Gordian Knot of Nonsense – Part 3. More Math (Sorry about that.)

“Make everything as simple as possible, but not simpler”

Albert Einstein

As usual, 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.

I would like to elaborate on my previous post, in which I presented a simple temperature vs. time function that causes KMVR2011′s model relating sea level rise rate to global temperature behave in a rather peculiar manner.  I am trying to find a balance between simplicity, clarity and thoroughness.  The level of mathematical literacy of my readers may vary widely, but this time around I need to employ some calculus.  If the equations bother you,  just consider the conclusions.

Starting with the conclusions

There exists a simple class of realistic temperature vs. time functions, which when applied to KMVR2011′s model yield results that disqualify it as representing a relationship between global temperature and sea level rise rate.  This class of temperature vs. time functions gives a family of curves for which it is guaranteed that the higher the temperature the lower the sea level rise rate.  This implausible effect is so severe that if forces rejection of the KMVR2011 model.

The Math

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 Ia.

Now, consider the following temperature evolution.  It is nearly the same as equation II from my previous post, but has an additional unitless constant, γ (a.k.a. “gamma”), in the exponential…

If equation II is inserted into equation I, then…

Rearranging terms in equation III gives…

H is the sea level.  dH/dt, the derivative of the sea level,  is the sea level rise rate.  d²H/dt², the second derivative of the sea level, is the rate at which the sea level rise rate changes.  That is, d²H/dt², is the acceleration.  If d²H/dt², is positive, the sea level rise rate is increasing.  Conversely, if d²H/dt², is negative, then the sea level rise rate is decreasing.  Taking the time derivative of equation IIIa gives…

Let’s also consider the difference in the sea level rise rates at some time, t, for different values of γ.  We can do this by analyzing the derivative of dH/dt (equation IIIa) with respect to γ.

What does the math tell us?

KMVR2011 does not conclude with specific values for their model constants and their time varying T₀(t).  Instead, they present probability density distributions for some constants, or combination of constants.  However, there are some definite constraints that can be noted about the variables and their relationships to each other.  These constraints are key to my conclusions.

Constraints:

1. a₁ + a₂ = a, where a1 and a2 are defined in KMVR2011 (see equation I, above) and a is defined in VR2009.  VR2009 found a = 5.6 mm/yr/K.
2. a₁ > 0 mm/yr/K  and a₂ > 0 mm/yr/K.  KMVR20011 states that the distribution of a₁ for their Bayesian analysis varied between 0.01 and 0.51 mm/yr/K.   Needless to say, if either of these terms were less than zero the KMVR2011 model would make even less sense that it does now.  That would be a road that the KMVR2011 authors do not want to travel.
3. b < 0 .  VR2009 found b = -49 mm/K.   KMVR2011 varied b about -49 mm/K with σ² = (10 mm/K)² for their Bayesian analysis.
4. C  > 0.  C is a unitless constant that I introduced, and for the purposes of this post I am constraining C to be greater than zero.
5. γ > 0.  γ is a unitless constant that I introduced, and for the purposes of this post I am constraining γ to be greater than zero.
6. Time, t, is restricted to about 1900 and later for my hypothetical temperature (equation II).  This insures that T(t) > T₀(t), which in turn insures that dT₀(t)/dt > 0.

The equations above, coupled with the listed constraints guarantee the signs of the derivatives shown in table 1, below.

Table 1. Derivatives of temperatures and second derivatives of sea level. Green “up arrows” indicate increasing values and red “down arrows” indicate decreasing values.

As you can see from table 1, it gets little confusing for 0<γ<1.  When a₁, a₂, b, C, and γ conform to the listed constraints, the signs of the various derivatives are known with certainty as long as…

But when …

at some point in time t- t’ will become large enough that d²H/dγdt will become positive.  When that time occurs depends on the choices of a₁, a₂, b  and γ.  If we choose a₁, a₂ and b to agree with VR2009 (recall a₁ +a₂ = a = 5.6 mm/yr/K, and b = -49 mm/K) and γ = 0.8, then d²H/dγdt will continue to be negative until t – t’ = 44 years.

The conclusion, again.

Equation 2, above, can be used to build realistic hypothetical temperature evolutions.  See figure 1, here, for some examples.  Remember, KMR2011′s model relates sea level rise to temperature, and when applied to these hypothetical temperatures it must yield realistic sea level rises.  It does not. 

Table 1 shows various aspects of temperature and sea level using my hypothetical temperature evolution and KMVR2011′s resulting sea levels.  Summed up succinctly, the table shows that with this combination the greater temperatures result in lower sea levels.  This implausible situation disqualifies KMVR2011′s model. 

Coming soon

I realize that a bunch of equations and a table do not give visceral understanding of this effect.  A graphical illustration of these points will be coming soon.

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Update (9/30/11)
Table 1 corrected.  Change makes no difference to conclusions.

Gordian Knot of Nonsense – Part 1. Rahmstorf and company strike again.

