METR 104:
Our Dynamic Weather

(Lecture w/Lab)
Key to Comments
on in-Class Quiz #3
Dr. Dave Dempsey
Dept. of Geosciences
SFSU, Spring 2012

I. Short Answer Question: The Earth's Average Temperature

(a)

  1. The warmer an object is, the more intensely it emits radiative energy. (See Thought Questions on the Earth's Average Temperature.)

  2. There is no physical law (or any basic law of radiation in particular) that says that an object emits as much radiative energy as it absorbs. There is a basic law of radiation that says that if an object absorbs a particular wavelength of radiation well, then it is also capable of emitting that wavelength well. (The intensity of emission of that wavelength depends on the object's temperature, though.)

  3. I'm not familiar with a "law of visible light".

  4. The Principle of Conservation of Energy (which we've used in the form of a heat budget relation) is not a basic law of radiation—it is more general than that (because it describes what happens when an object gains or loses heat by any of a number mechanisms, whether or not those mechanisms involve absorption or emission of radiation). Moreover, the Principle of Conservation of Energy does not by itself tell us what an object's temperature is—it tells us how and why (and how fast) and object's temperature changes. An object's temperature could be changing rapidly, slowly, or not at all, but any of these things could be happening if the object were very cold, or very hot, or somewhere in between—just knowing how fast the temperature is changing (and whether the temperature is increasing or decreasing) does not by itself tell us what the object's temperature currently is.

  5. One of the basic laws of radiation says that if an object absorbs a particular wavelength of radiation well, then it is also capable of emitting that wavelength well. (The intensity of emission of that wavelength depends on the object's temperature, though.)

  6. There is a basic law of radiation that says that most object emit radiative energy, of all wavelengths, all the time. From this we can conclude that most objects are continually losing heat by this particular mechanism (though they are no doubt gaining heat by one or more mechanisms at the same time, though the rates might not be the same—no guarantees about that!). However, this law by itself tells us nothing about an object's temperature.

  7. Material objects are made up of vast numbers of individual molecules and/or atoms of various types. When we talk about an object moving, we are talking about the bulk, collective motion of all of its molecules together (since that's what the object consists of in the first place). However, in addition to the bulk, collective motion of molecules (the molecules all moving around roughly together), the molecules are in random motion (vibrating, rotating, and in gases and liquids, moving from place to place). Temperature is a measure of the average energy of those random motions—not the energy of the bulk, average motion of all the molecules. Hence, the basic law of radiation that relates the intensity with which an object emits radiative energy to the object's temperature does not involve the bulk, collective motion of the molecules—just the random motions of the molecules of the object.

(b)

  1. For the earth as a whole, absorption of solar radiation is by far the dominant source of heat (235 W/m2). Because the earth's global, long-term average heat budget is nearly balanced, and because radiative emission to space is the only way by which the earth as a whole can lose heat, and because the intensity of radiative emission by an object depends on its temperature, we conclude that the earth must be at just about the right temperature to emit radiative energy to space as fast as it absorbs solar radiation (averaged over the whole globe for a sufficiently long time). That temperature happens to be close to 0°F.

    The atmosphere also absorbs solar radiation (though not as much as the earth as a whole does: 67 W/m2), but greenhouse gases and clouds in the atmosphere (especially in the lower troposphere) also absorb longwave infrared (LWIR) radiation emitted by the earth's surface (350 W/m2). The absorption of LWIR (especially in the lower troposphere) plus absorption of solar radiation (plus a little gain of heat by condensation of water vapor to make clouds and from the surface by conduction) greatly exceeds the absorption of solar radiation by the planet as a whole (519 W/m2 vs. 235 W/m2). Nonetheless, the long-term, global average heat budget for the atmosphere is also approximately balanced. The only way the atmosphere loses heat (on the average) is to emit radiative energy, so we conclude that the atmosphere must emit more radiative energy than the planet as a whole emits to space (519 W/m2 vs. 235 W/m2). A basic law of radiation that relates intensity of radiative emission to temperature requires that the atmosphere (notably the lower troposphere) must be warmer than the planet as a whole.

    The explanation for why the earth's surface is much warmer than the planet as a whole is basically the same, except that the surface absorbs LWIR radiation emitted downward by greenhouse gases and clouds instead of the other way around. The surface gains 492 W/m2 by solar absorption (168 W/m2) and by absorption of LWIR emitted downward by GH gases and clouds (324 W/m2). It loses most of this by emission of LWIR radiation (390 W/m2). Because the surface emits more than the the planet as a whole does to space (390 W/m2 vs. 235 W/m2), it follows from a basic law of radiation (the warmer a thing is, the more intensely it emits radiative energy) that the surface must be warmer than the planet as a whole.

  2. Nothing on earth reflects longwave infrared radiation. They absorb or emit it, or transmit it, but nothing reflects it.

