The Earth without Greenhouse Gases – Watts Up With That?

By Andy May

The overall greenhouse effect (GHE) is often defined as the difference between Earth’s average global temperature without greenhouse gases (GHGs) and with them. Greenhouse gases are all the gases that absorb some portion of the thermal energy emitted by Earth’s surface. The most important of these gases is water vapor, but there are minor GHGs like CO₂, ozone, and methane.

The calculation of Earth’s temperature without GHGs is usually done by unwrapping the planetary sphere and placing it in space at the average distance of the Earth from the Sun and having the whole of Earth’s surface illuminated by the Sun with one-fourth of the Sun’s power to account for the spherical Earth and the fact that half of Earth is always dark. This imaginary flat Earth does not rotate, and no part of it is ever dark. A description of the calculation can be read in Benestad, 2017. The global average temperature calculated with this scenario is around 255K (-18°C) and since this is about 33°C less than the current global average temperature of 15°C, the overall greenhouse effect is assumed to be about 33°C. Further discussion of this definition can be seen here.

In this post I will first list the problems with this “flat Earth” GHE model, then provide a model of a new GHG-free spherical rotating Earth. After this, I will list the assumptions used to create the spherical Earth model, the problems with it, and finally discuss what we learned making the model. Computer models are learning tools, they never give you a correct answer, but they do help you learn about the problems they were designed to investigate.

The flat-Earth no-GHG model ignores the fact that the Earth is a rotating sphere and that half of it is always in the dark. The dark side is always emitting energy, but receiving none, this is very different than the model in which the Sun is always shining directly overhead 24 hours a day with the same intensity everywhere. The proponents of the flat Earth model justify it because the satellite GHG emission temperature of the Earth is also 255K (Benestad, 2017), which using an average temperature vs height table (the U.S. or International standard atmosphere), works out to an average emission height of around five km. There is little discussion about how the energy gets from the surface, where most solar energy is absorbed to five km. But the transport is mostly done through convection and most of the GHG-emitted energy is carried to five km inside water vapor as latent heat. Above 5 km it is mostly emitted to outer space as the water vapor condenses to water droplets in the colder middle to upper troposphere.

By dealing only with averages, that is the average atmospheric temperature profile, the average surface temperature, the average emission height, the average emission intensity spectrum, etc. the flat Earth model can seem accurate and consistent with reality. But GHG emissions to space originate in a tropospheric layer from about two km to ten km, depending upon atmospheric and cloud conditions. Further, there is no consistent relationship between surface temperature and the temperature in the atmospheric GHG emission layer, it varies with weather conditions, cloudiness, season, and time of day. The surface operates mostly independently of the emission layer, they are separated by convection and weather.

In astronomy, the emission frequency and intensity of a planet is often assumed to be the planet’s “blackbody” frequency spectrum, which according to the Stefan-Boltzmann law defines a blackbody temperature. This is not the planet’s actual surface temperature; it is only a ballpark estimate. Planets are not perfect blackbodies. They are “gray bodies.” A black body emits all the energy that it absorbs, and since it has a constant temperature by definition, it emits all incident energy with a frequency spectrum that is determined by its temperature. Most importantly, a black body has no energy storage capacity or its total energy storage is constant and never changes.

Gray bodies on the other hand do not emit all the energy they absorb; they store some of it and emit the rest. Earth’s oceans have an enormous heat capacity and they store more energy than exists on the surface of Venus, but while Venus’ surface has a temperature of 464°C, Earth’s surface temperature is only about 15°C. This is because Earth’s oceans have a heat capacity of 5.4 x 10²⁴ Joules/K and Venus has no oceans. The lack of oceans and atmospheric water vapor combined with thick sulfuric acid cloud cover and a very dense atmosphere forces Venus to have a high surface temperature.

The goal of the flat-Earth model is to compute the greenhouse gas effect on Earth’s surface temperature. This requires an estimate of the surface temperature with no greenhouse gases. This is difficult, because it means no water, clouds, or water vapor and these are defining characteristics of Earth. The classic flat-Earth model assumes the planet retains its current albedo (reflectivity) of 0.3, meaning 30% of the incoming solar energy is reflected. But half or more of that albedo is due to clouds. Without water, there would be no clouds and Earth’s albedo would be more like the Moon’s, which has an albedo of 12%. We investigate both albedo estimates with the model.

Another important consideration is that GHGs emit energy to space, if there are no GHGs in the atmosphere it will emit little energy to space and act as an insulator. However, while GHGs capture surface emitted radiation in our atmosphere, the no-GHG atmosphere is transparent and all surface emissions will travel straight to space, Earth only retains the solar energy absorbed by the rocky surface.

