(1) (a) Based on the atmospheric absorption spectrum provided in class (supplemented by lecture about that figure), almost all of the absorption of solar radiation in the atmosphere by "air" is due to absorption of ultraviolet radiation by ozone plus near infrared (IR) radiation by water vapor (with minor contributions to the latter from carbon dioxide and other greenhouse gases). Almost no visible light is absorbed by "air", and the primary constituents of air (nitrogen, oxygen, and argon) absorb no solar radiation to speak of. (Oxygen absorbs a little bit of the shortest wavelengths of ultraviolet radiation, but not enough to mention here.)

             (b) We haven't been provided with enough information to tell what wavelengths of solar radiation that clouds tend to absorb. (The atmospheric absorption spectrum provided in class does not include clouds, which consist of tiny droplets of liquid water or ice crystals, not water vapor. In lecture the point was made that clouds are good absorbers of all wavelengths of longwave infrared radiation, but that doesn't tell us how well clouds absorb individual wavelengths of solar radiation.)

              (c)  Air molecules scatter blue wavelengths of visible light better than other wavelengths of visible light. That's why we see blue light coming from all parts of the sky, not just directly from the sun. From space, the earth has a bluish tint at least partly for this reason, too. Removal of blue wavelengths of visible light from the direct beam of the sun shifts its color to longer wavelengths, which on the average look yellowish. (On the moon, where there is no atmosphere and hence no scattering of visible light from the direct beam of the sun, the sun looks white and the sky looks black.)

             (d) On the average, the atmosphere absorbs a total of 16 units (almost entirely by ozone and water vapor plus a little by carbon dioxide) plus 4 units (by clouds) = 20 units of solar radiation (where these are "normalized" units). This amounts to 20% of the incoming solar radiation.

             (e) In the long-term, global average, of the solar radiation arriving at the "top" of the earth's atmosphere, 54% reaches the surface. The percentage of solar radiation reaching the surface at any particular place and time (of day and of year) will generally differ from this figure, of course, because the percentage depends both on weather conditions and on the sun angle (which determines how far through the atmosphere solar radiation travels before reaching the surface, and hence the likelihood that solar radiation will be absorbed, scattered back to space, or reflected back to space before reaching the surface). If we interpret the number "54" as a radiative flux instead of a percentage, recognizing that we would be expressing it in units about 3.42 times larger than Watts/m² (so that 100 of these units equals = 342 W/m², the long-term, global average flux on horizontal surfaces at the top of the atmosphere, which already takes into account the "spreading out" effect that lower sun angle has on the intensity of solar radiation on a horizontal surface).

          (2) (a) The earth reflects 30 (normalized) units of solar radiation back to space (6 scattered back by air molecules; 20 reflected by clouds; and 4 reflected back by the surface). This is 30% of the 100 normalized units of solar radiation arriving from the sun, so the albedo of the planet is 30%. (The "normalization" here would be division of all terms by the long-term, global average insolation on a horizontal surface at the top of the atmosphere.)

               (b) Clouds account for 2/3 of the reflection of solar radiation back to space by the planet (20 out of the 30 units reflected by the planet).

               (c) Albedo is defined as the amount (or alternatively, rate, or alternatively, flux) of solar radiation reflected from a surface as a fraction or percentage of the amount (or rate, or flux) of solar radiation striking the surface. In the long-term global average, the earth's surface reflects 4 (normalized) units of the 54 units striking it, so the average albedo of the surface (which is mostly ocean) is 4/54 = 0.074, or 7.4%.

               (d) Reflected solar radiation plays no direct role in a heat budget. It was never in the form of heat that the earth had but then lost—it's radiative energy that strikes the earth and reflects back to space, never having been absorbed. That is, it is neither a source of heat nor a sink of heat in the planet's heat budget.

