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Understanding Thermodynamic Diagrams for Atmospheric Processes

Explore the key characteristics, advantages, and transformations involved in using thermodynamic diagrams to depict atmospheric processes. Learn about different types of diagrams and their applications.

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Understanding Thermodynamic Diagrams for Atmospheric Processes

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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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