Summary to Date

These days, we are studying the carbon dioxide cycle of the Earth's atmosphere, with particular attention to the concentration of the radioactive isotope of carbon that we call carbon-14. But this blog began with posts like Reliability of Surface Temperature, in which we discussed source of error in the measurement of global surface temperature. In Absorption, Not Reflection, however, we discussed the manner in which the Earth's atmosphere absorbs and radiates heat.

We set out to understand the Earth's greenhouse effect, and in particular to estimate how increasing CO2 concentration might affect the Earth's temperature. To start with, we had to be certain of the mechanism by which an atmosphere might warm a planet. At that time, several semi-formal papers had attained popularity on the web with their claims that the atmospheric greenhouse effect is a violation of the laws of physics, and so must be impossible. Meanwhile, we found that we could make little sense of the available explanations of the greenhouse effect. We did not know what to think.

We performed a series of simple experiments and presented their results in posts such as Glass Houses and Cling-Film Diaper. In Radiative Symmetry we showed by means of a well-known thought experiment that all substances must radiate and absorb any given wavelength with exactly the same facility. We called this the principle of radiative symmetry and we subsequently found this principle to be fundamental to any examination of the greenhouse effect. In The Greenhouse Effect we described the simple process whereby an atmosphere warms a planet surface.

In Solar Heat we calculated how much heat arrives from the Sun, and in Earth Radiator we considered radiation and absorption by the Earth's oceans. In order to better understand the greenhouse effect in real planetary systems, we studied a series of simplified systems in posts like Extreme Greenhouse and Planetary Greenhouse. We talked of short-wave radiation from the Sun and long-wave radiation from the Earth. We realized that no atmosphere could be opaque to long-wave radiation all the way from the surface to the altitude at which it thins into insignificance. As the atmosphere thins, it becomes less opaque. We came to understand that thin gas at high altitude radiates heat through the even thinner gas at even higher altitudes.

We studied the variation in atmospheric pressure and temperature with altitude in Atmospheric Pressure and Atmospheric Temperature. We discovered that air rising from the Earth must expand, and in doing so, must cool. We studied the transport of heat by convection in Atmospheric Convection. To our satisfaction, we stumbled the source of work that powers the Earth's weather, as we describe in Work By Convection.

By now, we understood that we had to plot and quantify the absorption of long-wave radiation by atmospheric gases at various altitudes in order to calculate the heat escaping from the Earth by radiation into space. We sought out the absorption spectra of water vapor and CO2. In posts like Water Vapor Continuum and CO2 Continuum we discovered that the absorption of radiation by gases is far more complicated than we at first supposed. We eventually discovered the Spectral Calculator, a website that calculates the absorption spectra of gasses for you, using databases developed by astronomers. In The Earth's Atmosphere we divided the Earth's atmosphere into six layers and presented the infra-red absorption spectrum of each layer.

Armed with accurate spectra for a clear spring day, we wrote a computer program to calculate the heat escaping from the Earth as a function of surface temperature. We presented our program in Total Escaping Power. In subsequent posts, we refined the program and our spectra. At last, in With 660 ppm CO2 we were able to calculate the effect increasing CO2 concentration. We found that doubling the CO2 concentration decreased the heat escaping from the Earth by 2.2%. And we showed that one way to make up this loss would be for the Earth's temperature to rise by 0.55%, or 1.5°C, while all other atmospheric gas and vapor concentrations remained the same.

And so we discovered for ourselves the basis of popular claims to the effect that doubling the CO2 concentration will warm the planet by 1.5°C. We discovered first-hand that these claims are based upon the assumption that the only mechanism by which the Earth's climate can compensate for a 2.2% decrease in escaping heat is by raising the average temperature of the planet surface.

In High Clouds we used the same Total Escaping Power program to show that high, thin, clouds will tend to warm the Earth by 38°C. In Thick Clouds, we showed that low, thick, clouds will tend to cool the Earth by 96°C. In Self-Regulation by Clouds we concluded that there must exist an equilibrium between high clouds, low clouds, and clear skies that maintains the Earth's surface temperature within certain limits. Furthermore, it became clear to us that there are many mechanisms by which the climate can compensate for the effect of increasing CO2 concentration other than by a rise in the surface temperature. A slight decrease in the occurrence of high clouds, for example, would allow more heat to escape. Alternatively, a slight increase in the occurrence of low clouds would reduce the amount of heat arriving from the Sun.

