1. SO254: Vertical lapse rates and stability

2. Brief summary of vertical motion
Air parcels often rise and sink in the atmosphere
Common mechanisms that cause air to rise:
Divergence aloft
Differential cyclonic vorticity advection increasing with height
Warm-air advection
Positive buoyancy (parcel warmer than air around it)
Forced ascent
Air “pushed” up by surface convergence along a front, upslope along a mountain, or convergence associated with mesoscale circulations (like sea/land breeze, mountain/valley breeze)
Common mechanisms that cause air to sink:
Convergence aloft
Differential anticyclonic vorticity advection increasing with height
Cold-air advection
Negative buoyancy (parcel cooler than air around it)
Forced descent
Air “pushed” down by downslope along a mountain, convergence aloft

3. Brief introduction to vertical lapse rates
The first law of thermodynamics requires that as air parcels move up and down in the atmosphere, their temperature changes
Why? The parcels will expand (if rising) or contract (if sinking)
Expansion does work against the environment, reducing parcel temperature
Contracting parcels have work done on them by the environment, increasing parcel temperature
If the parcel is not saturated (RH < 100%, T > Td, w < ws, etc)
Its lapse rate is called the “Dry adiabatic lapse rate” and is written as:
If the parcel is saturated (RH = 100%, T = Td, w = ws, etc)
Its lapse rate is called the “Saturated adiabatic lapse rate” and is written as:

4. More on vertical lapse rates
The dry adiabatic lapse rate comes from the 1st law of thermodynamics:
where ΔT is the change in temperature, g is gravity (9.81 m s-2), Cp is the specific heat of air at constant pressure (1004 J kg-1 K-1), and Δq is the exchange of heat with the environment
Parcels generally ascend/descend adiabatically, with no exchange of heat with the environment
Thus Δq=0 and

5. More on vertical lapse rates
As air rises, it expands and cools
However, as we saw in the last chapter, the moisture content in an air parcel cannot (as a general rule) exceed the moisture capacity of an air parcel
This means that Td cannot be greater than T; w cannot be greater than ws; and e cannot be greater than es.
To maintain RH ≤ 100%, water vapor must condense into liquid (or deposit as a solid) while the air parcel rises
This phase change of water releases latent heat, Lv = 2260 kJ kg-1 (for evap/cond) and Ls = 334 kJ kg-1 (for freezing/melting).
As the parcel rises, the release of latent heat by water vapor molecules acts to slightly warm the air parcel (remember, the parcel is still cooling as it expands while rising)
Thus, the lapse rate for saturated air parcels is much less than for dry air parcels, or approx.
Saturated air parcels thus cool less than dry air parcels while they rise

6. Saturated adiabatic lapse rate
The saturated adiabatic lapse rate is approx. 5°C per kilometer. But that value varies, depending (primarily) on three factors
Temperature of the air
Vapor pressure of the air
Pressure of the air
Specifically, the saturated adiabatic lapse rate can be written as:
The value of the saturated adiabatic lapse rate varies from 4-7°C km-1, but a value of 5°C km-1 is reasonable for the lower troposphere

7. Example 1: calculating a parcel temperature as it rises
Let’s examine an air parcel that has a temperature of 32°C and a dew point temperature of 2°C at the surface (z=0 m).
At z=0, T=32°C and Td=2°C
Calculate the temperature of the air parcel at the following levels:
1000 m
2000 m
3000 m
4000 m
5000 m
Assume two things: dew point temperature does not change while air is unsaturated, and use -10°C km-1 for the dry adiabatic lapse rate ? ? ? ? ?

8. Example 1: calculate the air temperature of a parcel as it rises from 0 m to 5000 m
Let’s examine an air parcel that has a temperature of 32°C and a dew point temperature of 2°C at the surface (z=0 m).
At z=0, T=32°C and Td=2°C
Calculate the temperature of the air parcel at the following levels:
1000 m
2000 m
3000 m
4000 m
5000 m
Assume two things: dew point temperature does not change while air is unsaturated, and use -10°C km-1 for the dry adiabatic lapse rate

9. Example 2: Calculate the air temperature of a parcel as it goes over a mountain
z = 4000 m
T=?
Td=?
At the surface (z=0 m):
T=25°C
Td=10°C
Calculate the air temp and air dew point temp at 2000 m and m on the upslope mountain side of the mountain
Calculate the air temp and air dew point temp at 2000 m and 0 m on the downslope side of the mountain
Calculate the LCL height (in m)
z = 2000 m
T=?
Td=?
z = 2000 m
T=?
Td=?
z = 0 m
T=25°C
Td=10°C
z = 0 m
T=?
Td=?

