Derivation of Changes to Absorbed Sunlight

Note: This post is highly mathematical.

This particular piece of analysis is the least important of those in this round of analyses. So, feel free to skip it, if you’re just trying to sort out the overall logic.

Let’s develop a formula for small changes to the fluxed of absorbed solar irradiance expressed in terms of changes to albedo and total solar irradiance.

There are two distinct ways that we could analyze such changes:

Changes to global averages ― This involves looking at changes to the global average values that were used in the baseline global temperature formula.
Global averages of changes ― This involves looking at changes as they occur in particular places and then averaging those changes globally.

The two approaches offer different ways of viewing changes that are both valid, yet decompose the picture differently. I’ll derive results for both approaches.

Changes to global averages

Average absorbed solar irradiance can be expressed as

(1) $\begin{equation*} \ex{\Sn} = (1-\avg{\albedo}) \ex{\isi} \end{equation*}$

(2) $\begin{equation*} \avg{\albedo} = \exw{\albedo}{\isi} \end{equation*}$

Note that $\isi = (\isi/\tsi) \tsi$ where $\ex{\isi/\tsi} = \frac{1}{4}$ and where $\tsi(t)$ does not depend on location. So:

(3) $\begin{equation*} \ex{\isi} = \ex{\ex{(\isi/\tsi) \tsi}_g}_t = \frac{1}{4} \ex{\tsi}_t \end{equation*}$

Suppose there are small changes $\ex{\Sn} = \ex{\Sn}_0 + \Delta\ex{\Sn}$ , $\avg{\albedo} = \avg{\albedo}_0 + \Delta\avg{\albedo}$ and $\ex{\tsi}_t = \ex{\tsi}_{t,0} + \Delta\ex{\tsi}_t$ . Note that $\isi/\tsi$ is fixed by geometry and timing. We substitute these into the equation for $\ex{\Sn}$ , drop terms higher than first order in small quantities, and subtract off the equation for for $\ex{\Sn}_0$ . This process yields:

(4) $\begin{equation*} \Delta\ex{\Sn} = - \frac{1}{4} \ex{\tsi}_t \cdot \Delta\avg{\albedo} + \frac{1}{4} (1-\avg{\albedo}) \cdot \Delta\ex{\tsi}_t \end{equation*}$

Breaking the formula for $\Delta\ex{\Sn}$ into terms normalized by $\qg$ to make them into temperatures yields:

(5) $\begin{equation*} \Delta\ex{\Sn} = \qg\cdot \changetermg{\Delta\ex{\Sn}} \end{equation*}$

(6) $\begin{equation*} \changetermg{\Delta\ex{\Sn}} = - \changetermg{\Delta\avg{\albedo}} + \changetermg{\Delta\ex{\tsi}_t} \end{equation*}$

where

(7) $\begin{equation*} \changetermg{\Delta\avg{\albedo}} = \frac{1}{4\qg} \ex{\tsi}_t \cdot \Delta\avg{\albedo} \end{equation*}$

(8) $\begin{equation*} \changetermg{\Delta\ex{\tsi}_t} = \frac{1}{4\qg} (1-\avg{\albedo}) \cdot \Delta\ex{\tsi}_t \end{equation*}$

Global averages of changes

Absorbed solar irradiance can be expressed as

(9) $\begin{equation*} \Sn = (1-\albedo) \isi \end{equation*}$

Suppose there are small changes $\Sn = {\Sn}_0 + \Delta\Sn$ , $\albedo = \albedo_0 + \Delta\albedo$ and $\tsi = {\tsi}_{,0} + \Delta\tsi$ , with $\isi/\tsi$ unchanged. Then

(10) $\begin{equation*} \Delta\Sn = -\isi \cdot \Delta\albedo + (1-\albedo) (\isi/\tsi) \Delta\tsi \end{equation*}$

Averaging this geographically produces:

(11) $\begin{equation*} \ex{\Delta\Sn}_g = -\frac{1}{4} \tsi \cdot \exw{\Delta\albedo}{\isi/\tsi}_g+ \frac{1}{4} \left(1 - \exw{\albedo}{\isi/\tsi}_g\right) \cdot\Delta\tsi \end{equation*}$

Then averaging this over the time window yields

(12) $\begin{equation*} \ex{\Delta\Sn} = -\frac{1}{4} \ex{\tsi \cdot \exw{\Delta\albedo}{\isi/\tsi}_g}_t + \frac{1}{4} \ex{\left(1 - \exw{\albedo}{\isi/\tsi}_g\right) \cdot \Delta\tsi}_t \end{equation*}$

Breaking this up into normalized terms leads to:

(13) $\begin{equation*} \ex{\Delta\Sn} = \qp\cdot \changetermp{\Delta\Sn} \end{equation*}$

(14) $\begin{equation*} \changetermp{\Delta\Sn} = - \changetermp{\Delta\albedo} + \changetermp{\Delta\tsi} \end{equation*}$

where

(15) $\begin{equation*} \changetermp{\Delta\albedo} = \frac{1}{4\qp} \ex{\tsi}_t \exw{\exw{\Delta\albedo}{\isi/\tsi}_g}{\tsi}_t \end{equation*}$

(16) $\begin{equation*} \changetermp{\Delta\tsi} = \frac{1}{4\qp} \ex{\left(1 - \exw{\albedo}{\isi/\tsi}_g\right) \cdot \Delta\tsi}_t \end{equation*}$

Conclusion

That completes the analysis of formulas for $\Delta\ex{\Sn} = \ex{\Delta\Sn}$ in terms of changes in albedo and $\TSI$ .

Comparing the formulas for changes to global averages and global averages of changes:

They’re likely identical for all practical purposes, unless one is tracking rapid variations in $\TSI$ for some reason.
Technically, they appear slightly different if $\TSI$ varies within the time window. If I haven’t made any mistakes, the two formulations ought to be precisely equivalent, despite appearances. I’ve done some checking, but this bit of the analysis isn’t sufficiently high priority for me to vet the results as thoroughly as I might otherwise. Unless $\TSI$ is varying a lot within the time window, the nuances are unlikely to matter.