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Temperature
Pressure
The Troposphere
The Stratosphere
The Thermosphere
Molecular Temperature
The Ionosphere
Density
CHAPTER 2 The Atmosphere

The Troposphere


Meteorologists are concerned largely with the troposphere because it is within this layer that storms and clouds reside. In our discussions, we will used data from the Standard Atmosphere for the tropospheric layer. Within the troposphere the density of the atmosphere, atmospheric pressure and temperature all decrease systematically with altitude.

Temperature

FIGURE 20 Vertical Cross Section of Meridional Temperatures During Winter

(T in degrees C)

In the illustration above it is clear that everywhere below the tropopause temperatures decline with altitude at all latitudes (Figure 20). It is also evident that above the troposphere temperatures are nearly constant with height. It is significant that at around 20 km it is quite cold over the low latitudes compared to the polar regions. This pattern will prove to be very important in our study of the jet stream, which is located at the tropopause in the middle latitudes.

The rate of change of temperature with height is called the lapse rate and is symbolized as dT/dz. Often the upper case greek letter gamma (G) is used to denote a lapse rate. The change in temperature with height in the troposphere is rather linear and has a lapse rate of 6.4 oC/km using Standard Atmosphere data. So for every 1000 meters in elevation you should expect temperatures fall 6.4 oC. A graph of temperature against altitude is presented in Figure 21.

Temperature as a function of height may be determined by calling upon the Hydrostatic Equation ( Eq. 12), the Equation of State ( Eq. 13 and Eq. 15)and the First Law of Thermodynamics ( Eq. , Eq. 17 and Eq. 20).

(EQ 12)

(EQ 13)

The relationship between specific volume (a) and density (r) is given in Eq. 14 and in conjunction with Eq. 13 can be used to derive a second expression for the equation of state ( Eq. 15).

(EQ 14)

(EQ 15)

The First Law of Thermodynamics states that

(EQ 16)

(EQ 17)

The specific heat at constant volume (Cv) or at constant pressure (Cp) of one gram of a specific gas is the heat required to raise its temperature 1oC. The specific heat at constant volume is presented in Eq. 18, and the specific hear at constant pressure is presented in Eq. 19.

(EQ 18)

(EQ 19)

Using the expression for the specific heat at constant pressure, the first law of thermodynamics may be expressed in more specific terms for dU and dW ( Eq. 20).

(EQ 20)

If now we assume that there is not heat added to or removed from our gas we have defined adiabatic condition ( Eq. 21).

(EQ 21)

With the adiabatic assumption we can write the first law of thermodynamics as

(EQ 22)

With the hydrostatic equation ( Eq. 12) substituted into Eq. 22 we derive the following expression

(EQ 23)

Now solving for the rate of change of temperature with respect to height (dT/dz) we find the following relationship

(EQ 24)

As both gravity (g) and the specific heat at constant pressure are constants, we find that the rate of change of temperature with height under adiabatic conditions is also a constant (-9.8oC/km). If we calculate the temperature expected at 15,000 m using Eq. 24 we estimate the temperature there to be about -150oC. Actual measurements at 15,000 m indicates that a temperature of about -75oC is closer to reality. Why the difference between the calculation and the observation?

Pressure

The variation of pressure is negative (decreases) with altitude in the troposphere and is nearly linear up to about 6 km. After that rate of decreasing pressure(Figure 21, middle panel). At 5.5 km the pressure is 50.0 kPa or 500 mb. At this altitude half of the atmosphere is below and half is above. The rate of change of pressure with height from

FIGURE 21

Density, Pressure and Temperature Vs. Altitude (Standard Atmosphere data).

the surface to 5.5 km is about 9.1 kPa per km or 91 mb per km. The average pressure at mile-high Denver, Colorado is about 100 mb less than that at sea level. Given that the average sea level pressure is 1013.25 mb, the station pressure at Denver could be somewhere around 913 mb.

