Day and Night

In a previous post, we considered the dramatic change in the temperature of the moon as it revolves around the Earth. In the middle of its night, the moon cools to −190 °C. In the middle of its day, it warms to 100 °C. I was in Phoenix, Arizona last week. The sky was clear all day and all night, and the air was dry. In the middle of the night, the temperature dropped to 0°C. In the middle of the day, it rose to 20°C. The day-night variation on the moon is 290°C, but on the Earth it is only 20°C.

The moon's days are one month long. At night, the surface rocks have time to cool down, losing all the heat they gained during the day. They get so cold they draw heat out of the moon's interior, and this flow of heat from the depths of the moon is what stops the moon's surface from dropping below −190°C. During the day, the rocks heat up until they start to radiate almost as much heat as is arriving from the Sun. But some heat flows into the depths of the moon to make up for the heat drawn out during the previous night.

The moon has no atmosphere, so the only thing stopping it from cooling down to absolute zero at night is the heat capacity and thermal conductivity of its surface rocks. On Earth, we have the heat capacity of the atmosphere to keep the surface warm during the night. Above each square meter of the Earth's surface is 10,000 kg of air (see Atmospheric Weight). The heat capacity of dry air is is roughly 1 kJ/K/kg. So the air above each square meter of the Earth has heat capacity 10 MJ. The Earth and its atmosphere radiate something like 250 W/m² into space. Suppose this 250 W/m² were extracted uniformly from the entire column of air above each square meter of the surface. It would take 10 MJ ÷ 250 W = 10 hrs to cool the air by 1°C.

We see that the heat capacity of the atmosphere is significant in reducing the temperature changes caused by the setting of the sun or the formation of thick clouds. Indeed, the temperature changes we observe from day to night on Earth are more than ten times greater than our simple heat capacity calculation suggests. Perhaps if we consider the individual 3-km layers of the atmosphere, we will come up with a different answer.

As day turns to night, the Sun no longer warms the Earth, but the Earth and its atmosphere continue to radiate heat into space. The following table gives the escaping power in Watts per square meter from each 3-km layer of the Earth's atmosphere up to the tropopause, as calculated by our TEP2 program under conditions we describe here.

--------------------------------------------------
      Name    Temp       BB     Layer   Escaping  
--------------------------------------------------
   Surface    290.0    401.1    388.8     80.0 
      0-km    280.0    348.5    262.6     46.1
      3-km    270.0    301.3    184.8     39.7
      6-km    250.0    221.5    111.5     44.6
      9-km    230.0    158.7     50.7     26.4
     12-km    220.0    132.8     22.2     10.1
     15-km    220.0    132.8     14.4      5.5 
--------------------------------------------------
Total:                                   252.5
--------------------------------------------------

The surface of the Earth radiates 390 W/m². Of this, 80 W/m² escapes directly into space. The rest of it is absorbed by the atmosphere. But the atmosphere radiates heat towards the surface also. The 0-km layer (the first 3 km of the atmosphere) radiates 260 W/m² upwards and downwards. We may have thought only of the upward component in the past, but the downward component exists also, and is equal to the upward component.

Let us ignore the heat passing up through the atmosphere by radiation and convection, and assume that the Earth's surface starts to lose 80 W/m² when the sun sets. Suppose the surface itself is water 1 m deep. The heat capacity of water is roughly 4 kJ/K/kg and its density is 1,000 kg/m³. Beneath each square meter is fluid with heat capacity 4 MJ/K. This water will cool by only 1 °C in 10 hrs.

The 0-km layer radiates 46 W/m² directly into space. The mass of air in this first 3-km layer is a little less than 3,000 kg, with heat capacity roughly 3 MJ/K. At a loss of 46 W/m², this layer will cool down by only 0.5°C in ten hours.

The changes in temperature we observe at the surface of the Earth from day to night are much greater than we would expect from a consideration of the heat capacity of the atmosphere and total escaping power alone. What is going on when the sun sets, that makes the Earth's surface cool down so fast? We will hope to answer this question in our next post.

