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.)

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!).

As you learned in Lab Activity #11: Land/Water Contrasts, at middle latitudes, ocean surfaces warm and cool more slowly than land surfaces do both diurnally (that is, over the course of a 24-hour day) and seasonally. As a result, the diurnal and seasonal ranges in temperature of ocean surfaces is much smaller than it is on land surfaces.

1. [2 pts] On the attached figure, sketch an approximate emission curve for the ocean surface at the same latitude as Sacramento so that it highlights qualitatively several of the major differences between land and ocean surface temperatures over the course of the year that you discovered in the My World GIS surface temperature data set for 1987 shown in Lab Activity #11: Land/Water Contrasts.
   
   Your emission curve for the midlatitude ocean surface should reflect the following characteristics of the temperature observations (and hence radiative emission intensity):
   
   
   1. The maximum and minimum ocean surface temperatures lag behind those of land surfaces (in the My World data, by about a month).
      
      
   2. The maximum average temperature over land exceeds the maximum ocean surface temperature, while the minimum ocean surface temperature exceeds the minimum land surface temperature. These differences imply that ocean surface temperatures change more slowly (both warming and cooling) than land surfaces do.
      
   
   However, don't worry about trying to capture the following, additional two characteristics of the temperatures of midlatitude land vs. ocean surface in the Northern Hemisphere that we see in the My World GIS 1987 temperature data set:
   
   
   3. The maximum land temperature is only a couple of °C warmer than the maximum ocean temperature, whereas the minimum land temperature is about 20°C colder than the minimum ocean temperature.
      
      [This is in part because most land surfaces are higher than sea level, and temperatures generally decrease with increasing elevation. This effect shifts temperatures averaged over land systematically downward relative to temperatures averaged over the ocean. We want to highlight the effects of physical properties of water vs. rock/sand/soil on seasonal temperature variations over oceans vs. land, not the effects of elevation above sea level.
      
      In the My World data set for elevation of the earth's surface, the average elevation of midlatitude land areas is close to 1 km above sea level. In the long-term, global average, atmospheric temperature decreases by about 6.5°C per kilometer. If we use this average "temperature lapse rate" to correct the land-based temperatures to sea level, then the maximum land and ocean temperatures would differ by something like 8°C (instead of about 2°C) and the minimum temperatures would differ by only 14°C (instead of 20°C). Hence, although this correction doesn't eliminate the asymmetry between the maximum and minimum values over land and ocean, it does significantly reduce it.
      
      Moreover, the solar absorption and net radiative emission curve above is based on a calculation at one latitude and ignores the effects of non-radiative sources and sinks of heat, seasonal variations in downward emission of LWIR radiation from the atmosphere, and seasonal variations in cloudiness and surface albedo, any of which might differ over land and ocean, so you can't take the emission curve in the figure too literally as a representation of the data averaged over real-world midlatitude land areas.]
      
      
   4. The amplitude of the seasonal temperature variation over land is almost four times greater than it is over the ocean.
      
      [The emission intensity varies nonlinearly with absolute temperature, so you wouldn't expect the amplitudes of the radiative emission variations to differ by the same factor as the amplitudes of the temperature variations. However, there are other factors besides radiative emission and absorption that affect the observed temperature variations, so in this simple exercise we can't expect our simple radiative emission curve for the ocean to reflect this factor-of-four amplitude difference between land and ocean temperature variations while also trying to reflect characteristics (a) and (b) above.]
      
      
   Taking into account the adjustment described in (c) above, the midlatitude ocean surface and sea-level land surface are at about the same temperature in late April and in late October. Your radiative emission curve for the ocean surface should reflect this.
   
   
   
2. [3 pts] Based on your results, explain why the maximum and minimum temperatures of ocean surfaces lag behind the maximum and minimum temperatures land surfaces at midlatitudes. You should invoke the following:
   
   
   1. the heat budget equation;
      
      
   2. the reason why maxima and minima in temperature (and hence LWIR emission intensity, thanks to the Stefan-Boltzmann relation) occur when the solar absorption and LWIR emission curves cross (at least in the absence of other, variable sources and sinks); and
      
      
   3. ocean surface temperatures (and hence radiative emission intensity from the ocean surface) changes more slowly with respect to time than land surface temperatures do (as your graph should show).
      
      
   [Note that you do not have to explain why the ocean surface changes temperature more slowly than land surfaces do—that's an important question, but it is not the question being asked here.
   
   Note also that a common attempt to explain the different lags in the ocean and land temperature maxima and minima invokes the fact that it takes time for temperature to change in response to an imbalance between sources and sinks of heat, and ocean surfaces response more slowly than land surfaces do, even when ocean surfaces gain or lose heat more rapidly than land surfaces do (at least up to a point). However, this argument is not, by itself, a convincing or sufficient argument for why the temperature maxima and minima over land and oceans lag the maximum and minimum in solar absorption by different amounts. You will need to argue using the the three points listed above. It's easiest to start at a time when the land and ocean surfaces are about the same temperature, as they must be twice a year, and look ahead to where their respective maxima or minima occur, and why they occur when they do.]
   
   (Note that in the tropics, the seasonal variations in solar absorption are much smaller than at midlatitudes, so the month to month variations in monthly average surface temperature over both ocean and land surfaces are small, though variations over the course of a day are still much larger on land than on the oceans. At the highest latitudes in the Northern Hemisphere, the Arctic Ocean surface is frozen for much or all of the year (depending on location), so the surface temperature behaves differently from it does on a liquid water surface. Hence, we focus our attention on midlatitudes here.)
   
   
   
3. [1 pt Extra Credit] Under the simplifying assumptions made in the figure above, explain why it is impossible for the maximum temperature to occur at the same time as, or earlier than, the time of maximum absorption of solar radiation, and must occur later (that is, it must lag, as we see in the observations). Base your argument on the heat budget equation and the Stefan-Boltzmann Law.
   
   [Hint: Remember that the slope of the curve showing any quantity plotted vs. time is equal to the rate at which that quantity changes with respect to time.]