Analysis: Emissions boost factor from a temperature variation

Note: This page is highly mathematical.

Generic temperature variation

Suppose that the temperature T is given by:

(1)   \begin{equation*} T = T_0 \cdot \left(1 + \eta\left) \end{equation*}

where \eta is a temperature-variation with zero mean, so that \ex{\eta} = 0. Then, the temperature-variation emissions boost factor, \TVEB or \tveb, will be given by:

(2)   \begin{equation*} 1+\tveb = \frac{\ex{T^4}}{\ex{T}^4} = \ex{\left(1 + \eta\right)^4} \end{equation*}

(3)   \begin{equation*} \tveb = 4 \ex{\eta}+ 6\ex{\eta^2} + 4\ex{\eta^3} + \ex{\eta^4} = 6\ex{\eta^2} + 4\ex{\eta^3} + \ex{\eta^4} \end{equation*}

If \eta is small, i.e., |\eta| \ll 1, then this may be approximated by:

(4)   \begin{equation*} \tveb  \approx 6\ex{\eta^2} \end{equation*}

From this, it also follows that, if \eta is small, then:

(5)   \begin{equation*} M_\tveb = \frac{1}{\fourthroot{(1+\tveb)}} \approx 1 - \frac{3}{2} \ex{\eta^2} \end{equation*}

Contributions to emissions boost factor

Suppose we re-arrange one of the equations above to write:

(6)   \begin{equation*} \tveb  = \ex{\xloc\tveb(\theta,\phi,t)} \end{equation*}


(7)   \begin{equation*} \xloc\tveb =  6\eta^2 + 4\eta^3 + \eta^4  \end{equation*}

(8)   \begin{equation*} \eta =  \frac{T(\theta,\phi,t) - \ex{T}}{\ex{T}} =  \frac{T(\theta,\phi,t)}{\ex{T}} - 1 \end{equation*}

This allows us to think of \TVEB or \tveb as being made up of contributions from different times and places, contributions that are likely all positive (though I haven’t rigorously proved that \xloc\tveb must be positive). We might think of \xloc\tveb as the local TVEB Contribution, \mathrm{TVEBC}.

\xloc\tveb can alternatively be computed as:

(9)   \begin{equation*} \xloc\tveb = (1+\eta)^4 - (1 + 4\eta) \end{equation*}

Filling in the definition of \eta, this becomes:

(10)   \begin{equation*} \xloc\tveb(\theta,\phi,t)  =  \frac{T(\theta,\phi,t)^4}{\ex{T}^4} - 4 \frac{T(\theta,\phi,t)}{\ex{T}} + 3  \end{equation*}

If \eta is small, this may be approximated as:

(11)   \begin{equation*} \xloc\tveb(\theta,\phi,t) \approx 6 \left( \frac{T(\theta,\phi,t)}{\ex{T}} - 1 \right)^2 \end{equation*}

The quantity \mathrm{TVEBC} or \xloc\tveb is interesting insofar as it offers a way of attributing contributions to \TVEB or \tveb to different places and times (though \TVEB is ultimately a non-local quantity).

This local version of \TVEB could have been defined differently: adding or subtracting any multiple of \eta to the definition of \xloc\tveb would yield a quantity that averages to \tveb. So, it’s still unclear, as yet, if this way of defining \xloc\tveb is meaningful.

Sinusoidal time variation

Suppose that the temperature T varies as:

(12)   \begin{equation*} T = T_0 \cdot \left(1 + a\cdot\sin\left(\frac{2\pi t}{\tau}\right) \right) \end{equation*}

where T_0 and a are constants.

If a is small compared to 1, this leads to the approximate result:

(13)   \begin{equation*} \tveb = \frac{\ex{T^4 } }{T_0^4} -1 \approx 3 a^2 \end{equation*}

(14)   \begin{equation*} M_\tveb = \frac{1}{\fourthroot{(1+\tveb)}} \approx 1 - \frac{3}{4} a^2 \end{equation*}

As an example, suppose \ex{T}= 288 K and a\cdot\ex{T}= 5℃. Then this diurnal temperature variation would lead to a= 0.017, \tveb\approx 9e-4 and M_\tveb \approx 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, a has been decreasing. The above result indicates that the reduction in the amplitude of diurnal temperature variation, a, is not likely to contribute significantly to changes in the GMST (global mean surface temperature).