By Andy May
Originally posted November 2016, edited April 2017 to correct some errors.
We could define the total greenhouse effect (GHE) as the difference between the calculated blackbody temperature of the Earth, about 254K, and the actual average surface temperature, roughly 288K. Using this definition, the average total greenhouse effect on the Earth would be 34°C. The blackbody temperature is supposed to be the temperature of the planet without an atmosphere, but with the same Bond albedo (reflectivity). The Earth is not a blackbody because it stores heat, particularly in the oceans, but this is ignored by some. The Moon has a calculated blackbody temperature of 270K, no atmosphere, and an average temperature of 250K, so even the Moon is not a blackbody. Because the atmosphere is transparent to most solar radiation and the Earth’s surface is opaque, the surface of the Earth absorbs twice as much radiation as the atmosphere. Per the laws of thermodynamics, a planet must emit as much radiation as it receives. The Earth’s temperature controls the type of radiation it emits and it emits mostly in the thermal infrared, which is called outgoing longwave infrared radiation, usually abbreviated as “OLR.”
One can see many other definitions of the GHE. Common descriptions include both the effect and a cause. Some describe the GHE as a blanket, keeping some of the radiation emitted by the Earth from reaching outer space warming the Earth. Fourier proposed this idea in 1824. Others, like Arrhenius, propose that infrared-active gases (“greenhouse” gases) absorb radiation emitted by the Earth and re-emit some of it back toward the Earth’s surface. Some modern climate scientists like to define a theoretical “radiative forcing” as the GHE. This is defined in IPCC AR5 as the theoretical (but unmeasured) “change in net irradiance at the tropopause” over some period.
We prefer to use the term to describe the effect and leave the cause for later discussion. First, we need to make some reasonable assumptions about heat transfer in the atmosphere. Ultimately all heat energy emitted by the Earth’s surface must make its way to outer space. This is a reasonable assumption because the Earth’s surface temperature has been reasonably stable over geological time, only varying from about 282K to 302K for the last 500 million years per paleontological data. Our current temperature is about 288K, see Figure 1.
Figure 1 (source: Glen Fergus, click the name or figure for a high-resolution version)
The only heat transmission method that works in a vacuum is radiation, so the final step in the heat energy transfer process is radiative energy transfer. This last step into outer space is more likely to be from high in the atmosphere in the upper troposphere or in the stratosphere where the air is thin and at a cooler temperature. The space bound radiation is from infrared-active gases like CO2, H2O, and O3. Data from the Mars Global Surveyor probe confirms that this is the case. It was also found that only 17% (40 W/m2) of the radiation made it to outer space directly from the Earth’s surface according to Hugues Goosse’s textbook.
The main source of GHE heat energy is the surface of the Earth, which warms the atmosphere like a stove warms a pot of water. Except for the 17% of the emitted energy that makes it all the way to outer space as radiation in one hop, the remaining heat energy must be transported somehow to the upper troposphere or to the stratosphere or absorbed. James Hansen has estimated that, currently, the net absorbed radiation, by the Earth, mainly in the oceans, is 0.6 W/m2 per year, although our measurements are not accurate enough to measure such a small amount of power. The remaining energy eventually makes its way to space. The question is how does this happen and how fast?
There are four heat transfer mechanisms that work in the troposphere: radiation, convection, thermodynamic mechanical energy transfer, and conduction. If transferred by radiation, an infrared-active gas captures infrared energy and then re-emits it later, either up toward space or back toward the surface. The energy can be re-captured and emitted again and again until it reaches outer space. This is a relatively slow process that takes days or weeks. See figure 2.
Figure 2: atmospheric radiative cooling and warming (after Salby, 2012, Fig. 8.24)
Figure 2, from Murry Salby’s textbook (page 242), was created using a model of radiative heating and cooling in a clear atmosphere. Figure 2a shows the longwave infrared cooling rate in °C/day (X axis) versus height (Y axis). Figure 2b shows the shortwave (input from the sun) radiative heating of the atmosphere. The dotted line in 2a and 2b is the radiative cooling and heating provided by water vapor. Water vapor provides 80% to 90% of all radiative cooling in the troposphere. Water vapor is also the only significant shortwave absorber in the troposphere and provides almost all shortwave radiative heating (except for the surface). The water vapor shortwave heating is outweighed by the water vapor longwave cooling producing a net radiative cooling of 1°C per day in the troposphere. The cooling provided by carbon dioxide in the troposphere is negligible, but it is very significant in the stratosphere where the CO2 can radiate heat to outer space more readily due to a lower atmospheric density. The effects of other gases are also shown, ozone has a significant radiative heating impact in the stratosphere that is larger than the carbon dioxide cooling effect, producing net warming in the stratosphere.
