## Rahmstorf (2009): Off the mark again (part 7), constant sea level rise rate

September 7, 2010

One more go-around with Vermeer’s and Rahmstorf’s 2009 PNAS paper, “Global sea level linked to global temperature” (referred to as VR2009 for the rest of the post).

Recapping the first 6 parts…

Part 1, The basic problem.

Part 2, A little more detail on the math.

Part 3, A few examples that show some bizarre consequences that would result if Vermeer’s and Rahmstorf’s model were correct.

Part 4, Improbable parallel universes.

Part 5, Variation of gamma.  Fast increasing temperatures cause sea level rise rate to drop, while slowly increasing temperatures cause sea level rise rates to increase.

A look at Church and White sea level data.  This is the sea level data that is the foundation of VR2009.

Response to RealClimate comments.  A few poorly considered comments concerning this series about VR2009 showed up at RealClimate.  This is my response.

Chao’s artificial reservoir “correction” to sea level.  This “correction” to Church and White’s sea level data leads t0 a (supposed)larger sea level rise during the 20th century.  But this “correction” has some critical flaws.

Part 6, Vermeer’s and Rahmstorf’s model is applied to satellite sea level data and fails the test.

Part 6.5,  the gory mathematical details from part 6.

## Part 7

VR2009 relates sea level rise rate to temperature with the following model…

where H is the sea level and T is the global temperature.  VR2009 has already told us the values for a, b, and To (a = 05.6 mm*a-1K-1, b = -49 mm*K-1, and To = -0.41 K).

This can be rearranged to give temperature as a function of sea-level rise rate (see part 6.5 for the details)…

In parts 6 and 6.5 satellite sea level data, H,  was fit to a quadratic and its derivative, dH/dt, inserted into equation 2 to determine T.  A much simpler procedure would be to fit the satellite sea level data to a line, and use its derivative (i.e., its slope) for dH/dt in equation 2.

If you think that fitting the satellite sea level data to a simple line is unrealistic, you should take your criticism to  NASA and the University of Colorado…

Figure 1. Satellite measured sea level from the University of Colorado.

Following the lead of the University of Colorado,we can estimate the sea level rise rate for the last 17 years with a linear fit to the sea level data.  After looking at this data it does not seem far fetched for the sea level rise rate to still be about 3 mm/year for the next 5 years, 10 years, or even 15 years.  Letting dH/dt = C1 = 3 mm/year, then from equation 2…

All the terms in equation 3, except for C2 (the constant of integration), are already defined.  If initial conditions (that is, the temperature, T’,  at time, t’) are known, then equation 3 can be solved for c2…

Possible Temperature Evolutions

The following image shows a set possible temperature evolutions which all yield a 3mm/year sea level rise rate overlaying the GISS temperature.  The only difference between each curve is the choice of initial conditions used to set the value of c2.  It may seem strange that there are some cases where Vermeer’s and Rahmstorf’s model indicates that the temperature would have to continuously drop in order to maintain a sea level rise rate of 3 mm/year.  (It will seem less strange if you read part 3 of this series, particularly example 4.)  I want to stress this is not my invention, it is the natural consequence of their model with various choices of initial conditions.

Figure 2. Possible temperature evolutions based on Vermeer's and Rahmstorf's model and various choices of initial conditions.

So what is the best choice of initial conditions?  I propose three possibilities: the initial conditions that give temperatures closest to the GISS temperatures over the period for which satellite sea level data is available, as determined by a least squares fit; or, the initial conditions that give a 2010 temperature that is closest to the average GISS temperature for 2010 so far; or, 0.44 degrees at time 2001.5 (the same conditions used in part 6 and part 6.5).  All three are laid out in figure 3.

Figure 3. Three temperature evolutions based on Vermeer's and Rahmstorf's model and specific choices of initial conditions.