Adiabatic Magic

A while back, a visitor here tried to persuade us that heat flow through an atmosphere was not required to make it warmer at the bottom than the top. He was convinced that adiabatic circulation would always take place within the atmosphere, regardless of whether or not heat was flowing, and this circulation would always make the bottom warmer than the top. At Science of Doom, meanwhile, several authors try to convince us that the surface of Venus will be super-hot even if it is not being heated by the Sun. They describe some kind of adiabatic process, which we will call adiabatic magic, by which any atmosphere will always be warmer at the bottom than the top, even with no heat flow. Indeed, one author goes so far as to say that heat flow from bottom to top will actually decrease the temperature difference between the lower and upper atmosphere.

We are well aware that air taking part in atmospheric convection will cool down as it rises, and warm up as it falls, but this circulation is powered by heat absorbed from the planet and subsequently radiated into space by the atmosphere. It is not spontaneous or self-generated. The circulation requires that the planet surface be warmed by the sun and that the atmosphere be capable of radiating heat into space at higher altitudes. With no heat flow, there will be no circulation, and no temperature difference between top and the bottom of the atmosphere. In other words: there is no such thing as adiabatic magic, as we will now demonstrate.

The diagram below shows how we can use adiabatic magic to create a machine that is 100% efficient at turning heat into work. Such a machine is also known as a perpetual motion machine of the second kind.



In the center of the diagram is an air column. The action of gravity is indicated by the double-headed arrow with the letter g. The column is high enough that the weight of the air it contains causes the pressure to be greater at the bottom than the top. By adiabatic magic, the bottom of the column is hotter than the top, so T₁ > T₂. We exploit this temperature difference with a heat engine. Metal veins absorb heat from the air at base of the column. A metal rod, well-clad with thermal insulation, carries this heat to a heat engine. A heat engine is something like a steam turbine, which exploits the flow of heat from hot to cold in order to generate work. Heat flows out of the engine through another metal rod to top of the column. The heat flowing into the engine is q₁ and the heat flowing out is q₂. The work produced by the heat engine is W. We express the heat flow and work output in Watts.

The First Law of Thermodynamics requires that W = q₁ − q₂. The Second Law of Thermodynamics requires that the entropy of the heat going in must be less than or equal to the entropy of the heat going out. With the absolute temperature scale, this means that q₁ / T₁ ≤ q₂ / T₂. Subject to these constraints, we see that our heat engine can produce work by exploiting the temperature difference between the bottom and the top of the air column.

Of course, we are taking more heat out of the bottom of the column than we are restoring to the top, so the total heat of the column will tend to decrease. We're not sure what the adiabatic magic theory has to say about an air column as it cools, so we are going to avoid this issue by adding heat to the column from a large reservoir. We add exactly as much heat as the column is losing by the action of the heat engine. The heat we add is Q, so Q = q₁ − q₂ = W.

So now let us consider the air column and the heat engine acting together. They constitute a perpetual motion machine. We see that we have Q entering the machine and W leaving. The machine turns heat into work with 100% efficiency. It is a perpetual motion machine of the second kind.

The adiabatic magic idea is one of a class of impossible ideas: those that propose the spontaneous separation of heat into hot and cold. The above diagram could be adapted to disprove any such idea with equal facility. If someone claimed they had invented a new material that always became hotter at one end than the other, even with no heat flow from one end to the other, we could disprove his claim with the above diagram.

Because adiabatic magic is impossible, we see that a transparent atmosphere can be no warmer at the bottom than at the top. Transparent gas cannot radiate heat into space (see Radiative Symmetry). In the case of Venus, if nothing heats the surface, the gas at the surface will be no warmer than the gas at the altitude to which sunlight does penetrate.

Of course, our atmosphere does radiate heat into space, so the lower air is warmer than the upper air, in order to transport that heat. And Venus's atmosphere gets hotter all the way to the surface, which means it is transporting heat away from the surface and radiating it into space.

Condensation and Convection

One of our readers has been asking me about the role of condensation in accelerating convection, as described in papers like this one. So let us dicsuss the effect of evaporation and condensation in convection.

Consider a kilogram of air that has just been warmed at the surface of the Earth, and is now rising towards the tropopause as part of atmospheric convection. This kilogram has temperature 300 K, pressure 100 kPa, and volume 860 l (liters). Let us suppose this air contains 10 l of water vapor. This water vapor has mass 10 g and would occupy 10 ml (milliliters) if it condensed into liquid.

