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300 0 400 20 10 500 30 600 40 Pressure (mb) -1 0 -2 0 -3 0 -4 0 -5 0 200 -6 0 Thermodynamic Diagrams 700 800 900 1000 Prof. Fred Remer University of North Dakota Temperature (o. C)

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 Prof. Fred Remer University of North Dakota

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 Prof. Fred Remer University of North Dakota

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 Prof. Fred Remer University of North Dakota

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

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

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

Thermodynamic Diagrams • Most Simplistic Pressure (mb) 400 500 600 Temp. 700 800 900 1000 Prof. Fred Remer University of North Dakota Dew Point -20 -10 0 10 20 Temperature (o. C) 30

Thermodynamic Diagrams • Not very useful Pressure (mb) 400 500 600 Temp. 700 800 900 1000 Prof. Fred Remer University of North Dakota Dew Point -20 -10 0 10 20 Temperature (o. C) 30

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

300 0 400 20 10 500 30 600 40 Pressure (mb) -1 0 -2 0 -3 0 -4 0 -5 0 200 -6 0 Desirable Characteristics 700 800 900 1000 Prof. Fred Remer University of North Dakota Temperature (o. C)

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

300 0 400 20 10 500 30 600 40 Pressure (mb) -1 0 -2 0 -3 0 -4 0 -5 0 200 -6 0 Desirable Characteristics 700 800 900 1000 Prof. Fred Remer University of North Dakota Temperature (o. C)

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 – 90 o Optimum Prof. Fred Remer University of North Dakota

40 30 20 10 0 Pressure (mb) -1 0 -2 0 -3 0 -4 0 -5 0 -6 0 Desirable Characteristics Prof. Fred Remer University of North Dakota Temperature (o. C)

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

Coordinates • Known as Clapeyron Diagram • Small angle between T & q q 1 T 1 q 2 T 2 P Dry Adiabats 1000 mb a Prof. Fred Remer University of North Dakota

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

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

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

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

Equal Area Transformation • Differentiate s with respect to a and B • So. . . Prof. Fred Remer University of North Dakota

Equal Area Transformation • Differentiate p with respect to B • Differentiate A with respect to a Prof. Fred Remer University of North Dakota

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

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

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

Emagram • Using the Ideal Gas Law • Differentiate Prof. Fred Remer University of North Dakota

Emagram • Integrate Prof. Fred Remer University of North Dakota

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

Emagram • Substitute Prof. Fred Remer University of North Dakota

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

Emagram 400 mb o. C w = 800 mb o. C 40 o. C 20 q Pressure o. C 80 60 qe 600 mb 10 0 o C o. C 0 -2 1000 mb -40 o. C -20 o. C 0 o C Temperature Prof. Fred Remer University of North Dakota 20 o. C 40 o. C

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

Tephigram • T- f Diagram – Temperature = T – Entropy = f Prof. Fred Remer University of North Dakota

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

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

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

Tephigram • Solve for lna Prof. Fred Remer University of North Dakota

Tephigram • Solve for lna Prof. Fred Remer University of North Dakota

Tephigram • Select f(T) • Substitute Prof. Fred Remer University of North Dakota

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

Tephigram • Coordinates – Similar to Emagram Prof. Fred Remer University of North Dakota

-2 0 o C Temperature o. C 60 o. C 40 q 800 mb w o. C 20 Pressure qe 600 mb = o. C 0 1000 mb Prof. Fred Remer University of North Dakota 0 o C 400 mb -4 0 o C Tephigram

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

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

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

Skew-T Log-P • Multiply both sides by da Prof. Fred Remer University of North Dakota

Skew-T Log-P • Integrate • Ideal Gas Law Prof. Fred Remer University of North Dakota

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

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

300 0 400 20 10 500 30 600 40 Pressure (mb) -1 0 -2 0 -3 0 -4 0 -5 0 200 -6 0 Skew-T Log-P 700 800 900 1000 Prof. Fred Remer University of North Dakota Temperature (o. C)

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

Stuve Diagram • Coordinates Prof. Fred Remer University of North Dakota

Stuve Diagram 400 Pressure 500 600 700 800 900 -40 o. C Prof. Fred Remer University of North Dakota -20 o. C Temperature 40 o. C

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

Summary of Diagrams Prof. Fred Remer University of North Dakota