PhySci 134: Planetary Radiation Budget and the 1-Layer Greenhouse Effect

Gidon Eshel
401 Hinds
Dept. of the Geophysical Sciences,
5734 S. Ellis Ave., The Univ. of Chicago,
Chicago, IL 60637
(773) 702-0440, geshel@uchicago.edu

The simple principle on which this discussion is based is the requirement for a balanced heat budget at every point in space at all times,

\begin{displaymath}{\rm incoming\; energy}+{\rm outgoing\; energy}+{\rm storage} = 0.\end{displaymath}

One process the incoming and outgoing heat flux terms represent is the effect of vertical radiative fluxes (for example, heating by incoming solar radiation). In addition, in the real climate system these terms may represent many other processes, such as advective heat fluxes (cooling or heating by winds, such as, e.g., the cooling you feel while enjoying an ocean breeze in a hot summer day). For this discussion, however, we will consider only vertical radiative fluxes. The storage term represent heating or cooling of the environment in response to an unbalanced heat budget. For example, in the early morning in the summer, the ground is relatively cool, but subject to strong warming by solar radiation. Thus the local heat budget is one of a surplus; more energy is coming in than leaving the surface. In response, the ground heats up, thus storing more heat. This is what the storage term represents. The rate of heating or cooling in the presence of a given heat imbalance is not uniform, but rather depends on the properties of the environment. For example, a given surface heat flux imbalance will induce a much smaller temperature change over the ocean than over land, because the heat capacity of water is much larger than that of land. Regardless of its heat capacity, however, any environment can tolerate heat imbalances only very briefly; sooner or later other processes come into play, and restore balance. (For example, in the above case of the summer morning, the ground will quickly reach a high enough temperature that its own outgoing thermal (longwave) radiation will balance the incoming solar radiation, thus arresting the temperature increase.) Given this, we can simplify our discussion by assuming the surface has zero heat capacity. This is tantamount to simply disregarding the brief periods of heating and cooling, considering only the steady states achieved when

\begin{displaymath}{\rm incoming\; energy}=-\;{\rm outgoing\; energy}.\end{displaymath}


  
Figure 1: A schematic of the simplest model of planetary heat budget. The yellow Sl arrow represents the local incoming solar flux, while terrestrial longwave radiation is given in redish. See text for details.

Let's first consider the simplest possible case, of no atmosphere and $\alpha=0$ ($\alpha $ being the albedo, a measure of the fraction of the total incoming solar radiation that is reflected to space without ever entering the system). In this case ${\rm incoming}=-\;{\rm outgoing}$ requires that

\begin{displaymath}S_l=\sigma T_g^4\;\;\;\;\;\;\Rightarrow\;\;\;\;\;\; T_g=\sqrt[4]{{{S_l}\over{\sigma}}}\;\;{\rm in\;\;K}\end{displaymath}

where
  
Figure 2: A schematic of the meaning of Sl, the local solar input.

WHAT IS SL? As Fig. 2 shows, Sl is the solar input actually received by a given latitude. Because of the earth's tilted axis, Sl depends on the `latitude' with respect to the solar zenith angle ($\theta$ in the Figure), and not the geographical latitude, i.e.,

\begin{displaymath}S_l(\theta)=S_\circ {\rm cos} \theta,\end{displaymath}

where $S_\circ\approx 1380$ W m-2 is the solar `constant'. Because of the day/night cycle, and the fact that the solar influx may not assume negative values, we need to modify the above in order to hold throughout,

\begin{displaymath}S_l(\theta)=S_\circ {\rm max}\left(0,{\rm cos} \theta\right),\end{displaymath}

or

\begin{displaymath}T_g(\theta)=\sqrt[4]{ {{S_\circ {\rm max}\left(0,{\rm cos} \theta\right)}\over{\sigma}}}.\end{displaymath}


  
Figure 3: The equivalence of the sum of latitude-dependent insolation over the day half-sphere and the sum of latitude-independent $S_\circ $ over the disk with the same radius as the sphere.

BUT, at any given time, the sun shines only on half the planet, while terrestrial radiation is emitted outward from the entire Earth's surface. Fortunately, the total insolation received by the day half-sphere is exactly the same as the insolation that would have been received by a disk of the same radius, as Fig. 3 shows. Consequently, the total insolation received by the whole planet at a given time is $S_\circ\pi R^2$, where R is the Earth's radius. The outgoing longwave flux is given by $4\pi R^2\sigma T_g^4$. If we assume steady state (zero heat capacity), as before, the two ought to cancel out each other, i.e.,

\begin{displaymath}4\pi R^2\sigma T_g^4=S_\circ\pi R^2 \;\;\;\;\;\Rightarrow\;\... ...\Rightarrow\;\;\;\;\; T_g=\sqrt[4]{{{S_\circ}\over{4\sigma}}}.\end{displaymath}