Our rising volume of air expands and cools. Its pressure drops to 50 kPa and its temperature to 250 K. It expands to 1400 l. At this temperature, almost all of our water must condense into liquid. Before condensation, the water occupies 14 l. When it condenses it will turn into microscopic water droplets that occupy only 14 ml. Of course, these droplets will freeze afterwards, but let's ignore that for now, and consider what happens when the water first condenses.

Most likely, the condensation will occur gradually as the air expands, but we will imagine that it happens after the expansion is complete, so we can calculate the net effect of condensation upon the pressure of our gas. At the end of the expansion, our air contains supersaturated water vapor, like the gas in a cloud chamber.

Consider the first gram of water that condenses. Let us assume, for the time being, that this condensation happens instantly, so that our two cubic meters of air has no time to expand or contract during the process. That is to say: we assume the condensation takes place at constant volume.

Before condensation, this one gram occupied 1.4 l in our volume of 1400 l. After condensation, the water droplets occupy a combined volume of only 1 ml. The remaining air and water vapor expand into our 1400 l. As it expands, its pressure decreases by 0.1%.

When water condenses, it gives up its latent heat of evaporation, which is 2 MJ/kg. When the first gram of water condenses, it gives up 2 kJ of heat to our 1400 l of air. The heat capacity of air at constant volume is roughly 700 kJ/kgK, so our kilogram of air warms up by around 3 K. It's pressure increases by 3 K ÷ 250 K = 1.2%.

We see that the first gram of water condensation causes a net increase in pressure of 1.2% - 0.1% = 1.1% in our kilogram of air. Suppose the air all around is dry. It has not been warmed by condensation. It's pressure has not increased. Our two cubic meters pushes outwards with its greater pressure and expands until it's pressure is the same as that of the surrounding air. It will expand by 1.1%. Its 1 kg of mass will occupy the same volume as 1.011 kg of surrounding dry air. It will experience an upward buoyancy force of 11 g weight, or 0.11 N. This force will cause it to accelerate upwards at 0.11 m/s². Within less than a minute, it will be traveling upwards at a few meters per second. At that point, other forces will come into play to slow it down. We note that we allowed only 1 g of our water vapor to condense. If all 10 g condensed at once, our kilogram of air would experience ten times more lift.

We see that condensation does indeed cause air to rise more quickly than surrounding dry air.

Moist air will expand more than dry air as it rises towards the tropopause. Evaporation, on the other hand, causes air to rise more slowly. Air at the surface of a dry desert heats up fast in contact with the hot sand. It rises quickly. We get dust devils and sand storms. But air on the surface of the ocean does not heat up quickly. Water evaporates from the ocean, cooling the ocean surface and the air, so that the air lingers above the ocean, accumulating more and more water vapor. Eventually, the air will rise, if only because it becomes saturated with water vapor. But we see that water rising from the ocean is not as hot compared to the air above it as is air rising from the desert. The initial rise will be slow compared to the rise over a desert. But if the water starts to condense out of the moist air, and the moist air is surrounded by dry air, the moist air will begin to rise more quickly, sucking more moist air up beneath it.

Thus we expect low-altitude storms over a desert and high-altitude storms over an ocean.

PS. Another writer concludes that the upward force on a cubic yard of air due to condensation will be close to a thousand pounds. He says, "I’m not intending to do the calculations any further here, because it is basic knowledge." Had he performed the calculations, he would have discovered that the effect of water vapor contraction is opposite to his claims (the contraction tends to make the air more dense and therefore sink, while he claimed that the contraction decreases pressure and therefore creates lift) and that his forces were off by four orders of magnitude (the net lift is of order 0.1 pounds per cubic yard, not 700 pounds per cubic yard).

PPS: The debate on this subject continues over at Watt's Up With That?.

PPS. Look in the comments for an explanation of evaporation and condensation on a molecular level.

Venus

UPDATE: Initial version of this post used diatomic gas equation for the compression of CO2. Have now corrected this oversight, and find that our estimate of Venus's surface temperature is much improved.

