where td = (pcp)2D/b2 and t1 = pcp\JD/w/b. Upon comparison of the third line of this equation with the solution for the mixed layer model, it is seen that the solid case acts somewhat like a mixed layer model with frequency dependent layer depth ^JD/w. For low frequency forcing, wtd ^ 1, the surface temperature follows the instantaneous equilibrium, A = S1/b, just as for the mixed layer case. For high frequency forcing, the amplitude of the surface temperature fluctuation decays like 1/^/(w). This is slower than was the case for the fixed-depth mixed layer, since the layer determining the thermal inertia now gets thinner as frequency is increased. Note also that the phase lag of surface temperature relative to insolation differs from the mixed layer case. For the diffusion equation, the surface temperature lags the insolation by n/4 radians in the high frequency limit rather than n/2. The thinning of the active thermal layer keeps the surface temperature closer to instantaneous equilibrium than it would be in the fixed-depth case.
Apart from some exceptional circumstances, the thermal inertia of a solid surface has little effect on the seasonal cycle, though it can substantially moderate the diurnal cycle. This can be seen easily through the evaluation of tauD in a few typical cases. First we consider the case of Antarctic or Arctic ice-covered regions. The flux coefficient based on a linear OLR fit in the temperature range 240K to 270K is b = 2.16W/(m2K). Using the heat capacity and thermal diffusivity for water ice, given in Table 7.1, we find tauD = 11Earthdays. At latitudes somewhat away from the poles, the diurnal cycle of insolation becomes significant, particularly during the equinoxes. Since the time scale for the surface is shorter than that for the atmosphere, it would be more appropriate to use surface flux coefficients than OLR in analyzing the terrestrial diurnal cycle. As noted in Chapter 6, the turbulent heat transfer is strongly inhibited at night-time, when the boundary layer is statically stable. In this case, the flux coefficient is dominated by the radiative term 4aT3 based on surface temperature. For temperatures around 255K this yields an even shorter response time t = 4Earthdays. In the midlatitudes and Tropics, the estimate differs only in the use of the slightly larger values of b appropriate to the warmer temperatures, and the somewhat different thermal properties of rock or soil, but the result remains that td is on the order of a few days or less. For Mars, one may use b = 4aT3 based on T = 200K, given the thin atmosphere. This yields tauD = 15Earthdays, which is still not sufficient to appreciably affect the seasonal cycle. It is only at the extremely cold temperatures of Titan that the response time of a solid ice surface becomes significantly longer (roughly 1300 Earth days), but even there the effect is of little interest, owing to the much longer response time of Titan's atmosphere. In sum, a solid surface can generally be considered to be in equilibrium for the purpose of computing temperature fluctuations on the seasonal time scale.
It should not be concluded from the above estimates that the thermal inertia of solid surfaces is sufficient to eliminate the diurnal cycle. The variation of insolation between noon and nighttime is huge; On Earth, at a latitude where the Sun is overhead at noon, the amplitude of the variation is 1370W/m2, which leads to an undamped temperature fluctuation of 685K based on a flux coefficient b = 2W/(m2K). Even damped by a factor of 20, this amounts to a very considerable diurnal fluctuation. Similar considerations apply to the Martian diurnal cycle.
As a complement to the periodically forced solution, Figures 7.8 and 7.9 show the solutions for the diffusion equation in water ice which is initialized at a uniform temperature of 300K and allowed to cool without solar heating subject to a flux upper boundary condition. The heat loss from the surface was computed using an Earthlike OLR fit OLR(Ts) = 48.461 + 1.5866(Ts — 180) + .0029663(Ts — 180)2 A quadratic fit was used so that the the fit would remain accurate over a large temperature range. Except for the high initial temperature, which turns out to be inconsequential, this problem can be thought of as representing the cooling of the Antarctic ice cap after permanent winter night closes in. Figure 7.8 illustrates the progressive penetration of the surface cooling into the depth of the ice; at time t, the cooling has penetrated to a depth on the order of \f~Dt, where D is the thermal diffusivity of the ice. Figure 7.9 shows that there is an extremely rapid initial cooling, owing to the thin layer of ice affected at short times. After a half day, the temperature has already fallen below freezing. Therafter, the temperature drop becomes slower, as the depth of the ice layer involved becomes greater. The reduction in OLR as temperature drop also contributes to the reduction in cooling rate. Nonetheless, after two months, the temperature has fallen to 190K,
which is well below the 235K minimum temperature observed at the South Pole. Incorporation of the atmosphere's thermal inertia reduces the cooling rate somewhat, but does much increase the extremely cold temperature encountered at the end of the winter. Clearly, the Antarctic interior relies on heat transport from warmer latitudes to limit its winter temperature drop.
