P) between the top and bottom of our volume.We are now ready to explore the vertical structure of Earth's atmosphere. Various parts of the atmosphere are determined by the way in which temperature changes with height. This change is not uniform, but instead behaves in different ways in different layers of the atmosphere. These differences are associated with the chemistry, composition, and density of the atmosphere at different levels. Before embarking on this, though, we detour to explain these three important atmospheric physical quantities: temperature, pressure, and density. The latter two, pressure and density, are treated together since they are so closely related.
Temperature is a fundamental quantity for understanding the weather, radiation, and chemistry of the atmosphere. It is essentially a measure of the mean molecular motion of molecules. The faster they move and bump into each other, the warmer the temperature is. In order to measure temperature, a variety of ways have been devised and used throughout the last several hundred years. The United States conventional system is the Fahrenheit scale. On this scale, water freezes at 32°F and boils at 212°F. The temperature on a frigid day might hover around 0°F while on a hot summer day, we might find temperatures of 100°F. Thus, the Fahrenheit scale is relatively well suited for the temperature range we encounter in our everyday lives.
More prevalent around the world and more frequently used by scientists is the Celsius or Centigrade temperature scale. On this scale, water freezes at 0°C and boils at 100°C. The Celsius scale was originally defined using the temperatures at which water froze and boiled. Naturally, there is a conversion to easily translate temperature from Fahrenheit to Celsius.
3.1.1 Temperature conversion from Fahrenheit to Celsius -- To figure out that conversion, we first note that the difference between the boiling and freezing point of water on the Fahrenheit scale is 180 units while on the Celsius scale it is 100 units. This discrepancy immediately tells us that one Fahrenheit degree represents a smaller change in temperature than does one Celsius degree. In fact, a change of 180°F is required to represent the same temperature difference as 100°C does (that is, the temperature difference between the freezing and boiling points of water). Assuming the temperature scales are linearly related (which fortunately they are!) the conversion factor between Celsius and Fahrenheit is 180°F/100°C or 9/5. We must also note, however, that there is an offset between these scales: water freezes at 32° on the Fahrenheit scale and 0° on the Celsius scale. We must account for this offset by adding 32° to our Celsius temperatures to get Fahrenheit temperatures. So, in its final form, the conversion from Celsius to Fahrenheit is given by
T(F) = (9/5) T(C) + 32
3.1.2 The Kelvin scale -- Another temperature scale is the Kelvin scale. Systems cooled to 0 Kelvin (0 K) contain their minimum possible thermal energy. That is, it is the temperature at which all molecular motion of a substance ceases. This temperature is called absolute zero. It is not possible to reach 0 K precisely, but scientists have cooled substances to within a few hundredths of a degree of 0 K.
This scale is very useful scientifically because it provides a convenient transformation from temperature (K) to energy, and is widely used in atmospheric and space sciences.
3.1.3 Kelvin to Celsius temperature conversion -- Celsius temperatures can be easily converted to Kelvin. The conversion from Celsius to Kelvin is given by
T(K) = T(C) + 273.15
On the Kelvin scale, water freezes at 273.15 K and boils at 373.15 K. Notice that difference between the freezing and boiling points of water expressed in Kelvin and degrees Celsius are exactly the same --100 units! Changes in temperature expressed in Kelvin are the same as changes in temperature expressed in degrees Celsius.
Atmospheric pressure and density are two fundamental quantities utilized by atmospheric scientists. Both of these quantities decrease with altitude. By definition, pressure is the force exerted per unit area. Physicists therefore measure pressure in Newtons/m2, also known as a Pascal (Pa for short). Atmospheric pressure can be thought of most simply as the weight per unit area of the column of atmosphere above a given height. At the surface, this column includes all of the gases in the atmosphere, while at 10 km, it includes only those gases above 10 km. The atmospheric pressure at Earth's surface is 101,325 Pa, while at 30 km, the pressure falls to about 1,000 Pa. Atmospheric scientists frequently use the unit hPa (for hectoPascal, which is 100 Pa) for atmospheric pressure. In these units, the surface pressure is 1013.25 hPa. You will also find a third unit for pressure used in atmospheric science and meteorology -- the millibar (mb). One mb equals one hPa, so average sea level pressure in millibars is 1013.25 mb.
