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## Building a Planet Part 2: Greenhouse Effects

Introduction

In Part 1 a first, simple model of planetary temperature was discussed, all based on knowledge of how much starlight a planet receives ($Q$) and how “reflective” that planet is (i.e, its albedo, $\alpha$, an effect elaborated on in this post).

In the first post, an “effective temperature” was solved for of the form:

$\displaystyle T_{e}= [\frac{Q (1- \alpha)} {\sigma \mu}]^{0.25}$

where $\mu$ is a geometrical redistribution term that accounts for how well the input of stellar energy is evened out across the planet by rotation/thermal inertia and planetary motions. For a sufficiently rotating planet with an advecting atmosphere, it takes the value of ~4, but it would be more appropriate to take on a different local value on an airless body incapable of transporting much heat around (concerning the question of habitability, this could set up a regime in which water could exist in liquid form over parts of a planet but not others).

We now introduce the effect of an atmosphere that can interact with the radiation entering or exiting the planet. On Earth, that interaction is predominately in the infrared (the outgoing energy) via absorption/emission processes and is accounted for by trace gases in the atmosphere, which we refer to as greenhouse gases (water vapor, carbon dioxide, methane, nitrous oxide, etc); the interaction can also occur with aerosol particles in the atmosphere and with clouds, though these latter two also tend to scatter shortwave solar energy and cause a net cooling effect on Earth (there are exceptions to this, such as black carbon). Ozone interacts in the infrared as a greenhouse gas, but also absorbs incoming UV radiation high in the stratosphere to set up a persistent thermal inversion where temperatures increase with height.

On Venus, the predominant atmospheric constituent is CO2 which acts as a strong greenhouse agent, although contributions exist from sulfur dioxide, water vapor, and sulfuric acid based clouds. On Titan, one of Saturn’s moons, a greenhouse effect is arises due to N2, CH4, and H2. It is a widespread notion that symmetric, diatomic molecules cannot behave as greenhouse gases (as is the case with both main constituents of the Earth’s atmosphere- N2 and O2). In order to be a good infrared absorber, the molecule must 1) have quantum energy transitions whose energy corresponds to the infrared spectrum 2) a dipole moment, or a charge distribution such that a disproportionate amount of the electron clouds “negative charge” is clumped up to one side and the “positive charge” to another side. Symmetric, diatomic molecules do not exhibit a static electric dipole moment (as with water vapor) nor is there the possibility to vibrationally induce a temporary dipole moment, as in the case of CO2, where the “bending” mode allows for interaction of thermal radiation at ~15 micron wavelength.

However, in sufficiently dense atmospheres (as on Titan and on gas giant planets) collision-induced absorption leads to absorption features in diatomic gases. The collision-induced dipole that forms from colliding molecules forms transitorily, but can lead to broad absorption bands. Collision-induced absorption dominate the far-infrared spectra of Jovian planets. This is a direct radiative effect, but the diatomic molecules also have an influence in broadening the absorption lines of the typical greenhouse substances in our atmosphere. This pressure broadening effect becomes more important as the gas pressure increases, since collisions will be more abundant. The lack of any substantial atmosphere on Mars (1/100th the pressure of Earth’s atmosphere), despite consisting mostly of CO2, cannot generate a strong greenhouse effect because of this.

Figure 1 shows a typical thermal radiation spectrum emitted by the Earth (as seen from a viewer in outer space).

Figure 1) a. Earth spectrum in the thermal infrared showing upwelling energy, or outgoing longwave radiation (OLR) as seen from space b. Transmission as a function of wavelength. The absorption due to different compounds is shown.

The colored curve corresponds to the emission of Earth; the solid, light curves in the background correspond to the emission that would emanate from a “blackbody” (essentially a perfect radiator). Radiation is emitted across a spectrum of wavelengths, and the intensity at all wavelengths increases as the temperature is made higher (the family of curves shown on the diagram is the wavelength distribution of radiant intensity at various temperatures). The relevant equation to describe this relationship is the Planck law which can be integrated over all wavelengths and directions to yield $\sigma T^4$ .

