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

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, T_oo, a₁, a₂, b and τ are all constants and T_o(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.”

Figure 1. All of these temperature models satisfy equation II.

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.

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