Figure 3 can serve as a type of Rorschach test.  If you are a completely gullible global warming alarmist you will see rising temperatures, doom, and the end of the world.  If you are an anthropogenic global warming skeptic you will see a preposterous model.

If you are a global warming alarmist and mathematically illiterate, but still retains a shred of sanity, you will say to yourself “The author of this post must have gotten the math wrong. Even I know that it will take a hundred years for the temperature to go up 4 degrees, not a mere 15 years.”

If you are an anthropogenic global warming skeptic who is mathematically literate, you will say “this shows the vast unlikelihood that Vermeer’s and Rahmstorf’s model relating sea level rise to temperature is valid.”

## What would Vermeer and Rahmstorf say?

What would Vermeer or Rahmstorf say to this?  I have reason to believe that one or both of them have seen my criticisms, but they have chosen to remain silent.  But my guess is that they would point out that they use IPCC temperature scenarios and derive sea levels, and that those IPCC temperature scenarios do not show anything like the extreme temperature changes in 10 or 15 years that I show in figure 3.  But the sea level rise rates that they derive are much greater than 3 mm/year for the next 10 or 15 years.

How can that be?  It is a consequence of their model (see equation 1, above) having a negative b.  This causes times with high rates of temperature increase, dT/dt, to have lower sea level rise rates than times with the same temperature but low rates of temperature increase.  If this seems to go against your intuition, well, it went against their’s also.  Vermeer said..

I contacted Dr. Rahmstorf, proposing the idea: one would expect the ocean surface to warm up rapidly to completion, contrary to the deep ocean and the continental ice sheets. This would argue for a term, in addition to the secular a (TT0) term, of form b dT/dt…I downloaded Stefan’s script, modified it, did the first computations with the same real tide gauge and temperature data Stefan had used — surprise: negative b. Hmmm, strange. (emphasis added)

So, consider the situation.  Vermeer conceptualizes a model (equation 1, above).  It is extremely simple and relates the sea level rise rate (dH/dt) to a mere two terms, a(T-T0) and bdT/dt.  Since he conceptualized the model,  he must have had some idea of what the two terms that he chose meant.  He clearly had an expectation that as the temperature, T, went up the sea level rise rate would go up.   Similarly he expected that higher temperature rise rates dT/dt, would lead to higher sea level rise rates for a given temperature.   When he applied the data to the model he found that fully half of his conceptualization was wrong!

What’s a scientist to do?  Vermeer was faced with two possibilities. The first possibility:  his conceptualization was wrong, but that out of the infinite number of hypothetical models that might actually relate the sea level to the temperature, by sheer luck if he changed the sign of one of the terms in his wrong conceptualization he would end up with the right model.  The second possibility: he was just plain wrong.

## A Test

I always like testable propositions, and figure 3 could serve as a test for Vermeer’s and Rahmstorf’s model.  If the sea level rise rate stays at about 3 mm/year for the next 5 years then VR2009 would require the temperature to go up 0.4 degrees (based on the green curve in figure 3).  That is half the temperature rise of the last century compressed into 5 years!  If 3 mm/year is maintained and the temperature does not go up 0.4 degrees, you can toss their model out with the trash.

Similarly, if the sea level rise rate stays at 3 mm/year for the next 10 years, and the temperature does not go up a whopping 1.1 degrees, …well, adios VR2009.  15 more years at 3 mm/year would require a 2.3 degreee temperature rise.

I’ll be sure to check the numbers 5 years from now.

## More on Thermohaline Circulation

June 16, 2009

In a previous post “The Thermohaline Circulation Only Stops for Extreme, Unrealistic Models,” I compared the amount of fresh water used in “hosing experiment” models to drastically reduce the thermohaline circulation (THC, or Meridional Overturning Circulation, MOC) to the amount of water flowing over Niagara Falls, or flowing from all rivers into the Arctic,  or coming off of Greenland due to melting ice.