The most common rock on Earth’s surface is basalt. On our Earth we have a lot of water that chemically transforms basalt to mud or dirt. On GHG-free Earth there is no water, so I assumed the surface is bare basalt. It will broken up a bit by meteorites, but we will ignore that for this model. Rocks have a higher thermal inertia than dirt or mud and they retain absorbed heat longer.

Like most rocks, basalt is a pretty good insulator, but it does have a thermal diffusivity, which is the speed it transfers heat through its interior. Thermal diffusivity has units of m²/second. Anyone who has been in an old medieval church on a warm day knows that the thermal diffusivity of rock is low. Thus, when sunlight strikes basalt and warms it, some of the heat will penetrate into the basalt where it will be stored for a time, and the basalt will emit the rest of the solar energy through the GHG-free, transparent atmosphere to space. My GHG-free model explicitly takes thermal diffusivity into account.

In addition, the warm basalt will also pass some heat to the atmosphere through conduction. The atmosphere is GHG free and as a result it emits very little energy directly to space, so little we can ignore it for our model. However, the dayside of the rotating planet is around 170°C warmer than the night side, so winds will appear to transport excess thermal energy as sensible heat from the dayside to the nightside or from the tropics to the poles, these winds are likely to be quite fierce in the absence of water vapor which helps stabilize the weather on our planet due to its high heat capacity. For example, wind speeds in the water-free atmosphere of Venus reach 700 km/hour (430 mph). The winds carry excess heat from the hotter areas to the cooler areas and then some of the heat will be conducted down to the surface to be emitted to space or absorbed. For this reason, warming of the atmosphere in our GHG-free planet is assumed to be exactly counterbalanced by warming of the surface from the atmosphere.

The model assumes that the atmosphere is completely transparent to incoming sunlight and all the energy is absorbed by the basalt surface which warms according to the sunlight incident angle on the daytime half of the spherical surface. While some sunlight will be scattered by the atmosphere, this effect is ignored in the model. Figure 1 shows the temperature after one day of the GHG-free Earth according to the model. The east-west locations are arbitrarily centered on the equator at zero longitude and are meaningless due to GHG-Earth’s constant rotation. The north-south locations are meaningful and are representative of an Earth with no axial tilt. The lack of an axial tilt means that GHG-free Earth has no seasons.

Figure 1 illustrates that the surface warms in direct proportion to the radiation it receives, and the maximum radiation is received when the sun is directly overhead. The maximum insolation, after correcting for the lunar albedo of 12% is 1198 W/m² and occurs for a few minutes along the equator at the local noon. In figure 1 this is at longitude=0 and latitude= 0, that is the equator directly south of Greenwich England where the temperature reaches 381K (108°C) at noon. Earth rotates from west to east, which is why the temperature on the nightside is higher on the eastern end of the night side (right of figure 1) than on the western side (left of figure 1).

Because the thermal diffusivity of basalt is very low, about 9 x 10^-7 m²/sec (Robertson, 1988), it takes a while to warm the upper layers of the rock surface and reach a sort of equilibrium global average basalt temperature in this model. The day side both absorbs solar radiation and emits thermal energy. However, the night side receives nothing but still emits radiation due to its stored thermal energy. The night side emits less energy than the dayside due to its lower surface temperature.

Exactly how much is stored in the basalt in the daytime, versus emitted later in the day when the surface temperature is cooler, is unknown, but can be estimated using basalt’s thermal diffusivity. Diffusivity varies with temperature roughly according to the function plotted in figure 2.

Even though the diffusivity is lower at higher temperatures, the basalt initially stores more energy during the day than it releases at night. The model tells us that more total energy is emitted on the daytime side than on the nighttime side, but this is due to the higher daytime surface temperature. Thus, the model tells us that some of the energy that is diffused into the basalt during the daytime is taken to the nightside, the rest is emitted during the day at lower daytime temperatures encountered nearer to the edges of the dayside as the planet rotates.

On the nightside some of the daytime stored energy rises to warm the surface and is emitted to space. There are two opposing forces at work on the night side. The thermal diffusivity of basalt increases at lower temperatures, but the lower nighttime temperatures cause lower emissions of energy to space. Thus, there is a mismatch between daytime storage and nighttime emissions. This does not mean that the surface cools to absolute zero, that will not happen because the thermal inertia of basalt is too high and because the diffusivity is too low, but it does mean that a stable surface temperature takes a while to reach. I ran this model for 36,500 iterations or about 100 years. The global average surface temperature evolved as shown in figure 3.

As shown in figure 3, using the assumptions built into my model, the average surface temperature eventually stabilizes to a surface temperature cooler than today’s temperature. This result uses the lunar albedo, which would be similar to Earth’s albedo in the absence of water or water vapor. It also assumes a fudge factor for thermal inertia of 0.1. I tried various estimates of thermal inertia, including thermal effusivity (Sabol, Gillespie, McDonald, & Danillina, 2006) and the “R” insulation factor, and none of them worked well for various reasons. There are a lot of ways heat can be transferred, conduction, convection, and radiation and these vary with the local circumstances, so there is no general definition of thermal inertia. However, all reasonable assumptions show a high value of thermal inertia in the basalt which causes the temperature to decline after model initialization.

