If the CO2 increases exponentially, so (eventually) must the temperature.

While estimating the CO2 level is easy, it is the global temperature of our planet that actually drives weather. This is harder to measure because the measured temperatures vary with the time, place, weather and season. It is harder to predict because the average surface temperature depends on many factors, including the reflectivity of the upper atmosphere, and so on.

There are, however, two factors that make it easier to predict the global temperature going forward from the present time. The first is that model calculations indicate that the dominant cause for the rise in global temperature since 1975 has been the increase in CO2 contributed by human activities. The second factor to note is that the global temperature lags the CO2 level because of thermal inertia, largely due to the oceans. However, if the CO2 level rises exponentially, the temperature will also eventually rise exponentially, with the same growth rate, even if it lags the CO2 level.*

This means that we don't have to determine the growth rate for the temperature from the temperature curve itself. We can just use the same rate as for the CO2 curve. This is fortunate, because the temperature curve is more variable than the CO2 curve and it would not be as reliable to determine a growth rate from it alone. From the temperature curve itself, we only have to determine the time when it started to rise exponentially.

The figure shows the measured temperature (black line). This line is the temperature since 1975 smoothed using a moving five-year average. Even so, it is fairly variable. The temperature shown is with reference to the relatively flat average around 1910, the minimum last century. The red line shows an exponential match having the same growth rate as derived for the CO2 level in the previous post. A temperature curve having the same growth rate as the CO2 growth rate is clearly consistent with the temperature data.

It can be seen from this match that the rise in global temperature is expected to be about 2.5 degrees Centigrade (4.5 degrees Fahrenheit) by 2050 from the CO2 increase alone. This may not seem like much, until you realize that the warmest our Earth has been in the last 100 million years was only about 5 degrees Centigrade warmer than the present! Further, this much of a rise is enough to get us into the range of some possible catastrophic "tipping points," as discussed later.

Next: The rise in sea level from this temperature increase.

Details:

Some of the difficulties in predicting global temperature changes in the sort term may be seen in the news report, Will Global Warming Take a Short Break?, in Science News, May 5, 2008.

The measured temperature variation (black line) is from the 2001 report by the Intergovernmental panel on Climate Change with updated measurements. The red line is the best fit exponential (starting in 1975) that has the same time dependence as the CO2 level.

*To prove that the temperature rise must eventually become exponential with the same time scale, suppose the temperature rise is proportional to the CO2 level but with a lag time, T. The CO2 level itself is proportional to a constant plus the function exp(t/45), where t is the year. The temperature change will then be proportional to another constant plus a similar exponential term but with its time affected by the lag time, T. That means that the temperature response must contain a term like exp(t-T)/45). Because of the unique properties of the exponential function, the part involving T simply factors out as a constant equal to exp(-T/45), leaving the result that the rise in temperature must eventually become simply proportional to a constant plus a time-varying part containing the factor exp(t/45), the same factor as the variation in CO2 level. The only necessary assumption is that the overall system can be represented by a linear system in its first or lowest-level approximation, and nearly all systems can.

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