Rahmstorf (2009): Off the mark again (part 2)

To my readers:  Sorry about all the math.  I know it can seem opaque or tedious, but it will pay off in the end.  In a few days a I will post again with simple spreadsheets reproducing Vermeer and Rahmstorf’s relationship between temperature and sea level, using the same data they use, and extrapolating into the 21st century.  I will try to make these spreadsheets in a way that does not require an understanding of the math shown in this post.  The spreadsheets will have plots that make my point obvious, and the spreadsheets will be available for you to download for closer examination.

In my previous post I pointed out a simple expression for temperature that would show Vermeer’s and Rahmstorf’s (VR2009 for the rest of this post) proposed relationship between global temperature and sea level rise to be invalid.  In this post I will elaborate on that simple expression.

VR2009 propose the following relationship between global temperature and sea level rise rate…

where

H is the sea level
T is the temperature
T₀ is an “equilibrium temperature”
t is the time
a and b are constants

VR2009 find a, b and T₀  by fitting to smoothed versions of the GISS global temperature and Church and White’s sea level data with Chao’s correction for artificial reservoir storage.

Once a, b and T₀  are determined, any temperature scenario, like those proposed for the 21^st century by the IPCC, can be entered into the right side of the equation to calculate future sea level rise on the left side of the equation.  When VR2009 apply IPCC temperature scenarios they find 21^st century sea level rises up to 1.8 meters.

In my previous post I pointed out that given VR2009’s proposed relationship between sea level rise rate and temperature (eq. 1), the following temperature scenario…

…would result in a sea level rise rate of zero.

Here I elaborate on equation 2 to build a more useful example.  Consider the following time evolution of the temperature…

Inserting equation 3 into equation 1 gives…

where

a, b, and T_o are the same as equation 1
γ, gamma, is a unitless constant.
C is a constant.
t’ is time when equation 3 starts expressing the temperature.
T_offset is the temperature immediately before equation 3 starts expressing the temperature.

Here are a few points to ponder about equations 3 & 4.

1. VR2009 determined a to be a positive value and b to be a negative value.   I will use their values of a and b, and I will stick to cases where γ is positive.  Therefore, (-aγ/b) will always be positive.  Consequently, the exponential in equations 3 & 4 will always be increasing as time, t, moves forward.
2. The fact that the exponential in equations 3 & 4 will always be increasing in time, and by choosing C to be positive, insures that the temperature, T, of equation 3 will always be increasing.
3. For a given a and b (remember, I will use the a and b values derived by VR2009), the rate at which the temperature, T, in equation 3 increases is governed by γ.  A larger γ means a faster increasing temperature.
4. From equation 4 we see that when γ = 1, dH/dt = a(T_offset – T_o ), which is a constant (d²H/dt²=0).  That is, as long as γ = 1 the sea level rise rate will not change, no matter how high the temperature goes.  But if γ does not equal one, then the  sea level rise rate will change with time.  If γ<1, then the sea level rise rate will increase with time (d²H/dt²>0).  But if γ>1, then the sea level rise rate will decrease with time (d²H/dt²<0). 
5. Points 3 and 4 leave us with a curious situation.  That is, if γ>1, then according to VR2009’s model (equation 1) , the temperature would increase continuously, while the sea level rise rate would drop continuously.

It seems to me that the only hope for advocates of VR2009’s model at this point is to say something like this:

“Sure, Moriarty’s math may be correct, but Vermeer and Rahmstorf’s model can only be expected to be valid for reasonable, realistic temperature increases.  Moriarty’s temperature scenario (equation 3) is not reasonable or realistic.”

I would beg to differ with such a statement.  I will show in a future post that judicious selection of constants for equation 3 will yield temperature scenarios that are not much different from what we have seen for some periods of the 20th century, or would be perfectly reasonable for the 21st century.  When these temperature scenarios are applied to VR2009’s model, they will yield perverse results – making it necessary to scrap VR2009’s model.

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• Church, J. A., and N. J. White (2006), A 20th century acceleration in global sea-level rise, Geophys. Res. Lett., 33, L01602, doi:10.1029/2005GL024826. link
• M. Vermeer and S. Rahmstorf (2009), Global sea level linked to global temperature, PNAS 106, 21527-21532.  link
• B. F. Chao, Y. H. Wu, Y. S. Li., Impact of Artificial Reservoir Water Impoundment on Global Sea Level,320. no. 5873, pp. 212 – 214.  link
  

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