Including turbulent heat fluxes, the surface energy budget can be written
where Frad is the net radiative flux into the surface, given by
Without turbulent fluxes, the surface budget would be Frad = 0. Frad in isolation can drive the ground temperature to be either larger or smaller (and perhaps much larger or smaller) than the air temperature, according to the circumstances discussed in Section 6.2. Sensible heat flux always drives the ground temperature and air temperature to become identical, whereas latent heat flux drives the ground temperature to be colder than the air temperature, by an amount that depends on the boundary layer relative humidity. When all three fluxes act in concert the resulting behavior depends on the relative importance of the fluxes.
We'll begin our tour of the range of possible behaviors by discussing how the surface balance is accomplished for typical conditions in the Earth's tropical oceans. Take Tsa = 300K, CD = .0015, U = 5m/s and hsa = 80%. We'll assume the absorbed solar radiation (1—ag)Sg is 320W/m2, which is typical of clear-sky conditions over the tropical ocean. To determine the back-radiation, we need ea. At tropical temperatures in the moist case, this coefficient is not very sensitive to CO2, and has a value of about .9. The terms making up the surface balance are shown in the left panel of Figure 6.2. As noted previously, the equilibrium ground temperature would be exceedingly large without turbulent heat flux. In the figure, the no-turbulence equilibrium occurs where Frad crosses zero, at around 336K. Adding sensible heat flux to the budget makes the slope of the flux curve more negative, and brings the equilibrium ground temperature down to 316K. Adding in evaporation steepens the curve yet more, and brings the ground temperature down to 303K, which is only slightly warmer than the 300K temperature of the overlying air. At the equilibrium point, the dominant balance is between the evaporation (206W/m2) and the absorbed solar radiation (320W/m2), leaving only 114W/m2 to be balanced by the other terms. The sensible heat flux is weak because the ground temperature and air temperature are nearly identical, which also makes the net infrared cooling of the surface weak given that ea « 1.
Next we'll discuss a typical set of Earth polar or midlatitude winter conditions. We set the absorbed solar flux (1 — ag)Sg to 100W/m2, taking a low value on account of the high albedo of snow or ice and the reduced solar flux received at high latitudes. We'll set Tsa = 265K, in which case ea « .6 with 300ppmv of CO2 in the atmosphere. The remaining parameters are held at the same values used in the tropical case. The main differences from the tropical case are that in the
cold case the latent heat flux and the infrared back-radiation are weaker - the latter doubly so because of the lower air temperature and the lower ea. The right panel of Figure 6.2 shows that because of the weak solar radiation and the weak back radiation, the radiative equilibrium surface temperature is nearly 5K colder than Tsa, in contrast to the tropical case. The situation here is a less extreme version of the night-time radiative equilibrium temperature considered in Section 6.2.3. Since ea is fairly small the temperature plummets at night when (1 — ag)Sg = 0. In the present case, Sg doesn't vanish, but its weak value is insufficient to warm up the ground temperature to the point where it exceeds the air temperature. This is the typical daytime condition in high latitude winter over ice and snow. Warm air imported from low latitudes helps to keep Tsa from getting too cold in the polar and midlatitude winter, but the weak sunlight and weak back-radiation leave the ground colder.
Since the radiative equilibrium ground temperature in the cold case is colder than the air temperature, adding in sensible heat flux conveys heat from the atmosphere to the ground, warming the ground up to just over 263K. The sublimation is weak at such cold temperatures, and causes little additional change in the surface temperature. While the dominant balance in the tropical case was between solar heating and evaporative cooling, the dominant balance in the cold case is radiative, with slight modifications due to sensible heat flux. For any given air temperature, the amount by which the ground temperature departs from the air temperature depends on the absorbed solar radiation, but the sensible heat flux always pulls the ground temperature back towards equality with air temperature. For example, at higher latitudes or deeper in the winter or near sundown, we might take (1 — ag)Sg = 50W/m2. In this case the radiation-only ground temperature is 246.6K, which is substantially below the air temperature; however, addition of sensible heat flux brings the ground temperature back up to 260K. Nearer to noon, or as summer approaches, we might have (1 — ag)Sg = 150W/m2. In this case, the radiation-only ground temperature is 271.8K; again addition of sensible heat flux brings the ground temperature closer to air temperature, in this case by cooling the ground to 267.1K, rather than warming it. Note that in these calculations of the effects of sensible heat transfer, the drag coefficient CD was held constant. Incorporating the inhibition of turbulence in stably stratified layers has the potential to substantially reduce the warming effect of sensible heat fluxes, particularly when the absorbed solar radiation is weak, since the inversion is strongest in those cases. This is explored in Problem ??.
Next let's estimate the maximum daytime temperature over a subtropical desert on Earth. Solid surfaces like sand or rock take little time to reach equilibrium, and so the maximum temperature can be estimated by computing the equilibrium temperature at local solar noon. Using the present Earth solar constant and a relatively high albedo of .35 (typical of Sahara desert sand), the absorbed flux is about 890W/m2. Over the interior of a dry desert, there should be little moisture in the boundary layer, so set ea = .72 corresponding to a boundary layer relative humidity of 20%. Finally, we take Tsa = 300K. In these circumstances the radiative equilibrium ground temperature is a torrid 383K - hot enough to boil water. When sensible heat flux is added into the budget, heat is transferred from the ground to the air, moderating the surface temperature. Taking a relatively high drag coefficient CD = .003 on account of the roughness of land surfaces, the equilibrium ground temperature is brought down to 330K if the surface layer wind speed is 5m/s. The temperature approaches the radiative temperature as the wind is made weaker; for example when the wind is reduced to 2.5m/s the temperature increases to 349K. Consistent with these estimates, the hottest satellite-observed ground temperatures do indeed occur in subtropical deserts, and are near 340K. With a wind of 5m/s, making the ground moist and turning on evaporation brings the equilibrium temperature down from 330K to 306K. The general lesson is that dry surfaces heat up greatly during the daytime. Their maximum temperature can greatly exceed the overlying air temperature, especially when the wind is light. This can contribute to the urban heat island effect, since constructed environments often replace moisture-holding surfaces with low albedo impermeable surfaces like asphalt, which hold little water and dry out quickly. The surface heating also leads to amplified climate change over land, in circumstances where a formerly moist soil becomes dry, or vice versa.
We'll conclude this section with a discussion of the linearized form of the surface balance, which enables simple, explicit solutions for the temperature jump across the surface layer. Using the linearized flux coefficients defined previously, the temperature jump is simply
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