ERTH 535:
Planetary Climate Change
(Spring 2017)
Notes on
Lab Activity #4:
Intro to the Earth's Heat Budget
Dr. Dave Dempsey
Dept. of Earth & Climate Sci.,

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  5. The incoming solar radiation plot shows the intensity (flux) of solar radiation at top of atmosphere on a surface parallel to the earth's surface (that is, on a horizontal surface), averaged over a month (including day and night). The units are Watts per square meter.

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  7. The insolation at the top of the atmosphere (see graph below) is computed on surfaces parallel to the earth's surface (that is, horizontal surfaces), not directly facing the sun. Since the earth is curved, this introduces variations in sun angle (and hence variations in insolation) with latitude. (They are averaged over a month as well, including day and night. However, in March the number of hours of daylight averages close to 12 hours—actually a bit more in the Southern Hemisphere and a bit less in the Northern Hemisphere because the equinox on March 12 or so comes later than half way through the month, not in the middle of the month--but not much different from 12 hours anywhere. As a result, variations in sun angle with latitude dominate the variations in monthly average insolation with latitude.)

  8. Insolation at the top of the atmosphere vs. latitude in January, March, and July, 1987:

    Note that on this plot, the horizontal axis shows latitude, with latitudes in the Southern Hemisphere shown as negative values and Northern Hemisphere latitudes positive. Also note that the vertical axis doesn't start at 0 W/m2 but rather at about 50 W/m2.

    Features of this plot (which were already discussed in Lab #2: "The Seasons", for December, March, and June, which were similar to January, March, and July):

    1. Highest values in January and July weren't near the equator but rather at the poles. This is due to the fact that the data plotted are averaged over the whole month, including both day and night of each day. This means that not only sun angle affects the monthly-average insolation values, but also day length, and day length increases with increasing latitude in the summer hemisphere nearly all the way to the poles. The greater day length at high latitudes largely overcomes the effect of lower average sun angle during the day when computing the monthly average insolation.

    2. The monthly average insolation at the equator varies little with time of year. There are three reasons for this:

      1. The noon sun angle varies by only 23.5° over the course of the year at the equator. The noon sun angle is 66.5° above the northern horizon at the June solstice; it's 90°—that is, directly overhead—at the September equinox; it's 66.5° above the southern horizon at the December solstice; and it's back to 90° at the March equinox. Hence, the noon sun angle is never less than 66.5° above the horizon. (Figure 2-5 from Lab Activity #2: The Seasons illustrates this point.) The annual variation in sun angle increases away from the equator, to 47° (from 43° at the winter solstice to 90° at the summer solstice) at the Tropic of Cancer and Tropic of Capricorn.

        In contrast, at latitudes outside the tropics the amount by which the noon sun angle varies seasonally ranges from 23.5° at the poles, where it's never greater than 23.5° and is less than or equal to 0° for half the year, up to 47°, at latitudes from the Arctic Circle to the Tropic of Cancer and from the Antarctic Circle to the Tropic of Capricorn. For example, in San Francisco, which is located at 37.5°N, the noon sun angle varies from 29° at the December solstice to 76° at the June solstice (a range of 47°). (Figure 2-5 from Lab Activity #2: The Seasons also illustrates these points.)

        The monthly-average insolation includes insolation at all times of day, not just noon, but the noon sun angle determines the maximum insolation during the day and hence puts an upper limit on what the insolation on any particular day can be.

      2. The increase in the extent to which solar radiation "spreads out" as sun angle decreases, is smaller at higher sun angles than it is at lower sun angles (because the sine function that describes the relationship is not linear). That is, if we look at the increase in "spreading out" that occurs when the sun angle decreases by 1°, we see that when the sun angle is high, the increase in spreading-out is smaller for a decrease of 1° in sun angle than it is when the sun angle is low. As a result, variations in noon sun angle at the equator have a smaller effect on insolation than variations by the same amount at higher latitudes, where the sun angle is lower.