               (e) On the average, the earth's surface ultimately absorbs about half of the solar radiation that arrives at the top of the atmosphere. Most of the ultraviolet radiation in solar radiation is absorbed in the stratosphere by ozone, and some of the near-infrared radiation in solar radiation is absorbed by gases (especially water vapor) and clouds (I think), whereas little in the atmosphere absorbs visible light, so the majority of solar radiation absorbed by the earth's surface is probably visible light.

          (3) The earth's surface emits about 5% more radiative energy than arrives from the sun at the top of the atmosphere (105 emitted vs. 100 arriving) and over twice as much solar radiation as the surface absorbs (105 emitted vs. 50 absorbed).

          (4) (a) Less than 6% of the longwave IR radiation emitted by the earth's surface escapes to space (6 units out of 105).

               (b) ---

          (5) (a) ---

               (b) About 43% (64 units out of 64 + 85 = 149 units total) of the longwave IR radiation that the atmosphere emits escapes to space. The rest (85 units) is emitted downward and is absorbed by the earth's surface.

               (c) About 59% (38 units out of 38 + 26 = 64 units total) of the longwave IR radiation emitted by the atmosphere to space is emitted by air. According to one of the basic laws of radiation, only those objects/substances that are good absorbers of any particular wavelength of radiation are potentially good emitters of that wavelength (depending on the object's temperature). The good absorbers of longwave IR radiation in air are primarily water vapor, secondarily carbon dioxide, and to a lesser extent methane, nitrous oxides, ozone, etc., so these are the gases emitting longwave IR to space (as well as downward to the earth's surface). Clouds (which are also very good absorbers of all wavelengths of longwave IR radiation) emit the other 41% that the atmosphere emits to space.

               (d) At this point, the surface appears to be gaining 30 units of heat more than it is losing, the atmosphere is losing 30 units of heat more than it is gaining (quite a coincidence, that!), while the planet as a whole is gaining and losing heat equally (that is, the planet as a whole has a balanced heat budget). This should suggest to us that if the surface could give up 30 units of heat to the atmosphere, both of those "internal" budgets would balance without affecting the heat budget for the planet as a whole at all (because this heat wouldn't be gained or lost by the planet as a whole; what one part of the planet loses, another part gains, with no net effect on the the planet treated as a whole; the exchange is completely internal).

          (6) (a) We need to be clear here about the difference between transfer of energy (from place to place or object to object) and transformation of energy (from one form to another). These are not the same thing, though both can be involved in the same physical process.

"Sensible heat flux" is a fancy name for conduction, which is the transfer of heat from one object to another that is (a) in direct contact with it, and (b) at a lower temperature. Under these conditions, the molecules of the warmer object (which are in random motion) collide with the (slower-moving) molecules of the colder object; as a result the faster-moving molecules in the warmer object slow down while the slower-moving molecules of the colder object speed up. This means that the warmer object cools (loses heat) while the colder object warms (gains heat), as a result of the direct molecular collisions. (Moreover, the rate of transfer by conduction depends on the temperature difference between the two objects in contact; the larger the difference, the faster heat conducts.) In this process, heat doesn't change form (that is, heat is transferred from one object to another but is not transformed into some other form of energy). On the average (but certainly not all the time everywhere), the earth's surface is warmer than the atmosphere in contact with it, so on the average heat conducts from the (warmer) surface into the (cooler) atmosphere.

"Latent heat flux" occurs (mostly) when liquid water on the earth's surface (mostly in the oceans) evaporates. During this phase change, heat in the water is transformed into latent heat in the water vapor (which we can't feel and can't measure with a thermometer), so the earth's surface loses heat while the atmosphere gains latent heat. Later and (usually) elsewhere in the atmosphere, when the water vapor condenses to form tiny liquid droplets that form a cloud (or "deposits" to form tiny ice crystals to form a cloud), latent heat in the water vapor is converted into sensible heat in the air inside the cloud. In this process, the atmosphere loses latent heat and gains sensible heat. The net effect of the initial evaporation followed later (and elsewhere) by condensation to form a cloud, is a transfer of heat from the surface into the atmosphere. This transfer of heat involves two transformations of energy along the way.

