Also see the notes at the bottom of the discussion for HW #6. You may have to refresh the page to get the updates.  

Previously we discussed the atmosphere as a whole, using the ideal gas law and the hydrostatic and hypsometric equations. We also discussed horizontal motion (wind) in the atmosphere early in the semester. We now turn to a discussion of vertical motion in the atmosphere.

See your texts, Wallace and Hobbs Ch. 3 and Vasquez Ch. 3. You should be carefully reading Vasquez Ch. 3 at this point. Our goal is to understand the SkewT chart.

To facilitate our thinking about air moving vertically in the atmosphere, we introduce the idealized concept of an air parcel. An air parcel is assumed to be some amount of air where:

-- the mass of the air parcel is conserved
-- the air in the parcel is thermally insulated from its environment
-- is always at the same pressure as its surrounding air
-- the surrounding air is assumed to be in hydrostatic balance
-- the parcel is moving slowly enough that its kinetic energy is negligable

As the air parcel ascends or descends in the atmosphere, it is typical to assume the process is adiabatic which means the parcel exchanges no heat or mass outside of itself. This means it does not absorb or radiate heat. Any water droplets that may form from condensation are kept within the parcel.

Using the first law of thermodynamics, it can be shown that as a dry air parcel ascends or descends in the atmosphere adiabatically, it cools or warms at a constant rate with height, known as the dry adiabatic lapse rate. See Wallace and Hobbs for the derivation. This lapse rate is 9.8K per km.

The potential temperature is defined such that it is a constant property of air that is moving dry adiabatically. It is the temperature found when the parcel is moved dry adiabatically to the pressure of 1000mb (or 1000hPa). Poisson's Equation can be used to determine the potential temperature θ of an air parcel:

θ = T(1000/p)^0.286

where T is the temperature of the parcel in K and p is the pressure of the parcel in hPa (mb).

The SkewT chart is constructed with pressure as the Y axis and temperature as the X axis. In order to render thermodynamic relationships at "nice" angles, the pressure on the SkewT is plotted logarithmically, and the temperature is plotted as "skewed" lines, running at an angle from lower left to upper right.

On the SkewT the dry adiabats are plotted as a function of p and T and show the path a parcel of air would take when moving dry adiabatially. Note that the dry adiabats are also lines of constant θ -- constant potential temperature.

The potential temperature of any parcel can be found on the SkewT chart by finding the point of temperature and pressure of the parcel on the chart and following the dry adiabat that goes from that point to 1000mb.

The air in the atmosphere is rarely (or never) completely dry, so we next need to consider moisture in the atmosphere.

We start with the ideal gas law. The formulation of the gas law we are using for the atmopshere is:

p = ρR_dT

where ρ is the density of dry air and R_d is the gas constant for dry air and is 287.0 in SI units.

For water vapor -- water in its gaseous form -- the ideal gas law is:

p = ρ_vR_vT

where ρ_v is the density of the water vapor and R_v is the gas constant for water vapor and is 461.51 in SI units.

Note that the gas constant for water vapor is larger than the gas constant for dry air. The equations show that if we hold pressure and temperature constant, then water vapor is less dense than dry air.

By Dalton's Law, the total pressure of a gas is the sum of the partial pressures of the components of the gas. The partial pressure of water vapor in the atmosphere is commonly denoted by the variable e.

So the total pressure of a parcel of air is the sum of the partial pressures:

p = p_d + e

where p_d is the partial pressure of the dry air alone.

By combining the ideal gas equations for dry air and for water vapor, we can show that the density of moist air is:

ρ = (p-e)/(R_dT) + e/(R_vT)

where p-e = p_d

The virtual temperature is defined such that the density of moist air at temperature T is the same as the density of dry air at virtual temperature T^* (also written as T_v).

Using the equation above for density of moist air, we can see that:

p/(R_dT_v) = (p-e)/(R_dT) + e/(R_vT)

We can see that virtual temperature is a function of p, T, and e. We can further derive an approximation for the virtual temperature:

T_v ≅ T(1 + 0.61w)

where w is the mixing ratio, which will be described below.

The virtual temperature is often used to simplify thermodynamic equations (such as the hypsometric equation), as an alternative to specifying the behavior of water vapor in all the equations. The virtual temperature is always greater than or equal to the actual temperature, and in very moist air can be several degrees higher.