      3. Perhaps most important, though, is the fact that the number of hours of daylight is 12 hours all year long at the equator, whereas the number of hours of daylight varies with time of year more and more at higher and higher latitudes. The number of hours of daylight influences the monthly average insolation because the monthly average includes both daytime and nighttime hours.

    3. The maximum in the July curve (at high northern latitudes) is a little lower than the maximum in the January curve (at high southern latitudes), presumably because earth is a little closer to the sun in January than in July.

    4. There's a dip in the curve in the summer hemisphere in the higher latitudes; this arises from competing, nonlinear effects of day length and sun angle. (Need to see the actual equations to understand more clearly why this happens.)

    5. Insolation for March is asymmetric about the equator (that is, it's not centered on the equator with mirror images of itself on each side of the equator). Rather, it skews slightly higher toward the Southern Hemisphere side. This reflects the fact that the equinox occurs past the middle of the month (March 21 or 22). As a result, more days that come before the equinox are included in the monthly average than days that come after the equinox. In the Southern Hemisphere, the days that come before the equinox are summer days, when the days are longer than 12 hours and the sun angle is higher than on days that come after the equinox. In the Northern Hemisphere, the days that come before the equinox are winter days, when the days are shorter than 12 hours and the sun angle is lower than it is on days following the equinox. As a result the insolation values are greater in the Southern Hemisphere than they are in the Northern Hemisphere in the March monthly average. This is purely an artifact of the choice of averaging period. If we'd averaged over 30 or 31 days centered on the equinox, this asymmetry would essentially disappear.

  9. To achieve a balanced budget, outgoing longwave IR radiation (which the earth emits, and the energy in which is converted from heat, which the earth therefore loses) would have to be balanced by absorbed solar radiation (which is converted to heat in the earth). (The earth doesn't convert all of the energy arriving from the sun into heat because the earth reflects some of that energy back to space rather than absorbing it.)

  10. Absorbed solar radiation comes much closer to balancing outgoing longwave IR radiation than any of the other choices provided in (9). It's true that absorbed solar equals incoming solar radiation minus reflected solar radiation, but from the point of view of the heat budget equation it's more direct simply to say absorbed solar radiation must balance outgoing (emitted) longwave IR. There is no combination of all three that would make any sense, since absorbed solar and incoming minus reflected solar are redundant.

  11. In 1987, spot checks of several individual months and an average over the whole year showed absorbed solar radiation exceeding outgoing longwave IR by a small but not necessarily insignificant margin (around 8% for the year). If valid, this should result in an increase in global mean temperatures (though not necessarily at the earth's surface).

    There are some sources of error in the measurements, and missing data have been interpolated from neighboring areas where the values are measured, another source of error. It's possible that these errors might affect the outgoing longwave IR measurements more than absorbed solar measurements, in such a way as to underestimate the former. If true, the errors would exaggerate, or possibly even create, the ~8% imbalance that we've calculated based on the observations. (In fact, I think I can say with some confidence that the imbalance is much smaller than this, reflecting shortcomings in these data, though I can't prove that using information available only from My World GIS.)

  12. Outgoing (space-bound) longwave infrared (IR) radiation vs. latitude for January, March, and July 1987:

  13. Plots of zonally averaged absorbed solar and outgoing longwave IR (superimposed) for January, March, and July, 1987. (Absorbed solar shown using open squares; spacebound energy shown using black circles.)

    In March, there's a slight dip in absorbed solar near the equator, which doesn't appear in the incoming solar plots that we've made. Since the only difference between incoming solar and absorbed solar is the part of the incoming solar reflected back to space, there must be more solar radiation reflected back to space near the equator than at latitudes slightly farther north and south. In July there's also a slight dip, slightly north of the equator. In January we don't really see a dip near the equator, but there are flat spots (nearly dips), one just north of the equator and one just south.

