Explorations: Contributions to Global Warming

Note: this page offers miscellaneous, possibly evolving, information intended to support the emergence of insight regarding a particular topic.It’s a resource to support my own thinking, and may or may not be useful to others.

Based on my prior word (Analysis: Planetary Temperature – a Rigorous Formula and Analysis: Small Changes in Climate Variables), it’s possible to examine climate data and analyze what is contributing to observed changes in the mean global temperature.

Table of Contents


I’ve analyzed data from CERES, using their EBAF 4.1 dataset.

The EBAF dataset does not include emissivity data, so I’ve chosen to work with the “surface effective radiation emissions temperature” (or, more concisely, “surface effective temperature”), T_{se} = (\SLR/\sigma)^\frac{1}{4} (evaluated locally and then averaged globally), rather than actual temperature. Changes in the surface effective temperature are expected to closely track changes in actual temperature (differing only to the extent that emissivity is changing). We can look at emissivity changes at a later time.


Below is a plot of surface effective temperature over a 21-year period.

Global mean surface radiative effective emissions temperature

This temperature has been increasing at a rate that could be extrapolated to be 2.4 ℃/century.

Note that in this study all quantities are based on monthly data which has been smoothed using a 12-month rolling average to eliminate seasonal variations. Trends are evaluated via ordinary-least-squares linear regression against annual data.

Based on our planetary temperature formula, it’s possible to calculate the “equilibrium temperature,” which is the temperature that the planet is shifting towards. That is plotted below.

Equilibrium temperature to balance energy received from the Sun, given measured cooling efficiency

The calculated equilibrium temperature is increasing faster than the observed temperature, rising at a rate of 3.7 ℃/century.

This equilibrium value was calculated in order to balance heating from the Sun. If Earth experiences significant non-solar heating, then the total equilibrium temperature would be higher than the plotted value. (Mainstream climate scientists believe the rate of non-solar heating to be negligibly small, but some climate skeptics believe there is significant non-solar heating.)

Contributions to changes in temperature

Let’s look at how the different factors in the planetary temperature formula contribute to the observed changes in the global mean surface effective temperature. The contributions associated with each factor can be calculated separately, given that changes are relatively small. I’ve plotted the results below. To keep the results as informative as possible, I have separated the effects of clouds from non-cloud effects.

Non-cloud normalized Greenhouse effect

Increases in the non-cloud normalized Greenhouse effect contributed to warming at a rate of 3.2 ℃/century. As we’ll see, this was the largest contributor to warming.

The normalized GHE is the fraction of longwave thermal radiation power emitted by the surface which is not matched by thermal radiation emitted to space. The magnitude of the normalized GHE is determined by (a) the concentration of Greenhouse materials (water vapor, clouds, CO2, etc.) in the atmosphere and (b) the environmental lapse rate (the GHE would go to zero if temperature didn’t decrease with altitude).

The non-cloud normalized GHE would be expected to reflect increases in the concentration of CO2 plus any associated changes in water vapor and lapse rate.

Non-cloud albedo

Decreases in the non-cloud albedo (i.e., fraction of sunlight reflected back to space) contributed to warming at a rate of 1.1 ℃/century. This was the second-largest contribution to global warming.

The decrease in non-cloud albedo appears to have been primarily due to reduced sea ice at high latitudes. (See a later section of this report for evidence supporting that hypothesis.)

Cloud Greenhouse effect and albedo

Global cloud-cover decreased over the period being studied. This change in cloud-cover had two large effects—which largely cancelled each other out, resulting in a small net effect overall.

In particular, declining cloud cover reduced the cloud Greenhouse effect, contributing 1.6 ℃/century of cooling. But, that same declining cloud cover also reduced cloud albedo, allowing more sunlight to be absorbed, contributing 1.1 ℃/century of warming. The net effect was about 0.6 ℃/century of cooling. (There is some roundoff error in these numbers.)

Thus, during the studied period, clouds provided a minor negative feedback that moderated warming. (Mainstream climate models apparently anticipate cloud patterns shifting in coming decades so that cloud feedback may become positive overall, somewhat amplifying warming.)

Top-of-atmosphere mean solar irradiance

The intensity of the sunlight reaching Earth’s orbit declined very slightly over the study period, contributing about 0.1 ℃/century of cooling.

Temperature-variation emissions boost

The temperature-variation emissions boost factor (TVEB), \tveb^\prime = ([average of T_b^4]/[average of T_b]^4) – 1, is an effect that reflects the ability of variations in temperature to boost thermal emissions for a given average temperature. TVEB is mostly associated with the temperature difference between the tropics and the polar regions, though other temperature variations also contribute to TVEB. Over the period of study, changes in TVEB contributed to warming at a rate of about 0.1 ℃/century.

Energy storage and unrealized temperature changes

Satellites measure the net energy imbalance at the top-of-the-atmosphere, TEI, and the effects of this are plotted below.

TOA Energy Imbalance and associated unrealized temperature change

A positive value of TEI indicates that Earth’s stored thermal energy is increasing. This increase in thermal energy storage generally takes the form of ocean water warming or ice melting. If Earth was in thermal equilibrium, then the energy storage rate, \xglob Q_s, would be zero. If TEI is positive, that’s an indication that Earth is warming and not yet in equilibrium. It’s an indication of how much temperature change has been “deferred” and has not yet been realized. From the chart, it appears that about 1.3℃ of warming is currently unrealized. TEI is what accounts for the difference between the “solar equilibrium” T_b and the “observed” T_b, as previously plotted.

The increasing value of TEI corresponds to unrealized temperature change increasing at a rate of 1.3 ℃/century. This is warming that hasn’t shown up in T_b, but the warming shows up as an increase in the equilibrium value of T_b. Earth is continually shifting towards its equilibrium temperature, though it may take a long time to reach it (given how long it takes to warm even the upper layers of the oceans).