Pressure may be treated as a function of height in terms of atmospheric density:

(EQ 25)

Mean density of dry air is ~ 1.2 x 10-3 g cm-3. Take 1 m3 of air (100cm x 100cm x 100 cm or 106 cm3, i.e. (1.2 x 10-3)(106) g cm-3 cm3 = 1.2 x 103 g or 1.2 Kg (2.64 lbs). The density of air decreases with height until the mass of air in a given volume of air approaches 0. Therefore there is a gradient of density directed upward (toward lower density). Mass tends to move from regions of high density (high pressure) to regions of low density (low pressure) and so mass should move upward toward space due to this density or pressure gradient. On the other hand, gravity tends to cause mass to move toward the center of mass of the Earth. The acceleration of mass outward due to the pressure gradient and the acceleration of mass downward due to gravity balances exactly and the mass of the air tends to neither leak out to space or concentrate at the surface of the Earth.

FIGURE 22 A Column of Air Through the Atmosphere

The pressure (P1) at height h1 is the weight (W) of the atmosphere in the column above height level h1 and dh is h2 - h1. The volumn (V) in the column between h1 and h2 is

(EQ 26)

The mass of the volume (V) is

(EQ 27)

(EQ 28)

where g is the acceleration of gravity. The pressure at elevation level h2 (p2) is the pressure at elevation level h1, (the weight of the air above h1) minus the weight of the air in the volume (V).

(EQ 29)

(EQ 30)

Since P2 is smaller than P1 (pressure gets lower as you go up) the change in pressure with height is a negative number. If we replace the sympol of z for h we again have the hydrostatic equation.

(EQ 31)

By putting Eq. 29 into Eq. 30 we obtain

(EQ 32)

and

(EQ 33)

Eq. 33 tells us that the acceleration of gravity that attracts mass downward toward the earth is balanced by the acceleration of the decline in pressure with altitude. If we take a simple case of temperature constant with height (an isothermal atmosphere) we can integrate the above equation (recall that P = rRT).

(EQ 34)

(EQ 35)

(EQ 36)

Atmospheric pressure decreases exponentially with height in an isothermal atmosphere. Some comparisons of elevations and the pressures at those elevations is presented below.

This is how altimeters work, but altimeters must be corrected for temperature. Near sea surface level pressure decreases 1.2 mb for every 10 m increase in height. Below 100 km the atmosphere is well mixed (turbulence and convection) and the chemical species composition is constant. Hence, this layer is termed the homosphere. Above 100 km the atmosphere stratifies according to molecular weights. The layer of the atmosphere above 100 km is called the heterosphere.

The Troposphere

The troposhere extends from the surfact to an average altitude of about 10 km. Temperatures in the troposphere cool with height at a near constant rate of 6.5oC/km. Within this layer there are low, middle, and high altitude clouds. In the lower portion of the troposphere we find cumulus and stratus clouds that are made up of water droplets. Altocumulus and stratocumulus clouds in the middle troposphere are mixtures of water drops and ice crystals. Cirrus clouds occur in the high troposphere and consist of suspensions of ice crystals.

Cumulonimbus clouds (thunderheads) penetrate the entire troposphere and often penetrate into the stratosphere. Present day aircraft operate mostly in the upper troposphere at altitudes between 10 and 13 km. The temperature at the boundary between the troposphere and the stratosphre is around -50oC but may be significantly colder or warmer depending on geographic location and time of year.

The tropopause is the boundary between the troposphere and the stratosphere and may be as low as 7 km in altitude in polar regions and as high as 17 km in tropical latitudes. The stratosphere may be isothermal (no change in temperatuare with height) or may be characterized by an inversion in temperature (temperature increasing with height). Vertical motions in the stratosphere are inhibited owing to the fact that the stratosphere is a gaseous layer and not a material surface. Thus the stratosphere is a stable layer. Air motions in the stratosphere are largely horizontal rather than vertical. This lack of vertical motion allows pollutants (gas and dust particles) to become traped within the stratospheric layer. However, the pollutants may be brought to the surface by a process known as folding. folding is a process by which particulate matter subsides into the troposphere. Occassionally this particulate mater may reach the ground surface.

FIGURE 23 Folding or the downwelling of stratospheric air into the troposphere

The Stratosphere

The stratosphere extends from the top of the tropopause (around 10 km) to an altitude of 50 km. A maximum in the concentration of ozone (O3) density exists in the stratosphere between 20 and 25 km in altitude, and a maximum in ozone mixing ratio exists around 35 km in altitude. Temperatures in the stratosphere slowly increase with altitude. Why?

