Derivation of Global Temperature Change Formula

Note: This post is highly mathematical. See the list of Quantities and Symbols for further information on the symbols and notation used.

I previously derived a formula for the planetary surface temperature.[LINK] Here, I analyze what we can learn about the effect of small changes to that baseline situation.

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 will derive results for both approaches.

Most of the constituent pieces of the puzzle were worked out in prior analyses.

Changes to global averages

Let’s consider the energy balance equation for the top of the atmosphere (TOA):

(1) $\begin{equation*} \ex{\txh} = \ex{\Sn} - \ex{\OLR} \end{equation*}$

Here, $\OLR$ is the radiative flux of outgoing longwave radiation, $\Sn$ is net absorbed solar irradiance (i.e., sunlight not reflected back to space), and the TOA excess heating flux, $\txh$ , is the imbalance between power in and power out.

We’re interested here in what happens when the quantities of interest change by small amounts. So, in the equation above, there would be changes $\Delta\ex{\txh}$ , $\Delta\ex{\Sn}$ , and $\Delta\ex{\OLR}$ , and those changes would satisfy the equation

(2) $\begin{equation*} \Delta\ex{\txh} = \Delta\ex{\Sn} - \Delta\ex{\OLR} \end{equation*}$

Because we are ultimately interested in changes in the average global temperature, I’ll divide each term by a normalization factor, $\qg$ , defined as:

(3) $\begin{equation*} \qg = \frac{4 \ex{\OLR}}{\ex{\tsurf}} \end{equation*}$

where $\ex{\tsurf}$ is the average surface temperature. I’ll also make use of the previously derived formula[LINK] for $\Delta\ex{\OLR}$ . Then, rearranging the result to solve for $\Delta\ex{\tsurf}$ leads to the TOA energy balance equation being expressed as:

(4) $\begin{equation*} \Delta\ex{\tsurf} = \frac{\Delta\ex{\Sn} - \Delta\ex{\txh}}{\qg} - \changetermg{\Delta\avg{\ltr}} - \changetermg{\Delta\avg{\emis}} - \changetermg{\ex{\tveb}} \end{equation*}$

where

(5) $\begin{equation*} \changetermg{\Delta\avg{\ltr}} = \frac{\ex{\tsurf}}{\avg{\ltr}} \Delta\avg{\ltr} \end{equation*}$

(6) $\begin{equation*} \changetermg{\Delta\avg{\emis}} = \frac{\ex{\tsurf} }{\avg{\emis}} \Delta\avg{\emis} \end{equation*}$

(7) $\begin{equation*} \changetermg{\ex{\tveb}} = \frac{\ex{\tsurf}}{\ex{\tveb}} \Delta\ex{\tveb} \end{equation*}$

The above shows how temperature is affected by changes in absorbed solar irradiance $\Sn$ , longwave effective transmittance $\ltr$ (or absorbance $\lea=1-\ltr$ ), emissivity, or temperature change distribution $\tveb$ .

If it would be helpful, I’ve also derived a formula[LINK] for decomposing $\Delta\ex{\Sn}$ into changes in albedo and total solar irradiance.

The above expression for $\Delta\ex{\tsurf}$ is the main result of this change analysis, expressed in terms of changes in global averages.

Global averages of changes

Let’s consider the energy balance equation for the top of the atmosphere (TOA):

(8) $\begin{equation*} \txh = (1-\albedo) \isi - \OLR \end{equation*}$

Here, $\OLR$ is the radiative flux of outgoing longwave radiation, $\isi$ is the radiative flux of inbound shortwave radiation from the Sun, the albedo $\albedo$ is the fraction of sunlight reflect back to space, and the TOA excess heating flux $\txh$ is the imbalance between power in and power out.

Let’s examine what this equation can tell us about what happens as small changes are made to the baseline climate system.