Substituting the appropriate numerical values,

\begin{displaymath}T_g=\sqrt[4]{{{1380\;{\rm W}\;{\rm m}^{-2}}\over{4\times5.7\c... ... {\rm W}\;{\rm m}^{-2}\;{\rm K}^{-4}}}}\approx 279 \;{\rm K},\end{displaymath}

or $\sim6^\circ$C, or $\sim43^\circ$F. Until now we have assumed that there is no loss of incoming solar radiation whatsoever. This is, however, not so realistic, because in actuality some non-trivial fraction of the incoming solar radiation is reflected back to space right at the top of the atmosphere, before it had a chance to warm any element of the climate system. To crudely account for this effect, let's introduce now a nonzero albedo, $\alpha=0.3$, which means that only 70$\%$ of the solar radiation incident on the climate system is in fact absorbed by that system. In this case,

\begin{displaymath}4\pi R^2\sigma T_g^4=S_\circ(1-\alpha)\pi R^2 \;\;\;\;\;\Rightarrow\;\;\;\;\; 4\sigma T_g^4=S_\circ(1-\alpha),\end{displaymath}

or

\begin{displaymath}T_g=\sqrt[4]{{{S_\circ(1-\alpha)}\over{4\sigma}}} \approx 255\;{\rm K},\end{displaymath}

or $\sim -18^\circ$C, or $\sim 0^\circ$F. Thus a physically sensible albedo makes a huge difference.
  
Figure 4: The setup for an infinitely thin atmosphere that does not absorb solar radiation. Solar (shortwave) fluxes are shown in yellow, while longwave fluxes are purple. For technical reasons the albedo is denoted by a, not $\alpha $ as in the text.

NEXT, LET'S ADD AN ATMOSPHERE, but keep it simple by assuming it is thin and isothermal, and absorbs no solar radiation, as shown schematically in Fig. 4. First, consider the atmosphere's heat budget. Since we assumed no solar absorption, this budget comprises only longwave terms: the atmosphere is heated from below by the upwelling terrestrial longwave radiation, and is cooled by its own longwave loss to space (affectionately known in the trade as OLR, outgoing longwave radiation). The balance of these 2 terms is given by

\begin{displaymath}2\sigma T_a^4=\sigma T_g^4,\end{displaymath}

where Ta is the atmosphere's temperature, and we need not worry about geometrical terms [because both fluxes take place throughout their respective spheres, and those spheres' different radii are within 1 tenth of a percent of each other ($\sim 10$ km out of $\sim 6370$), i.e., they are well approximated as having the same radius.] In the above, the left hand side accounts for radiative cooling of the atmosphere, which happens both upward and downward at the same rate, hence the factor of 2. The right hand side quantifies the warming of the atmosphere from below by upwelling terrestrial radiation. The above equation yields

\begin{displaymath}T_a={{T_g}\over{\sqrt[4]{2}}} \approx 0.84 \;T_g.\end{displaymath}

This is monumentally important; the ground is warmer than the temperature at which the planet shines to space (the so-called brightness temperature) by a factor of $\sqrt[4]{2}\approx 1.19$. This factor is none other than the greenhouse effect of a 1-layer atmosphere! Now you have a better quantitative idea what the greenhouse effect is. If this appears too small to matter (after all, isn't $1.19\approx 1$?), recall the large number that the Earth's temperature amounts to in the Kelvin scale. For example, $T_a=260\;{\rm K}$ corresponds to $T_g\approx 309$ K, a very substantial difference that is similar in magnitude to the summer-winter temperature difference in Chicago!! The surface heat balance is given by

\begin{displaymath}\overbrace{{{1}\over{4}}S_\circ\left(1-\alpha\right)+\sigma T_a^4}^{\rm incoming}=\overbrace{\sigma T_g^4}^{\rm outgoing}\end{displaymath}

where the factor of 1 quarter is due to the geometry of the problem. Simplifying, we get

\begin{displaymath}{{S_\circ}\over{2\sigma}}\left(1-\alpha\right)=T_g^4 \;\;\;\... ..._g=\sqrt[4]{ {{S_\circ\left(1-\alpha\right)}\over{2\sigma}} }.\end{displaymath}

With top of the atmosphere solar flux $S_\circ=1380\;{\rm W}\;{\rm m}^{-2}$ and albedo $\alpha=0.3$, as before,

\begin{displaymath}T_g=\sqrt[4]{{{1380\;{\rm W}\;{\rm m}^{-2}\times 0.7}\over{2\... ...\;{\rm W}\;{\rm m}^{-2}\;{\rm K}^{-4}}}} \approx 303\;{\rm K},\end{displaymath}

while

\begin{displaymath}T_a=\sqrt[4]{{{1380\;{\rm W}\;{\rm m}^{-2}\times 0.7}\over{4\... ...\;{\rm W}\;{\rm m}^{-2}\;{\rm K}^{-4}}}} \approx 255\;{\rm K},\end{displaymath}

as before. Thus by simply introducing into the problem an atmosphere whose only contribution is to absorb upwelling terrestrial longwave radiation, we have elevated the ground temperature by roughly 48 K (or $86^\circ$F) a huge change indeed.
  