Photo-dissociation of O2 requires energy at a wavelength of 0.2424 mm or less. The absorption of photons of ultraviolet radiation from the sun by the oxygen molecule results in the production of atomic oxygen

(EQ 37)

where hv represents a photon of ultraviolet radiation. The atomic oxygen then may participate in two additonal chemical reactions.

(EQ 38)

(EQ 39)

These reactions require 3-body collisions, (represented by M) and may be any particle capable of absorbing the extra energy released by the reaction. O + O collide and unite. Howeve, the product (O2) is unstable and will dissociate unless it can release its energy to a 3rd particle in short time.

Mother of Pearl clouds (Nacreous clouds) are of unknown composition and are observed mainly in high latitudes (Scotland, Scandinavia). These clouds occur in the stratosphere and can be observed from the diffraction of sunlight from the spherical particles of which they are composed. The Mesosphere

The mesosphere is the layer of the atmosphere above the stratrosphere and extends from 50 to 100 km in altitude. The mesosphere is characterized by a strong temperature decrease with altitude. Clouds also may form in this layer. These clouds are called noctilucent clouds and are typically seen between 75-90 km altitude. They are bluish-silver in color and are most often seen around twilight at 5o to 13o below the horizon. These clouds are most common in summer at 50-75o N or S latitudes.

The Thermosphere

The thermosphere is the layer of the atmosphere abvoe 100 km altitude. Temperature increases rapidly in the thermosphere due to the absorption of high energy solar radiation at these altitudes. Auroras are one manifestation of this energy absorption. In an aurora, there is a radiant emission from nitrogen molecules (N2) which captured solar photons. The ionized nitrogen molecule plus atomic oxygen (O) from magnetic storms and the influx of charged particles form the sun (typical during periods of solar sun spots) produce the famous "draperies" and "curtains" of the aurora. In the northern hemisphere the aurora is called the Aurora borealis ,while in the southern hemisphere the aurora is called Aurora austrolis.

Molecular Temperature

It is useful to condider the meaning of temperature at the level of molecular processes. The following is the formula for molecular temperature

(EQ 40)

Tm is the kinetic temperature of the molecules arising from their velocities, Mois molecular weight of air at sea level (28.966), and M is the molecular weight of air at the altitude considered. Below 90 km Tm = T (i.e., Mo = M).Above 90 km Tm becomes increasingly greater than T, and Tm = 400 to 2000oC (note: temperature of sun = 6000oC). Above 90 km the diurnal variation is between 500 and 800 oC and the mean path length between molecular collisions is as much as 100 km. The heat transfer at these altitudes is very low owing to the fact that the pressure is very low. The temperature of the molecules at these elevations, like the temperatures on the surface of satellites, depends on the radiative balance.

The Ionosphere

The ionosphere is the upper most layer of the atmosphere. It consisted of ionized atoms ionized molecules, and free electrons. A gradient in free electrons exists with elevation. There are four sub regions of the ionosphere (D, E, F1 and F2) that due to their differing ionization play a central role in long-distance radio communication owing to their differing ionization.

Density

In our earlier discussion we noted that the distance between molecules of the air increased with altitude. Figure 21 (top panel) shows that density declines with altitude and that the change is relatively linear. If we rearrange the hydrostatic equation ( Eq. 10) such that density is on one side of the equal sign and all other terms on the other we can further improve our understanding.

(EQ 41)

Given the gravity of a planet and the rate of change of pressure with height, a specific density is required.

The relationships between temperature, pressure and density in the troposphere are shown in Figure 24. In Figure 24 the pressure axis is presented with highest pressure at the bottom and lowest at the top. In this way pressure serves as a surrogate for altitude in meters. In the troposphere temperature declines with pressure and altitude as detailed earlier. Density also declines with altitude and pressure. In addition, we find that density is highest at highest temperature. As we will see in later chapters atmospheric scientists often assume linear changes in temperature, pressure and density with height. The errors in making these assumptions of linearity are small and are ignored. The complexity of the equations we use is minimized in the assumjption of linearity without significant loss of information or precision. Simplicity in specifying equations is a pedological and practical necessity.

FIGURE 24 Temperature, Pressure and Density Co-variations


Climate Dynamics - 05 FEB 96
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