A second fundamental quantity for atmospheric scientists is density. Density is an amount of a material per unit volume. Atmospheric scientists use two types of density measurements: number density and mass density. These two measurements are related to one another, as we shall soon see, but the distinction between the two is important. The number density (n) is given simply by
n = Nv/V
where Nv is the number of molecules in a given volume of space and V is that volume. This quantity is frequently plotted for profiles of atmospheric constituents (like ozone). The mass density (usually referred to simply as density and denoted by d) is given by
d = mv/V
where mv is the mass in a given volume and V is that volume. If we know the mass (m) of each molecule in the volume V, we can easily translate from d to n through
d = mn
Pressure and density are related quantities. Near the surface, where the atmospheric pressure is the largest, we expect molecules to be squeezed most tightly together. Higher up, where pressure is lower, molecules are spread farther apart. The relationship between pressure (P) and density (d) in the atmosphere is given by a form of the ideal gas law
P = d RT
where T is the atmospheric temperature and R is the ideal gas constant (287 J/kg/K for dry air). This equation also often appears relating pressure (P) to number density (n):
P = n kT
where k = 1.38*10-23 J/K, also known as the Boltzmann constant.
Using our definitions of pressure and density, we derive the pressure at a given height in the atmosphere
P = Po exp(-z/H)
where Z is the height above the surface and H is called the scale height of the atmosphere (approximately 7 km or 4.3 miles). The derivation of this equation is given below in section 3.2.1.
3.2.1 Derivation of pressure and density as a function of altitude -- Derivation of pressure and density as a function of altitude begins in the simplest atmosphere we can imagine: one that is at rest, that is one without any winds to disturb anything. Let's examine a cubic volume V somewhere above the ground at an altitude h with a height h and a base with area A. What are the forces acting on such a volume? Well, there's no wind blowing, so there are no forces acting parallel to the ground. In fact, there are only two forces acting on V at all: the weight of the column of air above V acting to push V toward the ground and the pressure below V acting to push V upwards. Because V is at rest, these two forces must be in balance. Atmospheric scientists describe this balanced state as the hydrostatic equilibrium. This is a term which simply means that the atmosphere is in balance and at rest.
As for the force of gravity, note below that it is incorporated into the definition of weight. Weight is defined as mass multiplied by Earth's gravitational constant (about 9.81 m/sec2).
W=mg
The above expression implicitly assumes a unit area size. A more general expression of weight (W) per unit area (A) is
W/A=mg/A
Our first step is to note that the weight (force) acting on the top of V and on the weight (force) acting on the bottom of V is slightly different. This difference is because the weight at the top of V arises from the weight of all of the molecules of air in the column above V, while the weight at the bottom of V arises from all of the molecules of air in the column above V plus all of the molecules of air inside of V. The difference in the weight is exactly equal to the weight of the molecules in V itself.
The hydrostatic equilibrium condition means that the weight per unit area acting from above must be balanced by the pressure force per unit area from below. At the top of V, the pressure is simply W/A, while at the bottom of V, the pressure is W/A plus the weight per unit area of the molecules inside of V. Our weight per unit area is
W/A = mg/A,
which is which is just the pressure difference (P) between the top and bottom of our volume.
If we use a trick to multiply the rightside by 1 = h/ h, where h refers to the height of the cubic volume, we get
P = (mgh)/(Ah).
But
Ah = V,
so
P = mgh/V
Now we recognize that m/V is simply the density, d, which leaves us finally with
P = dg h
This equation is known as the hydrostatic equation and tells us how pressure changes with altitude. Of course, density (d) is also a function of altitude, so the equation is not really as simple as it might appear at first glance. If we use the ideal gas law, we can rewrite the hydrostatic equation as
P = (P/RT) g h
We've almost made it to our goal: the pressure as a function of altitude. Unfortunately, the next step requires a bit of integral calculus. For those of you familiar with calculus you already recognize the above equation. For those of you unfamiliar with calculus, you'll just have to trust us. Integration of the above equation yields:
P = Po * e(-z/H)
where Z is the height above the surface and H is called the scale height and is given by
H = RT/g
The scale height relates how quickly pressure falls off with altitude. More precisely, for each scale height you ascend above Earth's surface, the pressure falls off by a factor of e (2.728...). If we use the ideal gas law, we can also derive the density as a function of altitude as well. Figure 2.03 illustrates the vertical distribution of pressure and density in Earth's atmosphere.
We know from everyday experience that the temperature of Earth's atmosphere is not constant. Day-to-day changes in Earth's surface temperature can be quite significant (just ask anybody who has lived in the midwestern United States!). Not only does the temperature of the atmosphere vary with time, it also varies with height (which is why it is almost always colder on the mountain top than in the valley) and global position (which is why it is, in general, always warmer heading in the direction of the equator than it is heading in the direction of the poles).
Figure 2.04 illustrates the vertical temperature distribution through the atmosphere.
Notice that the temperature change with height is not uniform: in some places, it decreases with altitude, while in other places, it increases. For example, temperature decreases from the surface to about 10 km, rises from 15 km to 50 km, decreases again from 50 km to 85 km, and then rises again from 85 km to near 200 km. Above 200 km, the temperature remains fairly constant. Atmospheric scientists use the term lapse rate to describe the slope of the temperature profile, that is, the change in temperature with altitude.