We conclude this section with a few remarks on the special effects of snow and ice (whether from water, CO2 or some other substance) on the seasonal and diurnal cycle. Snow has a profound effect on the diurnal cycle, because of its very low thermal conductivity, which is nearly an order of magnitude lower than that of ice (see Table 7.1 for the case of water snow). The low thermal conductivity arises from the high proportion of the snow's volume which consists of air trapped in pores which are too small to allow the air to flow; since air itself has extremely low thermal conductivity, heat must primarily make its way through the contorted pathways of snow crystals in contact with each other. Other gases, trapped in snows made of other substance, have a similar effect. The low conductivity dramatically reduces the characteristic response time of the surface, even for a snow layer of modest thickness. In the Antarctic case discussed above, td drops to a mere 60 minutes for old snow, and 20 minutes for fresh snow. At night, the temperature of the snow surface plunges almost instantaneously to its equilibrium value. In the case of the Earth, the atmosphere has sufficient thermal inertia that it doesn't cool much at night, above the boundary layer. Given the suppression of turbulent flux in the stable nocturnal boundary layer, the night-time equilibrium temperature is maintained mainly by the downwelling infrared flux from the atmosphere, as discussed in Chapter 6. When the low level air temperature is 255K, the
Figure 7.9: Time evolution of surface temperature for the solution shown in Figure 7.8
downwelling infrared flux is about 120W/m2, maintaining a snow surface temperature of 214K. On present Mars, the atmosphere cools down markedly at night, and in any event is too thin to provide much downwelling flux, so it is less obvious what limits the night-time temperature drop over the CO2 snow fields that form in the winter hemisphere. One relevant consideration is that the flux coefficient b drops dramatically at very cold temperatures, leading to an increase of the relaxation time; when the surface temperature falls to 150K, td increases to 23 hours even over snow. However, at such low temperatures the saturation vapor pressure of CO2 is only 1.26mb, well below the ambient surface pressure. Hence, the night-time temperature minimum is likely to be governed by the latent heat release due to CO2 condensation, which sets in at surface temperatures near 160K.
Snow cover on any planet can change rapidly in the course of the seasons, and on Earth, sea ice cover similarly expands and retreats. Since snow and ice have higher albedo than the surfaces they generally cover, this has an important feedback effect on the seasonal cycle. It enhances the winter-time cooling once ice or snow begin to accumulate, delays the springtime warming, but then accelerates the warming once ice or snow begin to retreat. The albedo feedback of snow is especially pronounced, since snow has a much higher albedo than ice. For water snow, for example, the albedo of fresh snow averaged over the solar spectrum can exceed .85, whereas a typical albedo for sea ice is on the order of .6. The high albedo of snow, like its low diffusivity, arises from its highly porous nature which offers many opportunities for light to encounter discontinuities in index of refraction, leading to scattering. It is a generic property of the snow of any weakly-absorbing substance. Note that the concept of "sea ice" is peculiar to planets with water oceans. On a planet with a liquid methane or CO2 ocean "sea ice" would sink, and not have any chance to affect the surface albedo until the ocean were frozen to the bottom.
The presence of a solid phase on the surface of the planet also introduces a new form of thermal inertia, associated with the latent heat of phase change from the solid to liquid form. Where there is ice, whether it be in the form of sea ice or land glaciers, the surface temperature cannot rise above the triple point (the "melting point") until all the ice has been melted. The phenomenon is familiar from an experiment commonly performed in elementary school science classes, in which one tries to boil a pot of water containing ice cubes,and finds that the temperature doesn't start to rise above freezing until all the ice is gone. As an example, let us consider the melting of a 5m thick layer of ice with an albedo of .7. Based on the latent heat of melting, it takes 1.5 ■ 109 Joules to melt a square meter of this ice layer. In summertime at the pole, the ice absorbs 160W/m2 which would then take 110 days to melt the ice layer, even if all the absorbed solar energy is retained for melting, and none is lost by radiation to the atmosphere. Thus, a modest layer of ice can persist throughout a warm season, keeping the temperature from rising. One can clearly see this principle in operation in the summertime polar temperatures of the Earth, but it also operates in midlatitude areas subject to seasonal snow cover, in effect delaying the end of winter. The case of sublimation is somewhat similar, though subtler since there is no threshold temperature and the sublimated gas enters the atmosphere (with its stored energy), rather than flowing away in the form of rivers or ocean currents, as is the case for the liquid produced by melting. As discussed in Chapter 6, the latent heat flux due to sublimation (like evaporation) greatly increases the surface energy loss for any given temperature. This reduces the warming of the surface required to balance the summertime increase in absorbed solar radiation. Unlike melting, it does not generally cap the surface temperature at some fixed value, but it does reduce the warming below what it would be without the presence of the sublimating ice or snow.
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