If we assume that the troposphere is in radiative equilibrium, then heat energy from the surface would only travel to the upper atmosphere via radiation. In section 1.3.2 of David Andrews’ textbook An Introduction to Atmospheric Physics, he shows that a simple radiative model of the Earth results in a surface temperature of 286K (close to the observed 288K) and an atmospheric temperature of 245K. Dr. Roy Spencer has shown a similar model and result here. As David Andrews notes:
“This close agreement is somewhat fortuitous, however, since in reality non-radiative processes also contribute significantly to the energy balance.”
In other words, we know the troposphere is not in radiative equilibrium, so the close agreement of the radiative model temperature and the actual temperature of the Earth’s surface doesn’t prove that a radiative greenhouse effect causes (or sets) the Earth’s surface temperature.
The radiative models do not include convection, conduction or mechanical heat energy transfer. Global average atmospheric properties were used to compute Andrews’ surface temperatures and the curves in Figure 2. Local conditions at any place on the Earth would make the curves and values different. So, like “global average temperature” we are looking at a radiative model of global average conditions. Convection and thermodynamic mechanical energy transfer processes also work to cool the troposphere. Kiehl and Trenberth (1997) and Ramanathan and Coakley (1978) suggest that convection dominates heat transfer in the lower part of the troposphere.
The tropospheric lapse rate is the decrease of temperature with altitude. Currently the tropospheric observed average lapse rate is 6.5°C per kilometer. At this rate we reach the Earth’s calculated blackbody temperature at 5.2 kilometers on average, which we call the top of the atmosphere or the TOA (figure 3) for the purposes of this post. The Earth is not a blackbody, so the calculated blackbody temperature may have no physical meaning, but we will use the value for convenience. Above that altitude more radiation is emitted to outer space than retained by the atmosphere. As the lapse rate increases (the lower the line in figure 3, the higher the lapse rate), surface warming goes up and the TOA decreases in height. As the lapse rate decreases surface warming goes down and the TOA rises. This can be seen in Manabe and Strickler’s figure shown below as Figure 3.
Figure 3 (from Manabe and Strickler)
The actual surface temperature of the Earth is controlled primarily by convection and the lapse rate. That is, how rapidly the Earth’s OLR makes its way to outer space. If we add the troposphere to the Earth, without any water or other greenhouse gases, we would observe a dry adiabatic lapse rate of 9.8°C/km. Water absorbs shortwave radiation from the sun and longwave radiation emitted from the Earth’s surface, plus it carries latent heat of evaporation from the surface via convection. So, it works to cool the Earth’s surface and helps to lower the lapse rate in the troposphere to an average of 6.5°C/km. Other greenhouse gases and mechanical energy transfer also play a role in lowering the tropospheric lapse rate.
The dry adiabatic lapse rate is solely a function of gravity and the heat capacity (see equations 2.26 and 2.27 in Andrews’ textbook) of the troposphere, or put another way it is a function of pressure and heat capacity. We can see in Figure 2 that the radiative impact of 400 ppm of carbon dioxide is almost too small to see. But, the impact of evaporated water, mechanical energy transfer and convection is large. As described by Salby on page 90 of his textbook:
“The [interaction] of a displaced air parcel that does not interact thermally with its environment…is mechanical. … This also applies to an air parcel that is moist, but outside of a cloud. Inside a cloud, the simple behavior predicted by adiabatic considerations does not apply. It is invalidated by the release of latent heat.”
As water lowers the lapse rate, the surface temperature lowers and the TOA moves higher. This effect can be seen in the real world, since the tropopause is much higher over the equator than over the poles. The tropopause (where temperature stops decreasing with height) is also the point where nearly all the water vapor has condensed out as liquid water. There is very little weather above the troposphere.
The troposphere is not in radiative equilibrium and this is one of the reasons we have weather. Radiative equilibrium, in the troposphere is unstable because the radiative equilibrium lapse rate (see the lower line in figure 3) is below the dry adiabatic lapse rate (meaning it has a higher lapse rate). Any lapse rate higher than the gravity controlled dry adiabatic lapse rate of 9.8K/km is unstable and convection will start spontaneously. This is described in detail here in section 12.3. It is also explained well in Andrews’ textbook in section 2.5 and in Britannica.com. We will not explain further here other than to say a heated parcel of air will accelerate away from its position after moving slightly when the atmosphere is unstable. This sets up convection. A temperature inversion (think Denver or Los Angeles) where the temperature rises with altitude is stable, so convection is not initiated and air pollution is trapped. Another example is the stratosphere, it is stable and in radiative equilibrium because the stratospheric temperature increases with height, see figure 3.