In regions where the atmosphere is transparent to thermal radiation (e.g., between 8-12 microns, with the exception of ozone) the radiation is seen (from space) to emerge from the warm surface. In contrast, at opaque wavelengths (e.g, 15 microns) the radiation from the surface is absorbed by the atmosphere and will be shielded by the viewer in space. Radiation that interacts strongly with the atmosphere can only exit to space from the upper, thin layers of the atmosphere where it is quite cold. This effect is manifested as a “dip” in spectrum at those opaque wavelengths, which of course must physically correspond to a reduction in OLR at that temperature. In fact, regions of low OLR in the tropics for instance are often used as a proxy for areas of deep convection in satellite analyses, since the low thermal radiation comes from the cold, clouds tops. Figure 1b contains the same information, except showing the transmissivity ($t$) as a function of wavelength, which approaches zero as the opacity becomes large. $t$ can be related to a quantity called optical depth, $\tau$, which is a measure of the opacity. It depends on the density of the absorber and the pressure interval upon which a beam of light is traveling. At normal angle incidence, $t = exp(-\tau)$ where we remember that both quantities are wavelength-dependent.

Basic Radiative Transfer

Let us begin with the Earth-like case in which we add a substance, such as CO2, that is opaque in the infrared but transparent to incoming solar radiation. This addition of greenhouse gas will reduce the planetary outgoing longwave radiation (OLR) as seen from an observer in space. We define a greenhouse parameter, $G$:

$\displaystyle G = \sigma T_{s}^4 - OLR$

and now, the surface temperature becomes:

$\displaystyle T_{s}= [T_{e}^4 + \frac{G} {\sigma}]^{0.25}$

Figure 2 below shows the OLR of a planet that radiates as a blackbody (in the black line) according to $OLR = \sigma T^4$ and the red curve shows the same situation except with 400 parts per million (ppm) of carbon dioxide added to the atmosphere. The horizontal curve is a constant Earth-like value for the absorbed solar radiation, $Q(1-\alpha)/4$. The equilibrium temperature must correspond to the intersection of these curves. The addition of the greenhouse substance reduces the OLR at any given temperature, since some portion of the energy is now being blocked; alternatively, the intersection point must occur at a higher temperature.

Figure 2: Relationship between OLR and surface temperature for a blackbody (black curve) and with 400 ppm CO2 (red curve). The horizontal line is the absorbed solar radiation.

We can formulate the vertical temperature profile as a function of optical depth, $T(\tau)$, employing useful approximations. We will use the so-called Eddington approximation for a “grey” atmosphere (grey meaning that the absorption is wavelength-independent, which is clearly unrealistic, but a useful starting point for conceptualizing the problem). We have,

$\displaystyle T^4(\tau) = T_{o}^4 [1 + \frac{3 \tau}{2}]$
and, we can write the ground temperature, $T_{s}$, as function of the planetary emission temperature ($T_{e}$ as defined in the first link),

$\displaystyle T_{s}^4 = T_{e}^4 [1 + \frac{3 \tau*}{4}]$

where $\tau*$ is the column infrared grey optical depth. $T_{o}$ is the air temperature at $\tau = 0$, or the skin temperature of the planet.

When greenhouse gases are added to our atmosphere, the column optical-depth increases and the $\tau$ profile moves upward. Suppose, for instance, that the atmosphere was optically thick throughout the infrared and eventually became optically thin enough at some high altitude, such that the observer in space could only see to the $\tau=1$ level. The observer would be blind to all events happening below $\tau=1$ , much in the same way as we cannot “look into” the sun’s outer photosphere very far (as all radiation has been absorbed before exiting into space). As greenhouse gases are added, the height of the $\tau=1$ surface will move up, such that radiation does not escape to space until it reaches a higher altitude than before. Of course, where a $\tau$ surface is located is wavelength dependent. On the gas planets, it is not often desirable to know about the entire atmosphere right down to the interior of the planet, so for many purposes it is sufficient to consider radiation down to the point where the fluid becomes dense enough that it radiates like a blackbody. This acts like a “surface” (just in the same way that you don’t need to know the temperature profile right down to the core of the Earth in order to do atmospheric radiation).