Although different assumptions do change the ultimate equilibrium global average temperature, all reasonable values for thermal inertia result in a lower global average surface temperature than we have today. Figure 4 compares some of the scenarios I examined after 100-year runs. The model was run both with the lunar albedo of .12 and Earth’s current albedo (including the non-existent clouds) of 0.3. Without GHGs Earth would not retain its current albedo, but this is the traditional value used, so I ran it for comparison purposes. The two discussed thermal inertia values (0.1 and 0.15) are reasonable assumptions, since the actual thermal inertia of basalt is quite high, but even these values may be a little high. Higher values of this factor imply a lower inertia and lower values a higher inertia. Other values of inertia were investigated, but considered unlikely.

Thermal inertia is the resistance of a material to change its temperature. There is no formal or general way to describe thermal inertia, since it is very situation specific. Newton’s Law of Cooling works for small temperature differences but breaks down in situations like I’ve modeled here. Thermal effusivity, also called thermal responsivity, which is the square root of the product of thermal conductivity, density, and specific heat capacity didn’t work either. My final attempt was to use the “R” insulation factor, but it was another failure.

My analysis of these various inertia factors is attached as a spreadsheet, a link to download it is at the bottom of the post. Some sort of new measure of thermal resistance (inertia) will need to be developed for the situation I modeled. For now, I have created an assumed factor that is the proportion of thermal energy stored in the basalt that can make it to the surface in 12 hours and is free to be emitted as radiation. In the model this is called “inertia_f.” The values, 0.1 and 0.15, are reasonable considering the established values of thermal conductivity, density, and specific heat capacity of an average basalt.

Notice that all the temperatures calculated after 100 years are less than the current global temperature. Given the large uncertainty in the model, values that are greater than the current global temperature are possible, but values larger than 300K are thought to be extremely unlikely. In addition, the temperature difference between the dayside of the GHG-free rotating Earth and the nightside will never disappear, it is shown after 100 years in figure 5.

The assumptions used in the model

The model assumes that the only meaningful losses of energy are from the surface, although the atmosphere will emit a small amount of energy to space. These atmospheric emissions are ignored in the model.

The model assumes no axial tilt.

The model assumes a circular orbit.

The model does not consider convection, except to assume that it is net zero with regard to emissions to space. This is reasonable since we also assume the atmosphere is transparent to surface emissions.

The model assumes that any topography (mountains, valleys, etc.) do not affect heat transport by the atmosphere on a net basis.

The model assumes that the thermal diffusivity of the surface basalt follows a function of temperature as described in Robertson, 1988. The function is plotted in figure 2. Thermal diffusivity (α) follows the formula in equation 1:

Equation 1: α = where: k= thermal conductivity, ρ=density, and c_p is the specific heat capacity.

Thermal diffusivity increases at lower temperatures consistent with decreasing rock specific heat capacity and increasing thermal conductivity, see the attached spreadsheet for the details and units. The data available for thermal conductivity and specific heat capacity of basalt does not extend to the low temperatures encountered on the night side, so the values used in the model had to be extrapolated.

The thermal diffusivity of dry air is from 6 to 38 times higher than for basalt at the temperatures seen in this model. Thus, the surface heat flux will normally be from the basalt to the air if the temperatures are similar. But the lower value of six occurs at lower nighttime temperatures and if the overlying air is sufficiently warmer than the basalt there will be a flow from the air to the basalt. Thus, the expected high winds from the dayside to the nightside will matter and transport thermal energy to the nightside basalt to be radiated to space.

The most important assumption in the model is the assumed thermal inertia, which plays an important role in the temperature of the nightside. My calculation of retained heat carried from the dayside to the nightside is reasonable and justifiable, but the speed at which it is emitted to space on the nightside is somewhat speculative.

The calculated absorption of thermal energy by the surface on the dayside is very crude. I did use the thermal diffusivity of basalt in the calculation and assumed the remainder of the energy was emitted to space.

A uniform and smooth rocky surface was assumed for simplicity, which is unlikely. Without oceans, a rugged topography is likely, and it will guide the expected very high velocity winds in an atmosphere without water vapor. This will cause a more complicated and non-uniform surface temperature than shown in figures one and five. However, regardless of the complexity of the convection, it is reasonable to assume that net atmospheric transport of thermal energy is close to zero. The energy into the atmosphere comes from the basalt and the energy out of the atmosphere goes into the basalt. The real Earth has more control, by using energy storage in water and water vapor it has some control on both emissions and insolation by varying cloud cover, total atmospheric water vapor, and ocean storage, but this does not apply to a GHG-free Earth.