One of our readers suggested we consider the atmosphere of Venus. What a good idea. Let's see how well we can estimate Venus's surface temperature using our understanding of atmospheric convection and the greenhouse effect.

According to Wikipedia, Venus's tropopause is at an altitude of 65 km with temperature 243 K (−30°C) and pressure 10 kPa. The surface pressure, meanwhile, is almost a thousand times greater: 9,300 kPa, which is ninety-three times the surface pressure on Earth. Venus's atmosphere is made up almost entirely of CO2, but also contains 150 ppm of SO2 (sulfur dioxide), and this SO2 condenses into liquid droplets so that the atmosphere below the tropopause is filled with pale yellow clouds.

Venus reflects 90% of incident sunlight (it's Bond Albedo is 0.90). The remaining 10% is absorbed. We're not sure what fraction of the Sun's light reaches the surface of Venus, but our guess is 1%. The SO2 clouds refract and reflect light like our own clouds, but SO2 is a pale yellow liquid, not a clear liquid, and will absorb sunlight eventually.

According to Schriver et al., even a 0.5-μm film of SO2 ice will absorb over 20% of long-wave radiation, so a 10-μm droplet of SO2 liquid will absorb it all. The clouds of Venus are near-perfect absorbers of long-wave radiation, and near-perfect radiators too, just like our own water clouds. Unlike the Earth, however, Venus is always entirely covered with clouds. Neither the planet surface nor the lower atmosphere has any opportunity to radiate heat directly into space. Venus radiates heat directly into space only from its cloud-tops and upper atmosphere.

Although most of the sun's heat is reflected back into space, 10% is absorbed, and we estimate that around 1% reaches the surface of the planet. This heat will raise the temperature of the surface until it forces convection. Here is the convection diagram from our Atmospheric Convection post.



Gas warms at the surface. It rises, expands, radiates heat into space, falls, contracts, and warms again. As the gas expands, it gets cooler. As it contracts, it gets warmer. In our simplistic analysis, we assumed that the expansion and contraction were adiabatic, meaning they took place without any heat entering or leaving the gas. In reality, the gas radiates heat to nearby gas, absorbs heat from nearby gas, mixes with nearby gas, and generates heat through viscous friction. But our adiabatic assumption allowed us to estimate the temperature changes using the equation of adiabatic expansion and contraction. For an ideal diatomic gas, such as N2 or O2, p^−0.4T^1.4 remains constant during adiabatic changes, where p is pressure and T is temperature. For CO2, however, the equation is p^−0.3T^1.3 at 300 K and p^−0.2T^1.2 at 1300 K (see here for thermodynamic properties of CO2).

The Earth's tropopause is at altitude 15 km. According to our typical conditions, the tropopause is at 220 K with pressure 22 kPa, while the surface pressure is 100 kPa. Adiabatic compression implies that air descending from the tropopause to the surface will heat up to 340 K. But the Earth's surface is at only 280 K. Air descending 10 km from the tropopause to the surface appears to lose 20% of its heat while contracting to a third of its original volume.

The atmosphere of Venus will heat up as it descends from the tropopause to the surface, and it will be hottest if it does not lose any heat on the way down. The pressure rises from 10 kPa in the tropopause to 9,300 kPa at the surface. The temperature starts at 240 K. Adiabatic compression of an ideal diatomic gas would heat the gas to 1700 K. If we use p^−0.25T^1.25 as an approximation for CO2, we estimate a final temperature of 941 K.

According to Wikipedia, the surface temperature of Venus is actually 740 K. Carbon dioxide falling 65 km from the tropopause to the surface appears to lose 20% of its heat while being compressed into less than a hundredth of its original volume.

Despite the 20% difference between our calculations and our observations, we see that atmospheric convection makes the surface of Venus incredibly hot while leaving the surface of the Earth delightfully temperate.

If we recall our post on work by convection and another on atmospheric dissipation, we see that a strong convection cycle causes powerful weather. Venus's convection cycle produces an order of magnitude more contraction and expansion than the Earth's. We expect Venus's weather to be an order of magnitude more extreme. And indeed it is.

Self-Regulation by Clouds

On a typical spring day at latitude 30°N, our TEP2 program tells us that the formation of high clouds will tend to warm the Earth's surface by 38°C, while the formation of thick clouds will tend to cool the Earth's surface by 96°C.