10. Example 2: Calculate the air temperature of a parcel as it goes over a mountain
z = 4000 m
T= -2.5°C
Td= -2.5°C
At the surface (z=0 m):
T=25°C
Td=10°C
Calculate the air temp and air dew point temp at 2000 m and m on the upslope mountain side of the mountain
Calculate the air temp and air dew point temp at 2000 m and 0 m on the downslope side of the mountain
Calculate the LCL height (in m)
LCL height 1500 m
z = 2000 m
T = 7.5°C
Td = 7.5°C
z = 2000 m
T = 17.5°C
Td = -2.5°C
LCL 1500 m
z = 0 m
T=25°C
Td=10°C
z = 0 m
T = 37.5°C
Td = -2.5°
Notice how the entire parcel temperature depends only on its starting T and Td! (and the height)

11. Atmospheric stability
The stability of the atmosphere is a very complex topic
However, the general rule of thumb is basic: if air rising air parcels are warmer than their environment, they are unstable. If rising air parcels are colder than their environment, they are stable.
The atmosphere generally cools with height
Exceptions: inversions, and the stratosphere
To determine the stability of a layer of the atmosphere:
Compare the temperature of the environment to the temperature that an air parcel would have if it were rising through that environment
-- or --
Compare the lapse rate of the environment to the dry and saturated adiabatic lapse rates of air parcels

12. Examples of atmospheric stability
Case 1:
Parcel temperature 10°C. Environmental temperature 7°C. Unstable. Parcel will rise
Parcel temperature 10°C. Environmental temperature 13°C. Stable. Parcel will sink
Case 2:
Environmental lapse rate -4°C km-1. Stable. Rising parcels will be colder than the environment and will sink
Environmental lapse rate -12°C km-1. Unstable. Rising parcels will be warmer than the environment and will rise
Environmental lapse rate -8°C km-1. Conditionally unstable. Saturated air parcels will be unstable, but dry air parcels will be stable
Remember, dry adiabatic lapse rate is -9.8°C km-1 and the saturated adiabatic lapse rate is -5°C km-1.
If the environment cools more than 9.8°C per km, then both dry and saturated air parcels will be warmer than the environment (and that’s an unstable setup)
If the environment cools less than 5°C per km, then both dry and saturated air parcels will be cooler than the environment (and that’s a stable setup)

13. Stability example: back to the mountain
z = 4000 m
T= -2.5°C
Td= -2.5°C
Environment
z = 4000 m
T= -8°C
What should be the stability of these two layers?
z = 2000 m
T = 7.5°C
Td = 7.5°C
z = 2000 m
T = 17.5°C
Td = -2.5°C
z = 2000 m
T= 9°C
z = 0 m
T=25°C
Td=10°C
z = 0 m
T = 37.5°C
Td = -2.5°
z = 0 m
T= 25°C

14. Stability example: back to the mountain
z = 4000 m
T= -2.5°C
Td= -2.5°C
Environment
z = 4000 m
T= -8°C
Unstable
z = 2000 m
T = 7.5°C
Td = 7.5°C
z = 2000 m
T = 17.5°C
Td = -2.5°C
z = 2000 m
T= 9°C
Stable
z = 0 m
T=25°C
Td=10°C
z = 0 m
T = 37.5°C
Td = -2.5°
z = 0 m
T= 25°C

15. Stability: the skew-T diagram
The environmental temperature profile (and environmental lapse rates) can be highly variable!
One of the great tools available to meteorologists to plot the vertical changes in temperature is called the “Skew-T Log-P” diagram
On this diagram, the stability of a layer of air to rising (or sinking) air parcels can quickly be diagnosed
The diagram also offers lines to quickly trace out parcel temperatures
More details on the Skew-T diagram in the next lessons!
And you have an entire course on Atmospheric Thermodynamics (SO345) to learn lots more about stability and vertical profiles of temperature and moisture