Season Angle

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Season Angle

Figure 7.5: The seasonal and latitudinal distribution of daily-mean flux factor for four different values of the obliquity. In these plots, a circular orbit has been assumed. To obtain the daily mean energy flux incident on each square meter of the planet's surface, one multiplies the flux factor by the solar constant. For example, if the solar constant is 1000W/m2, the incident solar flux at the pole during the Summer solstice is about 700W/m2 if the obliquity is 45o.

Annual mean solar flux factor

Annual mean solar flux factor

Flux factor

Figure 7.6: The annual mean flux factor for various obliquities, assuming a circular orbit.

Flux factor

Figure 7.6: The annual mean flux factor for various obliquities, assuming a circular orbit.

humidity using the OLR results given in Chapter 4, the polar temperature rises to 237K, but the equatorial temperature becomes problematic: The annual mean equatorial solar flux is near or above the runaway greenhouse threshold discussed in Chapter 4, leading to extremely high or even unbounded equatorial temperatures. Lowering the relative humidity to 20% to reflect the fact that much of the tropical troposphere is very dry (recall Chapter ??) still leaves the tropics with temperatures in excess of 350K. Part of the problem lies in the neglect of albedo. Simply using the observed planetary albedo in the tropics gives the wrong answer, because almost all of the cloud albedo is offset by the cloud greenhouse effect in the present climate (Chapter ??). Using an albedo of .15, based on the observed tropical clear-sky albedo, reduces the equatorial solar absorption to 360W/m2, which is in balance with a tropical temperature of 318K assuming a relative humidity of 20%. This is still well in excess of the observed tropical temperature. In the real atmosphere, heat transports due to large scale atmospheric and oceanic motions remove some of the heat from the tropics and deposit it at high latitudes, reducing the tropical temperatures and increasing the polar temperatures. Since incorporation of ice-albedo effects would reduce the polar temperatures below the estimates given above, such transports are also needed to bring the polar temperatures up into the observed range. Some elementary models of heat transport will be discussed in Chapter 9, though a proper treatment of the subject must be deferred until Volume II, where the necessary fluid dynamical background will be developed.

If we put the same planet at the orbit of Mars the temperatures become 238K and 183K without any greenhouse effect. Since there is little water vapor feedback at such low temperatures, the greenhouse effect is less dramatic in this case. Addition of an Earthlike atmosphere with 300ppmv CO2 would increase the equatorial and polar temperatures to 256K and 192K, respectively, based on a linear OLR fit in the range 200K to 250K (specifically, OLR = 80.6+ 1.83(T-200)). Thus, a waterworld placed at the orbit of Mars would require a much stronger greenhouse effect than the Earth's to avoid succumbing to a snowball state.

The effects of obliquity on the seasonal and latitudinal pattern of insolation may be summed up as follows. Increasing obliquity increases the intensity of the seasonal cycle at mid to high

latitudes. The summer insolation gets steadily higher relative to the global mean, and a greater area of the winter hemisphere exposed to cold perpetual night or low insolation. Increasing obliquity also increases the annual mean polar insolation, though the way this affects polar climate depends on the thermal response time of the atmosphere-surface system. The increase might show up as very hot summers and bitterly cold winters, or as year-round warming, accordingly as the response time is fast or slow.

The preceding results on incident solar radiation have been derived in the absence of an atmosphere, but can still be used if there is in intervening atmosphere which may absorb or scatter solar radiation before it reaches the surface. The general geometry is illustrated in Figure 7.7. In this case, one suspends an imaginary sphere at an altitude above which the atmosphere is too thin to have a signficant effect on the solar radiation. The preceding results then give the solar flux entering each square meter of the surface of this sphere, and the angle at which the light enters the atmosphere. This is all that is needed as input to one-dimensional scattering models of the sort discussed in Chapter 5. One simply divides up the atmosphere into a series of patches near each latitude and longitude point, within which the properties are considered uniform, and applies a one-dimensional column model to each of these patches. As illustrated in 7.7, if the horizontal size of the patches is large compared to the depth of the atmosphere, the energy loss by horizontal scattering from one patch to another can be neglected, and each patch can be considered energetically closed, so far as radiation is concerned. The atmosphere has three effects, which can be inferred from the column radiation model: (1) Some of the incident solar radiation reaches the surface in the form of diffuse radiation at a continuous distribution of angles, rather than at the zenith angle, (2) Some of the solar radiation is absorbed in the atmosphere, rather than at the surface, and (3) some of the incident solar radiation is reflected back to space instead of entering the climate system. Of the three effects, it is the last - the effect of the atmosphere, including its clouds, on planetary albedo - that is most important for determining the climate. Diffuse radiation and atmospheric absorption do not change the amount of energy entering a column, but only the place and angle with which it enters. Often, this is of little consequence, so one can get a good estimate of the planet's temperature if one can obtain an estimate of the planetary albedo from one means or another.

The above reasoning can even to some extent apply to gas giant planets which have no surface. One can still define the imaginary sphere through which radiation enters the system, as before, but the problem comes in defining a characteristic depth scale. For the purposes of solar radiation, it suffices to consider the depth of atmosphere over which most of the solar radiation is absorbed, in effect a "photic" zone. This is typically shallow compared to the prodigious size of the gas giants. The full problem, including internal heat sources and dynamical motions, might require consideration of a deeper layer. Whatever the depth of this "active layer," the preceding reasoning applies provided one can sensibly model the large scale aspects of the climate on the basis of averaging over patches of horizontal extent that is large compared to the characteristic depth scale. For giant planets, as for the Earth or any other planet, the essential difficulty is that clouds, temperature, water vapor and other climate variables are manifestly not uniform over length scales comparable to or longer than the depth of the atmosphere. One makes progress by boldly assuming that one can represent the effects of these fluctuating quantities by their large scale averages. It is an assumption that is difficult or impossible to justify mathematically, and in some cases may not even be true. With the present state of the art, one can only make progress by proceeding on the basis of the averaging assumption, and seeing how things work out.

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