The first term is the temperature the planet would have in the absence of any atmosphere. As increases, the surface becomes warmer without bound. This constitutes our simplest quantitative model of the greenhouse effect for a temperature-stratified atmosphere. Note that the greenhouse warming depends on the lapse rate. For an isothermal atmosphere (R/cp = 0) there is no greenhouse warming. For fixed optical depth, the greenhouse warming becomes larger as the R/cp, and hence the lapse rate, becomes large. For Venus, the absorbed solar radiation is approximately 163W/m2, owing to the high albedo of the planet. For a pure CO2 atmosphere R/cp « .2304, for which r « .969. Then, the 737K surface temperature of Venus can be accounted for if = 156, which is a very optically thick atmosphere. This is essentially the calculation used by Carl Sagan to infer that the dense CO2 atmosphere of Venus could give it a high enough surface temperature to account for the then-mysterious anomalously high microwave radiation emitted by the planet (microwaves being directly emitted to space by the hot surface without significant absorption by the atmosphere).

Exercise 4.3.1 This exercise illustrates the fact that if the Earth's atmosphere acted like a grey gas, then a doubling of CO2 would make us toast! Using Eq. 4.35, find the rx that yields a surface temperature of 285K for the Earth's absorbed solar radiation (about 270W/m2 allowing a crude correction for net cloud effects). Now suppose we double the greenhouse gas content of the atmosphere. If the Earth's greenhouse gases were grey gases, this would imply doubling the value of rx from the value you just obtained. What would the resulting temperature be? Note that this rather alarming temperature doesn't even fully take into account the amplifying effect of water vapor feedback.

An examination of the radiative heating rate profile for the all-troposphere case provides much insight into the processes which determine where the troposphere leaves off and where a stratosphere will form. We'll assume that = 0 and that the turbulent heat transfer at the ground is efficient enough that Tsa = Tg. Consider first the optically thin limit, for which the grey gas version of Eq. 4.28 is

cos 0 y assuming the stated boundary conditions. Since the radiative heating rate is nonzero, the temperature profile will not be in a steady state unless some other source of heating and cooling is provided to cancel the radiative heating. According to Eq. 4.36, the atmosphere is cooling at low altitudes, where T > Tg/21, i.e. where the local temperature is greater than the skin temperature. The cooling will make the atmosphere's potential temperature lower than the ground temperature, which allows the air in contact with the ground to be positively buoyant. The resulting convection brings heat to the radiatively cooled layer, allowing a steady state to be maintained if the convection is vigorous enough. However, in the upper atmosphere, where T < Tg/21 the atmosphere is being heated by upwelling infrared radiation, and there is no obvious way that convection could provide the cooling needed to make this region a steady state. Instead, the atmosphere in this region is expected to warm until a stratosphere in pure radiative equilibrium forms. Indeed, the tropopause as estimated by the boundary between the region of net heating and net cooling is located at the point where T(p) equals the skin temperature; this is precisely the same result as we obtained in the steady state model of the tropopause for an optically thin atmosphere, as discussed in Section 3.6.

In the optically thick limit it is easiest to infer the infrared heating profile from an examination of the expression for net infrared flux, which becomes

K dp g Cp Kps Ps in the all-troposphere grey-gas case. Recall that this expression breaks down in thin layers within roughly a unit optical depth of the bottom and top boundaries. The formula shows that whether the bulk of an optically thick atmosphere is heating or cooling depends on the lapse rate. The formula is valid even if k depends on pressure and temperature. For constant k, if 4R/cp > 1 the optically thick net flux decreases with height, and most of the atmosphere is heated by infrared radiation, and hence we expect a deep stratosphere and shallow troposphere. If 4R/cp < 1, corresponding to a weaker temperature lapse rate, most of the atmosphere instead experiences infrared radiative cooling, so we expect a deep troposphere. For constant k, most gases would produce a deep stratosphere in the optically thick limit. This conclusion is greatly altered by pressure broadening. If k(p) = (p/ps)n(ps), then the appearance of the pressure factor in the denominator alters the flux profile. Specifically, we now only require that 4R/cp < 2 in order for the flux to increase with height, yielding radiative cooling throughout the depth where the approximation is valid. Most atmospheric gases satisfy this criterion, and therefore most gases ought to yield a deep troposphere in the optically thick limit. Real gases are typically optically thick at some wavenumbers but optically thin at others, and we shall see in Section 4.8 how the competition plays out

