1. How can the first law really help me forecast thunderstorms?
Adiabatic Processes
1000 mb
How can the first law really help me forecast thunderstorms?
Thermodynamics
M. D. Eastin

2. Adiabatic Processes Outline: Adiabatic Processes Poisson’s Relation
Review of The First Law of Thermodynamics
Adiabatic Processes
Poisson’s Relation
Applications
Potential Temperature
Dry Adiabatic Lapse Rate
Thermodynamics
M. D. Eastin

3. First Law of Thermodynamics
Statement of Energy Balance / Conservation:
Energy in = Energy out
Heat in = Heat out
Heating
Sensible heating
Latent heating
Evaporational cooling
Radiational heating
Radiational cooling
Work Done
Expansion
Compression
Change in
Internal Energy
Thermodynamics
M. D. Eastin

4. Forms of the First Law of Thermodynamics
For a gas of mass m Per unit mass
where: p = pressure U = internal energy n = number of moles
V = volume W = work α = specific volume
T = temperature Q = heat energy m = mass
Cv = specific heat at constant volume (717 J kg-1 K-1)
Cp = specific heat at constant pressure (1004 J kg-1 K-1)
Rd = gas constant for dry air (287 J kg-1 K-1)
R* = universal gas constant ( J K-1 mol-1)
Thermodynamics
M. D. Eastin

5. Types of Processes Isothermal Processes:
Transformations at constant temperature (dT = 0)
Isochoric Processes:
Transformations at constant volume (dV = 0 or dα = 0)
Isobaric Processes:
Transformations at constant pressure (dp = 0)
Adiabatic processes:
Transformations without the exchange of heat between the environment
and the system (dQ = 0 or dq = 0)
Thermodynamics
M. D. Eastin

6. Adiabatic Processes Basic Idea:
No heat is added to or taken from the system
which we assume to be an air parcel
Changes in temperature result from either
expansion or contraction
Many atmospheric processes are “dry adiabatic”
We shall see that dry adiabatic process play
a large role in deep convective processes
Vertical motions
Thermals
Parcel
Thermodynamics
M. D. Eastin

7. Adiabatic Processes p i f V P-V Diagrams: Isobar Isochor Adiabat
Isotherm V
Thermodynamics
M. D. Eastin

8. * NOT pronounced like “Poison”
Poisson’s* Relation
A Relationship between Temperature and Pressure:
Begin with:
Substitute for “α” using
the Ideal Gas Law
and rearrange:
Integrate the equation:
Adiabatic Form
of the First Law
* NOT pronounced like “Poison”
See:
Thermodynamics
M. D. Eastin

9. Poisson’s Relation A Relationship between Pressure and Temperature:
After Integrating the equation:
After some simple algebra:
Relates the initial conditions of temperature and pressure to
the final temperature and pressure
Thermodynamics
M. D. Eastin

10. Applications of Poisson’s Relation
Example: Cabin Pressurization
Most jet aircraft are pressurized to 8,000 ft (or 770 mb). If the outside air
temperature at a cruising altitude of 30,000 feet (300 mb) is -40ºC, what is
the temperature inside the cabin?
Thermodynamics
M. D. Eastin

11. Applications of Poisson’s Relation
Example: Cabin Pressurization
Most jet aircraft are pressurized to 8,000 ft (or 770 mb). If the outside air
temperature at a cruising altitude of 30,000 feet (300 mb) is -40ºC, what is
the temperature inside the cabin?
pinitial = 300 mb Rd = 287 J / kg K
pfinal = 770 mb cp = 1004 J / kg K
Tinitial = -40ºC = 233K
Tfinal = ???
Thermodynamics
M. D. Eastin

12. Applications of Poisson’s Relation
Example: Cabin Pressurization
Most jet aircraft are pressurized to 8,000 ft (or 770 mb). If the outside air
temperature at cruising altitude of 30,000 feet (300 mb) is -40ºC, what is
the temperature inside the cabin?
pinitial = 300 mb Rd = 287 J / kg K
pfinal = 770 mb cp = 1004 J / kg K
Tinitial = -40ºC = 233K
Thermodynamics
M. D. Eastin

13. Applications of Poisson’s Relation
Comparing Temperatures at different Altitudes:
Are they relatively warmer or cooler?
Bring the two parcels to the same level
Compress 300 mb air to 600 mb
300 mb
-37oC
600 mb
2oC
Thermodynamics
M. D. Eastin