               (b) Both conduction and evaporation followed by condensation should be highly variable from place to place and time to time, depending on the temperature difference between the surface and the air in contact with it. In fact, there commonly are occasions (though not enough to reverse the global average) when the earth's surface becomes colder than the air in contact with it (e.g., on many nights) and conduction becomes a sink for air (next to the earth's surface) and a source for the surface instead of the other way around. Evaporation varies considerably as well, depending on whether a source of water is present, how dry the air is, and how cold the air is.

               (c) The biggest source of heat for the surface is absorption of longwave IR radiation emitted downward by the atmosphere (85 units vs. 50 units of solar radiation absorbed). The surface gets rid of heat mostly by emitting infrared longwave IR radiation (105 units vs. 6 units lost by conduction to the atmosphere and 24 units lost to the atmosphere by evaporation and sublimation).

               (d) The biggest source of heat for the atmosphere is absorption of longwave IR radiation emitted upward by the earth's surface (99 units vs. 20 units of solar radiation absorbed, 24 units of latent heat transformed into heat during condensation to form clouds, and 6 units of heat conducted from the warmer surface). The atmosphere gets rid of heat by emitting longwave IR radiation, partly to space and partly back down to the surface (emission occurs in all directions possible).

               (e) The sensible and latent heat fluxes from the surface into the atmosphere are purely internal exchanges and don't affect the heat budget for the planet as a whole. As noted above, whatever the surface loses in these exchanges the atmosphere gains, so there is no net change for the planet as a while.

          (7) The planet as a whole absorbs 70 (normalized) units of solar radiation (or equivalently, 70% of incoming solar radiation). The planet's heat budget is approximately balanced, so it gets rid of 70 units as well. The only way it can get rid of energy is by emitting radiative energy to space. The Stefan-Boltzmann relation says that the emission flux (for a blackbody, which the earth is close to being for this purpose) depends on the temperature—the warmer an object is, the more radiative energy it emits—so the planet as a whole must be at whatever temperature is necessary to emit those 70 (normalized) units of radiative energy.

In contrast, the earth's surface absorbs 135 (normalized) units of energy (50 units of solar energy + 85 units of longwave IR radiation from the atmosphere). The heat budget for the earth's surface is also approximately balanced, so is must be getting rid of 135 units of energy. It gets rid of 30 units by conduction and evaporation, leaving it with 105 units to get rid of by emitting longwave IR radiation. The Stefan-Boltzmann relation says that the surface must be at a temperature sufficient to emit 105 (normalized) units of radiative energy, which has to be warmer than the planet as a whole (which emits only 70 units).

Bottom line: The surface has an extra, large source of heat that the planet as a whole doesn't have—absorption of longwave IR radiation emitted downward by greenhouse gases and clouds in the atmosphere. Given that the heat budgets for the planet as a whole and for the surface are both (approximately) balanced, and the planet as a whole absorbs less than the surface does and hence gets rid of less than the surface does by radiative emission, the Stefan-Boltzmann relation allows us to conclude that the planet as a whole must be colder than the surface.

          (8) Removing all of the "greenhouse" gases, including water vapor, carbon dioxide, methane, nitrous oxides, ozone, etc., would leave 98-99% of all air molecules still in place. (Air consists mostly of nitrogen and oxygen, with some argon, in the first place. Hence, we're talking about removing a small fraction of the constituents of the atmosphere, though since cloud droplets develop when water vapor condenses, so removing water vapor also would also remove the clouds, which have a major impact on the global heat budget.)

One consequence of removing these gases would be to eliminate most of the absorption of solar radiation by the atmosphere (which is mostly due to ozone, water vapor, and clouds). About 6 units would still be scattered back to space (nitrogen and oxygen, being the most abundant molecules in air, do most of the scattering in the first place), leaving 94 units to reach the surface. Of that 94 units, about 7.4% (the albedo of the surface), or about 7 units, would be reflected back to space, while the other 87 units would be absorbed by the surface. (This is quite a bit more than the 50 units that the surface absorbs currently.)