Figure 4.2 shows numerically computed profiles of net infrared flux (I+ —1_) without pressure broadening, for a range of optical thicknesses, with R/cp = 2. In this case, 4R/cp > 1, and we expect deep heating in the optically thick limit. For = 50 the profile does follow the optically thick approximate form over most of the atmosphere, and exhibits a decrease in flux with height, implying deep heating. There is a thin layer of cooling near the ground, where the optically thick formula breaks down. When = 10, the flux only conforms to the optically thick limit near the center of the atmosphere; there is a region of infrared cooling that extends from the ground nearly to 70% of the surface pressure. The case t^) — 1 looks quite like the optically thin limit, with the lower half of the atmosphere cooling and the upper half heating. Numerical computations incorporating pressure broadening confirm the predictions of the optically thick

L^etFiuxccompuedJ

0 20 40 60 80 100 120 Net Upward IR Flux (W/m2)

Figure 4.2: Net infrared flux (1+ — 1—) for the all-troposphere grey-gas model, for r^ = 1,10 and 50. In the latter two cases, the dashed line gives the result of the optically thick approximation. The surface temperature is fixed at 300K, and the temperature profile is the dry adiabat with

0 100 200 300 400 Net Upward IR Flux (W/m2)

0 20 40 60 80 100 120 Net Upward IR Flux (W/m2)

Figure 4.2: Net infrared flux (1+ — 1—) for the all-troposphere grey-gas model, for r^ = 1,10 and 50. In the latter two cases, the dashed line gives the result of the optically thick approximation. The surface temperature is fixed at 300K, and the temperature profile is the dry adiabat with

formula. Specifically, the boundary between cooling and heating is unchanged for optically thin atmospheres, but rises to p/ps = .24 for = 10 and to p/ps = .11 for = 50. The profiles are not shown here, but are explored in Problem ??.

The troposphere is defined as the layer stirred by convection, and since hot air rises, buoyancy driven convection transports heat upward where it is balanced by radiative cooling. Therefore, at least the upper region of a troposphere invariably experiences radiative cooling. In the calculation discussed above, the layer with cooling, fated to become the troposphere, occurs in the lower portion of the atmosphere. In Figure 4.2 one notices that the radiative cooling decreases as the atmosphere is made more optically thick, suggesting that tropospheric convection becomes more sluggish in an optically thick atmosphere, there being less radiative cooling to be offset by convective heating. However, one should note also that this sequence of calculations is done with fixed surface temperature, and that the OLR decreases as optical thickness is made larger. Hence, in the optically thick cases, it takes less absorbed solar radiation to maintain the surface temperature of the planet. There is less flux of energy through the system, and correspondingly less convection. Mars, at a more distant orbit than Earth, receives less solar energy; if Mars were given an atmosphere with enough greenhouse effect to warm it up to Earthlike temperatures, one would expect the radiative cooling in its troposphere to be less than Earth's, and one would expect the convection to be more sluggish.

The presence of a stratosphere causes the OLR to exceed the values implied by the all-troposphere calculation, since the upper portions of an atmosphere with a stratosphere are warmer than the all-troposphere model would predict. If the stratosphere is optically thin, it has a minor effect on the OLR; in essence, the all-troposphere OLR formula provides a good estimate if the effective radiating level is below the tropopause. If the stratosphere becomes optically thick, then the OLR is in fact determined by the stratospheric structure. Problem ?? explores some aspects of the effect of an optically thick stratosphere on OLR. Puzzling out the effect of the stratosphere on OLR is rather tricky, because the tropopause height itself depends on the optical thickness of the atmosphere. An optically thin atmosphere obviously can't have an optically thick stratosphere, but an optically thick atmosphere can nevertheless have an optically thin stratosphere if the tropopause height increases rapidly enough with rx. The grey gas radiative cooling profiles discussed above suggest that the stratosphere becomes optically thick when 4R/cp > 1. In contrast, for 4R/cp < 1 the radiatively cooled layer extends toward the top of the atmosphere in the optically thick limit,

164 CHAPTER 4. RADIATIVE TRANSFER IN TEMPERATURE-STRATIFIED ATMOSPHERES and hence the stratosphere could remain optically thin.

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