ERTH 535:
Planetary Climate Change
Spring 2017
Problem #3
(10 pts; Due Friday, March 9)
Dr. Dave Dempsey
Dept. of Earth & Climate Sci.,

The two curves on the accompanying graph, calculated using an idealized form of the heat budget equation (a mathematical expression of the principle of conservation of energy), 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 LWIR (terrestrial) radiation emitted by the earth's surface vs. time of year at Sacramento. ("Net" here means the difference between LWIR radiation emitted by the surface and LWIR radiation absorbed from the atmosphere, which emits both downward and upward (to space). To simplify things, we'll assume that the LWIR 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

  3. Time when IR emission is the most intense

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

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

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

  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)

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