1. Prof. Fred Remer University of North Dakota Thermodynamic Diagrams -20 -30 -40 -50 Pressure (mb) 1000 900 800 700 600 500 300 200 400 Temperature ( o C) 30 40 20 10 0 -10 -60

2. Prof. Fred Remer University of North Dakota Thermodynamic Diagrams ReadingReading –Hess Chapter 5Chapter 5 –pp 65 – 74 –Tsonis pp 143 – 150pp 143 – 150 –Air Weather Service, AWS/TR-79/006 –Wallace & Hobbs pp 78 – 79pp 78 – 79

3. Prof. Fred Remer University of North Dakota Thermodynamic Diagrams ObjectivesObjectives –Be able to list the three desirable characteristics of a thermodynamic diagram –Be able to describe how a transformation is made from p,  coordinates when designing a thermodynamic diagram

4. Prof. Fred Remer University of North Dakota Thermodynamic Diagrams ObjectivesObjectives –Be able to list the coordinates of each thermodynamic diagram –Be able to describe the advantages and disadvantages of each thermodynamic diagram

5. Prof. Fred Remer University of North Dakota Thermodynamic Diagrams Provide a graphical representation of thermodynamic processes in the atmosphereProvide a graphical representation of thermodynamic processes in the atmosphere

6. Prof. Fred Remer University of North Dakota Thermodynamic Diagrams Thermodynamic Processes?Thermodynamic Processes? –Isobaric –Isothermal –Dry Adiabatic –Pseudoadiabatic –Constant Mass

7. Prof. Fred Remer University of North Dakota Thermodynamic Diagrams Thermodynamic DiagramsThermodynamic Diagrams –Eliminates or simplifies calculations

8. Prof. Fred Remer University of North Dakota Thermodynamic Diagrams Most SimplisticMost Simplistic Temperature ( o C) Pressure (mb) 1020300-10-20 1000 900 800 700 600 500 400 Temp. DewPoint

9. Prof. Fred Remer University of North Dakota Thermodynamic Diagrams Not very usefulNot very useful Temperature ( o C) Pressure (mb) 1020300-10-20 1000 900 800 700 600 500 400 Temp. DewPoint

10. Prof. Fred Remer University of North Dakota Thermodynamic Diagrams Desirable CharacteristicsDesirable Characteristics –Area Equivalent Area enclosed by a cyclic process is proportional to energyArea enclosed by a cyclic process is proportional to energy

11. Prof. Fred Remer University of North Dakota Desirable Characteristics -20 -30 -40 -50 Pressure (mb) 1000 900 800 700 600 500 300 200 400 Temperature ( o C) 30 40 20 10 0 -10 -60

12. Prof. Fred Remer University of North Dakota Desirable Characteristics As many isopleths as possible be straight linesAs many isopleths as possible be straight lines

13. Prof. Fred Remer University of North Dakota Desirable Characteristics Pressure (mb) 1000 900 800 700 600 500 300 200 400 Temperature ( o C) 30 40 20 10 0 -10 -20 -30 -40 -50 -60

14. Prof. Fred Remer University of North Dakota Desirable Characteristics The angle between isotherms and adiabats be as large as possibleThe 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 environmentEasier to determine stability of the environment –90 o Optimum

15. Prof. Fred Remer University of North Dakota Desirable Characteristics Pressure (mb) Temperature ( o C) 30 40 20 10 0 -10 -20 -30 -40 -50 -60

16. Prof. Fred Remer University of North Dakota Coordinates Select so that it satisfies Area Equivalent characteristicSelect so that it satisfies Area Equivalent characteristic –Enclosed area is proportional to energy Use p & Use p & 

17. Prof. Fred Remer University of North Dakota Coordinates Known as Clapeyron DiagramKnown as Clapeyron Diagram Small angle between T & Small angle between T &  T2T2T2T2 T1T1T1T1 P  1000 mb 1111 2222 DryAdiabats

18. Prof. Fred Remer University of North Dakota Coordinates Equal Area TransformationEqual Area Transformation –Consider two other variables A & B P  A B

19. Prof. Fred Remer University of North Dakota Coordinates Equal Area TransformationEqual Area Transformation –Create a transformation from -p,  to A, B P  A B

20. Prof. Fred Remer University of North Dakota Equal Area Transformation A B P 

21. Prof. Fred Remer University of North Dakota Equal Area Transformation Closed integral cannot equal zero unless it is an exact differentialClosed integral cannot equal zero unless it is an exact differential

22. Prof. Fred Remer University of North Dakota Equal Area Transformation Differentiate s with respect to  and BDifferentiate s with respect to  and B So...So...