Suppose that the climate system depends on some variable $\chng$ , which could correspond to time, or concentration of a component of the atmosphere, or some complex mix of changes so long as those changes are parameterized by this single variable $\chng$ .

To see what happens as a small change is made in $\chng$ , let’s take the derivative of the TOA energy balance equation with respect to $\chng$ :

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

A small change, $\Delta\chng$ , will result in corresponding changes $\Delta\txh$ , $\Delta\albedo$ , $\Delta\isi$ ,and $\Delta\OLR$ . From the preceding equation, these small changes will be related by:

(10) $\begin{equation*} \Delta\txh = (1-\albedo) \Delta\isi - \isi\Delta\albedo - \Delta\olr \end{equation*}$

Averaging both sides of the equation over the surface of the planet (and possibly over some time window) yields:

(11) $\begin{equation*} \ex{\Delta\txh} = \ex{(1-\albedo)\Delta\isi} - \ex{\isi\Delta\albedo} - \ex{\Delta\olr} \end{equation*}$

To keep my work from getting unwieldy, I’ll use the notation $\qp\cdot\changeterm{\Delta X}$ to represent the energy balance change term for $\Delta X$ . The normalization factor $\qp\ = 4 \bex{\olr/\tsurf}$ has been previously[LINK] identified as being appropriate for turning each term $\changeterm{\Delta X}$ into a change term for the average global temperature.

Thus the TOA energy balance equation becomes:

(12) $\begin{equation*} \changetermp{\Delta\txh} = \changetermp{\Delta\isi} - \changetermp{\Delta\albedo} - \ex{\Delta\olr}/(4 \bex{\olr/\tsurf}) \end{equation*}$

where

(13) $\begin{equation*} \changetermp{\Delta\txh} = \ex{\Delta\txh}/(4 \bex{\olr/\tsurf}) \end{equation*}$

(14) $\begin{equation*} \changetermp{\Delta\isi} = \ex{(1-\albedo)\Delta\isi}/(4 \bex{\olr/\tsurf}) \end{equation*}$

(15) $\begin{equation*} \changetermp{\Delta\albedo} = \ex{\isi\Delta\albedo} /(4 \bex{\olr/\tsurf}) \end{equation*}$

In dealing with the terms related to solar irradiance, I’m going to do something that is’t important with regard to our conclusions (so feel free to skip over it), but which sharpens things conceptually and may make it easier to evaluate these terms. I will rewrite $\isi$ as $(\isi/\tsi)\tsi$ where $\tsi$ is total solar irradiance. This is helpful since $(\isi/\tsi)$ is the normal weight for calculating average albedo. It’s also independent of any changes in $\chng$ (since it’s just dependent on geometry and timing), which leads me to change the name of the term from $\changeterm{\Delta\isi}$ to $\changeterm{\Delta\tsi}$ . Also, $\tsi$ is independent of location, so we can pull it out of the global average and just leave it in the time average. (Probably $\tsi$ has no important temporal variations on time scales of relevance, but I want to avoid making assumptions.) Also note that the global average of $\isi$ is $\ex{\isi}_g = \tsi/4$ .

(16) $\begin{equation*} \changetermp{\Delta\albedo} = \frac{\ex{\tsi\ex{(\isi/\tsi)\Delta\albedo}_g}_t }{16 \bex{\olr/\tsurf}\ex{\isi/\tsi}_g} \end{equation*}$

(17) $\begin{equation*} \changetermp{\Delta\tsi} = \frac{\ex{\ex{(1-\albedo)(\isi/\tsi)}_g\Delta\tsi}_t}{16 \bex{\olr/\tsurf}\ex{\isi/\tsi}_g} \end{equation*}$

These expressions might look complicated, but they’re really just averaging the albedo ( $a$ and $\Delta a$ ) with the standard weighting, then dividing by $16 \bex{\olr/\tsurf}$ .