We were still far from being able to estimate the effect of increasing CO2 concentration upon the climate. In order to perform such an estimate, we would need to understand and quantify the interaction between water vapor, high clouds, and low clouds. We were not sure how to proceed.

We became diverted by our first-hand observation of a twenty-degree drop in temperature in the Arizona desert within a few hours of sunset. In Surface Cooling, Part I, we showed that this rapid cooling could not possibly be the result of air losing heat by radiation or conduction. In Surface Cooling, Part III we showed that the drop in temperature over the desert after sunset could be caused by cold air descending from above and replacing the sun-warmed air of the day. These and other posts like Night and Day and Venus forced us to think more clearly about convection and adiabatic expansion until we thought it might be possible to simulate both effects on an atmospheric scale with a simple, stochastic program.

We resolved to create an atmospheric simulation program that would model the transport of heat by convection and radiation, the evaporation of water, the formation of clouds, and the falling of rain within the Earth's atmosphere, or indeed any other atmosphere. We began to build our Circulating Cells program in small steps, each step resulting in a new version of the simulation. All our programs are free for our readers to download. Each is a text file that has instructions for execution in the comments at the top. It takes only a few minutes to get them running for yourself, and you can rum them on Windows, Linux, or MacOS.

We spent a couple of months working on the details of convection within our simulation. We found ourselves mystified by the forces that cause convection, as we present them in Impetus for Convection. Eventually, we came to understand the balance between thermal and gravitational energy that develops when convection stirs the atmosphere. We derived the relationship between temperature and altitude for a dry, moving atmosphere in Temperature, Pressure, and Altitude. We accounted for and explained the forces that cause convection in compressible and incompressible fluids in Impetus Dissected.

Once we were satisfied that our simulation handled convection properly, that we could relate the program iterations to the passage of time, and that all the heat entering the simulated system was accounted for by radiation from the top, we added blocks of either water or sand beneath the bottom gas cells, so as to simulate the planet surface. In Back Radiation we showed how the heat capacity and radiation produced by a semi-transparent atmosphere keeps the planet surface warm at night. In Island Inversion we see the surface of an island heating up ten times more than the surrounding ocean, while at night a layer of air a few hundred meters above the island is warmer, rather than cooler, than the air resting upon the island. Thus we see our simulation is consistent with our observations of surface cooling, including even temperature inversion.

Well-satisfied with our simulation of a dry atmosphere, we now turned to the simulation of a wet atmosphere, in which evaporation will cool the ocean and lead to the formation of clouds. To simulate cloud formation, we must have equations for the rate of evaporation from a water surface, the rate at which water vapor will condense out of rising air, the rate at which it will evaporate again in falling air, the cooling effect of evaporation upon the water surface, the warming effect of condensation upon the rising air, the amount of sunlight that will be reflected by existing clouds, and the amount of long-wave radiation that these same clouds will absorb and radiate. We obtained these relations in a series of posts Evaporation Rate to Consensation Rate.

We had not yet considered the downward drift of water droplets and the formation of rain. But after so many posts of mathematics and empirical relations, we returned to simulation. In Clouds Without Rain we presented a simulation of cloud formation and circulation without rain or snow. We start the simulation when the world is warm. Water evaporates. Hot air rises. It cools. Water vapor condenses into clouds. It reflects sunlight back into space. The world cools down. More clouds form. More sunlight is reflected. The world ends up freezing, and so we concluded that it is rain and snow that rids the sky of water vapor, saving the world from eternal frost.