    The January curve shows a dip in absorbed solar at around 80°S. As we'd expect in January, absorbed solar is zero at the highest northern latitudes, when it is dark all day at those latitudes. Similarly, the absorbed solar is zero at the highest latitudes in the Southern Hemisphere in July.

    Unlike the incoming solar plots for July and January, the peak in the absorbed solar isn't at highest latitudes; absorbed solar drops off considerably at higher latitude. This must also be due to variations in albedo, since reflected solar radiation is the only difference between incoming and absorbed solar radiation. What could be causing this reflection? We could benefit from some additional data to give us some ideas. Ice caps and sea ice at high latitudes? Clouds?

  14. Annual, zonal average absorbed solar and outgoing longwave IR radiation:

    There are dips in each plot slightly north of the equator. We conclude that the earth as a whole is slightly more reflective there than slightly farther north and south of that area, on the average (based on the absorbed solar plot) and is cooler there as well (even if it isn't cooler at the surface).

    There is a slight dip (Southern Hemisphere) or plateau (Northern Hemisphere) in absorbed solar in the lower high latitudes (around 70° in each hemisphere).

    Generally speaking, absorbed solar radiation is higher than emitted longwave infrared (LWIR) radiation at lower latitudes, but emitted LWIR drops off more slowly with increasing latitude than does absorbed solar, so that emitted LWIR is greater than absorbed solar at higher latitudes.

    Very clearly, in the annual average, absorbed solar and outgoing longwave IR fluxes do not balance at almost any latitude (the exceptions being where the two curves cross in the middle latitudes). At lower latitudes, the earth absorbs more solar radiation than it emits LWIR radiation. If these two radiative fluxes were the only source and sink (respectively) of heat, then at lower latitudes the earth would be warming up over the course of the year.

    At higher latitudes, in the annual average the earth emits more LWIR radiation than it absorbs solar radiation. If these two radiative fluxes were the only source and sink (respectively) of heat at those latitudes, then the earth would be cooling off over the course of the year at those latitudes.

    Since this particular year wasn't unusual, these results would apply to essentially every year. That is, we'd expect the lower latitudes to warm up each year and the higher latitudes to cool off each year. Only in a narrow range of latitudes at midlatitudes might we expect little change in temperature over the course of a full year (and hence from year to year).

    As a result of these changes in temperature, according to the Stefan-Boltzmann Law, we'd expect the amount of radiative energy emitted by the earth to increase at lower latitudes and decrease at higher latitudes. In contrast, we can't say with any assurance whether the absorbed solar radiation would change, or if it does change, how it would change, because we don't have a firm grip on how albedo might change as temperature changes. (It might increase where temperatures rise, because higher temperatures lead to more evaporation from the oceans, which might lead to more cloudiness, but this is relatively simple reasoning in a situation that could be more complex than that. Also, albedo might increase where temperatures fall below a certain threshold, where ice begins to cover unfrozen ocean or previously bare or vegetated ground around the edge of existing ice caps or sea ice, but wouldn't change where ice cover already exists.) If we assume that albedo doesn't change dramatically, then absorbed solar wouldn't change dramatically. As a result, the emitted LWIR radiation flux would increase at low latitudes until it equaled the absorbed solar, at which point the heat budget equation (from the Conservation of Energy) tells us that the temperature would stop changing (the budget would be balanced at that latitude). At higher latitudes, as the temperatures fell, LWIR radiative emission would decrease until it equaled absorbed solar, at which point the temperature there would also stop changing. When the radiative energy budget at each latitude reaches a balance, we'd have warmer temperatures at low latitudes and colder temperature at high latitudes than we observe today—much warmer and much colder, respectively, as it turns out. A much narrower range of latitudes than we have today would actually be habitable!

