### The components of the climate system

The climate system reflects an interaction between a number of critical sub-systems or *components*. In this course, we will focus on the components most relative to modern climate change: the atmosphere, hydrosphere, cryosphere, and biosphere. Please watch the following video to walk through the important aspects of these components.

### Atmospheric Structure and Composition

The atmosphere is of course a critical component of the climate system, and the one we will spend the most time talking about.

One key feature about the atmosphere is the fact that pressure and density decay exponentially with altitude:

As you can see, the pressure decays nearly to zero by the time we get to 50 km. For this reason, the Earth's atmosphere, as noted further in the discussion below, constitutes a very thin *shell* around the Earth.

The exponential decay of pressure with altitude follows from a combination of two very basic physical principles. The first physical principle is **the ideal gas law**. You are probably most familiar with the form $pV=nRT$ , but that form applies to a bounded gas, where the volume can be defined. In our case, the gas is free, and the appropriate form of the ideal gas law is

where $p$ is the atmospheric pressure, $\rho $ is the density of the atmosphere, $R=287\text{}J\text{}{K}^{-1}k{g}^{-1}$ is the gas constant that is specific to Earth's atmosphere, and $T$ is temperature.

The 2nd principle is the** force balance**. There are two primary vertical forces acting on the atmosphere. The first is gravity, while the other is what is known as the pressure gradient force — it is the support of one part of the atmosphere acting on some other part of the atmosphere. This balance is known as the *hydrostatic balance*.

The relevant pressure gradient force in this case is the *vertical* pressure gradient force. When we are talking about a continuous fluid (which the atmosphere or ocean is), then the correct form of force balance involves force per unit volume of fluid.

In this form, we have for gravity (the negative sign indicates a downward force):

where $g$ is Earth's surface gravitational acceleration ($9.81{\text{meters/second}}^{\text{2}}$ ).

The pressure gradient force has to be written in terms of a *derivative*:

The positive sign ensures that an atmosphere with a greater density below exerts a positive (upward) force.

In equilibrium, these forces must balance, i.e.

Now we can use the ideal gas law (eq. 1.) to substitute for ρ, the expression $\rho =p/RT$ , giving

or re-arranging a bit,

The term in parentheses can be treated as a constant (in reality, temperature varies with altitude, but it varies less dramatically than pressure or density, so it's easiest to simply treat it as a constant).

This is a relatively simple first order differential equation.

#### Self Check...

**Do you remember how to solve this first order differential equation from your previous math studies?**

Click for answer.

$$\mathrm{ln}p-\mathrm{ln}{p}_{0}=-\left(\raisebox{1ex}{$g$}\!\left/ \!\raisebox{-1ex}{$RT$}\right.\right)\left(z-{z}_{0}\right)$$

Or, we can exponentiate both sides to yield:

$$p={p}_{0}\mathrm{exp}\left[-\left(\raisebox{1ex}{$g$}\!\left/ \!\raisebox{-1ex}{$RT$}\right.\right)\left(z-{z}_{0}\right)\right]$$

The expression for atmospheric pressure as a function of altitude is:

where ${p}_{0}$ is the surface pressure, and ${z}_{0}$ is the surface height (by convention typically taken as zero).

This equation is known as the *hypsometric equation*.

The combination $\raisebox{1ex}{$g$}\!\left/ \!\raisebox{-1ex}{$RT$}\right.$ has units of inverse length, and so we can define a *scale height* (assuming a mean temperature $T=14\xb0C=287K$ ,

and write:

this gives an exponential decline of pressure with height, with the *e-folding height* equal to the *scale height*, representing the altitude at which pressure falls to roughly 1/3 of its surface value. At this altitude, which as you can see from the above graphic is just a bit below the height of Mt. Everest, roughly 2/3 of the atmosphere is below you.

#### Self-check...

**Using the hypsometric equation (9 above), estimate the altitude at which roughly half of the atmosphere is below you.**

Click for answer.

$$\raisebox{1ex}{$p$}\!\left/ \!\raisebox{-1ex}{${p}_{0}$}\right.=\mathrm{exp}\left[-\left(z-{z}_{0}\right)/{h}_{s}\right]$$

Take logs of both sides and rearrange:

$$z-{z}_{0}={\mathrm{-h}}_{s}\mathrm{ln}\left(\raisebox{1ex}{$p$}\!\left/ \!\raisebox{-1ex}{${p}_{0}$}\right.\right)$$

Take $\raisebox{1ex}{$p$}\!\left/ \!\raisebox{-1ex}{${p}_{0}$}\right.=0.5$ and use ${h}_{s}=8.4km$ from earlier: $$z-{z}_{0}=-8.4\mathrm{ln}\left(0.5\right)=5.8km$$

Let us look at the vertical structure of the atmosphere in more detail, define some key layers of the atmosphere:

*Dire Predictions: Understanding Climate Change, 2*

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Now, let us talk a bit more about the atmospheric *composition*:

The atmosphere is mostly nitrogen and oxygen, with trace amounts of other gases. Most atmospheric constituents are *well mixed*, which is to say, these constituents vary in constant relative proportion, owing to the influence of mixing and turbulence in the atmosphere. The assumption of a *well-mixed atmosphere* and the assumption of *ideal gas behavior*, were both implicit in our earlier derivation of the exponential relationship of pressure with height in the atmosphere.

There are, of course, exceptions to these assumptions. As discussed earlier, ozone is primarily found in the lower stratosphere (though some is produced near the surface as consequence of photochemical smog). Some gases, such as methane, have strong *sources* and *sinks* and are therefore highly variable as a function of region and season.

Atmospheric water vapor is highly variable in its concentration, and, in fact, undergoes phase transitions between solid, liquid, and solid form during normal atmospheric processes (i.e., evaporation from the surface, and condensation in the form of precipitation as rainfall or snow). The existence of such phase transitions in the water vapor component of the atmosphere is an obvious violation of ideal gas behavior!

Of particular significance in considerations of atmospheric composition are the so-called greenhouse gases (CO_{2}, water vapor, methane, and a number of other trace gases) because of their radiative properties and, specifically, their role in the so-called *greenhouse effect*. This topic is explored in greater detail later on in this lesson.