However, because nitrogen, oxygen, and argon don't absorb longwave IR radiation, they aren't capable of emitting it, either (a basic law of radiation), and clouds would be absent, too. Hence, the atmosphere no longer emits longwave IR radiation downward to the surface (or out to space). Hence, the surface gains a grand total of 87 units of heat, all by absorption of solar radiation. It wouldn't lose heat by evaporation any more (we're not allowing any water vapor in the air), but even if no heat conducted from the surface into the atmosphere (and it wouldn't, because if it did then the atmosphere's budget wouldn't balance) and if the surface had to radiate the entire 87 units to balance its budget (which it would in fact have to do), the Stefan-Boltzmann relation tells us that the surface would necessarily be colder than we observe it to be today (because the surface radiates 105 units today).

Bottom line: without greenhouse gases and clouds in the atmosphere, the surface would be much colder than it is today with greenhouse gases and clouds present. (The same is true if we imagined leaving clouds in place and removing only greenhouse gases, though the numbers would be somewhat different.)

          (9) (a) At first, when carbon dioxide is added to the atmosphere and mixes throughout the atmosphere (which doesn't take long—we're assuming here that it happens instantaneously), some radiative fluxes will change (as we'll argue below). However, changes in temperature require addition or removal of heat, and we're not doing that directly—we're just adding carbon dioxide.

Adding or removing heat from any part of the earth (atmosphere or the surface) requires both (a) an unbalanced heat budget and (b) a little time to pass. Hence, no significant temperature changes will occur instantaneously as a result of adding carbon dioxide to the atmosphere.

Moreover, it turns out that the wavelengths of radiation emitted by the surface directly to space are mostly in the atmospheric "window", and carbon dioxide (like other greenhouse gases) does not absorb those wavelengths. (Carbon dioxide and other greenhouse gases already absorb virtually all of the LWIR radiation emitted by the surface at wavelengths that they are capable of absorbing—only wavelengths in the atmospheric window make it directly to space from the surface, and only then when there are few or no clouds present.). Hence, the additional carbon dioxide would have little direct impact on the radiative energy emitted by the surface directly to space.

To restate this point: Because the radiative emission from the surface (105 units) doesn't change instantaneously (because the surface temperature doesn't change instantaneously), and because the portion of that radiation escaping directly to space (6 units) changes very little (because it's mostly in the atmospheric window, and carbon dioxide doesn't absorb those wavelengths), it follows that absorption in the atmosphere of LWIR radiation emitted by the surface (99 units) won't change (hardly at all) instantaneously, either.

Instead, the main "instantaneous" effect of adding carbon dioxide to the atmosphere would be:

• In the middle and upper troposphere, the additional carbon dioxide would absorb some of the LWIR radiation that is emitted upward by greenhouse gases and clouds lower in the atmosphere and that normally would escape to space; and
  
  
• as a result, less LWIR radiation emitted by GH gases and clouds in the lower troposphere would escape directly to space (because more of it is being absorbed by the additional GH gases higher up).
  
  
• The additional GH gases in the middle and upper troposphere that are absorbing more of the LWIR radiation emitted from GH gases and clouds lower down, are themselves emitting LWIR, both upward and downward, but they are colder than the gases and clouds below them (temperature in the troposphere decreases rapidly with increasing altitude), so they don't emit as much to space as the GH gases and clouds below them do.

This immediately unbalances the atmosphere's heat budget—it is gaining heat by absorption of solar radiation and by absorption of LWIR emitted by the surface just as fast as before, but it losing heat by radiative emission to space more slowly than before (because more of the emission that escapes to space is now coming from a part of the atmosphere that is colder than where that emission came from before). As a result of the imbalance, the atmosphere's temperature will begin to increase.