So far in this analysis I haven’t addressed the energy balance term $\ex{\Delta\olr}$ . Fortunately, I calculated that previously[LINK], yielding:

(18) $\begin{equation*} \bex{\Delta\olr} = 4 \bex{\olr/\tsurf} (\ex{\Delta\tsurf} + \changetermp{\Delta\ltr} +\changetermp{\Delta\emis} - \tvts) \end{equation*}$

Thus, the TOA energy balance equation becomes:

(19) $\begin{equation*} \changetermp{\ex{\Delta\txh}} = \changetermp{\Delta\tsi} - \changetermp{\Delta\albedo} - (\ex{\Delta\tsurf} + \changetermp{\Delta\ltr} +\changetermp{\Delta\emis} - \tvts) \end{equation*}$

Rearranging the energy balance equation to solve for the temperature change yields:

(20) $\begin{equation*} \ex{\Delta\tsurf} = \changetermp{\Delta\tsi} - \changetermp{\Delta\albedo} - \changetermp{\Delta\ltr} -\changetermp{\Delta\emis} + \tvts - \ex{\Delta\txh}/q \end{equation*}$

This equation is the main result of this analysis. It shows that the average surface temperature changes according to the sum of factors associated with various other specific types of change:

Changes in total solar irradiance. Increasing $\TSI$ increases temperature.
Changes in albedo. Increasing albedo decreases temperature.
Changes in the efficiency of atmospheric longwave radiation transport. Efficiency is measured by the ratio of how much thermal radiation is emitted at the top of the atmosphere compared to how much thermal radiation is emitted by the surface. Lower efficiency increases temperature.
Changes in emissivity. Increasing emissivity lowers temperature.
Changes in the distribution of temperatures around the globe or the size of temporal fluctuations. Increasing variations in temperature lowers the average temperature.
Changes in the energy imbalance at TOA. This offers only a temporary effect on temperature. An excess of energy flow in relative to energy flow out corresponds to a lower average temperature, but it also corresponds to heat building up and a rising temperature. So, any non-zero imbalance will tend to rectify itself, leading towards zero imbalance. Only the preceding 5 factors can alter the equilibrium temperature.

The specific form of each term is a direct consequence of:

the definitions of the various quantities;
inherent relationships between those quantities (arising from either their definitions or from energy conservation);
the Stefan-Boltzmann law.

Other factors can affect the average surface temperature only insofar as they alter one of the factors listed above.

This result (like others in this series) relies only on the Stephan-Boltzmann law, energy conservation, and tautological mathematical transformations.

(It is assumed that there is no significant source of ongoing heating other than the Sun; if there were, then the increase in that heat flux, divided by $q$ , would add an additional term to the sum controlling temperature.)

Applications of this formula will be valid as long as changes are small enough that it suffices to consider first derivatives only, with the effects of higher order derivatives being negligible.

For convenience, I gather together below the formulas for the various terms in this version of the master temperature change formula:

(21) $\begin{equation*} \changetermp{\Delta\albedo} = \frac{\ex{\tsi\ex{(\isi/\tsi)\Delta\albedo}_g}_t }{16 \bex{\olr/\tsurf}\ex{\isi/\tsi}_g} \end{equation*}$

(22) $\begin{equation*} \changetermp{\Delta\tsi} = \frac{\ex{\ex{(1-\albedo)(\isi/\tsi)}_g\Delta\tsi}_t}{16 \bex{\olr/\tsurf}\ex{\isi/\tsi}_g} \end{equation*}$

(23) $\begin{equation*} \changetermp{\Delta\ltr} = \frac{\ex{\slr}}{4\bex{\olr/\tsurf}} \exw{\Delta\ltr }{\slr} = \frac{\ex{\slr\Delta\ltr}}{4\bex{\olr/\tsurf}} \end{equation*}$

(24) $\begin{equation*} \changetermp{\Delta\emis}= \frac{ \ex{\olr/\emis}}{4\bex{\olr/\tsurf}} \exw{\Delta\emis}{\olr/\emis} = \frac{ \ex{(\olr/\emis)\Delta\emis}}{4\bex{\olr/\tsurf}} \end{equation*}$

