Note: This page is highly mathematical.
Generic temperature variation
Suppose that the temperature is given by:
where is a temperature-variation with zero mean, so that . Then, the temperature-variation emissions boost factor, or , will be given by:
If is small, i.e., , then this may be approximated by:
From this, it also follows that, if is small, then:
Contributions to emissions boost factor
Suppose we re-arrange one of the equations above to write:
This allows us to think of or as being made up of contributions from different times and places, contributions that are likely all positive (though I haven’t rigorously proved that must be positive). We might think of as the local TVEB Contribution, .
can alternatively be computed as:
Filling in the definition of , this becomes:
If is small, this may be approximated as:
The quantity or is interesting insofar as it offers a way of attributing contributions to or to different places and times (though is ultimately a non-local quantity).
This local version of could have been defined differently: adding or subtracting any multiple of to the definition of would yield a quantity that averages to . So, it’s still unclear, as yet, if this way of defining is meaningful.
Sinusoidal time variation
Suppose that the temperature varies as:
where and are constants.
If is small compared to 1, this leads to the approximate result:
As an example, suppose 288 K and 5℃. Then this diurnal temperature variation would lead to 0.017, 9e-4 and 1 – 2.3e-4, so that there the emissions boost would lead to a net temperature reduction (for the same average emissions) of about 0.065℃.
Studies show that global warming has been leading to night-time low temperatures increasing more than in day-time high temperatures. In other words, has been decreasing. The above result indicates that the reduction in the amplitude of diurnal temperature variation, , is not likely to contribute significantly to changes in the GMST (global mean surface temperature).