I was highly critical of Stefan Rahmstorf’s 2007 attempt to scare the world about sea level rise. He claimed that any temperature rise would result in an increase in the sea level rise rate that would take a millenium to dampen out. I pointed out at the time that this claim was contradicted by the very data that he used.

Rahmstorf”s original equation simple equated the sea level rise rate to the temperature above some equilibrium temperature.

where *H* is the sea level, *dH/dt* is the sea level rise rate (that is, the change in sea level per unit time, *t*), and *T _{0}* is some equilibrium temperature. According to this formula, when the temperature,

*T*is constant, the sea level rise rate is constant. I pointed out instances where the instrumental temperature was nearly constant, but the sea level rise rate dropped, contradicting his model.

Now, Martin Vermeer and Rahmstorf (referred to as “VR2009″ for the rest of this article) have given it another try. They have added another term, *dT/dt* which they say “corresponds to a sea-level response that can be regarded as ‘instantaneous.’”

where dT/dt is the change in temperature per unit time. They fit the GISS temperature data and tide gauge sea level data from Church and White (with a reservoir correction) to the equation to determine the best values for *a*, *b* and *T _{0}*. For the sake of argument I will accept VR’s values (

*a*= 0.56 +- 0.05 cm*a

^{-1}K

^{-1},

*b*= -4.9 +- 1.0 cm*K

^{-1}and

*T*= -0.41 +- 0.03 K).

_{o}Intuition might make you guess that the constant, b, would be a positive value, since it would seem that an increase in temperature would cause an increase in the sea level rise rate. But VR2009 found that the best fit to their data yielded a negative value for *b* (*b* = -4.9 +- 1.0 cm*K^{-1}).

They proffer two possible explanations for this counter-intuitive result. The first is that when temperature rises “higher evaporation from the sea and subsequent storage of extra water on land, e.g., in form of soil moisture” results in an initial drop in sea level rise rate. They correctly discount this explanation by pointing out that “It is hard to see how the very large amount of water needed to be stored on land could remain inconspicuous.”

Their preferred explanation is as follows….

“Thus, the most plausible physical interpretation of our statistical fit is that the negative value of

bresults from a positive ocean mixed layer response combined with a lag of over a decade in the response of the ocean-cryosphere system. Several mechanisms could be envisaged for a delayed onset of sea-level rise after warming. For example, mass loss of ice sheets can be caused by warm water penetrating underneath ice shelves, triggering their collapse and subsequent speed-up of outlet glaciers banked up behind the ice shelf (21). We cannot explore causes of delay in more detail here, but note the statistical result is robust irrespective of its causes.”

## description vs explanation

It is very important to understand that VR2009′s model (equation 2) is put forth as more than just a *description* of sea leve rise for the last 120 years. Rather it is an *explanation* for that sea level rise. If this point is obvious to you, then please forgive me for belaboring it and skip to the next section.

But the difference between a formula that *describes* and a formula that *explains* is essential to understand, For example, if you were to drop a rope on the ground in a random fashion you could come up with some kind of formula for the elevation of the rope at each point along the first half of its length. Perhaps you would fit the elevation to an 10th order polynomial that mimics the pattern that the first half of the rope made. But that formula would have no power to tell you the pattern of the second half of the rope. Your formula would be a *description* of the pattern of the first half of the rope, but not an *explanation*. Also, if you lifted your rope and dropped it in a new pattern, your formula would now be useless to predict the elevation of the first half of the rope again.

Now suppose you dropped your rope on the ground and noticed that the rope made contact with the ground along its entire length. You could come up with a formula giving the elevation of the first half of the rope, Z_{r}(x), as a function of the elevation of the ground, Z_{g}(x). Namely, Z_{r}(x) = Z_{g}(x). This simple formula does more than *describe* the elevation of the rope – it *explains* it. This simple formula has predictive power for the second half of the rope. That is, given the elevation of the ground you can predict the elevation of any point on the second half of the rope. This simple formula also lets you draw conclusions about what the elevation of the first half of the rope would have been if it had been dropped in a different pattern.