Rahmstorf and friends are at it again, but this time they have signed on a bigger fish: Michael Mann of hockey stick infamy.  Somehow it does not surprise me that this new serving of dribble comes to us via the Proceedings of the National Academy of Sciences.  Frankly, it grieves me to know that this is the state of the scientific culture in the US. 123

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.  This paper dishes up a third generation model relating sea level rise rate to temperature whose immediate ancestors are Rahmstorf’s 2007 model and Vermeer’s and Rahmstorf’s 2009 model.

With H being sea level and T being global temperature the models have evolved as follows.

Generation 1, form Rahmstorf’s 2007 ”A Semi-Empirical Approach to Projecting Future Sea-Level Rise“

Generation 2, from Vermeer and Rahmstorf’s 2009 ”Global sea level linked to global temperature“

And now, Generation 3, from KMVR2011

where

A cursory examination of equation I makes it plain the this new model is simply the cobbling together of  the VR2009 model (with a₁ and T_oo in this model being the same as a and T_o  respectively in VR2009) with an additional term,  a₂[T(t) - T₀(t)], taken from Jevrejeva (GRL, 37, 2010).  KMVR2011 sum up the meanings of each term in equation I as follows…

The first term captures a slow response compared to the time scale of interest (now one or two millennia, rather than one or two centuries as in [VR2009]). The second term represents intermediate time scales, where an initial linear rise gradually saturates with time scale τ as the base temperature (T0) catches up with T. In [VR2009], T0 was assumed to be constant. The third term is the immediate response term introduced by [VR2009]; it is of little consequence for the slower sea-level changes considered in this paper.

 In Rahmstorf’s 2007 model linking sea level rise rate to temperature there were only two constants (a and T_o) that needed to be determined.  The 2009 Vermeer and Rahmstorf (VR2009) model went a step further with three constants (a, T_o, and b) that needed to be determined.  The new KMVR2011 model advances the science with four constants (a₁, a₂, T_oo and b).  Count them!  But even more astonishing: this model requires not just solving for the four constants, but also a time varying function (T_o(t) )!

Back at the keyboard

I have had a leisurely summer, and have not written any blog posts for several months, but my eyes and ears have been open, and my pencil has scratched out a few equations.   This post represents the beginning of a new series on KMVR2011, which I will call the “Gordian Knot of Nonsense.”

This series will be interspersed with posts on other topics, so please check back occasionally for updates.

Tree-rings: Proxies for Temperature or CO2?

Recall step 2 of the  five super-simple steps for building a hockey stick:

Step 2. Select those time series that fit the instrumental (measured) temperature record of choice. Assume that since these time series match the measured temperature in some way, then they are, in fact, temperature proxies.

This step begs the question (in the classical logical sense) about the usefulness of tree-rings as proxies for temperature.  Surely tree-ring width is not solely dependent on temperature, is it? 

What about drought conditions?  Usually we think of droughts as coinciding with high temperatures.  Would higher temperatures cause tree-rings to be thicker even when the tree is being stressed or dying due to lack of water?  Of course not.

What about the abundance of CO2? 

Back when I was a college student I worked at Phytofarms of America in DeKalb, Illinois, USA, which grew the highest quality leafy vegetables in a giant indoor,  innovative, artificially lighted, hydroponic facility.  One of the keys to this industrial sized facility was elevated CO2.  Huge tanks of CO2 pumped up the indoor level to about 1000 ppm, or about 4 times the pre-industrial level.  The resulting veggies were expensive, but they were the best money could buy.

Is it possible that tree-rings are better proxies for atmospheric CO2 than for temperature?   As a simple test, I selected all the tree-ring proxies used for Mann’s 2008 version of the hockey stick and did a simple correlation to the Northern Hemisphere instrumental temperature record and to the atmospheric CO2 record.  The tree-ring data and the instrumental temperature record came from the NCDC archive for Mann’s paper. 

 The CO2 data is a combination of Mauna Loa data (1959 to present) and the Siple Station ice Core (1744 to 1953).  The Siple data was not in yearly increments, so I interpolated.   I also interpolated between the end of the Siple data (1953) and the beginning of the Mauna Loa date (1959).  The Mauna Loa data was truncated beyond 2006 so that the CO2 data would cover the same time domain as the instrumental record used by Mann.

Results

The first graph below (click to enlarge) shows the 30 tree-ring time series with the best correlations to the instrumental temperature record.  Each of these correlations has been matched with the correlation to the CO2 level.  The first thing to jump out is that for 23 out of 30 cases, the CO2 correlation is better than the temperature correlation!

The next graph is the reverse situation.  It shows the 30 tree-ring time series with the best correlations to CO2 in descending order.  As above, each of these correlations is matched with the correlation to the instrumental temperature record.  This time the thing that jumps out is that the correlation to CO2 is better in every single case.  And these correlations to CO2 are not just a little better, they are a lot better.