It’s this phenomenon of “energy storage” or “unrealized temperature changes” that leads to the 3.7 ℃/century of warming in the equilibrium temperature being reduced to 2.4 ℃/century of warming in the observed temperature.

Non-solar heating

Note that \TEI = \xglob Q_s - \xglob S_x, so that \TEI also reflects any non-solar heating. If the rate of non-solar heating \xglob S_x was non-negligible, it would have the following implications:

  1. What wouldn’t change: The effects of non-solar heating are reflecting in \TEI, and are already accounted for in the planetary temperature change calculation. If there is significant non-solar heating, that wouldn’t and couldn’t replace or change any of the contributions to temperature already calculated and charted.
  2. What would change: There would be a “TOTAL unrealized temperature change” distinct from and larger than the “TEI unrealized temperature change” as plotted above. How much larger depends on the size of \xglob S_x. Thus, if there is significant non-solar heating, we could expect long-term warming to be larger than the 3.1 ℃/century of warming that was calculated based on the rise in the solar equilibrium temperature. The solar-plus-nonsolar equilibrium temperature would be higher than the plotted solar equilibrium temperature.

Checking the math

The temperature change contributions were computed using approximations expected to be valid if the changes involved are relatively small. So, it’s worth checking the validity of this approximation.

Verification that sum of temperature change contributions matches actual temperature change

In the chart above I’ve plotted both the actual surface effective temperature, as well as the sum of the various contributions to temperature change, minus the unrealized temperature change. The two curves match perfectly at the level of visual inspection. This confirms the general validity of the approximations involved in decomposing the temperature change into various components.

Expected forcing due to Carbon Dioxide

The expected warming associated with the radiative forcing due to CO2 alone is plotted below, using NOAA GML data for atmospheric CO2 concentration.

It’s assumed that the associated radiative forcing is approximately logarithmic, so that \Delta F = 5.65 \ln(C/C_0) W/m^2 where C is the concentration of CO2 and C_0 is an arbitrary reference concentration. The number 5.65 is taken to match the estimate of CO2 forcing in the period 2011-2019 as estimated by IPCC 2021 WG1 p. 948.

For the period under study, my temperature change framework implies a short-term temperature contribution per radiative forcing of 0.300 ℃ W-1 m2. Combined with the preceding radiative forcing formula, this nominally implies a temperature contribution of 1.17℃ per doubling of the CO2 concentration (in the absence of any associated feedback effects).

Radiative forcing from carbon dioxide

Thus, the measured change in the concentration of CO2 was expected to give rise to warming at a rate of 0.96 ℃/century. IPCC WG1 p. 948 estimated the increase in radiative forcing from all persistent (i.e., non-water-vapor) Greenhouse gasses from 2011-2019 to be 23 percent larger than the forcing from CO2 alone. So, we can estimate the expected total direct warming rate from these gasses to be about 1.2 ℃/century.

Mainstream climate science predicts that water vapor and lapse rate should provide significant positive feedback, amplifying the warming effect of any increase in the concentration of CO2 and other persistent Greenhouse gases. The observations seem consistent with that narrative, insofar as persistent GHGs were expected to provide sufficient radiative forcing to directly account for 1.2 ℃/century of GHE warming, yet the data show that the non-cloud GHE contributed 3.2 ℃/century of warming. So, 2.0 ℃/century of non-cloud GHE warming must be attributed to “feedback” (i.e., changes in responsive variables such as water vapor concentration or the environmental lapse rate) increasing the GHE.

Location of changes in Non-Cloud Albedo

To help understand the significance in changes in Earth’s non-cloud albedo, I’ve plotted changes in the contribution to global albedo below.

The “contribution to global albedo” was defined as the local flux of shortwave radiation emitted to space at the top of the atmosphere divided by the global average downward shortwave flux at the top of the atmosphere. Fluxes were averaged over five-year periods. The 5-year-average fluxes were used to calculate the average regional contribution to global albedo during that 5-year period. This was done for the period 2001-2005 and the period 2017-2021, and the difference between these was computed and plotted on the map below.

Changes in the regional contribution to global albedo from the period (2001-2005] to the period [2017-2021]

As can be seen from the figure, there were notable reductions in albedo in the Antarctic and Arctic Ocean regions. Thus, the reduction in non-cloud albedo during the period under study appears to have been primarily due to reduced sea ice.

This tracks well with CERES SYN Edition 4.1 Terra-Aqua data on changes in snow/ice coverage, as seen below.

Changes in snow and ice coverage

Contribution of Aerosols

Although the CERES EBAF dataset does not include information on aerosols, this information is available in the CERES SYN1deg dataset. Below, I’ve charted the normalized Greenhouse effect and albedo associated with aerosols, according the the SYN data. These have been converted to temperature changes based on conversion factors derived from EBAF data. It is possible that this procedure of using two different datasets might introduce some error into the temperature conversions; however, I would expect the error in the temperature conversion factors to be small, since the two datasets are generally similar.

Temperature change contribution of aerosols (based on CERES SYN ed4.1 Terra-Aqua data and converted to temperature using CERES EBAF ed4.1 data)

Although aerosols contributed a minor amount to the Greenhouse effect, that contribution did not change enough over the study period to impact global temperature. The aerosol contribution to albedo increased slightly over the study period, contributing 0.3℃/century of cooling.

Recall that non-cloud albedo in general contributed 1.1 ℃/century of warming. If aerosols were contributing to cooling, this suggests that albedo changes due to changes in ice and snow extent likely contributed more than 1.1 ℃/century of warming.