The key number was one Sverdrup, or 1 million cubic meters of fresh water per second.  One Sverdrup of fresh water artificially dumped into the Labrador sea, for 100 years would have the feared effect.  But it turns out that one Sverdrup of fresh water is 350 times the amount of water flowing over Niagara falls, and about 300 times the amount of water from melting ice that flows off of Greenland.  It was seen that there is not plausible source for this amount of extra fresh water to be dumped into the arctic.

An interesting letter that appeared in Science a year ago gives a little more perspective,  So I have reproduced it in full here:

Freshwater Forcing: Will History Repeat Itself?

IN THEIR RESEARCH ARTICLE “REDUCED North Atlantic deep water coeval with the glacial Lake Agassiz freshwater outburst” (4 January, p. 60), H. F. Kleiven et al. present compelling evidence for an abrupt deep-ocean response to the release of freshwater from glacial Lake Agassiz into the northwest Atlantic about 8400 years ago. Such data are particularly important in evaluating the response in ocean models of the Atlantic Meridional Overturning Circulation (MOC) to freshwater forcing. For this event, the freshwater forcing was likely large but short; Clarke et al. (1) estimate that the flood had a freshwater flux of 4 to 9 Sv [Sverdrups] released in 0.5 years.

In this context, we are aware of no possible mechanism that might reproduce such a forcing in response to global warming, and all available model simulations, including those with estimates of maximum Greenland Ice Sheet (GIS) melting rates, indicate that it is very unlikely that the MOC will undergo an abrupt transition during the course of the 21st century (2). Multimodel ensemble averages under Special Report on Emissions Scenario (SRES) A1B suggest a best estimate of 25 to 30% reduction in the overall MOC strength (2). In one example, 14 coupled models simulated a 100-year 0.1-Sv freshwater perturbation to the northern North Atlantic Ocean—17 times the recently estimated melt rates from the GIS [Greenland Ice Sheet]—and the MOC weakened by a multimodel mean of 30% after 100 years; none of the models simulated a shutdown (3). Another model simulated greenhouse gas levels that increased to four times preindustrial values and then remained fixed; the resulting GIS displayed a peak melting rate of about 0.1 Sv, with little effect on the MOC (4). One model simulation uses the SRES  freshwater forcing as an upper-bound estimate of potential GIS melting. In this case, the MOC weakened but subsequently recovered its strength, indicating that GIS melting would not cause abrupt climate change in the 21st century (5). Accordingly, we urge caution in drawing comparisons of the abrupt change 8400 years ago to future scenarios involving, for example, the melting of the GIS and its relevance to human societies.

PETER U. CLARK1, THOMAS L. DELWORTH2, ANDREW J. WEAVER1
1Department of Geosciences, Oregon State University, Corvallis, OR 97331, USA.
2Geophysical Fluid Dynamics Laboratory/NOAA, Princeton, NJ 08542, USA.
3School of Earth and Ocean Sciences, University of Victoria, Victoria, BC V8W 3P6, Canada.

References
1. G. K. C. Clarke, D. W. Leverington, J. T. Teller, A. S. Dyke, Quat. Sci. Rev. 23, 389 (2004).
2. G. A Meehl et al., in Climate Change 2007: The Physical Science Basis. Contribution of Working Group I to the Fourth Assessment Report of the Intergovernmental Panel on Climate Change, S. Solomon et al., Eds. (Cambridge Univ. Press, New York, 2007), pp. 747–845.
3. R. J. Stouffer et al., J. Clim. 19, 1365 (2006).
4. J. K. Ridley, P. Huybrechts, J. M. Gregory, J. A. Lowe, J. Clim. 17, 3409 (2005).
5. J. H. Jungclaus, H. Haak, M. Esch, E. Roeckner, J. Marotzke, Geophys. Res. Lett. 33, 10.1029/2006GL026815 (2006).

So, the event that occurred 8400 years ago involved 4 to 9 Sverdrups of fresh water.  This is THOUSANDS of times greater than the flow of the Niagara Falls today.  It is THOUSANDS of times greater than the amount of fresh water flowing from melting Greenland ice today. It is multiples bigger than the entire fresh water budget into the Arctic.