It is typical that the radiative equilibrium profile described above introduces a strong temperature discontinuity between the surface and overlying air column. This results in convection that transports heat from the surface to overlying atmosphere. The concept of radiative-equilibrium is explored, for example, in Manabe and Wetherald, 1967 (see also Isaac Held’s useful summary). Typically, if the vertical temperature discontinuity becomes too large, we think of the atmospheric temperature profile being relaxed to some critical value called the lapse rate via convection, which has a value of $\Gamma = -g/c_{p} = -9.8 K/km$ on Earth (where cp is a specific heat value). The actual lapse rate on Earth tends to be closer to a moist adiabat, which is less steep (-6.5 K/km or so) due to the latent heat of condensate being released which partially offsets the cooling induced by a rising parcel of air as it expands under decreased pressure.

The following diagram, from Manabe and Strickler, 1964 (Figure 3 here, Figure 4 from the paper) shows a radiative-equilibrium profile along with a a dry and moist adiabatic lapse rate typical of the convecting part of Earth’s atmosphere. Lapse rates of this sort emerge in other planetary atmospheres as well.

Anti-Greenhouse Effects

It is possible to modify the picture somewhat by including the effects of solar absorption in the atmosphere. Earth’s atmosphere is ~80% transparent to incoming sunlight, though even the energy deposited throughout the troposphere doesn’t substantially impact the above argument, as the whole column is yoked together by convection, requiring us to think about the energy budget of the whole surface+troposphere column. However, it is possible to substantially decouple the surface from the atmosphere with enough atmospheric solar absorption, or to absorb the sunlight high enough (without impacting the reflection) such that there is no communication with the surface. On Titan, this “anti-greenhouse effect” partially compensates for the traditional greenhouse influence. It arises from the absorption of sunlight by haze and CH4 in the upper atmosphere (e.g., McKay et al (1991)). This haze is opaque at visible wavelengths but is virtually transparent to thermal wavelengths, in contrast to the greenhouse case of familiarity. This is also similar to the “nuclear winter” problem, and has also been proposed to be important on early, Archean Earth.

Following McKay et al (1999), with this type of solar absorption acting, the surface temperature can be expressed as:

$\displaystyle \sigma T_{s}^4 = \frac{Q (1-\alpha) (1-\gamma)}{4}[1 + \frac{3\tau*}{4}] + \frac{\gamma Q(1-\alpha)}{8}$

where $\gamma$ is a measure of the anti-greenhouse effect strength (i.e., the portion of solar energy blocked by the anti-greenhouse layer but not reflected back to space). In the extreme case where there is only a strong anti-greenhouse effect acting ($\gamma \rightarrow 1$) with no infrared opacity ($\tau* \rightarrow 0$), then the surface actually becomes colder than the emission temperature of the planet by a factor of (1/2)0.25. This can be imagined by envisioning a layer between the surface and space; the high-altitude layer absorbs all the solar energy, upon which it then emits half back to space and half to the surface as thermal radiation. The surface then radiates all of the emission to space, thus receiving an amount of radiation half of the incident value but returning it all to space.

One could also make the atmosphere isothermal via solar absorption, in which case the greenhouse effect diminishes due to the lack of temperature differential between the surface and top of the atmosphere (recalling the definition of $G$). Physically, one can see this based on the infrared spectrum plotted in Figure 1. If the upper atmosphere radiated to space at the same temperature as the surface, there would be no “dips” in the spectrum and no need for the atmosphere to heat up. When a “dip” occurs, it represents a portion of energy that would have otherwise escaped to space but no longer is. This means the planet is now taking in more energy than it is losing. That lost energy “removed” from the total OLR must be accompanied by an increase in column temperature. This increases the emission at other wavelengths by an amount equal to the area of the “bite.”

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### 2 Responses

1. on June 12, 2012 at 2:18 pm | Reply Edward Greisch

The math does not copy to NeoOffice.

• Edward- Sorry about this, but I’m not really sure what I can do about it. wordpress uses latex to implement math (but if anyone knows more about computers than me chime in). I’ve viewed this webpage successfully on a few computers with a few different web browsers (internet explorer, mozilla, safari) with no problems.