Figure 5: The situation with an imperfectly-absorbing atmosphere. For simplicity, the sphericity of the problem is suppressed, and enters the problem implicitly through the 0.25 factor modifying $S_\circ $, the top of the atmosphere solar flux.

UP TO NOW we have considered our skinny atmosphere to be a perfect absorber of upwelling terrestrial longwave radiation. Unlike other assumptions we have made along the way, this is not such a great one. The real atmosphere is in fact a partial absorber, not a perfect one. Let's now explore the effect this has on the planetary heat budget, and, in particular, on surface temperatures. To do that, let's introduce a new parameter, call it $\beta$, that quantifies just how effective the atmosphere is at absorbing longwave radiation (i.e., $\beta$ is the atmospheric longwave `leakiness' parameter). That is, if the atmosphere is radiated at from below at a rate of $\sigma T_g^4$, it absorbs (and thus heated by) only $\sigma T_g^4(1-\beta)$. The situation is shown in Fig. 5. With this configuration, the atmospheric heat budget is given by

\begin{displaymath}\overbrace{\sigma T_g^4(1-\beta)}^{\rm incoming}= \overbrace{2\sigma T_a^4}^{\rm outgoing}, \end{displaymath}

or, after some trivial gymnastics,

\begin{displaymath}T_a=\sqrt[4]{ {{1-\beta}\over{2}}}\;T_g.\end{displaymath}

To examine this result, let's first do what physicists always do with a new result, namely test its behavior under various relevant limits. Recall that before, when we assumed the atmospheric longwave absorption to be perfect, the corresponding expression was

\begin{displaymath}T_a=\sqrt[4]{ {{1}\over{2}}}\;T_g.\end{displaymath}

Thus the first requirement we insist our new result must satisfy is that as $\beta\rightarrow 1$ (i.e., as the atmosphere approaches zero longwave absorption, the situation we previously treated as `no atmosphere'), $T_a\rightarrow 0$, an indication that in such a case there is no greenhouse warming of the surface. Examining our new expression shows that it indeed satisfies this condition ( $\lim_{\beta\rightarrow 1} [(1-\beta)2^{-1}]^{1/4}=0$), which is reassuring. Next, let's consider the case of perfect longwave atmospheric absorption ( $\beta\rightarrow 0$) we have considered before. In this limit, our new expression approaches the old one, namely

\begin{displaymath}\lim_{\beta\rightarrow 0}\sqrt[4]{{{1-\beta}\over{2}}}= \sqrt[4]{{{1}\over{2}}},\end{displaymath}

which is what we required.
  
Figure 6: The dependence of ground temperature on albedo (vertical axis) and atmospheric longwave absorptivity (horizontal axis). the lower panel shows the values for 3 different albedo values (i.e., it shows a left-to-right cross-section of the upper panel at $\alpha =0.25$, $\alpha =0.50$ and $\alpha =0.75$).

Next, let's write down the ground's heat budget,

\begin{displaymath}\overbrace{{{1}\over{4}}S_\circ\left(1-\alpha\right)+\sigma T_a^4}^{\rm incoming}=\overbrace{\sigma T_g^4}^{\rm outgoing}\end{displaymath}

or

\begin{displaymath}{{1}\over{4\sigma}}S_\circ\left(1-\alpha\right)+ {{1-\beta}\over{2}} T_g^4= T_g^4.\end{displaymath}

After simplification, this becomes

\begin{displaymath}T_g=\sqrt[4]{ {{S_\circ\left(1-\alpha\right)}\over{4\sigma\left(1- {{1-\beta}\over{2}}\right)}}},\end{displaymath}

and the atmosphere's temperature is given by

\begin{displaymath}T_a=\sqrt[4]{ {{1-\beta}\over{1+\beta}} {{S_\circ}\over{4\sigma}} \left(1-\alpha\right)}.\end{displaymath}

The dependence of ground temperature Tg on albedo and atmospheric longwave absorptivity is given in Fig. 6. If you examine the blue curve in Fig. 6's lower panel (which represents a reasonable approximation of the Earth), you can immediately see that atmospheric absorptivity is crucially important. For example, changing $\beta$ from 0.4 to 0.6 yields a ground temperature change of about 8.5K. If this does not impress you, note that surface temperatures during the peak of the last glacial maximum, some 20 thousand years ago, were roughly 3-16K colder than now.