Note that in my previous post I referred to hosing experiments that pumped up to one Sverdrup of fresh water into the oceans.   The authors of the above letter refer to hosing experiments that used only 0.1 Sverdrups – yet they still point out how gigantic this is compared to actual sources of fresh water in the Arctic today.

So, when Al Gore ominously implies that that the Greenland Ice Sheet [GIS] is going to melt down and dump enough fresh water into the Atlantic Ocean to shut down the Thermohaline Circulation, remember the works of Clarke, et.al., in the above letter: “we urge caution in drawing comparisons of the abrupt change 8400 years ago to future scenarios involving, for example, the melting of the GIS [Greenland Ice Sheet] and its relevance to human societies.”

## The Thermohaline Circulation Only Stops for Extreme, Unrealistic Models

June 4, 2009

Gore gives a cartoon description of the ocean circulation system when he explains what has become known as the thermohaline circulation, or the meridional overturning circulation.  In his simplistic scenario the surface ocean current that flows north in the Atlantic, bringing warmth to northern Europe will be halted by melting ice from Greenland, subsequently throwing Europe into an ice age.

Here is Gore’s explanation in his own words from the Inconvenient Truth movie:

The Earth’s climate is like a big engine for redistributing heat from the equator to the poles.  And it does that by means of ocean currents and wind currents.  They tell us, the scientists do, that the Earth’s climate is an non-linear system – just a fancy way they have of saying that the changes are not all just gradual, some of them come suddenly, in big jumps… And so, all those wind and ocean currents that have formed since the last ice age and have been relatively stable – they’re all up in the air – they change.

And one of the ones they’re most worried about, where they’ve spent a lot of time studying the problem is in the the North Atlantic where the gulf stream comes up and meets the cold winds coming off the Arctic over Greenland and that evaporates the heat out of the gulf stream and the steam is carried over to western Europe by the prevailing winds and the Earth’s rotation.  But isn’t it interesting that the whole ocean current system is all linked together in this loop, they call it the ocean conveyor.

And the red are the warm surface currents, the Gulf Stream is the best known of them.  But the blue represent the cold currents running in the opposite direction…

Up in the North Atlantic, after that heat is pulled out, what’s left behind is colder water, and saltier water, because the salt doesn’t go anywhere. And so, that makes it denser and heavier.  And so that cold heavy dense water sinks at the rate of 5 billion gallons per second.  And then that pulls that current back south.

At the end of the last ice age as the last glacier was receding from North America the ice melted and a giant pool of fresh water formed in North America, and the Great Lakes are the remnants of that huge lake.  An ice dam on the eastern border formed, and one day it broke, and all that fresh water came rushing out, ripping open the St. Lawrence there, and it diluted the salty dense cold water, made it fresher and lighter so it stopped sinking, and that pump shut off.

And the heat transfer stopped.  And Europe went back into an ice age for another 900 to 1000 years.  And the change from conditions like we have here today to an ice age took place in perhaps as little as ten years time.  So that’s a sudden jump.  Now, of course, that’s not going to happen again because the glaciers of North America are not there… Is there any other big chunk of ice anywhere near there…?  Oh, yeah [Gore says ominously, as the image pans to ice covered Greenland] we’ll come back to that one…

Later in the movie Gore tells us that Greenland is rapidly melting.  The point being that it will provide a massive amount of fresh water that will stop the the thermohaline conveyor and  “would raise sea level almost 20 feet if it ‘went,'” Gore tells us.  He tells us about water seeping to the bottom of the ice sheets where it “lubricates where the ice meets the bedrock” causing the ice to slide toward the ocean.