The atmosphere is divided up into different layers based on the observed temperature structure. The observed motions (dynamics) in each layer are intimately associated with this temperature structure. As for composition, the atmosphere remains fairly uniform up to about 100 km in altitude. That is, the percent of oxygen, nitrogen, and argon is fairly uniform. As we have discussed, the trace gas species can be quite variable. These include water vapor, carbon dioxide, methane, and oxides of nitrogen. The layer below about 100 km is called the homosphere because the atmosphere is rather homogeneous in terms of its composition. Above this layer is the heterosphere, where the extreme thinness of the air leads to fractionation of the constituent gases. That is, lighter species move up while heavier ones do not.
3.3.1 Atmospheric static stability -- Any discussion of the atmosphere must include an overview of the concept of static stability. To begin with, the temperature is different at different levels in the atmosphere. The rate of change of temperature with height is called the lapse rate. In general in the lower atmosphere, temperature drops with heigh, so the lapse rate is negative. The resulting temperatures at different levels form a tempeature profile. Static stability measures whether an air parcel with some initial temperature when displaced upward in height will continue to rise, or whether the parcel will sink. It is thus a measure of the temperature of a parcel of air in relation to its surroundings. If it is warmer than its surroundings, it will rise. If it is cooler, it will sink. This is because parcels of air that are displaced upward to lower pressure naturally cool off, while downward displacements result in warmer temperatures. Examples of what can cause an air parcels to be displaced upward are mountain ranges and frontal boundaries.
A parcel of air that is warmed by the summer sun near the surface will be warmer than its surroundings, and hence it will rise. Likewise, warmer air ahead of a cold front will experience a forced ascent up and over the colder air. The warmer air rises until its temperature is equal to that of its surroundings and then it stops. Moisture often condenses out of the parcel, forming clouds and precipitation. The precipitation releases the "latent heat" inside the parcel- that is, the heat energy it originally required to evaporate the water into the parcel. Thus, the temperature of the parcel will change.
Figure 2.05 shows three different lapse rate conditions that can exist in the atmosphere: an "unstable" one where temperature falls off quickly with height, a "neutral" one where temperature is constant with height, and a "stable" one where temperature rises with height. When a parcel of air that was warmed near the surface is forced upwards because of a frontal boundary or a mountain range, depending on the type of lapse rate it encounters will determine if it will rise, remain steady, or sink. If the air parcel is warmer than its surroundings, such as occurs when the lapse rate is "unstable" and temperatures fall off quickly with height, it will rise. The parcel will rise, cooling as it does, until its temperature matches the temperture of its surroundings. On the other hand, if the air parcel is colder than its surroundings, such as occurs when the lapse rate is "stable" and falls slowly or even rises with height, it will sink back down. The parcel will sink, warming as it does, again until its temperature matches the temperature of its surroundings. The air parcel will always tend towards an equilibrium position. Once it finds this position, it will oscillate up and down about it at a known frequency.
3.3.2 Potential temperature -- As we've seen in section 3.2, pressure varies directly with temperature in the ideal gas law. Thus the temperature of an air parcel varies directly with pressure. The parcel cools as the pressure drops. Since different locations on earth have different elevations, atmospheric pressure in a place like New York City, located at sea level, is considerably higher than it is in a place like Denver, Colorado, located nearly a mile above sea level at edge of the Rocky Mountains. For purposes of analysis, meteorologists and other atmospheric scientists prefer to use a single pressure level to which temperatures at these different altitudes/pressure levels can be referenced. The chose pressure level is 1000 hPa or 1000 mb, which is just about standard sea level pressure. The temperature of the air in any parcel at any altitude above sea level is referenced to the 1000 hPa surface.
Potential temperature will be discussed in more detail in section 5.2.1 of this chapter.
The lowest region of the atmosphere is known as the troposphere. It is where life exists. In Figure 2.04, this is the region nearest the surface where temperatures are decreasing with altitude. The reason for this temperature pattern in the troposphere is because of how energy from the Sun is absorbed by Earth's surface and reemitted upward.
3.4.1 Thermal energy transfer -- Radiant energy from the Sun is known as shortwave radiation because it has such a small wavelength, and it is also very energetic (see Chapter 4). The bulk of the energy intercepted by the earth's surface falls in the visible region of the spectrum since the air lets these wavelengths of light through uninterrupted. This is absorbed by the surface, which is warmed by this energy. It is then reemitted by the surface at a much longer wavelength, in the infrared or thermal region, into the atmosphere. We can actually feel this infrared energy as warmth on our skin. This infrared energy is then repeatedly absorbed and reemitted in all directions by certain trace gases in the atmosphere, like CO2 and water vapor. The portion that is emitted upward gradually cools as the air density thins out (less efficient transport of heat). It is thus warmer nearer the surface in the troposphere.
