The radiative equilibrium solutions discussed in the preceding section are all unstable near the ground. As convection sets in, it will mix away the unstable layer and replace it by an adiabat; the well-mixed region is the troposphere. The change in lower level temperature profile, however, will alter the upward radiation which heats the stratosphere, and therefore cause temperature changes even above the layers reached directly by convection. When all this sorts itself out, how deep is the troposphere? This is is the problem of tropopause height, which we have already touched on briefly for grey gases. Here we will offer a taste of a few of the most important aspects of real gas behavior, and lay the physical basis the reader will need for further explorations with more comprehensive models. In this section we will assume, as in the all-troposphere model, that turbulent fluxes couple the ground so tightly to the overlying air that there is no discontinuity at the ground. This assumption will be relaxed in Chapters 6.
For a grey gas, the problem of finding the tropopause height is relatively simple. Since the radiative equilibrium profile depends only on OLR - and that only via a simple formula - one starts with the radiative equilibrium profile for the desired OLR, picks a guess for the tropopause pressure, and then replaces the temperature between there and the ground with the adiabat for the gas under consideration. One then computes the actual OLR for the resulting profile, and generally will find that it is generally somewhat different from the OLR assumed in computing the radiative equlibrium. To make the solution consistent, one then adjusts the tropopause height until the computed OLR including the troposphere is the same as the target OLR within some desired accuracy. This is a simple problem in root-finding for a function of a single variable (the tropopause pressure), and can be solved by any number of means, Newton's Method and bisection being among the most commonly employed.
For a real gas,the radiative equilibrium in the upper atmosphere depends on the spectrum of the infrared upwelling from below, so we no longer have the luxury of assuming that the stratospheric temperature profile remains fixed as we vary the estimate of the tropopause height. Instead, one must simultaneously solve for both the tropopause height and the corresponding equilibrium profile aloft. This is most easily done by a modification of the time-stepping method we employed to compute the pure radiative equilibrium solutions. As in that case, it is somewhat awkward to pick an OLR and find the corresponding ground temperature Tg. Instead, we fix Tg, and compute the corresponding OLR. This can be done for a range of ground temperatures, whereafter the ground temperature in equilibrium with any specified solar absorption can be determined. We are back in the familiar business of computing the OLR(Tg) curve, much as we did for the all-troposphere model, but this time taking into account the effect of a self-consistent stratosphere on the OLR.
The general problem of representing convection in climate models is a very challenging one, about which entire volumes have been written. (See the Further Readings section of Chapter 2.) For the problem at hand, there are a number of simplifying assumptions which allow us to avoid some of the more subtle aspects of the subject. First, we will be content to assume that convection instantaneously resets the profile to an adiabatic profile. Next, given where instability occurs in the pure infrared radiative equilibrium profiles, it is safe to assume that convection occurs in a single layer extending from the ground to the tropopause height, without any possibility of multiple interleaved internal convecting and radiative equilibrium layers. Further, we will only seek an equilibrium solution, without attempting to accurately represent the approach to equilibrium. Finally, we carry out the calculation by holding Tg fixed and allowing the rest of the atmosphere to relax to the corresponding equilibrium. Under these circumstances, the elimination of unstable layers by convective mixing can be carried out through the following simple modification to the pure radiative equilibrium time-stepping algorithm: One calculates the adiabat Tad(p) corresponding to the ground temperature Tg and surface pressure ps. Then at each timestep, wherever T(p) < Tad(p), the temperature is instantaneously reset to Tad. The rationale for doing this is that convection is a much faster process than radiative relaxation, and that wherever the temperature is below the adiabatic temperature, air parcels starting at the ground have enough buoyancy to reach that level, mixing air all along the way. The procedure also assumes that the turbulent coupling of the ground to the overlying air is so strong that ground and air temperature remain essentially identical at all times. The adjustment to the adiabat with surface temperature Tg in general increases the static energy (moist or dry, as appropriate) of the adjusted layer of air. This is not a source of concern if we are only using time-stepping to find the equilibrium state; if we were instead trying to represent the actual time-course of approach to equilibrium, a more sophisticated adjustment approach conserving static energy would need to be employed. Conservative adjustments would transport heat vertically by cooling the lower levels at the same time they are warming the upper levels.
Results for an Earthlike air/CO2 atmosphere are shown in Figure 4.45. The convection reaches to 380mb, where the temperature is about 200K; above that, the atmosphere is in radiative equilibrium, which defines the stratosphere. Note that the temperature continues to decline even above the maximum height reached by convection, because the infrared radiative equilibrium profile also has decreasing temperature, so long as part of the spectrum is optically thick. This is the case also for Earth's real stratosphere, even though ozone heating eventually causes the upper stratospheric temperature to turn around and begin to increase. Thus, one shouldn't take temperature decline as a signature of convection, and in a case where atmospheric heating causes upper stratospheric temperature to increase the temperature minimum will generally be above the top of the convective layer.