Then he shows a series of pictures purporting to show the amount of melting in Greenland.  Gore says…

“In 1992 they measured this amount of melting in Greenland … Ten years later this is what happened…And here’s the melting from 2005”

## Hosing Experiments

But what if…?  What if there were a huge amount of low density fresh water dumped into the North Atlantic where the high density water is supposed to be sinking, just like the giant Canadian lake crashing through the barrier of ice the Gore told us about?  This possibility is explored with computer models known as  “hosing experiments.”  In a hosing experiment a model that simulates the ocean and atmosphere circulation patterns is modified to artificially dump huge amounts of extra fresh water, as if from a giant hose, into some location in the ocean.   It has been found that when enough fresh water is forced in, the circulation can be slowed, but rarely stopped

How much fresh water do the hosing experiments use to nearly stop the thermohaline circulation?  Typically (or here), they use one million cubic meters of fresh water per second, for 100 years!!!  (One million cubic meters per second has its own unit name: One Sverdrup or 1 Sv).  How does 1 Sv compare to, say, the rate of water flowing over Niagara Falls?

168,000 cubic  meters of water fall over Niagara Falls every minute.  That is about 2,800 cubic meters of water per second.  So one Sverdrup of water is the same as about 350 Niagara Falls!  (1,000,000 / 2,800  = 357).  So, roughly speaking, if 350 Niagara Falls were dumped into the oceans around Greenland continuously for 100 years, then we could expect to see a significant slow down of the thermohaline circulation.

River systems discharging into the Arctic Ocean.

How does one Sverdrup compare to the freshwater discharge of ALL the rivers emptying into the arctic ocean?  One Sverdrup of fresh water amounts to nearly 32,000 km3 of water per year  (1 Sv  x 106 m3 s-1/sv x (86,400 s/day) x (365 day/year) = 31,536 km3/year).  The total fresh water discharge from all rivers into the arctic is only about 4,300 km3 per year.  So, typical hosing experiments that nearly stop the overturning circulation add a water volume about 7 times the amount of water from all rivers discharing into the Arctic Ocean combined.

Gore ominously implies that the amount of fresh water needed to turn off the overturning circulation is just waiting to pour off of  Greenland, due of course (drum roll), to CO2 induced anthropogenic global warming.   His pictures of Greenland, shown above, imply that about half of Greenland’s 2.8 million cubic kilometers of ice have melted in the 13 years between 1992 and 2005.  This is wildly misleading.  Only a miniscule fraction of the area shown in Gore’s Greenland images actually melts every year.   This is evidenced by mass balance studies, which show Greenland loses on the order of hundred cubic kilometers of ice every year,  which translates into a measly 0.003 Sverdrups.

100 km3 /year= 1011 m3/year

(1011 m3/year) / (365 days/year) / (86,400 seconds/day)
= 3 x 103 m3/second
= 0.003 Sv

Put another way, one Sverdrup of fresh water is 86.4 km3/day.  So the hosing experiments pouring in one Sverdrup put about as much fresh water into the ocean each day (86.4 km3) as Greenland provides in a year (100 km3).

But if Greenland actually started melting, by some extraordinary circumstance,  300 times faster, then it would yield 1 Sverdrup, or 1,000,000 cubic meters, of fresh water every second.  What would happen after 100 years of melting at that rate?  Well, that’s a trick question, because at a melting rate that gives 1 Sverdrup of freshwater Greenland would run out of ice in about 90 years.  This is because Greenland has only 2.85 million cubic kilometers of ice, and one Sverdrup of water is the same as about 31,500 cubic kilometers of water per year.  Ignoring the difference in density between ice and water, then 2.85 million cubic kilometers divided by 31,500 cubic kilometers per year gives 90 years.

## Conclusion

You don’t hear as much about the threat of the collapse to the thermohaline circulation today as you did a few years ago.  This is because it has become recognized as being a very far fetched possibility, even by most alarmists who want to maintain a shred of dignity.  But I have a feeling we will not see this wildly exaggerated threat removed from new editions of Gore’s “An Inconvenient Truth” anytime soon.