It is a property of water that at a particular temperature, there is a maximum pressure of water vapor, beyond wich the water vapor is condensed to liquid (or solid) water. This can be seen by performing the experiment of placing liquid water at the bottom of a closed container, and then waiting for the water vapor in the container to reach equilibrium with the liquid water. It is not intuitive, but the amount of water vapor is not dependent on what other gasses are in the container. The container could start as a vacuum, or it could have air pressurized to 1000 bars, and the partial pressure of the water vapor in equilibrium with the liquid water would be the same!

This is a very very important property of water. The partial pressure of the water vapor in equilibrium with liquid water is known as the saturation vapor pressure and is denoted by e_s

The Clausius-Clapeyron relationship theoretically describes the behavior of the satuation vapor pressure and shows that it is dependent only on temperature. There are a number of emperical formulas that give an approximation of the function e_s(T), such as:

e_s(T) ≅ 6.1094exp(17.625T/(T+243.04))

where T is in °C and the result is in mb. (This is the equation given in the Wikipedia entry on Clausius-Clapeyron. There are other equations available, depending on the error tolerance one wishes.) It is also common to look up values of e_s from tables. (If the dewpoint is known, it can be used in the above equation to get the value of e, the actual vapor pressure.)

The mixing ratio is defined as the ratio between the mass of water vapor and the mass of dry air for a specific volume of moist air:

w = m_v/m_d

Values of the mixing ratio are often given in g/kg -- be careful! here, when doing any computations the ratio needs to be dimensionless, eg. kg/kg. You may have to do the conversion. Typical values range from 2 or 3 g/kg to 20g/kg.

The saturation mixing ratio is the mixing ratio when the air is saturated with water vapor, is defined as the ratio between the mass of water vapor and the mass of dry air for a specific volume of moist air:

w_s = m_sv/m_d

An approximation for the saturation mixing ratio is given as:

w_s ≅ 0.622e_s/p

Note that, since e_s is a function of only temperature, this close approximation gives saturation mixing ratio as a function of pressure and temperature. Lines of constant satuation mixing ratio are plotted on the SkewT chart, usually as dashed lines that run from lower left to upper right.

At this point, we also note that relative humidity is defined as the ratio of mixing ratio to saturation mixing ratio:

RH/100 = w/w_s ≅ e/e_s

The dewpoint is the temperature to which air must be cooled at constant pressure to become saturated. In other words, the dewpoint is the temperature at which the air must be cooled for the saturation mixing ratio to equal the actual mixing ratio of the air. So if we have the temperature T, dewpoint T_d, and pressure p of the air, we can calculate the RH as the ratio of w_s at temperature T_d and pressure p over w_s at temperature T and pressure p. The SkewT chart can be used to obtain the needed mixing ratios. An important note: if you are doing this calculation for a station report, you use the station pressure -- not the altimeter; not the SLP. (If you know the altimeter and the elevation of the station, it is possible to compute the station pressure.)

In a similar way, the RH can be calculated from temperature and dewpoint using the vapor pressures directly, where e/e_s can be calculated as the ratio of e_s at the dewpoint over e_s at the air temperature.

Another important property of water is that it undergoes changes of phase at temperatures common in the atmosphere. This means that water changes state between gas, liquid, and solid.

If you have ice and warm it, when the ice reaches 0°C, it begins to melt. As you add heat, the ice temperature stays at 0°C until it is entirely melted. The amount of heat added to melt the ice but not affect the temperature is called the latent heat of melting. Similarly, if you freeze the ice, you take away heat. To convert liquid water to gas also takes heat, the latent heat of vaporization. When water condenses to liquid, it gives up heat, called the latent heat of condensation.

The heat released by the phase change of water from vapor to liquid is an important component of the energy that drives weather systems.

Consider a parcel of air that is rising through the atmosphere. As long as the mixing ratio of the parcel is below saturation, the parcel rises dry adiabatically. Once the parcel cools enough to reach saturation, water vapor begins to condense from the air if the air rises and cools further.

As the air rises and cools further, beyond the saturation point, we assume the water that condenses from the air is removed from the parcel. Thus the process is no longer adiabatic. The process is called pseudoadiabatic. The latent heat of condensation added to the air parcel results in the parcel cooling at a slower rate than the dry adiabatic rate.

Lines are plotted on the SkewT that show the rate of cooling of a saturated parcel as it is lifted. They are called moist adiabats. The moist adiabats are also lines of constant equivalent potential temperature.