Convection allows wind to carry away water vapor evaporated from bodies of water and the water vapor carries latent heat of evaporation with it. This cools the Earth’s surface and redistributes heat energy to cooler places like the poles. Thus, the OLR that is not radiated directly to outer space can be carried away from the surface as latent heat of evaporation or transported as internal kinetic energy by non-greenhouse gases. Latent heat is explained in Figure 4. The portion of the heat energy that is carried by atmospheric processes such as molecule to molecule radiative transfer, convection or thermodynamic processes is obviously slower than direct radiation from the surface to space and this difference in speed warms the Earth’s troposphere.
Figure 4 (credit splung.com)
When a solid, like ice, is heated to its melting temperature “B” the temperature stops rising and all additional input energy melts the ice. At energy point “C” all the ice has melted. Then additional energy begins increasing the temperature of the liquid until the boiling point “D” is reached. At this point the temperature stops increasing until the liquid is gone and the water has vaporized at point “E.” As you can see in the diagram it takes 5 times the energy to boil the water into a vapor than it takes to raise the temperature of the water from 0°C to 100°C. The energy required to boil the water is called the latent heat of vaporization or the enthalpy of vaporization. For water the specific latent heat of vaporization is 2,258 kJ/kg, which is higher than the value for most liquids. In addition, water vapor is the dominant radiative conveyor of heat from the surface to the top of the troposphere as shown in Figure 2.
When solar energy strikes the surface of an ocean or body of liquid water it will vaporize some of it, this will cause the body of water to lose heat energy and cool. The vaporized water will carry that energy as latent heat into the atmosphere. The surface air pressure is constant (ignoring the minor impact of humidity on air density) for a given elevation because it is only a function of average atmospheric density and gravity. So, the heating of the surface air causes it to expand per the physical gas laws. The expansion of the gas causes more collisions between gas molecules, resulting in an increase in internal energy and mechanical energy transfer. This is illustrated in Figure 5.
Figure 5 (credit purdue.edu)
The new energy is carried both up and outward, think of it as expansion, or an increase in the volume of an air parcel. The thermodynamic mechanical energy transfer can be compared to a sound wave where the individual air molecules don’t travel very far, but they transmit kinetic energy a greater distance very quickly. Connolly and Connolly have estimated that mechanical energy transfer by work of surface heat energy in the troposphere has a speed of at least 40 meters/sec.
Unlike energy transmission by radiation; conduction, thermodynamic and convective energy transfer all require mass. Conduction moves energy through the mass without requiring any mass movement. It seems that everyone on both sides of the debate thinks conduction through air is insignificant because air is such a good thermal insulator. The heat conductivity of humid air is only 0.028 W/m/K and at this rate moving any significant amount of heat to the top of the atmosphere would take months or years.
So, we are left with convection and thermodynamic heat transfer. As a surface air parcel warms and expands due to heat radiated by the Earth’s surface, it works to push air upward and outward while cooling and seeking equilibrium (See figure 6). This, in addition to any radiative greenhouse cooling or heating, sets up convection. Most of the molecules (oxygen and nitrogen) can only transfer heat energy mechanically by colliding with their neighbors and transmitting their internal kinetic energy to another molecule.
Water vapor can transport latent heat of evaporation a considerable distance. The heat is released to the surroundings whenever the water vapor condenses into water droplets and falls as rain. Water vapor can be carried by convection as high as the tropopause in thunderstorms before it condenses.
When the water vapor gets that high, it is a short trip to the stratosphere where carbon dioxide can absorb it and later emit it to outer space. Heat transfer from the upper troposphere to the stratosphere is normally through radiation or thermodynamic (mechanical) processes. In the stratosphere, which is close to radiative equilibrium (see Figure 3), radiative transfer dominates.
Connolly and Connolly have shown that the troposphere, tropopause and the lower stratosphere appear to be in thermodynamic equilibrium internally and with each other. Over warmer areas, like the western equatorial Pacific Ocean, the tropopause is very high, up to 18 km. This lowers the lapse rate and it provides more cooling to the Earth’s surface. Newell and Dopplick have suggested that this extra cooling effect acts like a thermostat and effectively limits the surface temperature to about 30°C. This is supported by paleontological evidence that suggests the Earth’s equatorial temperatures have not changed much in thousands of years. Rosenthal, et al. determined from foraminifer fossils that sea surface temperatures near Indonesia have varied less than 0.5°C (+-0.35°) in the last 9,000 years. Hoffert, et al. have shown that even 65 to 140 million years ago (the Cretaceous geological period) when carbon dioxide levels were eight times what they are today, the equatorial sea surface temperatures did not exceed the evaporatively buffered limit of 303K (30°C).
