 ## Gordian Knot of Nonsense – Part 2. A simple hypothetical temperature model

September 5, 2011

I will refer to “Climate related sea-level variations over the past two millennia” (Andrew C. Kemp, Benjamin P. Horton, Jeffrey P. Donnelly, Michael E. Mann, Martin Vermeer, and Stefan Rahmstorf, PNAS, 2011)  as KMVR2011.

As I noted in the previous post, the KMVR2011 model is the progeny Vermeer and Rahmstorf’s 2009 PNAS model and Rahmstorf’s 2007 Science model.

Here is the KMVR2011 model where Where H is the sea level, T(t) is the global temperature, Too, a1, a2, b and τ are all constants and To(t) is a to-be-determined time varying function related to T(t) as defined by equation 1a.

Now, consider a temperature evolution of the following  form, where t’ is a constant… Note the following points about equation II

• KMVR2001 stipulate that a1 + a2 = a, where a is defined in VR2009.
• VR2009 says a = 5.6 ± 0.5 mm/year/K  > 0
• Therefore a1 + a2 > 0
• b is defined in VR2009, where they claim that b =  -49 ± 10 mm/K.
• Therefore, b < 0
•  Therefore, -(a1 + a2)/b > 0
• Since -(a1 + a2)/b > 0, then the exponential in equation II increases with increasing t.
• So, if C is chosen to be positive, then T(t) is increasing with increasing t.

Does equation II present a realistic temperature evolution?  Figure 1 shows some simple examples compared to the GISS global temperature.  Figure 1 uses a =  a1 + a2 = 5.6 ± 0.5 mm/year/K and b =  -49 ± 10 mm/K, but it would look the same, qualitatively, for any choice of a and b used in the KMVR2011 Monte Carlo simulations used to populate their data for their “Bayesian updating.”

What happens to the sea level rise rate, dH(t)/dt, when equation II is inserted into equation I? All the terms on the right side of equation III are constants except T0(t).    d2H(t)/dt2 is the rate at which the sea level rise is increasing or decreasing.  So given a time evolution in the form of equation II… We know that a2 is greater than or equal to zero. (If a2 = 0, then the KMVR2011 model becomes exactly the same at the VR2009 model.)   What about dT0(t)/dt? (i.e. How does the equilibrium temperature change with time?)

Consider equation Ia.  Notice that T0(t)  is always trying to “catch up” with T(t).  That is, if T(t) > T0(t), then T0(t) is increasing.  Conversely, if T(t) < T0(t), then T0(t) is decreasing.

Since we are told that the world is now at unprecedented high temperatures compared to the last millennium, then by implication T(t) > T0(t) for the present day.  This obvious point is confirmed for the last 100 years by KMVR2011 figures 4A & 4C.  Consequently, dT0(t)/dt must be increasing with time for the present day and for the entire last century.

## Pulling it all together

Given the KMVR2011 model described by equations 1 and 1a, and given a temperature evolution described by equation II (and as illustrated by the various model temperatures in figure 1, and “Model temperature 2” in particular), then  d2H(t)/dt2 is negative.  That is, the sea level rise rate is guaranteed to be decreasing.  This is a rather bizarre result that is a consequence solely of the design of KMVR2011’s model.  It is not some math trick or mistake.  KVMR2011 should have been able to anticipate this problem, since it parallels very closely a similar problem with VR2009.  And I know Mr. Rahmstorf was reading my blog.

KMVR2011 would likely argue that my equation II cannot represent a realistic temperature scenario, and that their model can only work for realistic temperature scenarios.   Figure 1, model temperature 2, above refutes such a claim.  We are left with the following situation: When a simple temperature scenario with a rapidly rising temperature that is similar to the last half of the 20th century is applied to KVMR2011’s model, it yields a decreasing sea level rise rate.  This point alone should be enough to raise the eyebrows of  KMVR2011’s readers.

But there is much more to come.

1. • 