The Earth is a Thermometer

I allude, of course, to one of my earliest posts on the subject of the short-term (years to decades) variability of the climate, particularly in the northern mid-latitudes, i.e. where I live – wherein it was revealed that the Earth is, in fact, a giant fridge!

I’ve recently been pondering on the question of the extent to which short-term changes in the Earth’s climate would show up in sea-level measurements. Serendipitously, my Inbox beeped a few days ago (OK, it didn’t make a noise, I’m just thinking of the movie treatment!) to tell me that a piece, Science Story: the Making of a Sea Level Study, by Martin Vermeer, had just been posted to Realclimate.

Vermeer’s essay is fascinating on at least two levels. It gives an insight into the sociology of science, and it describes an interesting finding. I’d urge everyone to read the piece carefully. But just in case some don’t want to, I’ll summarise what it says.

First, though, let’s just show what we’re trying to explain.

Observed sea-level rise

The IPCC has (of course!) put together a graph of how sea-level has risen since 1880:

Annual average global mean sea-level (IPCC Fig.5.13)

What bothers me about this particular graph is that the rate of sea-level rise was apparently only marginally affected by the slight cooling of global temperatures over the period from the early 1940s to the late 1970s. This cooling is usually explained as resulting from aerosols from industrial pollution, so it might be suggested that only land areas were affected. But the data do not bear this out. See the IPCC graphs in my previous post discussing the Atlantic Multidecadal Oscillation (AMO) natural cycle.

Causes of sea-level rise

The IPCC has summarised research on ice-melt and the heat in (and hence expansion of) the oceans to put together a breakdown of the different causes of sea-level rise (all figures in mm/yr):

This data reflects a lot of uncertainty, particularly as to whether Antarctica is gaining or losing ice! What surprised me, though, was that thermal expansion only accounts for just over half the sea-level rise over the rapid warming period 1993-2003 and somewhat less before that. It’s therefore a little odd that so many people are going round saying that most of the sea-level rise is due to thermal expansion.

So how is the data explained?

Vermeer describes how he was able to refine the mathematical description of sea-level rise:

Rahmsdorf’s first approximation

The story starts with a paper by Stefan Rahmstorf in 2007, referred to as R07, which suggested that the rate of sea-level rise is proportional to the difference between the actual (average surface global) temperature and an equilibrium temperature.

Rahmstorf’s approximation makes a lot of sense for sea-level rise caused by melting ice. You can imagine that the amount of ice melting each year is proportional to the difference between the temperature that year and an equilibrium temperature when no ice would melt (or the same amount would be formed as melts).

But ice melt is not the only cause of sea-level rise. However, imagine that the temperature at the surface of the sea is above equilibrium. The sea will continue to take up heat for many years, because it takes so long for the depths to warm up. Rahmsdorf’s approximation also includes this gradual take up of heat (though it’s all lumped together with ice melt in one term). There’s a slight problem, though, that Rahmstorf doesn’t account for the heat taken up immediately by the surface ocean as temperature rises. Also, Rahmstorf’s model doesn’t allow for the gradual slow-down in heat uptake as the depths warm up.

Vermeer’s (and Rahmstorf’s) second approximation

Vermeer came along and, working together with Rahmstorf, as described at Realclimate, took account of the uptake of heat by the surface ocean as temperature rises. The rate of sea-level rise therefore becomes proportional to the difference from an equilibrium temperature plus a quantity determined by the rate of temperature rise. i.e. for the arithmetically inclined, the rate of sea-level rise, dH/dt = a(T-Te) + b(dT/dt) where T and Te are the average surface temperature at a given time and at equilibrium and a and b constants to be determined.

A few comments:
1. Note how the surface ocean acts as an extension of the atmosphere for heat, similar to the way it does for CO2.
2. Vermeer still does not model any slowing of heat uptake (and hence sea-level rise) with time. For CO2, the Bern carbon cycle model focuses solely on this aspect (i.e. the Bern model is of a pulse upwards in atmospheric CO2 asymptotically returning to equilibrium). Curious how the work on heat uptake has gone in an entirely different direction to that on CO2 uptake.
3. Note that you can’t conceive of the ocean responding to a temperature rise. Rather, the overall average global surface temperature is largely determined by that of the ocean surface. One way to think about it is that excess heat absorbed by the ocean causes both the sea-level and the surface temperature rise.

Vermeer’s negative b problem

Vermeer’s model works well, he says, when tested against climate model data, and historical data for the last millennium. Unfortunately, though, with observed data from 1880 onwards, he can only make a fit with negative b. That is, it seems that the top 100m of water loses ~4.9cm/C (or K, i.e. degree of temperature) rather than (as determined from the computer models) ~2.5cm/C.

It simply cannot be the case that b is negative. Clearly, what’s needed is a model that’s even more complex, and adds up all the terms – what happens to the ocean surface waters immediately the temperature rises, the ice melt, soil moisture and so on.

Vermeer suggests implausibly that either a temperature rise has an initial negative response on sea-level – which makes no sense (note that the increase in atmospheric water vapour is an order of magnitude less than the surface water thermal expansion at ~2mm/C) – or “a positive, but time-lagged sea-level response”. This simply can’t be what’s really happening. I prefer logic to screeds of math, but as pointed out here more rigorously, if b is negative, that would imply that a halt in rising temperatures would cause a sudden increase in the rate of sea-level rise! And you can see clearly from the IPCC’s sea-level graph, shown above, that volcano induced cooling caused by Agung in 1963 and El Chichon in 1982 (Pinatubo in 1991 is not quite so clear, and other short-term variability in the data is likely due to El Nino events or, more precisely, the El Nino Southern Oscillation or ENSO) led to actual sea-level falls (presumably the cooling was more than ~0.1C which would cause a 2.5mm sea-level drop, based on Vermeer’s estimate for the true value of b, exceeding the average rate of sea-level rise, 2.46mm/yr, according to Chao et al, see below). b can’t be negative. Period.

