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

where

(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).