ERTH 535:
Planetary Climate Change
(Spring 2018)
Solutions to Problem #3 Dr. Dave Dempsey
Dept. of Earth & Climate Sci.,

The two idealized curves on the accompanying graph show:

  1. Solid curve: the daily average intensity of solar radiation absorbed by the earth's surface at different times of the year at Sacramento, California (about 38°N latitude);

  2. Dashed curve: the daily average net intensity of terrestrial radiation emitted by the earth's surface at different times of the year at Sacramento. ("Net" in this case means the difference between terrestrial radiation emitted and terrestrial radiation absorbed. The terrestrial radiation that is absorbed at the surface comes from the atmosphere, which emits terrestrial radiation downward as well as upward to space. To simplify things, we'll assume in this exercise that the terrestrial radiation emitted downward by the atmosphere is constant during the year, so the net emission varies only because emission does.)

The beginnings of individual months of the year are numbered along the horizontal axis of the graph, starting arbitrarily with 0 (which isn't necessarily January).

For the purposes of this assignment, assume that the earth at Sacramento is gaining and losing heat (on the average) only by absorbing and emitting radiation (a simplification that is only partly true but is good enough to get some interesting results!).

Below is a list of events that occur at either (a) specific times of year or (b) ranges of times during the year. The specific time of year, or range of times, when each event occurs can be determined using information on the graph. For each event on the list, determine the approximate time of the year (for example, "late June", or "mid April") or the range of times during the year (for example, "late June to late December") when the event occurs. (You need not give answers within any particular month more precise than "early", "late", or "mid" month; the graph doesn't permit answers more precise than that. You should be able to determine which month is which, so refer to months by name rather than by number.)

Depending on the question, you might be able to determine each answer by (1) simply reading the graph directly; (2) applying common sense and/or information that you've already learned this semester; or (3) applying one or two of the following three principles to interpret information on the graph:

  1. A basic law of radiation, the Stefan-Boltzmann relation: The warmer most objects/materials are, the more intensely they emit radiative energy.

  2. An object's temperature will be increasing if it is gaining heat faster than it is losing it; it's temperature will be decreasing if it is losing heat faster than it is gaining it; and it's temperature won't be changing if it is gaining and losing heat at the same rate.

  3. The bigger the net rate of heat gain or loss is (that is, the bigger the difference between the rates at which an object simultaneously gains and loses energy by separate mechanisms), the faster the temperature will change.
Note that the latter two principles above really just follow directly from a simplified, semi-general statement of the conservation of energy applied to the heat budget for the earth's surface:
Rate at which
an object's

(i.e., is
Rate at which
the object's
heat content
= Rate at which
the object absorbs
solar radiative energy
Net rate at which
the object emits
radiative energy

For events 4 through 7 below, there there are two different ways to get an answer. Each of the two ways of answering uses (directly or indirectly) one of the three physical principles listed above. For one way of answering, you'll need information from only one of the two curves on the graph. For the other way of answering, you'll need information from both curves. (Hint: If you read the principles above closely, you'll note that the two that are based on the conservation of energy require information from both curves.) You'll find that using information from both curves generally gives you a more precise answer than using information from only one curve.

For events 4 through 7, identify the time of year or range of time of year when each occurs, using both ways of answering the question. (Of course, since the results of both approaches should agree, this happens to provide a way to check your answers.)

For each event (1 through 7), briefly summarize how you got your answers. For those questions (that is, questions 4 through 7) for which it is possible to answer in two different ways (by applying different physical principles), your summary should include how you applied each principle.

  1. The winter and summer solstices and the spring and autumn equinoxes (based in part on the graph, not based solely on prior knowledge of the date)

  2. Time when solar radiation is the least intense

    Mid or late December, when the solar absorption curve is at a minimum. (Of course, by definition this occurs on the day of the winter solstice.)

  3. Time when IR emission is the most intense

    Late July (month #8) or early August (month #9), the apparent peak of the emission curve.

  4. The single warmest or coldest time of the year (choose one)

    Warmest time of year:

    (a) Late July (month #8) or early August (month #9). The Stefan-Boltzmann law of radiation connects emission to temperature, so the time when emission is greatest must be the time when temperature is greatest.

    (b) Late July. Before the warmest time of year the temperature must be increasing, and afterwards it must be decreasing. (This is the nature of the maximum value of any quantity.) In this case, according to the heat budget equation, for the temperature to be increasing absorption must exceed emission, and for temperature to be decreasing emission must exceed absorption. Hence, the absorption curve must lie above the emission curve leading up to the single warmest time of year, and the opposite must be true immediately after the warmest time. It follows that the warmest time of year must be at the precise moment when the two curves cross.

    Coldest time of year:

    (a) Early or mid February (months #3 and 15), based on similar reasoning as above.

    (b) Very early February, based on similar reasoning as above.

  5. Range of times when temperature is increasing or the range of times when it is decreasing (choose one)

    Increasing temperature:
    (a) The temperature must be increasing from its minimum until its maximum. Based solely on the emission curve , which the Stefan-Boltzmann law of radiation relates to temperature, that period appears to be roughly from early or mid February to late July or early August.

    (b) According to the heat budget equation in this case, the temperature will be increasing when absorption exceeds emission. This occurs from early February to late July, when the absorption curve lies above the emission curve.

    Decreasing temperature:
    (a) Based reasoning similar to that above, this would occur from late July or early August until early or mid February.

    (b) Based on reasoning similar to that above, this would occur from late July to early February.

  6. Single time (approximately) when temperature is increasing the fastest

    (a) Based on the emission curve and the Stefan-Boltzmann law of radiation only, this occurs when the slope of the emission curve is greatest (since the temperature would be increasing the fastest where the emission is increasing the fastest). This is not easy to tell from the emission curve very precisely, but it looks like it occurs somewhere in late April or May.

    (b) According to the heat budget equation in this case, the temperature should increase the fastest where the difference between absorption and emission is the greatest. By marking the (vertical) distance between the absorption and emission curves on a piece of paper, the maximum separation between the curves appears to be in late April or early May.

  7. A time when (avg. daily) temperature isn't changing (that is, isn't increasing or decreasing) (if there is more than one, choose one)

    (a) The temperature must stop changing at the times when it is maximum or minimum. Before each such point the temperature changes in one sense (that is, with one sign, positive or negative) and afterwards changes in the other sense, so the temperature goes from increasing to decreasing or vice versa at maximum and minimum points. To do this (that is, change the sense with which it is changing), the temperature must momentarily stop changing at the maximum and minimum points. The answers were given in (4)(a).

    (b) According to the heat budget equation in this case, the temperature should stop changing when absorption equals emission, which happens when the curves cross. The answers were given in (4)(b).

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