(25) $\begin{equation*} \tvts = -\exw{(\Delta\tsurf - \ex{\Delta\tsurf})}{\olr/\tsurf} = -\frac{(\Delta\tsurf - \ex{\Delta\tsurf})(\olr/\tsurf)}{\olr/\tsurf} \end{equation*}$

(26) $\begin{equation*} \qp = 4\bex{\olr/\tsurf} \end{equation*}$

The notation $\ex{X}$ represents the average of $X$ over the globe and within some time window. A subscript of g or t indicates just doing one of these averages. The notation $\exw{X}{W}$ reflects the average of $X$ weighted by $W$ , computed as $\ex{XW}/\ex{W}$ .

The quantities associated with each symbol are identified here[LINK], where one can also find the formulas for computing averages.

Conclusions

I’ve used the equation for energy balance at TOA to develop a first-order equation equation relating different quantities. I’ve done this using both a change-of-global-averages approach and a global-average-of-changes approach.

The results are similar, but have small differences. While one uses a normalization proportional to $\ex{\OLR}/\ex{\tsurf}$ , the other uses a normalization proportional to $\ex{\OLR/\tsurf}$ .

Some of the differences might be due to subtleties like taking the delta of a weighted average, $\Delta\exw{X}{Y}$ not being the same as the weighted average of a delta, $\exw{\Delta X}{Y}$ . There are additional terms that distinguish the two of these.
Maybe the differences are the result of our having thrown out higher order terms.
Maybe there’s a subtle error in my analysis of one of the cases. Or, maybe it’s something else I haven’t yet thought of.

In view of the appearance of differences, I’d suggest using these difference-based change formulations in contexts where small errors don’t matter much; for more exacting work, the planetary temperature formula (without the assumption of small changes that I’ve used here) should be something that can be relied upon.

Note that if there are no changes in emissivity or temperature distribution, then:

(27) $\begin{equation*} Delta\ex{\tsurf} = -\frac{\ex{\tsurf}}{\avg{\ltr}} \Delta\avg{\ltr} = \frac{\ex{\tsurf}}{1-\avg{\lea}} \Delta\avg{\ltr} \Delta\avg{\lea} \end{equation*}$

UNFINISHED:

This equation specifies that the temperature change associated with a TOA “forcing” $\ex{\Delta\olr}$ is characterized by an instantaneous sensitivity $4\bex{\olr/\tsurf}$ . (The actual temperature response could be modified by subsequent feedbacks.)

For Earth at present, $4 \ex{\olr}/\ex{\tsurf}$ ≈ 4⋅(240 W/m²)/(288 K) = 3.3 W m^-2 K^-1. I haven’t yet evaluated how much $4\bex{\olr/\tsurf}$ differs from $4 \ex{\olr}/\ex{\tsurf}$ .

This response is what the climate scientist refer to as the “Planck response” or “Planck feedback” (which is the response to radiative forcing at TOA before any feedbacks come into play). They denote this response by the symbol $\alphap$ and use a sign convention which makes $\alphap$ negative. IPCC reports that a “crude estimate” of the value of $-\alphap$ is 3.3 W m^{^-2} ℃^-1 and indicate that better estimates say that the response is likely within the range from 3.1 to 3.3 W m^{^-2} ℃^-1 (IPCC 2021 AR6 WG1 report p. 968). This is a good match to what I’ve independently derived in the analysis here.

Within the framework I’m offering, those better estimates presumably are equivalent to determining the value of $4\bex{\olr/\tsurf}$ as opposed to relying on $4 \ex{\olr}/\ex{\tsurf}$ as an approximation.

(Given that there are essentially no assumptions made in the analysis provided above, this close correspondence with what other investigators have reported shouldn’t be surprising.)