In my opinion, the GHG-free model has a fairly narrow range of plausible outcomes. Some model runs (not all runs are shown) result in global average temperatures a little above freezing, but global average surface temperatures higher than today are considered unlikely. Temperatures much lower than 235K (albedo=0.3 and Inertia_f=0.05) are also unlikely. Using this model, the total overall greenhouse gas effect is likely between 15 and 53°C. Thus, the flat Earth overall greenhouse effect is within the plausible range seen using this rotating spherical Earth model. While the influence of water vapor, ice, and water on the climate of Earth is readily seen, the influence of the other greenhouse gases is harder to detect.

The flat Earth greenhouse effect model is designed to simply compute the difference between the apparent blackbody temperature of the Earth as seen from space from the current global average surface temperature. Yet, Earth is clearly not a blackbody and the blackbody temperature as seen from space is not a surface temperature. The GHG radiation detected from space is emitted mostly by water vapor from 2 to 10 km in the atmosphere (see figure 4 here), plus some minor emissions from other greenhouse gases from various other altitudes. Surface radiation in the GHG frequencies cannot make it all the way to space from the surface. At sea level, a greenhouse gas is 50,000 times more likely to dissipate the energy from an absorbed photon via collisions with other molecules as re-emit it, so convection must first transport the thermal energy from the surface to an altitude where it can be radiated to space.

The rotating spherical Earth GHG-free model described in this post is more realistic than the flat Earth model, but it still has problems. As George Box famously wrote in 1976, “all models are wrong.” The model is not definitive, but my preferred model run has an albedo of 0.12 and an inertia_f of 0.1, the result of this run is shown in figure 3. It results in a GHG-free surface temperature of 265K (-8°C) which is smaller than the flat Earth model and shows a smaller overall greenhouse gas effect. However, this result is still uncertain. The main uncertainty in the model is in the thermal properties of the rocks on the surface, in particular the poorly defined “thermal inertia,” which was assumed.

This is a very simple model, more of a proof of concept than an actual model. It can be improved. Adding an axial tilt so GHG-free Earth has seasons might be interesting, so would adding some orbital eccentricity. But the most significant add would be a well-defined and appropriate function for basalt thermal inertia. Perhaps some petrophysicist out there has an idea of how to do that? We can only hope. Comments on the appropriateness of assuming the atmosphere is thermally net neutral would be interesting to read. In any case this is certainly an improvement on simply subtracting the average satellite measured black body temperature from the current average surface temperature to compute a possible greenhouse gas effect.

One additional point, I dislike the tendency of climate modelers to ignore surface thermal properties when modeling Earth’s climate. The surface, whether it is an ocean or land or a combination, is not a thermally static “slab.” In the real Earth, the surface, both the ocean and the land, have a large store of thermal energy and that storage changes with time (May & Crok, 2024) & (Crok & May, 2023), it definitely plays a role in long-term climate and should be taken into account.

To download the model, which is written in R, click here.

To download the thermal diffusivity spreadsheet click here.

Benestad, R. E. (2017, May). A mental picture of the greenhouse effect. Theoretical and Applied Climatology, 128, 679-688. Retrieved from https://link.springer.com/article/10.1007/s00704-016-1732-y

Box, G. E. (1976). Science and Statistics. Journal of the American Statistical Association, 71(356), 791-799. Retrieved from http://www-sop.inria.fr/members/Ian.Jermyn/philosophy/writings/Boxonmaths.pdf

Crok, M., & May, A. (2023). The Frozen Climate Views of the IPCC, An Analysis of AR6. Andy May Petrophysicist LLC.

Halbert, D., & Parnell, J. (2022). Thermal conductivity of basalt between 225 and 290 K. Meteorit Planet Sci, 57, 1617-1626. doi:10.1111/maps.13829

Hartlieb, P., Toifl, M., Kuchar, F., Meisels, R., & Antretter, T. (2015). Thermo-physical properties of selected hard rocks and their relation to microwave-assisted comminution. Minerals Engineering, 91, 34-41. doi:10.1016/j.mineng.2015.11.008

May, A., & Crok, M. (2024, May 29). Carbon dioxide and a warming climate are not problems. American Journal of Economics and Sociology, 1-15. doi:10.1111/ajes.12579

Robertson, E. C. (1988). Thermal Properties of Rocks. Reston: USGS. Retrieved from https://pubs.usgs.gov/of/1988/0441/report.pdf

Sabol, D. E., Gillespie, A. R., McDonald, E., & Danillina, I. (2006). Differential thermal inertia of geological surfaces. Proceedings of the 2nd annual international symposium of recent advances in quantitative remote sensing, torrent, Spain, (pp. 25-29).