By Al Tekhasski, Rev.0.4 11/26/2010
Greenhouse gas effect (GHGE) is an effect on a scale of a planet. It should not be confused nor conflated with glass panes or polymer films over a bed of plants. It does not matter how bad the name is; it is what it is. We can call it “greenfruugh effect,” or “greencheese effect,” the name does not make any difference. In addition, the existence of greenhouse gas effect must be differentiated from how its magnitude actually varies (or not) with change in GH gases concentration, which is sometimes dubbed as “enhanced greenhouse effect” and is the cornerstone of AGW theory. This article deals only with existing effect as it currently is. Changes in GHGE should be considered in a different article. For conceptual simplicity, the effect of condensable substances (like water vapor) is omitted as well.
As usual, there are several ways to define an effect. I will use a top-down definition approach.
Definition: Planetary Greenhouse Gas effect is the difference between effective emission temperature of a planet and global average temperature of its surface.
Definition: Effective Emission temperature is the temperature that a blackbody planet would need in order to emit the same amount of IR radiation as it could be measured from outer space by remote observer. This concept came from astrophysics, and it is a measurable quantity. It is just a different representation of total outgoing radiative energy of a warm body.
To measure this “temperature”, one needs to register outgoing longwave radiation (OLR) from the object, then calculate corresponding surface flux knowing the distance to observer, effective diameter of emitting body, and assuming spherical symmetry. Then calculate the number as (OLR/sigma)^0.25, where sigma is the classic Stefan-Boltzmann constant. In astrophysics this “temperature” is just a convenient ballpark number to compare stars, and somewhat less convenient when comparing heat balance of a planet.
[Footnote: because planets usually don’t have substantial internal heat source, and rely on external radiation to heat its surface, so (a) some overlap with “pumping” spectrum can occur, and (b) surface temperature loses spherical symmetry.]
Definition: Global Average [Surface] Temperature is an arithmetic mean of daily mean temperatures taken at a grid of stations around the globe. Climatology considers this temperature as a proxy for warming. This quantity can be formally calculated for any spatial distribution of grid stations, but it has the intended meaning only if the surface is totally spherically symmetrical, meaning that the actual temperature is the same at any point on the globe, be it at equator or at poles (in which case it would suffice to take measurement only at one point).
Definition: Daily mean temperature is an arithmetic mean of minimum and maximum temperatures of air recorded 2 meters above the surface at a certain arbitrary location on the globe. Strictly speaking, this temperature also has no definite meaning as a proxy for energy exchange without additional assumptions about actual shape of diurnal temperature variations. Most temperature records came from min-max thermometers, so the actual time when the min or max occurred is unknown.
Now we have the definitions, and even methods how to calculate these quantities. In populist literature it is commonly estimated that the average radiation flux density into Earth “low troposphere” is 240W/m2 of shortwave radiation after accounting for 30% reflection called “albedo”. When a planet is in stationary state (considering that the planet is more or less in this condition for millions of years), then an external observer should expect the planet to emit the same 240W/m2 of total emission, but now in IR, as it is appropriate for emissions from warm bodies.
For the expected stationary state with pass-through flux of 240W/m2, the effective emission temperature (Te) calculates as about 255K. However, the statistics of global average surface temperature (Ts) comes up as 288K. Climatology calls this formal difference of 33K as “greenhouse effect.”
3. Mechanism of greenhouse gas effect
To simplify things and illustrate the main physics of how the effect works, lets assume that air is really filled with GH gases and other particles/aerosols, such that it looks (absorbs and emits) as a blackbody in IR range when air density is high enough. But at the same time the air is still fully transparent to SW radiation of Sun to let the energy in. To have physically meaningful temperatures, we have to assume spherical symmetry. This is the usual (but not spelled out) assumption in all classic classroom calculations of GHGE. It is obvious that this condition would be difficult to satisfy with one-sided heater position of the Sun even if the planet rotates. To get uniform heating, the Sun (or Earth) must rotate like crazy in randomly changing directions, or the surface must have infinite thermal conductivity. Alternatively, the SW radiation could be replaced with a uniformly distributed electric heater with power density of 240W/m2 everywhere on the surface.
Lets hypothetically assume that there is so much absorbing material in the air that it is completely non-transparent to IR at ground level. For simplicity, the effect of water evaporation and vapor condensation will be temporarily omitted. Also I will assume that we start from a planet that is somewhat colder than the stationary state would otherwise require. Then the following would occur:
(1) Sun radiation (minus whatever was reflected back) hits ground surface, and gets fully absorbed. Surface heats up. We assume the globe-averaged radiation flux from Sun as 240W/m2.
(2) Surface transfers the heat to air (and into ground) by all means: conduction, evaporation (which we ignore in this example), convection. Radiation does not play any essential role at the surface because whatever is being radiated by surface is equally compensated by back radiation from IR-dark air. Likewise, upflux and downflux of radiation at any virtual horizontal surface above the ground cancel each other. In astrophysics is it called “Rosseland approximation,” in engineering they call it “diffusion approximation.” The radiation “diffuses” through air similar to regular temperature or trace GH gas, except when the air becomes very thin, literally and optically. The ground surface continues to heat up because the thermal conductivity of air is low, and net IR radiation at the surface is nearly zero.
(3) In the field of gravity, the atmosphere density gets gradually thinner with height. Therefore at some point it cannot be considered as being IR-dark; Therefore the IR dark absorbing media has a top somewhere, which is called “radiative TOA,” top of atmosphere, or “effective emission layer.” Figure 1 illustrates the concept:
(4) At the radiative TOA, there is only one sink of energy to outer space – radiation. The amount of emitted radiation depends on local temperature of this layer in accord with Stefan-Boltzmann law.