The OLR for all three calculations is similar: 296.4 W/m2 for the radiative-convective model vs. 295 W/m2 for pure radiative equilibrium and 296.6 W/m2 for the all-troposphere model. Thus, if the atmosphere is maintained by 295 W/m2 of absorbed solar radiation, neither the formation of a troposphere by convection nor the formation of a stratosphere by upper level radiative heating has much effect on the surface temperature, though the effects on the atmospheric profile are considerable.
Figure 4.45: Radiative-convective equilibrium for an Earthlike dry atmosphere with 300ppmv CO2. The left panel shows the temperature profile in comparison with pure radiative equilibrium and the dry adiabat. The right panel shows the radiative heating rates for the radiative-convective solution and for the all-troposphere model. In all cases the ground temperature is 280K.
The heating rate profile shown in the right panel of Figure 4.45 sheds some light on the basic mechanism maintaining convection in the troposphere. The entire troposphere is subject to radiative cooling. This feature is guaranteed by the construction of the solution, since any positive heating would warm the atmosphere, causing it to exceed the adiabatic temperature and shutting off convection. Suppose convection has just occurred and reset the temperature to the adiabat. Then, in the next small interval of time, radiative cooling will cause the temperature to fall below the adiabat, triggering convection once more, which adds back the heat lost by radiative cooling and restores the adiabat. The heat is supplied by parcels of air that pick up heat from the ground, become buoyant, and carry the heat upward to the level where it is needed. The thermal balance in the troposphere is between radiative cooling and convective heating. The addition of atmospheric solar absorption to the troposphere would have no effect on the tropospheric temperature or the tropopause height, so long as it doesn't turn the net radiative cooling at any level into a net radiative heating. Short of that happening, the sole effect of tropospheric solar absorption is to reduce the convective heating, and hence the frequency or vigor of convection. However, since infrared cooling is weakest just below the tropopause, solar absorption near the tropopause level can easily move the tropopause downward.
Using Figure 4.45 we can compare two simple estimates of the tropopause height with the actual value. Looking at where the adiabat intersects the pure radiative equilibrium, we find an estimate of 205mb. This is somewhat too high in altitude, since the formation of the troposphere changes the upward flux and warms the stratosphere. The other way to estimate tropopause height is to look at the heating profile computed for the dry adiabat, shown in the right panel of the Figure. Identifying the region of radiative heating with the stratosphere yields an estimate of 325mb, which is closer to the true value, but still too high in altitude. In the light of the discussion surrounding Fig. 4.2, we can say that the real gas atmosphere behaves rather like a pressure-broadened grey-gas atmosphere with optical depth somewhat greater than unity.
How does our computed tropopause height compare to the Earth's actual tropopause? We have defined the tropopause as the height reached by convection, and in comparing this with atmospheric soundings one needs to recall that even above the convective region, the temperature continues to decrease with height, because temperature goes down with height even in pure infrared radiative equilibrium; in the Earth's atmosphere, the temperature eventually begins to increase with height because of the effects of atmospheric solar absorption. Hence, the temperature minimum seen in Earth soundings, which is sometimes loosely called the tropopause, is always somewhat above the convectively defined tropopause (see Problem ?? ). Still, if we take the position of the temperature minimum in tropical soundings as an estimate, we note that the tropopause estimated in the preceding calculation is considerably lower in altitude (higher in pressure) than the observed value of about 100m6. What is it that makes the tropical tropopause so much higher?
The main factor governing the tropopause height is the lapse rate. If the lapse rate is weaker, then one has to go to higher altitudes in order to intersect the radiative equilibrium profile. In the warm tropics, the moist adiabat has significantly weaker gradient than the dry adiabat. In fact, a radiative-convective calculation based on the radiative effects of a dry CO2/air atmosphere, but employing the moist adiabat in the temperature profile, yields a tropopause height of 130mb when the surface temperature is 300K. This is quite consistent with the observed tropopical tropopause height. This suggests that the effects of moisture on lapse rate are more important than the radiative effects of tropospheric moisture in elevating the tropical tropopause. In other words, the main reason the Earth's present tropical tropopause is higher in altitude than the midlatitude tropopause is that the tropical lapse rate is weaker, owing to the greater influence of moisture for Earthlike tropical temperatures. The additional optical thickness due to the extra water vapor in the tropics plays at most a secondary role. For colder surface temperatures, the moist adiabat deviates less from the dry adiabat, so the tropopause height approaches the lower altitude found in the dry calculation. For example, with a surface temperature of 280K, a calculation adjusting to the moist adiabat yieldes a tropopause pressure of 250mb. This reasoning suggests that the tropopause height should be lower in the midlatitudes, and indeed observations show this to be the case. Optical thickness can indeed affect the tropopause height, but it is not the main player on Earth. The calculations referred to here are carried out in Problem ??
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