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

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

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

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 (8.3143 J K-1 mol-1) Thermodynamics M. D. Eastin

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

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

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

* 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: http://en.wikipedia.org/wiki/Simeon_Poisson Thermodynamics M. D. Eastin

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

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

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

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

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

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

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

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

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? MAYBE Thermodynamics M. D. Eastin

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

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

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

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

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

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

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

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