  3. Arguments that claim that the lower troposphere and/or the earth's surface are warmer than the planet as a whole because the lower troposphere and earth's surface are gaining heat faster than they are losing it, simply don't work. Why? Because in the long-term, global average, the lower troposphere and the earth's surface have approximately balanced heat budgets—that is, in the long-term, global average, each gains about as much as it loses! (The energy budget diagram shows that, as we demonstrated in class.) [Note, however, that because humans are gradually changing the composition of the atmosphere by injecting carbon dioxide and other greenhouse gases, the greenhouse effect is getting gradually stronger, which is creating a slight imbalance between the rates of heat gain and loss in the atmosphere and ultimately in the surface, too. The net effect is to create a slow, gradual warming of the lower troposphere and the surface—that is, global warming. However, although global warming is something to be quite concerned about, enhancement of the greenhouse effect by humans does not explain why the surface and lower troposphere are so much warmer than the planet as a whole as seen from space. The planet has been that way to some degree or another for billions of years, because the greenhouse effect has been operating to some degree or another for most of the earth's 4.6 billion year history!)


II. Short Answer Question: Heat Budget for the Daily Temperature Cycle

(a)

  1. The relevant basic law of radiation is that the warmer an object is, the more intensely it emits radiative energy. One of the two curves on the figure shows the intensity of radiative emission from the earth's surface, which we can use to identify the time of day when it is warmest.

  2. There is no "conservation of radiation" law.

  3. What is the "absorption law"? How does it address the question posed?

  4. Is "energy transference" a law? What law? "Energy cannot be lost" is not a law as written, though "energy can't be created or destroyed" is the simple way of expressing the Principle of Conservation of Energy. It's certainly an important physical principle, but it isn't a basic law of radiation. (Radiation is only one form of energy, whereas the Principle of Conservation of Energy applies to all forms of energy.

  5. The Principle of Conservation of Energy relates the rate at which an object's heat content (and hence temperature) changes to the sum of the rates at which it gains and loses heat by various mechanisms. It is fundamentally important for understanding how and why temperature changes, and using information from both the absorption curve and the emission curve in the figure, this principle can indeed tell us when the warmest time of day is. (It is at one of the two times of day when the two curves cross—the one where solar absorption has been exceeding radiative emission before then the opposite thereafter, so that the temperature was increasing to that point and decreasing afterwards, which is the very essence of a maximum in temperature). However, the Principle of Conservation of Energy is not specific to radiative energy—it is much more general than that. That is, it is not a basic law of radiation, which is what the question asks for. In contrast, the basic laws of radiation are specific to radiative energy only, and it is also possible to determine the warmest time of day from the figure more simply by applying a basic law of radiation.

  6. There is no law that says that an object can absorb as much radiation as it emits. There is a basic law of radiation that says that if an object is capable of absorbing a particular wavelength of radiation, then it is also capable of emitting that wavelength (though according to another basic law of radiation, the intensity with which the object emits that wavelength, like other wavelengths that it can emit, depends on its temperature, regardless of how much radiative energy the object is absorbing).

  7. The sun does emit longwave infrared radiation, and the earth absorbs that radiation, but the amount is very, very small compared to the visible light, shorter wavelengths of infrared radiation, and ultraviolet radiation that arrive from the sun (and collectively define "solar radiation"). Hence, longwave infrared radiation from the sun is not even mentioned in energy budgets for the earth.

  8. What are "solar radiation laws"? We haven't talked about anything like that, and I don't know what they would be, if they exist.

  9. "The rate of absorption of LWIR warms the surface" is not a basic law of radiation.

  10. There is no basic law of radiation that says that the warmest time of day occurs when the absorption of solar radiation is the greatest. In fact, observations the daily temperature cycle (as shown by lots of meteograms) and our computer model of the daily temperature cycle both show that this is very rarely true. (Now, if we average over a long enough time—a whole day or many days, a season, or a year, say—there is indeed a connection between temperature and the intensity of solar radiation absorption, but no basic law of radiation by itself tells us that—we also need the Principle of Conservation of Energy and observations that the heat budget roughly balances when averaged over time. But for the daily temperature cycle, the heat budget doesn't balance during most of the day, which of course is why the temperature changes over the course of a day in the first place, and the relation between solar absorption and the warmest time of day is not what we expect!)

  11. A basic law of radiation says that the warmer an object is, the shorter are the wavelengths at which it emits most of its radiative energy. This law explains why the sun emits mostly visible light, short wavelengths of infrared radiation, and ultraviolet radiation, whereas the earth, which is much, much cooler than the sun, emits most of its radiation in the form of longwave infrared radiation. The temperatures of the earth and the sun are so different that there is very little overlap between the types of radiation they emit, which motivates us to define "solar radiation" and "terrestrial radiation" (or shortwave radiation and longwave infrared radiation, respectively) as distinct (separate, non-overlapping) parts of the electromagnetic radiation spectrum. However, this basic law tells us nothing about when the warmest time of day is based on information in the figure provided. Neither does another basic law of radiation, which says that most things emit radiative energy, of all wavelengths, all the time. Note too that radiative energy and heat are quite different types of energy, so I don't understand what "emitting shortwaves reveals itself as heat" means.

