Pure radiative equilibrium for a grey gas atmosphere

For the temperature profiles discussed in Sections 4.3.2 and 4.3.3, the net infrared radiative heating computed from Eq. 4.14 is nonzero at virtually all altitudes; generally the imbalance acts to cool the lower atmosphere and warm the upper atmosphere. In using such solutions to compute OLR and back-radiation, we are presuming that convective heat fluxes will balance the cooling and keep the troposphere in a steady state. The upper atmosphere will continue to heat, and ultimately reach equilibrium creating a stratosphere, but in the all-troposphere idealization we presume that the stratosphere is optically thin enough that it doesn't much affect the OLR.

Now, we'll investigate solutions for which, in contrast, the net radiative heating vanishes individually at each altitude. Such solutions are in pure radiative equilibrium,, as apposed to radiative-convective equilibrium. First we'll consider the case in which the only radiative heating is supplied by infrared; later we'll bring heating by atmospheric solar absorption into the picture. Pure radiative equilibrium is the opposite extreme from the all-troposphere idealization, and tells us much about the nature of the stratosphere, and the factors governing the tropopause height.

Assuming the atmosphere to be transparent to solar radiation, pure radiative equilibrium requires that the frequency-integrated longwave radiative heating H vanish for all t . From the grey gas version of Eq. 4.14, we then conclude that 1+ — 1— is independent of t. Applying the upper boundary condition, we find that this constant is I+(tto), which is the OLR. Now, by taking the difference between the equations for I+ and I_ we find

ar which gives us the temperature in terms of (I+ +1_). Next, taking the sum of the equations for I+ and I_ yields dT(I+ + I_) = -(I+ - I_) (4.41)

This is easily solved by noting that — (I+ — I_) = const = -OLR. In consequence,

where we have again used the boundary condition at tto. This expression gives us the pure radiative equilibrium temperature profile T(t). In pure radiative equilibrium, the temperature always approaches the skin temperature at the top of the atmosphere, where t = tto. This recovers the result obtained in the previous chapter, in Section 3.6. When the atmosphere is optically thin, — t is small throughout the atmosphere, and the entire atmosphere becomes isothermal with temperature equal to the skin temperature. When the atmosphere is not optically thin, the temperature decreases gently with height, approaching the skin temperature as the top of the atmosphere is approached.

Eq. 4.42 also gives us the upward and downward fluxes, since we now know both I+ — I_ and I+ +1_ at each t. In particular, the downward flux into the ground is

I_(0) = 1((I+ + I_) — (I+ — I_)) = 2((1 + t^)olr — OLR) = 1 t^olr (4.43)

For an optically thin atmosphere, the longwave radiation returned to the ground by the atmosphere is only a small fraction of that emitted to space. As the atmosphere becomes optically thick, the radiation returned to the ground becomes much greater than that emitted to space, because the radiative equilibrium temperature near the ground becomes large and the optical thickness implies that the radiation into the ground is determined primarily by the low level temperature. If we assume the planet to be in radiative equilibrium with the absorbed solar radiation (1 — a)S, where a is the albedo of the ground, then OLR = (1 — a)S and the radiative energy budget of the ground is aT44 = (1 — a)S + I_(0) = (1 — a)S • (1 + 1 tto) (4.44)

where Ts is the surface temperature. This, together with the temperature profile determined by Eq. 4.42, determines what the thermal state of the system would be in the absence of heat transport mechanisms other than radiation. For an optically thin atmosphere, the surface temperature is only slightly greater than the no-atmosphere value. As becomes large, the surface temperature increases without bound. Note that, while this formula yields a greenhouse warming of the surface, the relation between surface temperature and is different from that given by the all-troposphere radiative convective calculation in Eq. 4.35, because the pure radiative equilibrium temperature profile is different from the adiabat which would be established by convection.