23. Prof. Fred Remer University of North Dakota Equal Area Transformation Differentiate p with respect to BDifferentiate p with respect to B Differentiate A with respect to Differentiate A with respect to 

24. Prof. Fred Remer University of North Dakota Equal Area Transformation So…So…

25. Prof. Fred Remer University of North Dakota Equal Area Transformation Specify B, can determine ASpecify B, can determine A Equal Area maintainedEqual Area maintained

26. Prof. Fred Remer University of North Dakota Emagram Energy per Unit Mass DiagramEnergy per Unit Mass Diagram Set B = TSet B = T

27. Prof. Fred Remer University of North Dakota Emagram Using the Ideal Gas LawUsing the Ideal Gas Law DifferentiateDifferentiate

28. Prof. Fred Remer University of North Dakota Emagram IntegrateIntegrate

29. Prof. Fred Remer University of North Dakota Emagram Once again, the Equation of StateOnce again, the Equation of State Take the natural logarithmTake the natural logarithm

30. Prof. Fred Remer University of North Dakota Emagram SubstituteSubstitute

31. Prof. Fred Remer University of North Dakota Emagram Select f(t) such thatSelect f(t) such that Finally … coordinates A & B are …Finally … coordinates A & B are …

32. Prof. Fred Remer University of North Dakota Emagram 1000 mb 800 mb 600 mb 400 mb -20 o C 0oC0oC0oC0oC 20 o C 40 o C -40 o C Pressure Temperature -20 o C  = 0 o C 20 o C 40 o C 60 o C 80 o C 100 o C eeee w

33. Prof. Fred Remer University of North Dakota Emagram Area proportional to energyArea proportional to energy Four sets of straight (or nearly straight) linesFour sets of straight (or nearly straight) lines 45 o angle between adiabats and isotherms45 o angle between adiabats and isotherms

34. Prof. Fred Remer University of North Dakota Tephigram T-  DiagramT-  Diagram –Temperature = T –Entropy = 

35. Prof. Fred Remer University of North Dakota Tephigram CoordinatesCoordinates –Similar to Emagram –Different constant of integration

36. Prof. Fred Remer University of North Dakota Tephigram Evaluate f(T) using Potential TemperatureEvaluate f(T) using Potential Temperature Ideal Gas LawIdeal Gas Law Substitute for pSubstitute for p

37. Prof. Fred Remer University of North Dakota Tephigram Take the natural logarithmTake the natural logarithm

38. Prof. Fred Remer University of North Dakota Tephigram Solve for ln Solve for ln 

39. Prof. Fred Remer University of North Dakota Tephigram Solve for ln Solve for ln 

40. Prof. Fred Remer University of North Dakota Tephigram Select f(T)Select f(T) SubstituteSubstitute

41. Prof. Fred Remer University of North Dakota Tephigram SubstituteSubstitute Since g(T) = -f(T)Since g(T) = -f(T)

42. Prof. Fred Remer University of North Dakota Tephigram CoordinatesCoordinates –Similar to Emagram

43. Prof. Fred Remer University of North Dakota Tephigram 1000 mb 800 mb 600 mb 400 mb Pressure -40 o C Temperature -20 o C 0oC0oC0oC0oC  = 0 o C 20 o C 40 o C 60 o C eeee w

44. Prof. Fred Remer University of North Dakota Tephigram Area proportional to energyArea proportional to energy Four sets of straight (or nearly straight) linesFour sets of straight (or nearly straight) lines –Isobars Curved! 90 o angle between adiabats and isotherms90 o angle between adiabats and isotherms

45. Prof. Fred Remer University of North Dakota Skew-T Log-P Modified EmagramModified Emagram –Isotherm-Adiabat angle 90 o Set B = -R lnpSet B = -R lnp

46. Prof. Fred Remer University of North Dakota Skew-T Log-P But...But... So...So... Becomes..Becomes..

47. Prof. Fred Remer University of North Dakota Skew-T Log-P Multiply both sides by d Multiply both sides by d 

48. Prof. Fred Remer University of North Dakota Skew-T Log-P IntegrateIntegrate Ideal Gas LawIdeal Gas Law

49. Prof. Fred Remer University of North Dakota Skew-T Log-P Select f(lnp)Select f(lnp) K = arbitrary constant

50. Prof. Fred Remer University of North Dakota Skew-T Log-P CoordinatesCoordinates –Similar to Emagram

51. Prof. Fred Remer University of North Dakota Skew-T Log-P Pressure (mb) 1000 900 800 700 600 500 300 200 400 Temperature ( o C) 30 40 20 10 0 -10 -20 -30 -40 -50 -60

52. Prof. Fred Remer University of North Dakota Skew-T Log-P Area proportional to energyArea proportional to energy Three set of straight linesThree set of straight lines –One set of gently curved lines –One set of curved lines Adiabat-isotherm angle about 90 oAdiabat-isotherm angle about 90 o

53. Prof. Fred Remer University of North Dakota Stuve Diagram CoordinatesCoordinates

54. Prof. Fred Remer University of North Dakota400500 600 700 800 900 Pressure Temperature 40 o C 20 o C 0oC0oC0oC0oC -20 o C -40 o C Stuve Diagram

55. Prof. Fred Remer University of North Dakota Stuve Diagram Not area equivalentNot area equivalent Four sets of lines that are straight or nearly straightFour sets of lines that are straight or nearly straight Adiabat-isotherm angle 45 oAdiabat-isotherm angle 45 o

56. Prof. Fred Remer University of North Dakota Summary of Diagrams

57. Prof. Fred Remer University of North Dakota Pressure (mb) 1000 900 800 700 600 500 300200400