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.

Please see this index of my posts concerning KMVR2011. Check back occasionally because the list of posts is slowly growing.

*T*_{o}(t), the “equilibrium temperature”

Recall the KMVR2011’s model includes a moving target “equilibrium temperure”, *T*_{o}(t), given by equation* Ia*

The “equilibrium temperature” can be determined by inserting the temperature history or scenario into equation *Ia* and solving the resulting differential equation for *T*_{o}(t). Figure 1, below, shows an equilibrium temperature found by KMVR2011 when Mann’s Global EIV land and ocean temperature is used.

Figure 1. this is figure 4C from KMVR2011

In my previous post I laid out a formula (equation *II*, previous post) for temperature vs. time that will cause the KMVR2011 model to yield an unrealistic sea level rise rate for a realistic temperature. In this post I will take the necessary step of finding the “equilibrium temperature” that results when my hypothetical temperature scenario is inserted into KMVR2011’s equation *Ia*. In a subsequent post I will show how my hypothetical temperature scenario and its resulting equilibrium temperature affect the sea level rise rate as calculated by the KMVR2011 model.

### Quick and to the point

Here is *T*_{o}(t).

If you are not interested in the details, you can just take my word it and stop reading here. Otherwise, continue on the following sections.

### “Reasonable” temperature scenarios

Even the best possible model could not be expected to give reasonable results if the input is nonsensical and it would not be a fair test of the model. That is why, for the moment, I am choosing to apply hypothetical temperatures for the past (1960 to 2000) to the KMVR2011 model. In that way the reader can compare my temperature scenarios to the same data used by KMVR2011 for that period and decide if my scenarios are “reasonable”.

The following graph shows five different temperature scenarios created by my temperature formula. Each of these scenarios is identical, except for the choice of *γ (gamma)*.

Are these “reasonable” temperature scenarios? Are they a fair test of the KMVR2011 model? Let’s compare them to Hansen’s GISS instrumental temperature data and to Mann’s (Mann is the “M” in KMVR2011) own Global EIV, Land and Ocean temperature reconstruction for the same period…

*T*_{o}(t) from my hypothetical temperature scenarios

If you agree that my temperature scenarios are reasonable, then without further ado, here is the derivation of *T*_{o}(t).

Let

Inserting equation* II* into equation *Ia* gives

Letting

Then

Solving the differential equation in *IIIa* gives

The constant of integration, *C*_{2}, can be found by choosing a known *T*_{o}(t) at some time,* t’…*

…and solving for* C*_{2}

Now, simply substitute equation *VI* in equation *IV* for *C*_{2}

Coming Soon

Sea level rise rates from the KMVR2011 model when my simple, reasonable temperature scenarios and the corresponding KMVR2011 “equilibrium temperatures” are used. I think you will find it interesting.

**Update 11/27/11**

The term (ατ + 1) were corrected to (ατ – 1) in equations (IV) through (VII). This was a typographical error and all calculations had been done with the correct term.