1. MET 61 1 MET 61 Introduction to Meteorology MET 61 Introduction to Meteorology - Lecture 3 Thermodynamics I Dr. Eugene Cordero San Jose State University Reading: W&H Chapter 3: pg 63-74 Class Outline: Sensible and latent heat First law of thermodynamics Lapse rate Potential temperature

2. MET 61 2 MET 61 Introduction to Meteorology

3. MET 61 3 MET 61 Introduction to Meteorology

4. MET 61 4 MET 61 Introduction to Meteorology

5. MET 61 5 MET 61 Introduction to Meteorology Dry and moist air Universal gas constant: R*=8314 J K -1 kmol -1 Molecular weight: M Problem: Calculate (estimate) the ratio of R d /R v

6. MET 61 6 MET 61 Introduction to Meteorology

7. MET 61 7 MET 61 Introduction to Meteorology Dry and moist air Universal gas constant: R*=8314 J K -1 kmol -1 Molecular weight: M

8. MET 61 8 MET 61 Introduction to Meteorology Dry and moist air Universal gas constant: R*=8314 J K -1 kmol -1 Molecular weight: M M d =29, M w =18

9. MET 61 9 MET 61 Introduction to Meteorology Equation of State Ideal Gas Law; relates the thermodynamic states of a gas  For moist air one can use the virtual temperature; T v =T(1+0.61r) r- water vapor mixing ratio Virtual temperatures allows for the use of R for dry air in ideal gas law. p-Pressure (Pa)  - density (kg m -3 ) R - Gas Constant for dry air (287 J K -1 kg -1 ) T - Temperature (K)

10. MET 61 10 MET 61 Introduction to Meteorology Equation of State Ideal Gas Law; relates the thermodynamic states of a gas  For moist air one can use the virtual temperature; T v =T(1+0.61r) r- water vapor mixing ratio Virtual temperatures allows for the use of R for dry air in ideal gas law. p-Pressure (Pa)  - density (kg m -3 ) R - Gas Constant for dry air (287 J K -1 kg -1 ) T - Temperature (K)

11. MET 61 11 MET 61 Introduction to Meteorology Moist Air Where, e – water vapor pressure  v – specific volume of water vapor R v =461 J K -1 kg -1 The ideal gas law can be applied to water vapor such as:

12. MET 61 12 MET 61 Introduction to Meteorology Moist Air Where, e – water vapor pressure  v – specific volume of water vapor R v =461 J K -1 kg -1 The ideal gas law can be applied to water vapor such as:

13. MET 61 13 MET 61 Introduction to Meteorology Partial Pressure  Dalton’s law of partial pressure: –Total pressure exerted by a mixture of gases is equal to the sum of the partial pressures of the individual molecules. –

14. MET 61 14 MET 61 Introduction to Meteorology Partial Pressure  Dalton’s law of partial pressure: –Total pressure exerted by a mixture of gases is equal to the sum of the partial pressures of the individual molecules. –Partial pressure of gas if pressure exerted is gas alone occupied volume.

15. MET 61 15 MET 61 Introduction to Meteorology

16. MET 61 16 MET 61 Introduction to Meteorology Virtual Temperature  Moist air has (higher/lower) molecular weight than dry air  Thus, R v > or < R d  In principle, R v is hard to calculate since one needs to know how much water vapor is in the air.  Thus, we can use R d, along with a fictitious temperature, T v, called the virtual temperature.

17. MET 61 17 MET 61 Introduction to Meteorology Virtual Temperature  Moist air has (higher/lower) molecular weight than dry air.  Thus, Rv > or < Rd  In principle, R v is hard to calculate since one needs to know how much water vapor is in the air.  Thus, we can use R d, along with a fictitious temperature, T v, called the virtual temperature.

18. MET 61 18 MET 61 Introduction to Meteorology Virtual Temperature Consider temperature T, volume V, density  and mass, m d and m v, representing the mass of dry air and water vapor respectively. The total density is thus: Where  ' represents the partial densities. Now, use the idea gas law for dry air and water vapor

19. MET 61 19 MET 61 Introduction to Meteorology Virtual Temperature Consider temperature T, volume V, density,  and mass, m d and m v, representing the mass of dry air and water vapor respectively. The total density is thus: Where  ' represents the partial densities. Now, use the ideal gas law for dry air and water vapor:

20. MET 61 20 MET 61 Introduction to Meteorology Virtual Temperature Noting that the partial pressure relationship requires: One can combine the above equations to form:

21. MET 61 21 MET 61 Introduction to Meteorology Virtual Temperature Noting that the partial pressure relationship requires: One can combine the above equations to form: Do this on your own time!