Ferenc Miskolczi (2014) has written:
“As long as the Earth has unlimited water supply (in the oceans) with its three phases permanently present in the atmosphere and two phases on the ground surface, the stability of the planetary climate will be controlled by the equations (see paper, page 19). These two equations, together with the Clausius-Clapeyron equation, will regulate the transfer of the latent heat through the boundary layer in such a way that the net amount maintains the planetary radiative balance.”
Clearly, convection and thermodynamic heat energy transport dominate in the low latitudes. Radiation probably plays a larger role at the poles where absolute humidity is lower. In the colder polar regions, especially during the winter, there is a net loss of radiant energy directly to outer space and the tropopause is close to the surface, sometimes as low as 8 km.
In figure 3 we can see that the radiative equilibrium curve terminates at the surface at a temperature of roughly 332K or 78°C higher than the calculated blackbody temperature, much warmer than we observe today. Convection and thermodynamic mechanical heat energy transfer short-circuit the radiative process in the troposphere and carry heat energy (often as latent heat) to the stratosphere more quickly than radiative transfer can.
Since radiative equilibrium is unstable in the troposphere, thermodynamic mechanical energy transfer sets up convection and convection causes our weather.
Thermodynamic mechanical heat energy transfer is quicker than convection and convection is quicker than radiative transport (Salby, page 238). Evaporated water condenses along the saturated adiabat and cannot normally pass the tropopause which is where the saturated-adiabatic and the radiative-equilibrium profiles cross. The mechanical/thermodynamic mechanism will transfer energy first due to its speed, but how far is in question due to the probable rapid energy loss with distance. Connolly and Connolly believe that thermodynamic energy transfer (pervection) is effective up to 40 kilometers or so. Convection (including enthalpy) is the second fastest and may transport most of the energy, especially over long distances, due to its efficiency.
Conclusions and summary
The dry adiabatic lapse rate of 9.8K/km in the troposphere is determined by gravity and the heat capacity of the air. Because this is higher than the current equilibrium lapse rate, the troposphere is unstable, and convection starts spontaneously. The actual lapse rate, currently an average of 6.5K/km, is due to convection and thermodynamic gas expansion “short-circuiting” the radiative cooling process. The TOA (“top of atmosphere”) height, also called the effective radiating level, and the actual lapse rate may determine the surface temperature.
In the absence of greenhouse gases, like water vapor and carbon dioxide, a greenhouse effect will still develop and will cause a surface temperature larger than the blackbody temperature. But, there is not enough data to be sure what the GHE would be absent all greenhouse gases. Thus, we cannot determine, at this time, the relative importance of radiation and thermodynamic processes. Even the blackbody temperature itself probably has no real physical meaning.
Convection (especially of water vapor) is probably the most efficient method of long distance heat energy transfer and probably transports the most heat energy in both the troposphere and in the oceans. Convection can transport water vapor great distances before it condenses to liquid water. Convection does change as temperature changes and it is a good thermostat as noted by Newell and Dopplick. Convection of water vapor in the troposphere where almost all our weather occurs is especially important. Some long term climatic changes may be a result of convection cycles in the world ocean.
Except for conduction, radiation is the slowest and least efficient mechanism. It is also the least likely mechanism, except for conduction, for transporting large amounts of heat in the troposphere. Although, in the stratosphere and above, radiative transfer, especially by carbon dioxide, dominates the transport of energy from near the tropopause to outer space. The greatest direct-to-outer-space radiation (OLR) losses occur in the winter polar areas and in dry desert areas like the Sahara. Because these areas lose more heat energy than they gain from the sun, there is a net convective flow of heat energy into them.
Speed and efficiency in heat transfer matter, heat energy is always likely to take the most efficient path to a cooler place or to outer space. We need good long term quantitative measurements of these processes under a variety of conditions to be sure how surface heat makes its way, via each process, to the stratosphere and higher. The speed and efficiency of the transfer of surface heat energy to outer space determines the lapse rate and the surface temperature. The slower the process is, the warmer the surface is. The faster the process is, the cooler the surface is and the smaller the lapse rate.
The Earth’s surface absorbs most of the sun’s energy that is not reflected from the Earth. It then radiates this energy to the atmosphere. Most of the emitted energy that makes it to outer space is radiated by water vapor and carbon dioxide high in the troposphere or in the stratosphere. But, the energy must make it to the upper atmosphere through the troposphere which is not in radiative equilibrium, nor is it stable. Radiative energy transfer is slow relative to convection and thermodynamic mechanical energy transfer and is probably not the dominant heat transfer mechanism in the troposphere. How much heat energy each mechanism carries in the troposphere is unknown right now.
Previous versions of this post were reviewed by Dr. Ronan Connolly, Dr. Michael Connolly and Javier. Their criticism and comments were invaluable, but any errors are the author’s alone.