Over the past few weeks here in Boston, the warmest days have been those with high, thin clouds, and the coolest have been those with low, thick clouds. But the difference between the warm days and the cold days is only 10°C. Despite alternating clouds, rain, and sunshine, the temperature at a particular location anywhere in the world is almost always within 10°C of its monthly average at mid-day, and these monthly averages vary by less than 1°C from one year to the next.

How can it be that the Earth's surface temperature remains so stable when clouds have such tremendous power to heat and cool the surface? Some kind of self-regulation of cloud formation and evaporation must be taking place. Indeed, at first glance it seems to us that warmth will promote evaporation from the oceans and lead to the formation of the low, thick clouds that cool the Earth, while cold will suppress evaporation, encourage atmospheric convection, and lead to the formation of the high, thin clouds that warm the Earth. So each type of cloud tends to promote the formation of the other, which is a basis for self-regulation.

An example of self-regulation is the action of a thermostat in a house's heating system. When the heater turns on, the house warms up. If the heater stays on, the house will get too hot. The thermostat turns off the heater when the house is warm enough. The house cools down. If the heater stays off, the house will get too cold. The thermostat turns on the heater when the house has cooled enough. The difference between the turn-on and turn-off temperatures is the hysteresis. The temperature half-way between is the set-point.

The heating system in our house can heat the inside to 30°C when the outside is at 0°C. With the heater off, the house will eventually cool to 0°C. But our house is always close to 20°C inside. That's because the set-point of our thermostat is 20°C and its hysteresis is only 1°C.

High clouds at latitude 30°N in spring would heat the Earth to 55°C if the day were long enough and no change to the high clouds took place. Thick clouds would cool the Earth to −79°C. But the mid-day temperature remains within ±5°C of 17°C. The set-point of cloud self-regulation is roughly 17°C and its hysteresis is 10°C. The self-regulation reduces a maximum possible variation of 134°C to an observed variation of 10°C. That's a factor of thirteen reduction through self-regulation.

Our TEP2 program tells us that if we double the CO2 concentration in the atmosphere, the Earth's surface will tend to warm up by 1.5°C. Given the strong self-regulation of warming and cooling by clouds, how much warming will we actually observe if we double the CO2 concentration? It may be that the self-regulation of clouds is entirely independent of the heat retained by extra CO2, so that doubling the CO2 will increase the average temperature of the Earth's surface by a full 1.5°C.

But cloud self-regulation is not independent of the heat absorbed by extra CO2. According to our calculations, doubling the CO2 concentration of the atmosphere causes the atmosphere to retain an additional 5.0 W/m² of heat. According to our typical atmospheric conditions, air close to the surface contains 1% water vapor, which amounts to roughly 10 g/m³. When this water condenses into water droplets, thus forming a cloud, the water vapor gives up 20 kJ/m³ of latent heat. If this condensation takes place over 1000 s (about fifteen minutes) in a column 1000 m high, it generates 20 kW/m², which is four thousand times the heat retained by our extra CO2. The heat retained by extra CO2 and the heat produced by condensation are exactly the same type of heat. They are indistinguishable. We think it likely, therefore, that cloud self-regulation will reduce the effect of extra CO2 in the same way that it reduces the effect of cloud variation: by a factor of thirteen. In that case, doubling CO2 concentration will warm up the world by only 0.1°C.

In the presence self-regulation, however, we cannot assume even that CO2 doubling will have a net warming effect. Our calculations show that the extra CO2 causes the heat to be retained mostly between altitudes 3 km and 6 km. In an earlier post we showed how turning on an electric heater in the same room as your home's thermostat will cause the the rest of the house to cool down. It could be that the heat retained by CO2, by virtue of being concentrated in the middle-troposphere, affects cloud self-regulation in such a way as to cause a slight cooling of the surface of the Earth below. Such an outcome seems unlikely, but it is possible.

Another possibility suggested by documents like this one is that the 1.5°C warming from CO2 doubling will in fact be amplified by the self-regulation of clouds, so that we see a 3°C or even 4°C warming due to CO2 doubling. We can think of no examples of such behavior by self-regulating systems, nor do we understand the arguments presented by climatologists who propose such amplification.

