Note that this reduces to the previously derived neutrally stratified result when Ri = 0. As the surface layer is made more stable, the drag coefficient goes down monotonically, and approaches zero as Ri ^ Ric.
Now let's do a few examples illustrating the effects of stable surface layer physics on turbulent heat flux. First consider a melting slab of sea ice or glacier ice, in an environment where the air temperature is 280K. Since the ice is melting, the ground temperature is pinned at the freezing point, namely 273.15K. Since the air is warmer than the ice, there will be a flux of sensible heat from the air to the ice, which will help sustain melting. At these temperatures, it is safe to neglect the contribution of water vapor to buoyancy. Suppose that the wind is 5m/s, the air temperature and wind have been specified at 10m above the ice surface, and the roughness height is .005 m. Then, for a neutrally stratified boundary layer CD = .0028 and the sensible heat flux would be 125'W/m2. With the specified parameters, the Richardson number is 0.1, and incorporation of buoyancy effects on the turbulence bring CD down to .0007, and the sensible heat flux falls to 32 W/m2. If we increase the air temperature to 285K, the Richardson number increases to .17, and the increasing stability reduces the sensible heat flux to a mere 4.8W/m2, whereas neutral surface layer theory would have led us to expect a substantial increase in flux.
Exercise 6.4.1 How long would the sensible heat fluxes computed above take to melt through a 5 meter thick layer of water ice, if all the energy is used to melt ice?
Next, let's consider the effect on the night-time temperature inversions appearing in cold, dry climates such as Antarctica or the tropics on Snowball Earth. The radiative balance for such cases was discussed towards the end of Section 6.2.3. For example, in Earthlike conditions with Tsa = 240K, the equilibrium surface temperature is 177K if the only coupling of ground to atmosphere is via infrared. Using the same assumptions as in the previous example, except for the new temperatures, neutral surface layer theory would predict a sensible heat flux of over 1500 W/m2, which of course would mean that the turbulent transfers would keep the inversion from getting nearly as strong as it would be in the purely radiative case. However, with such a large temperature difference, the Richardson number is 1.4, which leads to a complete suppression of turbulence and allows the extremely strong inversion to be realized, if there is enough time for the surface to cool down to equilibrium. An interesting aspect of this problem, however, is that the sensible heat flux is not a monotonic function of Tsa - Tg. As Tg is decreased from Tsa, the flux first increases, reaches a maximum, then decreases to zero as the critical Richardson number is approached. This means that there is the possibility of multiple equilibrium states - one with turbulence and heat flux, and another with turbulence suppressed. Whether or not this happens depends on the slope of the radiative flux, but even when there aren't multiple equilibria, there tend to be abrupt transitions between turbulent and non-turbulent states as a control parameter such as solar absorption is continuously varied. This behavior is explored in Problem ??.
The inclusion of a light, condensible substance like methane or water vapor has an important effect because it allows the surface layer to remain neutrally stratified even when the ground temperature is significantly lower than the air temperature at the upper edge of the surface layer -provided that the air there is appreciably undersaturated. To get a feel for the numbers, let's do an example involving water vapor in air. Suppose that the air temperature is 300K and the relative humidity is 70%. Then, using the formulae for the Richardson number and for buoyancy, we find that the surface layer is neutrally buoyant (Ri = 0) when Tg = 299K. Without the effect of water vapor on buoyancy, the Richardson number would be 0.013 assuming u = 5m/s, so in this case the suppression of turbulence caused by neglect of the moisture contribution to buoyancy is small. The effect increases sharply at higher temperature, though. For Tsa = 340K, the surface layer remains neutral down to 335.5 K, and the Richardson number without the moisture contribution to buoyancy would be 0.05. The maintainence of buoyancy by light vapor will figure importantly in our estimates of precipitation rates on hot planets, in Section 6.8.
Finally, let's take a quick look at the unstable case, where the surface layer is positively buoyant. In this case, the buoyancy-driven turbulence adds to the mechanically driven turbulence due to wind shear across the surface layer. Buoyancy-driven turbulence is particularly important when the mean winds at the top of the surface layer are weak. When u = 0 the neutral theory would predict that there are no turbulent fluxes, but if the surface layer has upward buoyancy, then convection should in fact be able to sustain turbulence. The case u = 0 with upward buoyancy is the free convection limit. In this limit, there is no longer an intrinsic velocity scale separate from that defined by the buoyancy scale, and there is no longer any intrinsic length scale such as enters the Monin-Obukhov theory. Instead, we can define a velocity scale a/ft*zi, which is the order of magnitude of the upward velocity attained by a buoyant plume when it reaches the top of the surface layer. Since there is no longer a characteristic length scale, the buoyancy profile ft(z) is logarithmic, just as for the neutral case, and this allows us to relate the buoyancy gradient to the difference in buoyancy between the upper and lower edge of the surface layer. Using the characteristic velocity scale, the buoyancy flux in the free convection limit can be written
w'ft' = (ln(Zl/Z ))2 (a ■ ft(zi)ln(zi/z*))1 (ft(z*) — ft(zi)) = Cn,neutUfreeAft (6.32)
where CDneut is the usual neutral drag coefficient, Ufree is a characteristic buoyancy velocity and a is a nondimensional constant whose value has been empirically determined to be about 15. To apply this result to the flux of other quantities, such as latent or sensible heat, we use the same drag coefficient and Ufree, but replace Aft with the difference in the quantity whose flux we wish to obtain. Note that because of the definition of Ufree, the buoyancy flux scales like the | power of buoyancy. However, by casting the flux formula in the above form we can see that it is just like the neutral case, but with a buoyancy velocity replacing the mean wind. As an example, consider a dry case with ground temperature of 305K, air temperature 300K, Earth gravity, a surface layer thickness of 10m and a roughness length of .001m. With these parameters Ufree = 15m/s, so buoyancy driven turbulence becomes a significant player when the mean wind is 15m/s or weaker. Based on this estimate, it is clear that convection will lead to large sensible and latent heat fluxes whenever the ground temperature tries to get much bigger than the overlying air temperature. The upshot is that it is quite easy for the ground to get a lot colder than the overlying air, because of the inhibition of turbulence in stable conditions, but it is harder for the ground to get much hotter than the overlying air.
The general unstable case with nonzero mean wind can be treated similarly to the way we treated the stable case, though it is necessary to adopt different scaling functions for wind and momentum and the form of the functions is sufficiently complicated that a Newton's method iteration is generally needed in order to solve for Zi. In addition, the scaling functions most commonly in use do not, in fact, reduce to the correct free-convection limit when the mean wind becomes weak. This point, together with a resolution of the problem, is discussed in the paper by Delage and Girard given in the Further Readings section for this chapter. A simple expedient for dealing with the general unstable case, however would be to compute the turbulent fluxes for both the free-convection and neutral limits, and then take whichever of the two is greater. This procedure by definition gives the correct free convection limit, and also eliminates the chief shortcoming of the neutral theory, namely the spurious vanishing of fluxes when mean winds become very weak. One can easily implement this formulation by computing fluxes using the usual neutral CD, but replacing u with Ufree when u < Ufree.
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