14. Applications of Poisson’s Relation
Comparing Temperatures at different Altitudes:
Are they relatively warmer or cooler?
pinitial = 300 mb
pfinal = 600 mb
Tinitial = -37ºC = 236 K
Tfinal = 288 K = 15ºC
Note: We could we have chosen
to expand the 600 mb parcel
to 300 mb for the comparison
300 mb
-37oC
600 mb
2oC
15oC
Thermodynamics
M. D. Eastin

15. Potential Temperature
Special form of Poisson’s Relation:
Compress all air parcels to 1000 mb
Provides a “standard”
Avoids using an arbitrary pressure level
Define Tfinal = θ
θ is the potential temperature
where: p0 = 1000 mb
1000 mb
Thermodynamics
M. D. Eastin

16. Applications of Potential Temperature
Comparing Temperatures at different Altitudes:
An aircraft flies over the same location at two different altitudes and makes measurements of pressure and temperature within air parcels at each altitude:
Air parcel #1: p = 900 mb
T = 21ºC
Air Parcel #2: p = 700 mb
T = 0.6ºC
Which parcel is relatively colder? warmer?
Thermodynamics
M. D. Eastin

17. Applications of Potential Temperature
Comparing Temperatures at different Altitudes:
Air Parcel #1: p = 900 mb
T = 21ºC = 294 K
Air Parcel #2: p = 700 mb
T = 0.6ºC = K
The parcels have the same potential temperature!
Are we measuring the same air parcel at two different levels? MAYBE
Thermodynamics
M. D. Eastin

18. Applications of Potential Temperature
Potential Temperature Conservation:
Air parcels undergoing adiabatic transformations
maintain a constant potential temperature (θ)
During adiabatic ascent (expansion) the parcel’s
temperature must decrease in order to preserve
the parcel’s potential temperature
During adiabatic descent (compression) the parcel’s
temperature must increase in order to preserve
Constant θ
Thermodynamics
M. D. Eastin

19. Applications of Potential Temperature
Potential Temperature as an Air Parcel Tracer:
Therefore, under dry adiabatic conditions, potential
temperature can be used as a tracer of air motions
Track air parcels moving up and down (thermals)
Track air parcels moving horizontally (advection)
Constant θ
Constant θ
Thermodynamics
M. D. Eastin

20. Dry Adiabatic Lapse Rate?
How does Temperature change with Height for a Rising Thermal?
Potential temperature is a function of pressure and temperature: θ(p,T)
We know the relationship between pressure (p) and altitude (z):
We can use this hydrostatic relation and
the adiabatic form of the first law to obtain
a relationship between temperature and
height when potential temperature is
conserved (dry adiabatic lapse rate)
Hydrostatic
Relation
(more on this later) z
Dry Adiabatic Lapse Rate?
Adiabatic Form
of the First Law T
Thermodynamics
M. D. Eastin

21. Dry Adiabatic Lapse Rate
How does Temperature change with Height for a Rising Thermal?
Begin with the first law:
Substitute for “α” using
the Ideal Gas Law
and rearrange:
Divide each side by “dz”:
Substitute for “dp/dz”
using the hydrostatic
relation and re-arrange:
Thermodynamics
M. D. Eastin

22. Dry Adiabatic Lapse Rate
How does Temperature change with Height for a Rising Thermal?
Substitute for “ρ” using
the Ideal Gas Law
and cancel terms:
We have arrived at the Dry Adiabatic Lapse Rate (Γd):
Thermodynamics
M. D. Eastin

23. Application of the Dry Adiabatic Lapse Rate
Example: Temperature Change within a Rising Thermal
A parcel originating at the surface (z = 0 m, T = 25ºC) rises to the top of the
mixed boundary layer (z = 800 m). What is the parcel’s new air temperature?
Mixed Layer
Constant θ
Thermodynamics
M. D. Eastin

24. Adiabatic Processes Summary: Adiabatic Processes Poisson’s Relation
Review of The First Law of Thermodynamics
Adiabatic Processes
Poisson’s Relation
Applications
Potential Temperature
Dry Adiabatic Lapse Rate
Thermodynamics
M. D. Eastin

25. References Thermodynamics M. D. Eastin
Petty, G. W., 2008: A First Course in Atmospheric Thermodynamics, Sundog Publishing, 336 pp.
Tsonis, A. A., 2007: An Introduction to Atmospheric Thermodynamics, Cambridge Press, 197 pp.
Wallace, J. M., and P. V. Hobbs, 1977: Atmospheric Science: An Introductory Survey, Academic Press, New York, 467 pp.
Thermodynamics
M. D. Eastin