Rahmstorf (2009): Off the mark again (Part 6 and a half). Gory details

This is to fill in some of the details of the math used to reverse Vermeer’s and Rahmstorf’s model ( Global sea level linked to global temperature, 2009, PNAS) to give temperature as a function of sea level.  In my previous post I skipped over these details for the sake of brevity.

In various posts I have used Vermeer’s and Rahmsorf’s (VR2009 for rest of this post) model that relates sea level rise rate to global temperature to see how different temperature scenarios would result in sea level rise rates, according to VR2009.  In my previous post I inverted their model to calculate the temperature from satellite sea level data.  Here are the details of that inversion…

The Math

Starting with VR2009’s model…

Re-arranging equation 1 gives…

If we assume that dH/dt is a known function of t, then equation 2 is a first order linear differential equation.  So, multiplying both sides of equation 2 by exp(at/b) gives..

The left side of equation 3 can be re-written…

If both sides of equation 4 are integrated, then…

Solving for T….

Remember, we are assuming that dH/dt is a known function of t.   We will get that function by taking the derivative of a quadratic fit of satellite derived sea level data.   That is, the satellite sea level data will be fit to…

So, substituting equation 7 into equation 6 gives….

We can solve the integral on the right side of equation 8 as follows…

Substituting equation 9 into equation 8 gives…

So far, so good.

a, b, and T₀ are given by VR2009.  c₁ and c₂ are determined from the best fit of H to a quadratic.  That leaves only c₄ as an unknown.  If initial conditions (that is, the temperature, T’,  at time, t’) are known, then equation 10 can be solved for c₄…

Initial conditions

There are a variety of sources for getting initial conditions (temperature, T’,  at time, t’) to calculate c₄ in equation 11.  However, the final results of the temperature that comes from equation 10 is highly sensitive to c₄, which is in turn highly sensitive to the chosen initial conditions.  VR2009 used the GISS global temperature to derive their model, so we will first consider the GISS Monthly Mean Surface Land/Ocean Temperature Anomaly (which covers 1996 to the present).

For example, we could choose t’ = 1998.12 with T’ = 0.8 °C during the peak of an extreme El Nino.  In this case equation 10 would give a temperature rise of about 2 °C between 1996 and 2010.  Or we could choose t’ = 1999.38, when the global temperature was 0.21 °C  (according to GISS).  For this choice the temperature drops from about 1996 to 2001, and then rises about 0.5 °C by 2010.

These extreme initial condition choices seem to yield extreme results.  Perhaps it would be better to choose smoothed temperature data.  The following plot shows an overlay of the plain GISS Monthly Mean Surface Land/Ocean Temperature Anomaly, and with a 7 month running average (generated by me), as well as the GISS Annual Mean Land/Ocean Surface temperature anomaly.  It seems prudent to select an initial time where the monthly data, the 7 month smoothed data, and the annual mean data are all about the same, as marked in the image, below.

I decided to pick T’ = 2001.5 and t’ = 0.44 °C (GISS monthly T = 0.51 °C, GISS monthly T with 7 month average = 0.46 °C, GISS yearly average t = 0.48 °C, and GISS 5-year mean T = 0.44 °C).

Applying my choice of initial conditions to equation 11 to determining c₄, inserting the result into equation 10, and plotting T vs. t from equation 10 for 1996 to the present gives the following result.

So, Vermeer’s and Rahmstorf’s model requires an unrealistic temperature rise from 1996 to the present to reproduce the sea level rise rate over that period.  The decade from 2000 to 2010 (during which the GISS data shows a more or less constant) would have required a temperature increase of almost 0.7 °C, according to VR2009.

This is just another reason to reject VR2009.

Try it yourself.

I have given any interested person with a basic understanding of calculus and differential equations everything needed to reproduce my results.  My conclusions are sound.  VR2009 is unrealistic.

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