GLOBAL warming theory is not that complicated. According to Michael Hammer it can be incorporated into a simple model which readers can build themselves on an excel spread sheet.
Mr Hammer gives detailed instructions and formulae to enter for his DIY climate change model. The model then calculates the temperature rise from carbon dioxide plus water vapour forcing and allows for the time constant in the system.
A temperature rise by 2070 of 3C for a business as usual scenario, however, gives model output that is not compatible with what we know of past climates. So, what is wrong with Mr Hammer’s model?
A SIMPLE MODEL OF THE IMPACT OF CARBON DIOXIDE ON GLOBAL TEMPERATURE
By Michael Hammer
The basic claim of the anthropogenic global warming theory is that rising carbon dioxide levels increase the retained energy and thus causes the surface of the Earth to warm. A further claim is that this warming in turn causes water vapour concentrations in the atmosphere to rise causing even more energy retention in a positive feedback loop.
The relationship between concentration and energy retention is very widely accepted to be logarithmic in nature. Thus the additional forcing from carbon dioxide is proportional to the logarithm of the concentration change. In the case of water vapour the relationship is also logarithmic but the relationship between temperature and concentration is exponential. Thus the relationship between temperature and additional water vapour forcing is the logarithm of an exponential which is linear.
A third point of significance is the claim that the ocean has enormous thermal mass and that this imposes a long time constant on temperature changes. Since the water vapour level depends on current temperature this time constant also affects the water vapour feedback.
IPCC in their 4th assessment report claimed that the increase from 280 to 390 ppm had increased direct forcing by 1.77 watts /m2. This translates to 3.7 watts /m2 per doubling. Thus the forcing from carbon dioxide becomes;
Forcing = 3.7 * log (current CO2 level/280)/log(2) which simplifies to
Forcing = 12.3 * log (current CO2 level/280)
This is the equation defining the first box.
IPCC also claim the direct effect of doubling CO2 is a 1 degree increase in temperature hence the climate sensitivity is 1/3.7 C per watt or 0.27 C per watt.
This is the climate sensitivity term.
Dressler has recently claimed the water vapour forcing is 2 watts /m2 / C however this does not include cloud feedback. To include that we will have to make some estimate of this value so this becomes a parameter we adjust in our model.
The general form of the equation for a simple time constant is
New output = K* new input + (1-K)* old output where K = 1/time constant.
Technically this is known as an infinite impulse response digital filter. It is probably not obvious why this should give a time constant but rather than explain filter theory here, those interested are urged to try the equation for themselves and confirm that it does indeed give a response of the form (1- e-time/time constant).
The time constant is also not known with any precision so again we will have to make some estimate. A second adjustable parameter in our model.
Mauna Loa data gives the carbon dioxide levels for the period from 1959 to 2008 and this conforms extremely well to the equation
CO2 concentration for year X = 0.012029X 2 – 46.287X + 44828
Where X is the year starting at 1959
Because X is very large the equation is very sensitive and does require the number of decimal places given for coefficients to give the correct answer. A less sensitive equation giving the same answer is
CO2 concentration = 0.012 * X2 + 0.843 * X + 315.2
Where X = (year – 1959)
These equations predict that carbon dioxide will reach 560 ppm in 2071 which is exactly in line with IPCC claims.
The model can be created and run on an Excell spreadsheet. Each column gives a different parameter as outlined below. Successive rows gives results for successive years and we will use 111 rows to cover the years 1959 to 2070. Since we have two parameters we can adjust we will enter these parameters in cells A1 and A2 as outlined below
Set A1 to the estimate of water vapour feedback coefficient (value entered by modeller)
Set A2 to the estimated time constant (value entered by modeller)
Set B2 = 1/A2 giving the value K from above
Then starting in row 4
A4 = 1959 (the year) incrementing by 1 for each row downwards ie: set
A5 = A4 +1
B4 = 0.012 * (A4-1959)^2 + 0.843 * (A4-1959) + 315.2 the carbon dioxide level for that year using the above (less sensitive) formula
C4= 12.3 * log ( B4/280) the CO2 forcing function for that year
D4 = C4 + $A$1 * F3 the total forcing function including water. Column F is the estimate of temperature rise for that year
E4 = 0.27 * D4 the total forcing divided by the climate sensitivity
F4 = $B$2*E4 + (1-$B$2)*F3 our time constant. Each value in column F gives the predicted temperature (above baseline) for the year as specified by the column A value for the same row.
Set F3 = 0 as an initial condition. This is only to ensure a starting point for the time constant filter.
Now fill down columns B to F from row 4 . Fill down column A from row 5 (we set A5=A4+1). Do this for total of 111 rows so that A115 = 2070.
This is our model. It embodies the IPCC data and should account for both the direct forcing from CO2 and the water vapour feedback effect claimed by IPCC.
IPCC claim 0.5 C of warming up to 2008 and a further 3C of warming by 2070 for a total of 3.5C above 1900 temperatures.
Each change to A1 or A2 will automatically run the model giving the new temperature predictions. Try first a water vapour feedback value of 2 watts/m2 / C as claimed by Dressler (A1=2). Adjust A2 to match the 0.5C rise in 2008 (I found a time constant of around 20 years ie: A2=20). Note the total rise in 2070 – about 1.42C or 0.92C above today’s temperatures. I could not find any choice of time constant which gave 0.5C in 2008 and 3.5C in 2070 (3C rise above 2008 value).
In order to fit both the 2008 claim and 2070 prediction it is necessary to increase the water vapour positive feedback coefficient to around 7 watts/m2 / C with a time constant of around 45 years. However, there is a problem with such a scenario. It means a 1C rise in temperature causes water vapour feedback to increase retained heat by 7 watts/m2. For a temperature sensitivity of 3.7 watts/m2 / C that implies a further rise of 1.9C. This is almost double the positive feedback needed for thermal runaway even without any external forcing at all. Clearly, if that were the case, the climate we have had over the last several million years would have been impossible. It would have boiled the oceans billions of years ago, irrespective of carbon dioxide levels.
If as many predict, cloud feedback is negative and possibly strongly negative, say -2 watts/m2 /C or more, then a1 = 0 or negative. Even with a very short time constant a value of A1 = 0 gives a maximum temperature rise by 2008 of 0.45C and by 2070 a further 0.53C. Still stronger cloud feedback reduces these numbers even further. Programming this model only takes a few minutes, I urge people to do that and explore the range of outputs.
The model suggests that a temperature rise by 2070 of 3C for a business as usual scenario is not compatible with a past climate that has been stable enough for life to have continued to flourish. Question, where is the error in this model?
For those tempted to dismiss this on the basis that the model is too simple, consider first what such a statement implies. It suggests that the other factors not considered above in fact account for most of the predicted temperature rise. If so then the factors that are considered above are minor players.
More posts by Michael Hammer here: http://jennifermarohasy.com/blog/author/michael-hammer/