In future posts we hope to explore the sources of self-regulation by clouds, and perhaps construct a simple model to investigate how the average temperature of the self-regulated system will be affected by such things as CO2 concentration and the availability of water. Before that, however, we will test our climate model by applying it to day-night variations, and to the planet Venus.

Thick Clouds

We have been using our computer program, TEP2 to estimate how much the Earth's surface will tend to warm up if we make certain permanent changes to its atmosphere. In With 660 ppm CO2 we found that doubling the atmosphere's CO2 concentration from 330 ppm to 660 ppm tends to warm the Earth's surface by 1.5°C. In High Clouds we found that the formation of high, thin clouds in a clear sky will tend to warm the Earth's surface by 38°C.

Today we consider the effect of thick clouds. We start with the Earth on a typical clear spring day at latitude 30°N, as represented by our EA1.txt absorption spectra. Our TEP2 program tells us that the total escaping power is 252.5 W/m². We allow a thick cloud to form from altitude 3 km to 12 km. In Clouds we argued that a cloud this thick will reflect 90% of the Sun's radiation and absorb 100% of the Earth's radiation. We simulate such a cloud by setting the absorption of our 3-km, 6-km, and 9-km layers to 1.0 for wavelengths 5 μm to 60 μm to produce EA4. We apply TEP2 to EA4 and obtain the following output.

------------------------------------------------
      Name    Temp       BB     Layer   Escaping
------------------------------------------------
   Surface    290.0    401.1    388.8      0.0
      0-km    280.0    348.5    262.6      0.0
      3-km    270.0    301.3    292.0      0.0
      6-km    250.0    221.5    214.0      0.0
      9-km    230.0    158.7    152.3    120.8
     12-km    220.0    132.8     22.2     10.1
     15-km    220.0    132.8     14.4      5.5
------------------------------------------------
Total:                                   136.5
------------------------------------------------

This output is exactly the same as the one we obtained for high clouds. Both the high clouds and the thick clouds have the same effect upon the total escaping heat. The difference between high clouds and thick clouds is that thick clouds block 90% of the Sun's heat from reaching the Earth, while high clouds block only 10%.

As we did in High Clouds, let us suppose that the Earth radiates as much heat as it absorbs when its surface and atmosphere are as described by EA1. The total power escaping the earth is 252.5 W/m², so the total power arriving from the Sun must be 252.5 W/m² also. In the conditions described by EA4, 90% of the heat arriving from the Sun is reflected back into space, so only 25.2 W/m² is absorbed by the Earth. But the total power escaping the Earth is 136.5 W/m². When these large clouds form, the Earth will start to radiate 111 W/m² more than it absorbs. It will cool down. In our TEP2 program, we decrease the temperature of the Earth and its atmosphere until we arrive at the following output.

------------------------------------------------
      Name    Temp       BB     Layer   Escaping
------------------------------------------------
   Surface    194.3     80.8     76.0      0.0
      0-km    187.6     70.2     59.0      0.0
      3-km    180.9     60.7     56.4      0.0
      6-km    167.5     44.6     40.8      0.0
      9-km    154.1     32.0     28.7     22.3
     12-km    147.4     26.8      4.5      2.9
     15-km    147.4     26.8      2.1      1.1
------------------------------------------------
Total:                                    26.3
------------------------------------------------

Here we see the total escaping heat is 26.3 W/m², which is pretty close to the 25.2 W/m² arriving from the Sun through the thick clouds. The Earth's surface has cooled by 96°C.

According to our simulation, the formation of a 9-km thick layer of cloud at altitude 3 km has a cooling effect sixty-four times stronger than the warming effect of doubling the atmosphere's CO2 concentration.

UPDATE: When we look at discussions like this one about clouds, we see that climatologists recognize the power of clouds to cool and warm the Earth. But we don't see them giving us values in Centigrade in the same way that they do for a hypothetical doubling of CO2 concentration.

High Clouds

Sitting on a plane over the Atlantic Ocean last month, I was at 13,000 m looking down on a continuous layer of thin, high cloud. Through occasional openings, the ocean below appeared brightly-lit despite the clouds above it. Today we estimate the effect of such a cloud layer upon total escaping power using our TEP2 program.

