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Thermodynamic Diagrams

-30. -40. -20. -50. -60. 200. -10. 300. 0. Pressure (mb). 400. 10. 20. 500. 30. 600. 40. 700. 800. 900. 1000. Temperature ( o C). Thermodynamic Diagrams. Thermodynamic Diagrams. Reading Hess Chapter 5 pp 65 – 74 Tsonis pp 143 – 150 Air Weather Service, AWS/TR-79/006

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Thermodynamic Diagrams

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  1. -30 -40 -20 -50 -60 200 -10 300 0 Pressure (mb) 400 10 20 500 30 600 40 700 800 900 1000 Temperature (oC) Thermodynamic Diagrams

  2. Thermodynamic Diagrams • Reading • Hess • Chapter 5 • pp 65 – 74 • Tsonis • pp 143 – 150 • Air Weather Service,AWS/TR-79/006 • Wallace & Hobbs • pp 78 – 79

  3. Thermodynamic Diagrams • Objectives • Be able to list the three desirable characteristics of a thermodynamic diagram • Be able to describe how a transformation is made from p, a coordinates when designing a thermodynamic diagram

  4. Thermodynamic Diagrams • Objectives • Be able to list the coordinates of each thermodynamic diagram • Be able to describe the advantages and disadvantages of each thermodynamic diagram

  5. Thermodynamic Diagrams • Provide a graphical representation of thermodynamic processes in the atmosphere

  6. Thermodynamic Diagrams • Thermodynamic Processes? • Isobaric • Isothermal • Dry Adiabatic • Pseudoadiabatic • Constant Mass

  7. Thermodynamic Diagrams • Thermodynamic Diagrams • Eliminates or simplifies calculations

  8. Pressure (mb) Temperature (oC) Thermodynamic Diagrams • Most Simplistic 400 500 600 Temp. 700 800 Dew Point 900 1000 -20 -10 0 10 20 30

  9. Pressure (mb) Temperature (oC) Thermodynamic Diagrams • Not very useful 400 500 600 Temp. 700 800 Dew Point 900 1000 -20 -10 0 10 20 30

  10. Thermodynamic Diagrams • Desirable Characteristics • Area Equivalent • Area enclosed by a cyclic process is proportional to energy

  11. -30 -40 -20 -50 -60 200 -10 300 0 Pressure (mb) 400 10 20 500 30 600 40 700 800 900 1000 Temperature (oC) Desirable Characteristics

  12. Desirable Characteristics • As many isopleths as possible be straight lines

  13. Temperature (oC) Desirable Characteristics -30 -40 -20 -50 -60 200 -10 300 0 Pressure (mb) 400 10 20 500 30 600 40 700 800 900 1000

  14. Desirable Characteristics • The angle between isotherms and adiabats be as large as possible • Sensitivity to the rate of change of temperature with pressure in the vertical • Easier to determine stability of the environment • 90o Optimum

  15. Temperature (oC) Desirable Characteristics -30 -40 -20 -50 -60 -10 0 Pressure (mb) 10 20 30 40

  16. Coordinates • Select so that it satisfies Area Equivalent characteristic • Enclosed area is proportional to energy • Use p & a

  17. Coordinates • Known as Clapeyron Diagram • Small angle between T & q q1 T1 q2 T2 P Dry Adiabats 1000 mb a

  18. P A a B Coordinates • Equal Area Transformation • Consider two other variables A & B

  19. P A a B Coordinates • Equal Area Transformation • Create a transformation from -p, a to A, B

  20. P a Equal Area Transformation A B

  21. Equal Area Transformation • Closed integral cannot equal zero unless it is an exact differential

  22. Equal Area Transformation • Differentiate s with respect to a and B • So ...

  23. Equal Area Transformation • Differentiate p with respect to B • Differentiate A with respect to a

  24. Equal Area Transformation • So…

  25. Equal Area Transformation • Specify B, can determine A • Equal Area maintained

  26. Emagram • Energy per Unit Mass Diagram • Set B = T

  27. Emagram • Using the Ideal Gas Law • Differentiate

  28. Emagram • Integrate

  29. Emagram • Once again, the Equation of State • Take the natural logarithm

  30. Emagram • Substitute

  31. Emagram • Select f(t) such that • Finally … coordinates A & B are …

  32. Emagram 400 mb 100oC 80oC qe 60oC 600 mb 40oC Pressure 20oC w q = 0oC 800 mb -20oC 1000 mb -40oC -20oC 0oC 20oC 40oC Temperature

  33. Emagram • Area proportional to energy • Four sets of straight (or nearly straight) lines • 45o angle between adiabats and isotherms

  34. Tephigram • T- f Diagram • Temperature = T • Entropy = f

  35. Tephigram • Coordinates • Similar to Emagram • Different constant of integration

  36. Tephigram • Evaluate f(T) using Potential Temperature • Ideal Gas Law • Substitute for p

  37. Tephigram • Take the natural logarithm

  38. Tephigram • Solve for lna

  39. Tephigram • Solve for lna

  40. Tephigram • Select f(T) • Substitute

  41. Tephigram • Substitute • Since g(T) = -f(T)

  42. Tephigram • Coordinates • Similar to Emagram

  43. Tephigram Temperature -20oC -40oC 400 mb 60oC 0oC qe 40oC 600 mb Pressure 20oC 800 mb w q = 0oC 1000 mb

  44. Tephigram • Area proportional to energy • Four sets of straight (or nearly straight) lines • Isobars Curved! • 90o angle between adiabats and isotherms

  45. Skew-T Log-P • Modified Emagram • Isotherm-Adiabat angle 90o • Set B = -R lnp

  46. Skew-T Log-P • But... • So ... • Becomes ..

  47. Skew-T Log-P • Multiply both sides by da

  48. Skew-T Log-P • Integrate • Ideal Gas Law

  49. Skew-T Log-P • Select f(lnp) K = arbitrary constant

  50. Skew-T Log-P • Coordinates • Similar to Emagram

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