In version 9.1 of our simulation, the water droplets in our simulated clouds sink towards the ground because of their own weight. In Slow-Sinking Clouds we allowed our water droplets to descend at 3 mm/s. In Fast-Sinking Clouds they descended at 300 mm/s. In both cases there emerged a striking and stable equilibrium between the rate at which water sank from the sky and the rate at which it evaporated from the sea. In Negative Feeback we show that this equilibrium is so stable that doubling the power arriving from the Sun causes only a 4°C change in the surface temperature of our planet. If we ignore the equilibrium, and estimate the effect of doubling Solar power using black-body calculations alone, we would expect the surface to warm up by 50°C.

Although our simulation was still primitive, it did show us that the balance between evaporation and precipitation brings stability to our climate. If we ignore this equilibrium when we estimate the effect of increased Solar power or atmospheric CO2, we are likely to over-estimate the effect by an order of magnitude. If that is the case, then the effect of doubling CO2 concentration is not the 1.5°C we calculated in With 660 ppm CO2, but smaller.

In Rain we discussed for the first time the source of rain. It turns out that the microscopic droplets of clouds hardly ever combine to form rain. Instead, rain begins as snow, and clouds turn into snow in what is called the Bergeron Process. We proposed a simple Evaporation Cycle for our simulation, by which water evaporating from the surface eventually returned to the surface as rain. We implemented this cycle in CC10, which we introduced in Simulated Rain.

Clouds reflect sunlight, which our simulation already accounted for, but they are also opaque to long-wave radiation. In Up and Down Radiation we discuss the absorption and emission of radiation by clouds. Up until this point, we had avoided addressing this phenomenon in our cloud experiments by making the simulated atmosphere itself opaque to long-wave radiation. In CC11 we implement the up-welling and down-welling long-wave radiation absorbed and emitted by the planet surface, the atmospheric gas, and the clouds. We restored our atmospheric gas to a state of 50% transparency to long-wave radiation, and ran our simulation again.

In Lapse Rate, we showed that our simulation now produced a surface temperature of 15°C, very close to the Earth's 14°C. As we ascended from the surface, the temperature drop was no longer the 0.01°C/m we obtained with our dry atmosphere, but 0.008 °C/m, which is closer to the Earth's lapse rate of roughly 0.0065 °C/m. We confirmed that the simulation was reaching thermal equilibrium with the incoming solar power: the heat radiated into space was equal to the solar power reaching the surface.

We now ran our simulation for several weeks to see what state it would be in for a wide range of solar powers. We started with 100 W/m² and ended with 1300 W/m². We compiled our data and arranged in the following way. First, we recognize that the only heat arriving from space and absorbed by the entire system is the solar power that penetrates to the surface. All the rest of the solar power is reflected into space. Thus the state of the atmosphere is depends upon the surface temperature, which dictates its radiation and evaporation, and in turn the temperature of the tropopause. Each time our simulation reaches equilibrium at a certain temperature, we have a measurement of the fraction of solar power penetrating to the surface and the total power escaping into space by radiation.

In Equilibrium Point, we assume a fixed solar power of 350 W/m², which corresponds to that of the Earth, and we plot the solar power penetrating to the surface, and total power escaping into space, versus temperature. We see that the equilibrium temperature of our simulation for 350 W/m² solar power is the temperature at which these two lines intersect: 16°C (1°C higher than for the first version of C11).

And so we found ourselves in a position to estimate the effect of doubling the CO2 concentration in the atmosphere. In Anthropogenic Global Warming, we bring together our study of the absorption spectrum of gases in the Earth's atmosphere and the measurements we obtained with our simulation, and we provide a graphical presentation of the anthropogenic global warming effect. Our graphical presentation shows not only a 1.6°C increase that would occur with constant cloud cover, but also a 2.7°C increase if we allow cloud cover to increase and ignore the increased reflection of solar power by these clouds. When we take account of the reflection of solar power by clouds, we see an increase of 0.9°C.

We conclude that a doubling of the Earth'e atmosphere's CO2 concentration would cause the average global surface temperature to rise by roughly 0.9°C. We suspect that the reason most contemporary climate models predict a rise of around 3°C is because they have ignored the drop in penetrating solar power that accompanies an increase in cloud cover. Thus these models take account of the warming effect of clouds, but not the cooling effect. The cooling effect, our simulation tells us, is far stronger than the warming.