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Adiabatic Processes. 1000 mb. How can the first law really help me forecast thunderstorms?. Adiabatic Processes. Outline: Review of The First Law of Thermodynamics Adiabatic Processes Poisson’s Relation Applications Potential Temperature Applications Dry Adiabatic Lapse Rate

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## Adiabatic Processes

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**Adiabatic Processes**1000 mb How can the first law really help me forecast thunderstorms? M. D. Eastin**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**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**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**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**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**Adiabatic Processes**P-V Diagrams: Isobar p i Isochor Adiabat f Isotherm V M. D. Eastin**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**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**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**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**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**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**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**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**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**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**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**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**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**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**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**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**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**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

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