The planet as a whole is losing less energy to space than before, so the planet's budget is also unbalanced. Interestingly, unlike the actual temperature of any part of the planet, the effective radiating temperature of the planet decreases (instantaneously) at this time! That's partly a consequence of the definition of the effective radiative temperature of an object—namely, the temperature that a blackbody would have to have to emit radiative energy as intensely as the object (for example, the earth). Because the planet as a whole emits less radiative energy to space immediately after greenhouse gases are added, it's effective radiative temperature must immediately decrease, even though no actual temperatures of any part of the planet have changed yet!

               (b) After a short time, the atmosphere will warm (because it's emitting less LWIR radiation to space than it's gaining by absorption of solar radiation and LWIR radiation from the surface and by conduction and latent heat fluxes from the surface). And the planet as a whole must be gaining heat (because it's losing less by emission than it is gaining by absorption of solar radiation).

As a result of the atmosphere warming, the Stefan-Boltzmann Law tells us that it will emit more radiative energy, both to space and back to the earth's surface. As it emits more, the atmosphere's heat budget will begin to come back toward balance and its temperature will eventually stop changing, but at a higher value than before.

As a result of increased downward emission by the warming atmosphere, the surface will start gaining more heat, which unbalances its heat budget. The surface will therefore warm up, and Stefan-Boltzmann Law tells us that the surface will therefore emit more. The surface temperature, and hence surface radiative emission, will increase until the increased emission balances the increased absorption, at which point the surface temperature will stop changing, but at a higher value than before.

Some of the increased radiative energy emission from the surface will be at wavelengths in the "atmospheric window" and will escape directly to space. Combined with the increase in spaceward emissions from the warming atmosphere, this increase in radiative emission to space eventually brings the planet's heat budget back into balance.

As the space-bound emissions from the warming atmosphere and surface increase to what they were (combined, though not individually) before the carbon dioxide was added, the effective radiating temperature of the planet increases to its original value, too. Although the total amount of radiation escaping to space ultimately doesn't change, more comes from the surface (through the atmospheric window) and less from the atmosphere than before. (Less comes from the atmosphere because atmospheric emissions to space will be coming from GH gases higher in the atmosphere, where it is colder, than before, not because the atmosphere as a whole has become colder.)

The net result: The atmosphere's temperature increases and the surface temperature increases—but the effective radiating temperature of the planet doesn't change! (Of course, this assumes that the albedo of the planet doesn't change—a big "if"!)

Although it sounds counter intuitive that the effective radiating temperature of the planet ultimately doesn't change (assuming unchanged albedo), remember that the effective radiating temperature depends on the radiative emission rate, and when the planet's heat budget is balanced, the radiative emission rate of the planet depends only on solar absorption rate, which in turn depends on the flux of incoming solar radiative energy and the albedo of the planet. Hence, as long as the albedo doesn't change, the solar absorption rate won't change, and as long as the planetary heat budget ultimately rebalances, the radiative emission flux won't differ from where it started, and hence the effective radiating temperature won't be any different from it was before carbon dioxide was added.

However, in it's new balanced state, the planet as a whole would be losing more heat by emission of longwave IR radiation from the surface directly to space and less by emission from the atmosphere to space, than it did before we added carbon dioxide to the atmosphere. Although the lower and middle troposphere have warmed, some of the additional radiation that GH gases and clouds emit from those parts of the atmosphere are absorbed by the increased GH gases above that level, where it is colder, and those gases now emit a higher proportion of the LWIR emitted by the atmosphere to space than before. The "effective" altitude of emission to space from the atmosphere (a kind of weighted sum of emissions directly to space from all altitudes combined), which is near mid-troposphere, becomes a little higher (and hence colder, and hence less intense) after carbon dioxide is added and a new balance is achieved. This is compensated by the increase in spacebound LWIR radiation emitted from the surface through the "atmospheric window", which is relatively intense because the surface is warmer than the atmosphere. The net result: the total intensity of LWIR emitted by the planet to space ends up the same as before (assuming no change in planetary albedo—again, a big "if").

We call this global-average warming of the earth's surface and the lower and middle atmosphere global warming. It arises from an enhancement of the greenhouse effect, which temporarily disturbs the initially (approximately) balanced heat budget for the planet and its components.