    We don't in fact observe this to happen—the plots above don't change anything like that much. Hence, either there is something wrong with these data, or our assumption that these two radiative fluxes are the only sources and sinks of heat at each particular latitude must be wrong, or both. (The data are not perfect, but they are actually reasonable and the imbalances are so large that we can rule out the first possibility.)

  15. The (tedious) calculation of the annual, area-weighted global average flux of longwave IR radiation emitted to space, using My World GIS spacebound energy data measured by satellites, was done for you. So was the annual, area-weighted global average flux of longwave IR radiation emitted by the earth's surface, based on surface temperature data in My World. The results were, approximately:

    These are not very close to each other. If the My World data are reasonable and our calculations are anywhere close to being correct, then an awful lot of radiative energy emitted by the earth's surface doesn't make it out to space. What can be happening to it?

  16. Applying the Stefan-Boltzmann Law to the annual, area-weighted global average spacebound energy flux (above) to calculate an effective radiating temperature, gives us a value of about 254.5 K. (This is the single temperature that a blackbody would have to have to emit radiative energy with the same global average intensity as the earth does to space, namely 237.7 W/m2 as calculated above.) This translates to -18.7°C, or -1.7°F. That is, as viewed from space, the earth (interpreted as a blackbody) appears to have an effective (radiating) temperature of -1.7°F.

    Calculating an annual, area-weighted, global average surface temperature using My World GIS gives us a value of about 287.8 K (14.6°C, 58.3°F). Hence, the global-average surface temperature is about 60°F (33.3°C) warmer than the effective radiating temperature of the earth as viewed from space. If the data are reasonable, then this is a large discrepancy. What can account for it? (Note that the discrepancy in planetary vs. surface temperature and in spacebound vs. surface radiative emission fluxes are at least consistent with each other: the surface is warmer and it emits radiative energy more intensely than is/does the planet as a whole, as viewed from space. This consistency follows trivially from our use of the Stefan-Boltzmann relation to connect emission fluxes to temperature, of course.)

    We also note that the greatest global average emission fluxes from the planet as a whole and from the surface (and the corresponding temperatures) occur in August and July, respectively, among the months when the earth is farthest from the sun (aphelion itself is in early July). Moreover, the surface emission flux and surface temperature, and emission flux from the planet as a whole (and the corresponding effective radiating temperature), are lowest in January, when the earth is closest to the sun. The differences are not very large compared to seasonal variations at middle and high latitudes in any one hemisphere, as we might expect because the two hemisphere's seasons are 6 months out of phase and hence tend to cancel each other out in the global average, but the reason(s) for the timing of the maxima and minima in emission fluxes are not obvious—what might account for them?

    We do note that the pattern of variations over the course of the year, although rather small (as we'd expect from the global average, where the out-of-phase variations of the two hemispheres tend largely to cancel each other out), seems similar to Northern Hemisphere temperature patterns rather than Southern Hemisphere patterns.

    One hypothesis: In Lab Activity #1 we calculated the July minus January difference in temperature and discovered that the difference is much greater over land than over oceans outside the tropics. The Northern Hemisphere has about three times more land area than the Southern Hemisphere, and much of that land area is at middle latitudes and high latitudes (where the July minus January temperature difference is much larger than in the tropics). Hence, it seems qualitatively reasonable that the difference in land area between the hemispheres would mean that in the hemispheric average, the July minus January difference would be larger in the N. Hem. than in the S. Hem., and so in the global average, the intra-annual variations global average spacebound emission flux would be biased toward the N. Hem.

    Two possible shortcomings of this hypothesis: (1) although it is qualitatively reasonable, we would have to test it quantitatively to see if it could account for the observed variations over the course of the year in spacebound LWIR radiative emissions; and (2) although we can relate radiative emission flux from the earth's surface to the temperature of the surface, a large portion of the radiative energy emitted by the surface doesn't escape to space (as we discovered in #15 above), and we don't know whether the radiation that the earth does emit to space comes from the earth's surface or not. We therefore would need to test the hypothesis further.

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