Let us place a 1-km thick layer of cloud in our 9-km atmospheric layer. This layer extends from altitude 9 km to 12 km. In our previous post we showed that a 1-km thick layer of cloud absorbs 100% of incident long-wave radiation, but only 10% of incident short-wave radiation. In our atmospheric absorption spectra we simulate such a cloud by setting the absorption of our 9-km layer to 1.0 for wavelengths 5 μm to 60 μm. We start with our EA1 spectra (typical, clear, spring day at 30°N) and produce EA3. We submit EA3 to TEP2 and obtain the following table of escaping power in W/m². We add the final column by hand, which gives the change in escaping power caused by the introduction of the cloud.

-------------------------------------------------------
      Name    Temp       BB     Layer   Escaping  Change
-------------------------------------------------------
   Surface    290.0    401.1    388.8      0.0   -80.0
      0-km    280.0    348.5    262.6      0.0   -46.1
      3-km    270.0    301.3    184.8      0.0   -39.7
      6-km    250.0    221.5    111.5      0.0   -44.6
      9-km    230.0    158.7    152.3    120.8    94.4
     12-km    220.0    132.8     22.2     10.1    0.0
     15-km    220.0    132.8     14.4      5.5    0.0
-------------------------------------------------------
Total:                                   136.5 -116.0
-------------------------------------------------------

No radiation escapes into space from the Earth's surface or from the first three layers of the atmosphere. A total of 210.4 W/m² is absorbed by the lower surface of the cloud layer. The cloud layer itself, however, is a fine radiator of heat, and radiates 120.8 W/m² into space.

We note that the black-body radiation for our 9-km layer is 158.7 W/m² (BB column) while the heat that TEP2 calculates by numerical integration is only 152.3 W/m² (Layer column). Our spectra extend to 60 μm, but the cool cloud (−43°C) radiates some power in the range 60-100 μm. We will accept this 4% error and move on.

The cloud causes the total escaping power to drop by 116 W/m², or 46%. We now ask how much warmer the Earth's surface will have to get in order to make up a 46% drop in total escaping power caused by the formation of a 1-km layer of cloud at altitude 10 km. The 1-km cloud absorbs some of the heat arriving from the Sun, and we must take this into account somehow when we perform our estimate.

Let us suppose that the Earth radiates as much heat as it absorbs when its surface and atmosphere are as described by EA1. Because the Earth and its atmosphere radiate a total of of 252.5 W/m², we suppose that a like amount of heat arrives from the Sun and lands upon the Earth's surface. When we place a 1-km layer of cloud in the way, we estimate that 10% of the Sun's heat will be reflected back out into space. When the cloud layer forms, the heat arriving from the Sun decreases to 227 W/m² and the heat escaping from the Earth decreases to 136 W/m².

By experimenting with TEP2, we find that a 13% increase in the absolute temperature of the Earth and its atmospheric layers will produce the following result.

------------------------------------------------
      Name    Temp       BB     Layer   Escaping
------------------------------------------------
   Surface    327.7    653.9    629.9      0.0
      0-km    316.4    568.3    413.4      0.0
      3-km    305.1    491.3    284.7      0.0
      6-km    282.5    361.1    169.3      0.0
      9-km    259.9    258.7    250.4    201.4
     12-km    248.6    216.6     34.6     14.8
     15-km    248.6    216.6     23.5      8.7
------------------------------------------------
Total:                                   224.9
------------------------------------------------

The total escaping power is now 225 W/m², which is close enough to the 227 W/m² arriving from the Sun. When a cloud forms at altitude 10 km, the heat escaping from the Earth will balance that arriving from the Sun only if the Earth's surface warms by 38°C. If, instead of a cloud forming at 10 km, we double the CO2 concentration from 330 ppm to 660 ppm, the same calculation suggests the Earth will warm by only 1.5°C.

According to our simulation, the formation of a 1-km thick layer of cloud at altitude 10 km has twenty-five times the warming effect as doubling the atmosphere's CO2 concentration.

UPDATE: Michele points out that high clouds will contain microscopic ice particles instead of water droplets. Ice, like water, is also opaque to long-wave radiation and transparent to short-wave radiation.