3.4.2 Convective instability -- Air that is warm air tends to rise because it is more buoyant than colder air. In the troposphere, in general, warmer air lies beneath colder air. Such a temperature structure is unstable as the warmer air rises. When large regions of warmer, more buoyant air rise up and mix into colder regions, we say we have convective overturning. This convective activity consists of rising and sinking motions, much like the bubbles in a pot of boiling water. Convective instability plays a major role for most of the observed weather. The troposphere, therefore, is the region of the atmosphere in which Earth's weather, in all its wonderful variety, occurs. One consequence of these instabilities and weather systems is the mixing of trace gases with lifetimes of at least 1 month relatively well throughout the troposphere.
3.4.3 The tropopause --The altitude where temperature begins to rise again is termed the tropopause. The temperature lapse rate ceases to be negative. Recall that this is the temperature of air if adiabatically compressed to 1000 mb. The tropopause is usually found around 12 km, although it is somewhat higher in the tropics and lower in the polar regions. The height of the tropopause also varies with season owing to changes in the atmospheric circulation. Figure 2.06 illustrates the average height of the tropopause for January and July plotted as a function of latitude.
In the Northern Hemisphere midlatitudes (30°-50°N), we see that the tropopause height is lower in January (winter) than July (summer). In the Southern Hemisphere poleward of 50°S latitude, the height is lower in summer than in winter. The causes for these differences lie in the seasonal changes in the atmospheric circulation. Regardless of its altitude, however, the tropopause represents the boundary between the troposphere and the next higher region in the atmosphere known as the stratosphere.
The stratosphere is defined as the region of the atmosphere directly above the troposphere in which temperatures increase with altitude. From Figure 2.04 we see that this occurs between about 12 km and 50 km. In the stratosphere, the lowest temperatures are found at the bottom and the highest at the top.
3.5.1 Convectively stable, vertically stratified -- Because temperature rises with height in the stratosphere, the condition of warmer air above colder air exists. Such a condition is convectively stable. Vertical motions are therefore suppressed, leading to vertical stratification of the air masses it contains; hence the name stratosphere. This increase in temperature with height--the definition of an inversion--acts as a global cap on the weather. Convective motions and deep cumuliform clouds are limited to the height of the tropopause. Air parcels rising up from the surface through the troposphere hit the tropopause and flatten out almost as if it were a rigid lid. Consider how thunderstorm tops flatten out into the characteristic "anvil heads" when they bump up into the tropopause. This is because the cloudy air parcels in the thunderstorm suddenly find themselves in an environment warmer than their internal temperature and they cease to rise, flattening out instead.
It should be pointed out, though, that while vertical motions are often quite small in the stratosphere, horizontal motions can be quite rapid.
3.5.2 Ozone and stratospheric temperature profile -- Given that air density falls off with height and that thermal emissions occur from the surface, we should expect temperatures to decrease steadily with altitude. Increasing temperatures, such as occur in the stratosphere, are therefore somewhat of a curiosity and are a result of the presence of ozone at these altitudes.
It is the presence of ozone, which absorbs certain shortwave radiation from the Sun, that heats the stratosphere, and indeed, it is this ozone layer that is basically responsible for the existence of a "stratosphere." Ozone molecules absorb the very energetic solar radiation that is found at wavelengths even shorter than that of visible light. This energy has wavelengths less than violet light (the shortest wavelength visible light color), and so it is called ultraviolet or UV radiation. Though only a minor constituent even in the stratosphere, ozone at this height strongly absorbs solar UV radiation and reemits in all directions as thermal longwave radiation, much like the surface of Earth does with visible light.
The impact on the temperature structure is readily observable. Figure 2.07 illustrates the impact of ozone on the temperature in the stratosphere. Notice the high correlation between the altitudes at which ozone increases and the altitudes at which the temperature increases in the stratosphere. This is not a coincidence, but instead is a direct result of ozone.
The location of maximum temperature near 50 km is known as the stratopause. Again, this boundary separates the stratosphere from the next higher region of the atmosphere, the mesosphere.
In the mesosphere, temperature again decreases with altitude reaching a minimum at the mesospause near 85 km. Temperature increases above 85 km in the region known as the thermosphere. Above 200 km temperature is roughly constant with altitude, showing only a strong diurnal cycle. The lowest temperatures in the entire atmosphere are found at the mesopause during the summer at high latitudes and can be as low as 130 K (-226°F). The highest temperatures in the atmosphere can be found in the thermosphere, where 2000 K is sometimes encountered. Although at these altitudes, the sparse population of molecules forces us to abandon the familiar concept of temperature with which we live at the surface. Intense solar radiation with wavelengths between 100 and 200 nm (see Chapter 4) is absorbed between 85 and 100 km in the thermosphere by molecular oxygen (O2), while radiation with wavelengths shorter than 100 nm is absorbed above 100 km.
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