Let's now compare the surface temperature with the temperature of the air in immediate contact with the surface. From Eq. 4.42 we find that the low level air temperature is determined by aT(0)4 = (1 — a)S • (± + 2) Taking the ratio,

Thus, the surface is always warmer than the overlying air in immediate contact with it. In the previous chapter, we saw that this was the case for pure radiative equilibrium in an optically thin atmosphere, but now we have generalized it ot arbitrary optical thickness. In the optically thin limit, the formula reduces to our earlier result, T(0) = 2-1/4Ts. In the optically thick limit, Ts — T(0) = 4Ts/t0, whence the temperature jump (relative to surface temperature) falls to zero as the atmosphere is made more optically thick. As we already discussed in Section 3.6, cold air immediately above a warmer surface constitutes a very unstable situation. Under the action of diffusive or turbulent heat transfer between the surface and the nearby air, a layer of air near the surface will heat up to the temperature of the surface, whereafter it will be warmer than the air above it. Being buoyant, it will rise and lead to convection, which will stir up some depth of the atmosphere and establish an adiabat - creating a troposphere.

In pure radiative equilibrium, the surface heating inevitably gives rise to convection. However, it is also possible that the temperature profile in the interior of the atmosphere may become unstable to convection, even without the benefit of a surface. This is a particularly important possibility to consider for gas giant planets, which have no distinct surface to absorb solar radiation and stimulate convection. To determine stability, we must compute the lapse rate dT/dp in radiative equilibrium, and see if it is steeper than that of the adiabat (moist or dry) appropriate to the atmosphere. Taking the derivative of 4.42 with respect to optical thickness, we find jrp

whence, using d/dp = (dT/dp)(d/dT), we find d ln T 11 dT ,

Stability is determined by comparing the slope of the adiabat to the radiative-equilibrium slope we have just computed. For the dry adiabat, the atmosphere is stable where

The factor p appearing on the right hand side of this equation guarantees that the upper portion of the atmosphere will always be stable, unless dT/dp blows up like 1/p or faster as p ^ 0. Moreover, optically thin atmospheres are always stable throughout the depth of the atmosphere. This is so because the denominator is close to unity in the optically thin limit, while —pdT/dp = Kp/(g cos theta) < t0 ^ 1. Since optically thin atmospheres are nearly isothermal in pure radiative equilibrium, it is hardly surprising that they are statically stable.

In the case of constant absorption coefficient k, we have pdT/dp = —Kp/g cos 0, which is just t — T0. Thus, the stability condition becomes

The right hand side has its maximum at the ground t = 0, and the maximum value is | t0/(1+t0). The more optically thick the atmosphere is, the more unstable it is near the ground. For large optical thickness, the stability criterion becomes the remarkably simple statement 4R/cp > 1. Dry Earth air, with R/cp = 2/7, just misses being unstable by this criterion, and pure noncondensing water vapor is almost precisely on the boundary. Pure noncondensing CO2, NH3 and CH4 just barely satisfy the condition for instability near the ground when the atmosphere is optically thick.

Typical atmospheres, however, will be more unstable than the constant k calculation suggests. As will be explained in Section 4.4 ,collisional broadening typically causes the absorption coefficient to increase linearly with pressure, out to pressures of several bars. With collisional broadening, k(p) = K(ps) ■ (p/ps) for a combination of well-mixed greenhouse gases with pressure-independent concentrations. In this case —pdr/dp = 2tto at the surface, so that the dry adiabatic stability condition in the optically thick limit becomes 2R/cp > 1. The extra factor of 2 compared to the case without pressure broadening destabilizes all well-mixed atmospheres, provided they are sufficiently optically thick near the ground. The maximum value of the stability parameter occurs for mono-atomic gases like Helium, which has 2R/cp = .8 and easily meets the criterion for instability.

Other processes can destabilize the atmosphere as well. For example, on the moist adiabat the slope —d ln T/dlnp is always less than the dry adiabatic slope, which deepens the layer within which the radiative-equilibrium atmosphere is unstable. In addition, a sharp decrease of optical thickness with height tends to destabilize the atmosphere, particularly if it occurs in a place where the atmosphere is optically thick. This happens whenever there is a layer where the concentration of absorbers decreases strongly with height. Because of Clausius-Clapeyron, this situation often happens in the lower portions of atmospheres in equilibrium with a reservoir of condensible greenhouse gas (water vapor on Earth or Methane on Titan, for example). In this case, the con-densible substance has a destabilizing effect through its influence on the radiative equilibrium, as well as through its effect on the adiabat. Condensed water is a good infrared absorber, so radiative equilibrium drives cloud tops to be unstable. In contrast, the atmosphere is stabilized where the infrared absorber concentration increases strongly with height; this situation is less typical, but can happen at the bottom of a water cloud.