22. MET 61 22 MET 61 Introduction to Meteorology Virtual Temperature The ideal gas law can now be written as: T v is called the virtual temperature, which is used for moist air. In this way, the total pressure and density of the moist air are related by the ideal gas equation through the virtual temperature. The virtual temperature can also be approximated as:

23. MET 61 23 MET 61 Introduction to Meteorology Virtual Temperature The ideal gas law can now be written as: T v is called the virtual temperature, which is used for moist air. In this way, the total pressure and density of the moist air are related by the ideal gas equation through the virtual temperature. The virtual temperature can also be approximated as:

24. MET 61 24 MET 61 Introduction to Meteorology Hydrostatic Balance (I) Pressure decrease produces a vertical pressure gradient force. Vertical pressure gradient force is balanced by gravity. p+  p p z

25. MET 61 25 MET 61 Introduction to Meteorology Hydrostatic Balance (I) Pressure decrease produces a vertical pressure gradient force. Vertical pressure gradient force is balanced by gravity. p+  p p z

26. MET 61 26 MET 61 Introduction to Meteorology Hydrostatic Balance (I) Pressure decrease produces a vertical pressure gradient force. Vertical pressure gradient force is balanced by gravity. p+  p p z

27. MET 61 27 MET 61 Introduction to Meteorology Hydrostatic Balance (II) The weight of the atmosphere can be expressed as:

28. MET 61 28 MET 61 Introduction to Meteorology Hydrostatic Balance (II) The weight of the atmosphere can be expressed as:

29. MET 61 29 MET 61 Introduction to Meteorology Geopotential (I) And so, Geopotential is defined as the amount of work required to lift (against gravity) a parcel from sea level. The change in geopotential is:

30. MET 61 30 MET 61 Introduction to Meteorology Geopotential (I) And so, Geopotential is defined as the amount of work required to lift (against gravity) a parcel from sea level. The change in geopotential is:

31. MET 61 31 MET 61 Introduction to Meteorology Geopotential (II) And the geopotential height, labeled as Z is then Geopotential at some height z is thus,

32. MET 61 32 MET 61 Introduction to Meteorology Geopotential (II) And the geopotential height, labeled as Z is then Geopotential at some height z is thus,

33. MET 61 33 MET 61 Introduction to Meteorology Hypsometric Equation From geopotential height relationship, one can derive (see W&H) an expression for the thickness (Z 2 -Z 1 ) Thickness of atmosphere relates to difference between two atmospheric layers; z t (m) = thickness between two pressure levels. H is the scale height.

34. MET 61 34 MET 61 Introduction to Meteorology Hypsometric Equation From geopotential height relationship, one can derive (see W&H) an expression for the thickness (Z 2 -Z 1 ) Thickness of atmosphere relates to difference between two atmospheric layers; z t (m) = thickness between two pressure levels. H is the scale height.

35. MET 61 35 MET 61 Introduction to Meteorology Activity 2 (Due Feb 7 th ) 1.Derive the hypsometric equation starting from hydrostatic balance. Show all steps and make sure you understand how this works. 2.How much oxygen is there at the top of Mt. Everest compared to San Jose? 3.Look at the 200hPa height field from the GFS model analysis. Describe the map and what the units mean and then explain the general distribution of the height field. 4.Exercise 3.24 5.Exercise 3.27 6.Exercise 3.29

36. MET 61 36 MET 61 Introduction to Meteorology Water Vapor The amount of water vapor present in the air can be expressed in a variety of ways: Mixing Ratio: Where m v is the mass of water vapor; m d is the mass of dry air: Units for r are typically given in: (g of water vapor/ kg of air) If there is no condensation or evaporation, then the mixing ratio of an air parcel is a conserved quantity. Note: symbol r is also commonly used for the mixing ratio

37. MET 61 37 MET 61 Introduction to Meteorology Water Vapor The amount of water vapor present in the air can be expressed in a variety of ways: Mixing Ratio: Where m v is the mass of water vapor; m d is the mass of dry air: Units for r are typically given in: (g of water vapor/ kg of air) Typical values; 1-5 g/kg midlatitudes and up to 20 g/kg in the tropics If there is no condensation or evaporation, then the mixing ratio of an air parcel is a conserved quantity. Note: symbol w is also commonly used for the mixing ratio

38. MET 61 38 MET 61 Introduction to Meteorology Virtual Temperature The virtual temperature is the temperature that dry air would need to be so that its density would be the same as air moist air. Because the density of moist air is less than dry air, as the air become more moist, the virtual temperature must increase. Example problem: What is the virtual temperature of air that is 50° F and has a mixing ratio of 5g/kg?