(b)

  1. In this idealized example, the warmest time of day is at about 15:30 (3:30 pm). That's when the surface emits radiative energy the most intensely, which according to a basic law of radiation occurs when the surface is warmest. (Solar noon is when the sun is highest in the sky and hence when insolation is greatest and when the surface absorbs solar radiation the most intensely. However, at solar noon, solar absorption exceeds radiative emission in this case, so the surface is gaining heat faster than it is losing it. From the Principle of Conservation of Energy, we conclude that at solar noon, the temperature must still be rising and so can't have reached its warmest point yet.)

(c)

  1. A basic law of radiation that says that the warmer an object is, the more intensely it emits radiative energy, tells us that where the radiative emission is decreasing, so must the temperature. From the graph, we can see that radiative emission is decreasing from its peak at about 3:30 pm (15:30) to about 7:30 am (07:30). (Note that this is also the period during which radiative emission exceeds solar absorption, so the surface is losing heat faster than it is gaining it. The Principle of Conservation of Energy tells us that the temperature must be doing down during this period. Hence, we have two different principles telling us the same thing!)

(d)

  1. According to the Principle of Conservation of Energy (expressed as a heat budget), an object will cool when it loses heat faster than it gains heat. (Most of the time, objects are losing heat and gaining heat by one or more different mechanisms at the same time, so we have to keep track of both gains and losses to determine whether an object is warming up, cooling off, or neither.) In the idealized situation shown by the figure, the surface cools off at night because it is losing heat by (net) emission of longwave infrared radiation faster than it is gaining heat by solar absorption. (Of course, at night solar absorption is zero.) Note that it is not enough to say that it cools off at night simply because there is no solar absorption—by itself, "lack of absorption" is not a mechanism of heat loss but rather simply the absence of a mechanism of heat gain. (Those aren't the same thing!) In the case of the earth's surface at night (and during the day, for that matter), (net) radiative emission certainly is a mechanism of heat loss, and is responsible for cooling of the surface at night (and in late afternoon, too).

  2. According to the figure, at night the surface cools for the reason explained in (d)(1) above. In the idealized situation described by the figure, there is no conduction of heat between the surface and air in contact with it. (See the introduction to the question.) However, if we took conduction into account, we would find that the surface cools faster at night than the air does, so the surface eventually becomes colder than the air next to it and heat starts to conduct from the air into the now cooler surface. The air next to the surface therefore loses heat and hence cools, while the surface gains heat but not enough to offset the net radiative emission from the surface—the gain of heat by conduction just slows down the radiative cooling of the surface somewhat. As a result, we see that between day and night (at least when there are few if any clouds), the biggest change in the temperature of the atmosphere typically occurs next to the earth's surface, and it is caused by conduction of heat into the atmosphere from the sun-warmed surface (during the daytime) and out of the atmosphere into the radiatively net cooling surface (at night). In most of the rest of the atmosphere, there is relatively little solar absorption, and LWIR absorption and emission very roughly balance, often leaving conduction of heat into and out of air next to the surface as the biggest factor causing atmospheric temperature changes. In any case, conduction of heat between the surface and the atmosphere definitely cannot explain why the surface cools at night—loss of heat by net radiative emission does.

(e)

  1. At solar noon and for a few hours afterwards (until 3:30 pm or so, in this idealized example), solar absorption is still greater than (net) radiative emission, even though solar absorption is going down during that time. According to the Principle of Conservation of Energy (expressed as a heat budget relation), when heat is gained faster than it is lost, the temperature must go up. Hence, the temperature continues to go up after noon, until solar absorption has fallen enough, and radiative emission has risen enough, for the two to become equal. At that point (again, around 3:30 pm in this example), the temperature stops going up. (It then starts going down, because solar absorption drops below radiative emission.)

  2. Does the greenhouse effect have any impact on the timing of the daily peak in temperature (which typically happens sometime in the afternoon, and in the idealized example shown here at around 3:30 pm)? Not particularly. Greenhouse gases and clouds emit radiative energy upward (to space) and downward (to the surface) all the time, and the variation in intensity of this emission is relatively small compared to emission from the surface because the temperature of the atmosphere doesn't vary as much as the surface temperature does (because the atmosphere doesn't absorb as much solar radiation as the surface does during the daytime)—that is, the surface absorbs longwave infrared radiation emitted downward by greenhouse gases and clouds at a relatively steady rate over the course of a day (unless there is a change in the extent of cloudiness, say). Variations in the temperature of the surface over the course of a day are driven mostly by variations in solar absorption and emission of radiative energy (and the latter depends on the temperature of the surface).

    Recall that in Lab #2, Part V, we ran a computer model of the daily temperature cycle without any greenhouse effect (just solar absorption and LWIR emission), and the temperature peaked in mid-afternoon (that is, it continued to go up for several hours after solar noon, even though solar absorption was declining during that period). In Lab #2, Part VI, we added a greenhouse effect, which had negligible effect on the timing of the peak temperature. That behavior of the model was realistic—that is, the atmosphere would do the same thing.

  3. Keep in mind when an object emits radiative energy, that energy comes from heat inside the object, and so radiative emission is a mechanism by which an object loses heat. This certainly cannot explain why an object might warm up! That would require a mechanism of gaining heat that is at least as great as the loss of heat by radiative emission.

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