## VR2009′s simple temperature rise scenario

If realistic data is applied to a model that is purported to explain a phenomenon, and the result is obviously unrealistic, then that model must be rejected. In this section I will explain how VR2009 apply a realistic temperature scenario to their model, namely a linear increasing temperature, to explain the effect of the counter-intuitively negative value of the model parameter, *b*. Their result is satisfying. But in the next section I will apply another realistic temperature scenario to their model, and the result will be outrageously bogus. This will force the rejection of their model and its predictive power for the 21st century.

VR2009′s preferred explanation for a negative b is that it is related to a physical phenomenon that results is a time lag between a temperature increase and an increase in sea level. They consider a hypothetical case where the temperature, *T*, undergoes a steady linear rise starting at time *t=0*. That is, *T=ct*, where *c* is a constant. In this case, dT/dt = (d/dt)ct = c. For the sake of simplicity, they assume that a temperature scale has been chosen such that *T _{0}=0*. Then…

Integrating gives…

The following image illustrates how equation 4 leads to an initial drop in sea level with an increasing temperature for their hypothetical linearly increasing temperature.

So, VR2009 have a model equation which counter-intuitively shows an initial drop in sea level rise with rising temperature. It is backed up by the just-so story about “warm water penetrating underneath ice shelves” of which they “cannot explore causes of delay in more detail here.” Well, very nice.

## An alternative temperature rise scenario

VR2009 used a simple linear increase in temperature to show it really makes sense that the constant, b, in their sea level rise rate model (equation 2) is negative. But their model must work with any realistic temperature increase scenario. So I propose at different type of temperature increase to test the saneness of their model.

Consider the case where…

Now, my hypothetical temperature is a bit more complicated than VR2009′s version (*T=ct*), but it is just as realistic, depending on the choice of *C*. The image below shows and overlay of GISS temperature change from 1950 to 2000 (with a 15 year smoothing similar to that used by VR) , VR’s hypothetical temperature scenario and my hypothetical temperature scenario. All three scenarios are examples of realistic temperature changes. All three have been adusted to about the same starting time and temperaure for the sake of comparison. For the VR2009 hypothetical, *c*=0.01 K^{-1}a^{-1}. For the my hypothetical, *C* = 0.002 K, and *a *and *b* are the same as VR2009 derived for their model, namely, *a *= 0.56 cm*a^{-1} K^{-1}, *b* = -4.9 cm*K^{-1}. For all three cases the temperature increase after fifty years is about half a degree.

What happens when we insert my hypothetical temperature (equation 5) into VR2009′s model (equation 2)? Inserting equation 5 into equation 2 gives…

Carrying out the derivative on the right of equation 6 and collecting terms yields…

## Surprise, Surprise!

The hypothetical temperature increase that I propose looks more like what we have seen in the last 50 years than the simple linear increasing temperature that VR2009 used for their illustrative purposes. My hypothetical temperature scenario could be used to create a much more brutal “hockey stick” than anything proposed by Michael Mann by simply increasing the constant, *C*. Yet in VR2009′s model it always results in a sea level rise of zero! You could choose the constant *C* to yield a realistic 1 degree, or an unrealistic 20 degrees, or 100 degrees in 100 years – it doesn’t make any difference - VR2009′s model would still give a sea level rise of zero. My simple test shows that VR2009′s model must be rejected.

This is not a trick of math. VR2009 claim that their model (equation 2) should work with any temperature increase scenario. In fact, they applied their model to 342 future temperature scenarios derived from “six emission scenarios, three carbon cycle feedback scenarios, and 19 climate models.” But my hypothetical temperature increase scenario shows that their model *describes* sea level change in the 20th century, but does not *explain* it – despite their vague just-so story about time lags due to water penetrating under ice shelves, blah blah blah. Consequently, it does not have predictive power for the 21st century.

But that doesn’t stop VR2009 from using this model to scare people about anthropogenic climate change induced sea level rise in the 21st century.