Climate Change/Science/Atmospheric Balance

One may wonder where exactly is the top of the atmosphere, and with good reason. We know that the atmosphere consists primarily of the gaseous envelop around Earth, and that pressure decreases with height, according to they hydrostatic approximation. Does the atmosphere end only when the pressure reaches a vanishingly small value? No, but there is not a good definition of the top of the atmosphere, and it changes with sub-discipline. For our purposes, we can usually take the top of the atmosphere, often abbreviated TOA, as somewhere in the low to mid-stratosphere, or even simply the tropopause. In this chapter, we can imagine it is the level at which the downward shortwave radiative flux is negligibly different from the solar constant and where there is negligible downward longwave flux (that due to the sun, which is small).

Now that we have an idea of what the TOA is, we can ask why it might be useful.

First consider conservation of energy in an equilibrium system. This could be a tank of water with a heating lamp above it all enclosed in a box. It could be a simple blackbody system, or any isolated system. Conservation of energy means that the total amount of energy does not change, which is equivalent to saying that any energy that is input to the system must be balanced by an outward flux of energy. In the case of Earth (as the "system"), this means that the energy coming in (the sunlight) must be balanced by outgoing radiation. We know the solar constant ($$1367 W m^{-2}$$), so if we integrate over the Earth's surface, we know how much incoming energy there is. This incoming energy, sometimes called solar insolation or downward shortwave radiation, needs to be balanced. Why? Well, if it is not balanced by an equal loss of energy, then the temperature of the system must change (this is the 1st law of thermodynamics). Wien's law tells us that the wavelength of peak emission from a blackbody is inversely related to the temperature, and for normal Earth-like temperatures that puts the emission in the infrared part of the electromagnetic spectrum. Thankfully, this light is invisible to humans, and because the wavelength is longer than visible light (solar or shortwave), the terrestrial infrared radiation is often referred to as longwave radiation. The amount that is radiated to space (which differs from that emitted by the surface because of the greenhouse effect) is often called outgoing longwave radiation (OLR). The OLR (which is equal to the net longwave at TOA) balances the net shortwave at the top of the atmosphere when the system is in equilibrium.

Is the net shortwave radiation at TOA equal to the incoming shortwave? The answer is no. The net shortwave, which when averaged over suitable time and over the global, is the source of energy to the climate system, but not all the solar insolation is absorbed by the earth. Let's not beat around the bush. What happens to incoming solar radiation when it arrives in the atmosphere? There are really just three paths a photon (a "particle" of light) can take. First it can be absorbed, either in the atmosphere or at the surface. Absorption means that the energy associated with the photon is imparted to some atom or molecule, resulting in a higher energy level in that particle. Second, the photon can be reflected, which means that the path of the photon is reversed. More generally, we should say that the photon can be scattered, with some probability of being scattered back in the direction it came from, but we do not need to deal with scattering right here. Third, the photon can continue unimpeded, ultimately reaching the surface and being absorbed or reflected; while the photon is traveling through a medium without interacting, it is said to be transmitted. To study climate, one need not (usually) worry about individual photons, but the effects of the light in aggregate. Since we now know what can happen to each photon individually, we can sum over all the photons that make up the solar insolation such that $$F_\downarrow = A + R + T$$, where F is the total downward shortwave flux, A is the fraction of light absorbed by the atmosphere, R is the fraction reflected back to space before reaching the surface, and T is the light transmitted to the surface.

From our understanding of the downward shortwave flux, we can continue the analysis by considering the surface. The amount of light absorbed by the surface is not exactly equal to the transmitted light, T. Why? Well, the surface can be highly reflective. For example, snow and ice reflect up to 80% of incident light, while open ocean surfaces reflect almost none. The reflectivity of the surface is usually called the albedo, denoted $$\alpha$$, and is simply the fraction of incident light that gets reflected. Knowing that the surface has a given albedo, we can now say that the amount of light absorbed at the surface must be equal to $$F_{\mathrm{absorbed}} = (1 -\alpha)^{-1} (F_\downarrow - A - R)$$. This says the absorbed light at the surface is equal to the transmitted insolation that is not reflected by the surface. Note that the albedo is constrained by definition to always be between 0 and 1, with typical global average of about 0.3.

It should also be noted that the shortwave light reflected by the surface does have a chance of being reflected (by clouds or particulate matter) or absorbed by atmospheric constituents. However, for most discussions of climate, and for the purposes here, we will neglect this process. Furthermore, we can (to a reasonable approximation) assume that the atmosphere is transparent to shortwave radiation, meaning there will be no absorption. This simplifies our previous expressions by eliminating the term A. To further simplify our notation, we can say that the total "planetary albedo" is the sum of the atmosphere albedo (later will will call this the cloud albedo) and the surface albedo, $$\alpha_p = \alpha_s + \alpha_c$$. These simplifications allow us to write the "net shortwave at TOA" as $$F_{net} = F_\downarrow - F_\uparrow = (1 - \alpha_p)F_\downarrow$$

As mentioned above, the radiative flux from the surface acts approximately according to the Steffan-Boltzmann and Wien's laws of blackbody radiation. Even if we take the total energy from the solar constant spread over the full surface area of Earth, the emission must be in the infrared. The calculation is left as an exercise.

When we assume that the surface absorbs some fraction of the incident shortwave, $$(1 - \alpha_s)F_{sfc}$$, and that the temperature reaches equilibrium, the emitted flux is then $$Q_{sfc} = \sigma T_{sfc}^4$$. Of course, in reality there is some emissivity associated with different surface types, but we neglect that here. Once emitted, these photons face similar consequences as the down-welling shortwave radiation. The difference in the longwave is primarily that the atmosphere is much more opaque in the infrared than visible, so the absorption can not be neglected. Various atmospheric constituents absorb infrared energy, then emit at a wavelength commensurate with the temperature of that part of the atmosphere. This is the natural greenhouse effect, and the active gases are often referred to as greenhouse gases; primary among these are water vapor, carbon dioxide, and methane.

The consequences of the natural greenhouse effect are crucial for life on Earth. In the absence of an atmosphere, the longwave radiation emitted to space would be exactly equal to the shortwave absorbed, and the surface temperature would be a chilly 255 K. Because greenhouse gases absorb infrared radiation, they act to warm the planet. How? We can think of the effects in two ways. First, the gases are heated by the absorbed radiation, and then radiate isotropically (equally up and down), sending energy back toward the surface to act as an extra energy source. Second, the absorption and subsequent emission by greenhouse gases changes the effective emission temperature of Earth (as seen from space). This second effect is a useful way to understand the greenhouse effect, and can be easily applied to changing climates. As a thought experiment, consider all the absorption by greenhouse gases happening in a thin layer of the atmosphere, which can effectively be thought of as a thin shell around the Earth. From space, the emission from the planet will be coming from an elevated level, with a much colder temperature than the surface. Of course, that means that the flux from that surface will be less than the incoming flux of solar insolation. The only way the climate system can achieve equilibrium, which is required by conservation of energy, is for the lower levels to warm, emit more energy as longwave radiation, which in turns warms the atmosphere, and changes the effective emission height and temperature. This adjustment continues until the shortwave and longwave budgets are balanced at the top of the atmosphere.

The figure below shows a rough cartoon of the radiative balances in the atmosphere. So far, we have focused only on the clear-sky scenario (on the left of the cartoon). Later we will consider the modifications that arise in the presence of clouds, and we will also explore the implications of changing the atmospheric composition.