I think the real reasons for a poor fit of Vermeer’s equation to recent data are some or all of:
1. The equation is too simple (as already mentioned).
2. The data is poor – note how much of the sea-level rise is unexplained in the bottom-up analysis in Table 5.3 from the 4AR, quoted above.
3. The data is accurate, but there are causes of sea-level change, not directly attributable to global warming, which are not properly accounted for (which at least partly explain the error bars in Table 5.3).

Let’s explore other causes of sea-level change:

The dam data

Vermeer describes how he improved the statistical fit of his model to the data by including an analysis of water stored in reservoirs since the start of the era of serious dam-building post WWII, as described in a paper by Chao et al. Now, I’m quite prepared to accept that dam-building accounts for short-term variability and that adjusting for reservoir water allows Vermeer to achieve a better fit.

But the improved fit doesn’t necessarily mean that the parameters a and b are any more correct.

My problem with the Chao paper is that it downplays the reverse effect – water released rather than water stored. There are several sources of this.

First, though, let’s get our units sorted. Chao suggests that reservoirs have stored around 11,000 km3 of water, equivalent to a sea-level drop of around 30mm. Let’s call it 3mm/1000 km3. What else could cause the release of 1000 km3 water or more?

Silt and seepage

Chao et al add about 3000 km3 of water as seepage from dams, raising water tables near reservoirs. Fine, but surely this water might have “seeped” from rivers if it hadn’t been trapped behind dams?

Chao et al also ignore the silt behind dams, arguing (contrary to other sources they reference) that the silt would have flowed to the sea with the water. Some might, but much might have been deposited on river flood-plains – that of the Nile, for example.

Use of fossil water

I really don’t see how reservoir water can be reduced from the sea-level rise when the use of water from aquifers is not added to sea-levels. For example, Fred Pearce writes in “When the rivers run dry” (p.79) that;

“Overall total pumping in India, China and Pakistan probably exceeds discharge by 150-200 cubic kilometres a year. The boom has so far lasted twenty years…”

That’s 3-4,000 km3 just there.

Pearce also describes (p.81ff) how at least 1000km3 has been taken from the High Plains aquifer in US.

And fossil water is being used all over the world.

The Aral Sea

The 4AR IPCC report (section 5.5.6, p.419) covers most of sources of sea-level rise I’m covering here, but for some reason doesn’t mention the Aral Sea.

There’s another 1000 km3.

And there are other. less dramatic examples: the Dead Sea, Lake Chad…

The Long Tail

Then there are wetlands and other ecosystem changes that have the effect of removing water, mainly from soils. Let’s do a tiny bit of math:
– humans have affected roughly 100m km2 of land. (2/3 of the land area – I’m just doing order of magnitude stuff here). 1cm of water over that area is 1000 km3.
– or take of the order of getting towards 10m km2 that we could estimate has been deforested or degraded over the 20th century. Say 30cm of water over that area is another 3000km3.
– similarly, maybe 1m km2 turned to desert, with perhaps 1m of water lost from the soils, depending what we started with. That’s another 1000km3.
– or drained wetlands, probably a few 100,000 km2. But several metres of water could be lost – that is maybe another 1000km3 in total. (Plus sandstorms could end up releasing mass to the sea, raising sea-level).

Then we have fossil-fuel burning which releases water as well as CO2 (some of which increases sea-level when it is taken up by the oceans). Maybe 500km3 in total (25Gt CO2/yr, only ~1/4 taken up in the oceans but ~8Gt H2O depending on the fuel all taken up ultimately).


Vermeer’s model of the effect of temperature on sea-level may well be a step forward. However, it requires modelling against properly adjusted data. Chao et al find a virtually straight line relationship i.e. roughly constant 2.46mm/yr sea-level rise (difficult to reconcile, at least by sight, with the slower rate of sea-level increase from 1940s to ~1980, after adjustment, shown in Vermeer’s paper), which is implausible. Maybe with proper adjustment, the parameter for Vermeer’s short-term warming term will return to the more plausible positive value of ~2.5cm/C.

The whole story shows some of the problems with the scientific process:
– it seems the Chao et al reservoir adjustment has been included simply because it is clearly quantified. The aquifer and ecosystem water storage changes need to be quantified and added to Vermeer’s and other models. Collectively, it’s likely they are more significant than reservoir water storage. The IPCC just assesses research that’s been done. It would be helpful if a list of topics that need to be urgently researched were included in each report.
– it is essential that mathematical modelling of data is not permitted to gain a life of its own. The true functions are much more complicated than Vermeer’s equation, which should really be divided into at least 3 functions: ice melt due to difference from an equilibrium temperature (itself increasing as ice melts); surface sea water expansion; the effect of slower deep water warming (perhaps similar in form to the ice melt). But, quite clearly, the surface waters expand as their temperature rises. If other effects of increased temperature act to reduce sea-level, then these must be modelled as separate terms.
– Vermeer (and Rahmstorf, Chao et al, various reviewers and so on) have put in a huge amount of effort producing papers for publication, but I just wonder if it might not have been a more effective use of time to have had a more open dialogue about the general approach to be followed. In other words, is the peer-review filter letting through enough of the right sort of thinking at the right time? Do we need better fora – somewhere between Realclimate and journals – for scientific debate at a higher level?