The numerator in this expression is the energy imbalance the surface would have if the ground temperature were equal to the overlying air temperature. It can be either positive or negative and its sign determines the sign of AT, since all three terms in the denominator are positive. The denominator is a stiffness coefficient. For any given magnitude of the numerator, the denominator determines how much the ground and air temperature differ. In other words, when the denominator is large, the ground and air temperature are very tightly coupled, but when the denominator is small, they can vary independently. The coupling constants will prove useful in making simple models of the seasonal and diurnal cycle of temperature, as we shall do in Chapter 7.

Table 6.2 shows some typical coupling coefficients for Earth and Titan. Since these are derived by linearizing around Tg = Tsa, the effects of buoyancy on CD to not affect bsens. Moreover, since both methane and water vapor are positively buoyant in the background gas, the surface layer is in the unstably stratified regime, so that suppression of turbulence does not enter into the picture. The unstable buoyancy effects do cause CD to increase slightly with Tg, and this would slightly increase bL. Note that bsens is nearly independent of temperature; the slight variation is due to the effect of the composition on mean specific heat and on surface layer density. The radiative coupling coefficient bir increases gently with temperature, but the latent heat coefficient bL increases sharply, owing to the exponential behavior of Clausius-Clapeyron. For Earth, the sensible heat transfer dominates the coupling in cold conditions, which apply near the poles in

T |
bir |
bsens |
bL | |

Water+Air |
250. |
3.54 |
21.00 |
2.76 |

Water+Air |
280. |
4.98 |
18.89 |
19.72 |

Water+Air |
300. |
6.12 |
17.99 |
57.95 |

Water+Air |
320. |
7.43 |
17.84 |
147.0 |

CHa + N2 |
85. |
0.14 |
95.07 |
161.36 |

CHa + N2 |
90. |
0.17 |
92.55 |
287.29 |

CHa + N2 |
95. |
0.19 |
91.88 |
365.75 |

Table 6.2: Some typical surface flux coupling coefficients. The "Water+Air" cases are done under Earthlike conditions, with a 1 bar Earth air noncondensible background. The CH4 + N2 cases are done under Titanlike conditions, with a 1.5 bar noncondensible N2 background. Both cases were done with 70% relative humidity at the top of the surface layer, U = 10m/s and CD held fixed at 0.0015 . Units for all the coupling coefficients are W/m2K

Table 6.2: Some typical surface flux coupling coefficients. The "Water+Air" cases are done under Earthlike conditions, with a 1 bar Earth air noncondensible background. The CH4 + N2 cases are done under Titanlike conditions, with a 1.5 bar noncondensible N2 background. Both cases were done with 70% relative humidity at the top of the surface layer, U = 10m/s and CD held fixed at 0.0015 . Units for all the coupling coefficients are W/m2K

climates like the present and globally for Snowball conditions. As temperature increases, latent heat fluxes come to increasingly dominate the coupling. In tropical conditions for the present Earth climate, evaporation accounts for 71% of the total coupling coefficient of 82.1 W/m2K. To get an idea of how tightly coupled the ground temperature is to the overlying air temperature in this case, we note that an increase of 40 W/m2 in absorbed solar radiation at the ground (arising perhas from a drastic decrease in cloudiness) could be accomodated by an increase of ground temperature by under a half a degree. In these circumstances, the most effective way to increase the surface temperature is not to alter the surface energy budget, but rather to increase the temperature of the atmosphere. This is the main way increases in greenhouse gases increase the ground temperature. As temperature increases beyond modern tropical values, evaporation becomes even more dominant, and coupling becomes even tighter.

In the Titan case, the infrared coupling is almost completely insignificant. It is interesting, however, that the sensible heat coupling coefficient is quite significant, owing to the high density of Titan's atmosphere. We have already noticed that on Titan evaporation easily dominates the weak absorbed solar radiation. Evaporation makes the numerator of Eq. 6.35 strongly negative if the relative humidity is appreciably less than 100%, which drives a temperature inversion at the surface. The dominant balance in this case is between evaporative cooling of the ground and transfer of sensible heat from the warmer atmosphere and the colder ground. In this situation, the suppression of turbulence by stable boundary layer effects, can play an important role in determining the strength of the inversion (see Problem ??). The Earth's tropics is not hot enough to be in this regime, but on a much hotter Earth the increase of water vapor would lead to very similar effects.

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