Adiabatic Processes

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

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

4. Forms of the First Law of Thermodynamics For a gas of mass m For unit mass where: p = pressure U = internal energy V = volume W = work T = temperature Q or q = heat energy α = specific volume n = number of moles 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 (8.3143 J K-1 mol-1) 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) 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 M. D. Eastin

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

8. 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: http://en.wikipedia.org/wiki/Simeon_Poisson 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 oftemperature and pressure to • the final temperature and pressure 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? 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 = ??? 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 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 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 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 • θis the potential temperature • where: p0 = 1000 mb 1000 mb 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? 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 = 273.6 K • The parcels have the same potential temperature! • Are we measuring the same air parcel at two different levels? M. D. Eastin

18. Applications of Potential Temperature • Potential Temperature Conservation: • Air parcels undergoing adiabatic transformations • maintain a constantpotential 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 • the parcel’s potential temperature Constant θ 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 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 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: 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): 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 M. D. Eastin

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

25. References 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. M. D. Eastin