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^{-1}K^{-1}*,

*b*= -49 mm*K

^{-1}, and

*T*= -0.41 K).

_{o}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…

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 = C _{1} = 3 mm/year*, then from equation 2…

All the terms in equation 3, except for C_{2} (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 *c _{2}*…

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 *c _{2}*. 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.

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 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(T–T_{0}) term, of formbdT/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: negativeb.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.