Surface Cooling, Part III

UPDATE: The original version of this post included an argument by Second Law of Thermodynamics that, although correct, did not point out the actual physical mechanism by which desert air cools at sunset. I cut out this argument and added a description of the physical mechanism as suggested by my father.

The air above the desert cools by 20°C at night. But the heat capacity of the air above the desert is so great that it is impossible for it to cool so quickly by radiation into space or by conduction to the ground. We ended our previous post with the suggestion that the drop in temperature over the desert at night is caused not by the cooling of air, but by the replacement of warm air by cold air.

In Biggles Flies South, the author tells us that flying over the desert during the day is exhausting and uncomfortable because the air is so "bumpy". The best time to fly is the early morning, when the air is still and there is daylight to see by. Our understanding of atmospheric convection suggests that the air above the desert will be a succession of convection cycles, something like the drawing below, where small features of the surface end up dictating the boundaries of the cycles, and the cycles are angled sideways by a prevailing wind.



All this convection is caused by the sun heating the desert sand until it warms the air and causes the air to rise. But when the sun goes down, the sand cools off quickly. The circulation will eventually come to a stop, but it's momentum will keep it going for a while. When it is near to stopping, air is descending slowly from above, and comes to a stop. The sand is no longer hot, and no longer warms the air.

When the sun goes down over the desert, the day-time convection of the atmosphere slows to a halt. The slowing cycle brings cold air down from above. This cold air is not warmed by the hot sand as it would be during the day. The temperature drops by ten or twenty degrees in an hour. For the rest of the night, the air is still and cools slowly by radiating into space. Perhaps the temperature will drop by another one or two degrees by dawn.

Surface Cooling, Part II

In our Surface Cooling, Part I, we saw that the air above the ocean cools by less than 1°C at night. The air above the desert in Arizona, however, cools by 20°C at night. In Night and Day we saw that the heat radiated into space by the Earth is barely sufficient to cool the atmosphere by 1°C at night, let alone 20°C, so we are left wondering how the air above the desert can cool down so fast.

The table below presents the thermal properties of some common surface materials. We will make use of this table in future posts. Values for composite materials like wet soil are approximate. We picked values that are representative from resources like this, this, and this. Today we will use only the properties of dry sand. In the first column we have the thermal conductivity in Watt per meter per Kelvin (W/mK = W/m°C). A layer of material with thermal conductivity α, thickness d, temperature T_b at the bottom and T_t at the top, will conduct α(T_b−T_t)/d for each square meter of its surface area. A 10-cm thick layer of dry sand that is 20°C cooler on top than bottom will conduct 0.20 W/mK × 20 °C / 0.1 m = 40 W/m².

           Conductivity  Density   Capacity 
Material     (W/mK)     (kg/m^3)  (J/kgK)
Air           0.024        1.2       1000
Asphalt       0.75        1300        920
Copper         390        8900        420
Ice            2.2         910       2000
Mercury        8.7       14000        140
Sand, Dry     0.20        1600        830
Sand, Wet      4.0        2000       2500
Sandstone      1.7        2500        800
Soil, Dry      0.4        1300       1400
Soil, Wet      4.0        1500       2500
Snow, Light   0.10         100        420
Steel           43        7800        460
Water         0.58        1000       4200
Wood, Oak     0.17         770       1700

The second column gives the density in kilogram per cubic meter (kg/m³). A 10-cm layer of dry sand has mass 0.1 m × 1700 kg = 170 kg/m². The third column gives the specific heat capacity in Joule per kilogram per kelvin ( J/kgK = J/kg°C). A layer of material with specific heat capacity C, density ρ, and thickness d will require Cρd Joules of heat per square meter of surface area to raise its temperature by 1°C. A 10-cm thick layer of dry sand requires 800 J/kgK × 1700 kg × 0.1 m = 136 kJ/m² to warm up by 1°C. Alternatively, we must remove 136 kJ/m² to cool it by 1°C. If this layer loses loses 80 W/m², it will take half an hour to cool down by 1°C.

Our SC1 program simulates the cooling of the surface of the Earth at night. You can download it and run it yourself, by following the instructions in the code. The program works by dividing the surface material into thin layers and calculating the temperature of each layer in a sequence of small time steps. We assume the material is at a uniform temperature when it begins to cool, and that the heat loss from the upper surface is a constant 200 W/². In earlier versions of the code, we calculated the surface loss as a combination of radiation into space and convection with the surface air, but our results were pretty much the same, so we reverted to the assumption of constant loss to keep things simple.

The following graph shows what our simulation program tells us happens to dry sand at various depths in millimeters when it starts to cool from the surface at 200 W/m². We see the surface cools by 13°C in 1000s while the sand 40 mm down cools by roughly 0.1°C.



Materials with greater conductivity cool more slowly because heat rises from below to keep the surface warm. If we apply our simulation program to sandstone or asphalt, we find the surface cools down ten times more slowly than the surface of dry sand. So it is above dry sand or soil that we expect to see the greatest drop in surface temperature at night.

Our calculations show that a sandy surface can cool by 20°C at night. They show that one meter down, the heat of the day will hardly affect the temperature of the sand. But they still don't explain why the temperature above the desert sand cools by 20°C at night. The surface of the sand cools down because it loses heat to the air and to outer space. If anything, the cooling sand warms the air. We already know that the heat radiated by the air itself is not sufficient to cause such cooling. Indeed, our calculations show that it is impossible for air above the desert to cool by 20°C at night. And yet we know that a thermometer 2 m above the surface of the desert can register a drop of 20°C.

We conclude that the air above the sand at night cannot be the same air above the sand during the day, and indeed this conclusion leads us to a most satisfactory explanation for cold desert nights.

Surface Cooling, Part I

In Day and Night we found that the heat capacity of the atmosphere keeps the Earth warm at night. There are ten tons of air above every square meter of the Earth, and this ten tons has heat capacity 10 MJ/°C. When the atmosphere radiates 200 W/m² into space, which is typical of a clear sky, its average temperature drops by less than 1°C in twelve hours.

The heat capacity of water, meanwhile, keeps a pond warm at night. In a 1-m deep pond, there is a ton of water beneath every square meter of surface, and this ton has heat capacity 4 MJ/°C. If the water surface radiates 80 W/m² into space, which is typical when the sky is clear, this radiation will cool the pond by less than 1°C in twelve hours. The water surface will, of course, be losing more heat by atmospheric convection and by evaporation. But in the ocean, where the top ten meters of water are well-mixed by waves, even a surface loss of 800 W/m² will cool the water by less than 1°C at night.

Here is a twenty-day recording of air and water temperature from an ocean buoy, which we obtained from the National Data Buoy Center. The data is here.



We see that the air temperature varies by 5°C from one week to the next, but by less than 1°C from day to night. The water itself, meanwhile, is hardly affected by the weekly variations, and not at all by the daily variations.

Why, then, does the temperature in Arizona drop by 20°C at night? The surface of Arizona is sand and rocks. These solid, opaque materials transport heat within themselves by conduction alone. During the day, their surfaces get hot. During the night, they cool down. In our next post, we will present the thermal properties of some common surface materials, along with a computer program to calculate how fast these surface materials will cool at night. We will find out whether the thermal properties of sand can explain why the desert is hot in the day and cold at night.

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.