 16, 2009 at 1:50 am | Reply

    Even apart from the obvious–that sea level must rise more at the equator than the poles due to Earth’s rotation, there is the fact that the geoid is not a truly spherical potential at Earth’s surface due to mountain ranges, etc.. Then there is the fact that as ice melts, the geoid also changes. I think there was a paper on this about the turn of the millennium.

  • David B. Benson // December 16, 2009 at 1:53 am | Reply

    Pterosaur // December 16, 2009 at 1:20 am — Try learning a bit about geodesy; the study of the geoid.

    Moving mass away from Greenland or Anarctica results in changing, slightly, the rotational properties of the globe and hence just how much the seas rise where. For example, an assumed melting of a substantial part of the West Antarctic Ice Sheet would result in much higher than average sea level rise off the east coast of the United states; it becomes the White Houseboat in Washington, D.C.

  • Pterosaur // December 16, 2009 at 3:01 am | Reply

    Thanks for the links, apologies to those who may have thought I was trolling – I had made the most of the points referred to, but was unable to find specific references which I like to provide to denialists.

    If indeed you still think I’m trolling, check out my contributions at :

    Gutter Trash

  • Gavin's Pussycat // December 16, 2009 at 8:37 am | Reply


    what you’re looking for is the ‘fingerprint effect’:

    The article is here:

  • Ray Ladbury // December 16, 2009 at 2:50 pm | Reply

    Pterosaur, I think we accepted your bona fides from the beginning. You don’t sound like a troll. It is not intuitively obvious that sea level rise should not be uniform. Sincere questions are always welcome here–and people generally put considerable effort into finding help.

    Tamino has declared this blog to be a stupid-free zone, which is why we don’t see much of certain commenters who used to haunt these boards.

  • Pterosaur // December 16, 2009 at 9:25 pm | Reply

    Thanks guys – and the “stupid free” zone is certainly a welcome relief :-)

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