(5) The surface continues to absorb all Sun radiation, and its temperature rises and rises.
(Some people do not appreciate this effect of unbounded rise in temperature. To convince themselves, they should try a simple experiment – take a 100W light bulb, and cover it with Styrofoam, and see what would happen with bulb temperature.)
(6) Rising bottom temperature (together with Coriolis forces) eventually triggers a massive instability in atmosphere (due to buoyancy of warm air), and various weather patterns stir the air until some sort of dynamic equilibrium occurs that is called “convective equilibrium.” The convective equilibrium forms a gradient of temperature due to thermodynamics of air mass movement known as “lapse rate.” As result of this “lapse rate,” temperature of the “effective emission layer” at TOA is generally lower than the ground temperature.
(7) The process of ground temperature rise (and corresponding intensification of convective patterns) continues until the TOA warms up to a temperature of about 255K, and thus would emit about 240W/m2 in IR. This is a special point because this amount of OLR is equal to incoming solar flux, so the system reaches a steady state frequently called as “equilibrium.” This “equilibrium” should not be confused with thermal equilibrium, and many classic theorems of Thermodynamics may not hold here.
The difference between the average ground temperature (Tsurface, Ts) and the temperature at radiative TOA is the GHGE. Figure 2 illustrates this geometrical relationship.
Please note that the radiative TOA is not the same as tropopause, it is about half way in between. The height of effective emission layer doesn’t have an easy definition and is usually estimated by backward calculations. It goes like this: the temperature of this layer must be 255K. Assuming wet-adiabatic lapse rate (Lr) of 6.5K/km and allegedly known “surface temperature” of 288K, the emission layer must reside at 33/6.5 = 5km. Interestingly, this estimation suspiciously coincides with the height of layer where most atmospheric water is residing.
As we can see, there is no warming going from “cold body” to “hotter body,” and no violation of any known law of thermodynamics takes place. All it is a flow-through heat transfer system: SW radiation hits and heats the ground; ground transfers this heat into air. The air carries the heat and expands when parcels float up, and cools off quasi-adiabatically. The top air is still warmer than outer space, so it will cool off by radiation.
This concludes the continuing passage of energy across the weather system. Without the escape route for energy the surface would heat up without a limit. Fortunately, the energy can and will escape from the top in the form of IR radiation, where the air is generally colder than at the surface, all due to the lapse rate. This creates and maintains the difference between bottom and top temperatures, just as any imperfect heat transfer system would have. In engineering this effect is associated with “thermal resistance” (of atmosphere) to heat flux, just as in ordinary electrics.
In the above construction, the ground temperature is initially undefined and is unknown. However, in process of convective stirring the surface temperature automatically adjusts itself to a value that provides sufficient and necessary heat transfer from the surface to air (and eventually to TOA) via various direct paths and feedbacks. The only formal boundary condition for the entire system is that the overall vertical profile of air temperature must reach the Te=255K at the radiative TOA. The height H of this outer boundary is determined by optical properties of air. Thus the formula for sustained ground temperature Ts is:
Ts = Te + Lr*H
The thermal link between ground surface and TOA is complex and involves radiation and convection (and latent heat transport, which is temporarily excluded from this illustrative example). However, it is incorrect to say, “convection is dominant”; it is formally not. But it does not matter because convection continuously re-balances the fluxes such that the final value of lapse rate is maintained the same regardless of what the other heat transfer paths are. This process is complicated, it depends on complex boundary conditions with many parameters, it is turbulent, and we cannot calculate this process with contemporary computing technology, and probably never will.
Planetary greenhouse gas effect is a self-sustained phenomenon when hydrodynamics of near-surface layer heated from SW radiation drives an atmosphere into convective equilibrium forming a mechanically controlled lapse rate. The control feedback mechanism is such that the surface temperature rises until the entire system meets its outer boundary condition of OLR = 240W/m2 at a certain height defined by IR opacity of air. Therefore, the entire effect is a product of two inseparable components – (1) presence of atmospheric lapse rate, and (2) finite opacity of air in IR region. Because the lapse rate is a necessary element, the effect cannot be reproduced under room conditions unless the gravity acceleration is somehow scaled proportionally.
It should be obvious that gradual changes in our initial extreme assumptions about optical air properties don’t fundamentally change the magnitude and sign of GHGE. The above construction would undergo some adjustments for realistic properties of air like “IR window” and other non-blackbody deviations in emission/absorption spectrum. Inclusion of water cycle will deform the value of lapse rate from dry 9.5K/km to observed wet rate of 6.5K/km, but the described essence of GHGE would remain the same. Latent heat transformations and atmospheric IR window would add additional escape routes for energy flux across the bulk of atmosphere, but these “resistive” paths would not change much the average magnitude of lapse rate because it is the hydro-mechanical feedback process that keeps it constant.
The GHG effect is straightforward under the initial assumptions of full spherical symmetry and very optically dense atmosphere that emits as a blackbody. In reality, the one-sided heating creates longitudinal temperature gradients and other non-homogeneities in polar-wise heat transport related to cellular structure of global circulation in atmosphere, which highly complicates things. The absorption-emission properties of rarefied gases are far from blackbody, and gray (or band-) averaged approximation may not correctly represent changes in radiative fluxes when mixing ratio of GH gases changes.
The mechanism above was derived from general concepts presented in the following sources: