1. ThermodynamicsM. D. Eastin Water Vapor in the Air How do we compute the dewpoint temperature, the relative humidity, or the temperature of a rising air parcel inside a cloud? Here we investigate parameters that describe water in our atmosphere

2. ThermodynamicsM. D. Eastin Outline:  Review of the Clausius-Clapeyron Equation  Review of our Atmosphere as a System  Basic parameters that describe moist air  Definitions  Application: Use of Skew-T Diagrams  Parameters that describe atmospheric processes for moist air  Isobaric Cooling  Adiabatic – Isobaric processes  Adiabatic expansion (or compression)  Unsaturated  Saturated  Application: Use of Skew-T Diagrams  Additional useful parameters  Summary Water Vapor in the Air

3. ThermodynamicsM. D. Eastin Basic Idea: Provides the mathematical relationship (i.e., the equation) that describes any equilibrium state of water as a function of temperature and pressure. Accounts for phase changes at each equilibrium state (each temperature) Review of Clausius-Clapeyron Equation Sublimation Fusion Vaporization T C T (ºC) p (mb) 3741000 6.11 1013 221000 Liquid Vapor Solid V P (mb) Vapor Liquid and Vapor T e sw Sections of the P-V and P-T diagrams for which the Clausius-Clapeyron equation is derived in the following slides

4. ThermodynamicsM. D. Eastin Mathematical Representation: Application of the Carnot Cycle… where: T= Temperature of the system l = Latent heat for given phase change dp s = Change in system pressure at saturation dT = Change in system temperature Δα = Change in specific volumes between the two phases Sublimation Fusion Vaporization T C T (ºC) p (mb) 3741000 6.11 1013 221000 Liquid Vapor Solid Review of Clausius-Clapeyron Equation

5. ThermodynamicsM. D. Eastin Computing saturation vapor pressure for a given temperature: Version #1: Assumes constant latent heat of vaporization ( l v = constant) Less accurate at extreme temperatures Version #2: Accounts for temperature dependence of the latent heat [ l v (T)] Most accurate across the widest range of temperatures Review of Clausius-Clapeyron Equation

6. ThermodynamicsM. D. Eastin Our atmosphere is a heterogeneous closed system consisting of multiple sub-systems We will now begin to account for the entire system… Review of Systems Water Vapor e, T, ρ v, m v, R v Open sub-system Ice Water p i, T, ρ i, m i Open sub-system Dry Air (gas) p d, T, ρ d, m d, R d Closed sub-system Liquid Water p w, T, ρ w, m w Open sub-system Energy Exchange Mass Exchange

7. ThermodynamicsM. D. Eastin Our Approach: Apply what we have learned thus far:Equation of State First Law of Thermodynamics Second Law of Thermodynamics Phase and Latent Heats of water Clausius-Clapeyron Equation Learn how to compute:Basic parameters that describe moist air Each parameter using standard observations and/or thermodynamic diagrams (Skew-Ts) What do we regularly observe? Total Pressure (p) Temperature (T) Dewpoint Temperature (T d ) or Relative Humidity (r) Moist Air Parameters

8. ThermodynamicsM. D. Eastin 1. Equations of State for Dry Air and Water Vapor: Water vapor in our atmosphere behaves like an Ideal Gas Ideal Gas → equilibrium state between Pressure, Volume, and Temperature Recall: Water vapor has its own Ideal Gas Law Basic Moisture Parameters Dry Air (N 2 and O 2 )Water Vapor (H 2 O) p d = Partial pressure of dry air ρ d = Density of dry air T = Temperature of dry air R d = Gas constant for dry air ( Based on the mean molecular weights ) ( of the constituents in dry air ) = 287 J / kg K e = Partial pressure of water vapor (called vapor pressure) ρ v = Density of water vapor (called vapor density) T = Temperature of water vapor R v = Gas constant for water vapor ( Based on the mean molecular weights ) ( of the constituents in water vapor ) = 461 J / kg K

9. ThermodynamicsM. D. Eastin 2. Mixing Ratio (w): Definition: Mass of water vapor per unit mass of dry air: We can use the Equation of States for dry air and water vapor with Dalton’s Law of partial pressures to place mixing ratio into variables we either observe or can calculate from observations: How do we find “e” from observations? Basic Moisture Parameters

10. ThermodynamicsM. D. Eastin 2. Mixing Ratio (w): How do we find “e”? Our integrated Clausius-Clapeyron equation Use T d in place of T to find the vapor pressure (e) where:e has units of mb T d has units of K Needed Information for Computation: Observed variables:p, T d Computed variables:e Physical Constants:R d, R v, l v Units:g/kg Basic Moisture Parameters

11. ThermodynamicsM. D. Eastin 3. Saturation Mixing Ratio (w sw ): Definition: Mass of water vapor per unit mass of dry air at saturation Can be interpreted as the amount of water vapor an air parcel would contain at a given temperature and pressure if the parcel was at saturation (with respect to liquid water) How do we find “e sw ” from observations? Basic Moisture Parameters

12. ThermodynamicsM. D. Eastin 3. Saturation Mixing Ratio (w sw ): How do we find “e sw ”? Our integrated Clausius-Clapeyron equation Use T to find the saturation vapor pressure (e sw ) where:e sw has units of mb T has units of K Needed Information for Computation: Observed variables:p, T Computed variables:e sw Physical Constants:R d, R v, l v Units:g/kg Basic Moisture Parameters

13. ThermodynamicsM. D. Eastin 4. Specific Humidity (q): Definition: Mass of water vapor per unit mass of moist air: where: It is closely related to mixing ratio (w): Since both q << 1 and w << 1 in our atmosphere, we often assume Basic Moisture Parameters

14. ThermodynamicsM. D. Eastin 5. Relative Humidity (r): Definition: The ratio (or percentage) of water vapor mass in a moist air parcel to the water vapor mass the parcel would have if it was saturated with respect to liquid water Using the Ideal Gas laws for dry and moist air: Note: How do we find “e” and “e sw ” from observations? Basic Moisture Parameters

15. ThermodynamicsM. D. Eastin 5. Relative Humidity (r): Finding “e” and “e sw ”: where:e and e sw have units of mb T d and T has units of K Needed Information for Computation: Observed variables:T d, T Computed variables:e, e sw Physical Constants: l v, R v Units:% Basic Moisture Parameters

16. ThermodynamicsM. D. Eastin Skew-T Log-P Diagram Isotherm (T=-10ºC) Saturation Mixing Ratio (10 g/kg) Pressure (200 mb) Dry Adiabat (283K) Pseudo-Adiabat (283K)

17. ThermodynamicsM. D. Eastin The Skew-T Log-P Diagram The lines of constant saturation mixing ratio are also skewed toward the upper left These lines are always dashed and straight, but may vary in color Our Version: Pink dashed Lines

18. ThermodynamicsM. D. Eastin Example: Typical surface observations at the Charlotte-Douglas airport in March: p = 1000 mb T = 25ºC T d = 16ºC Find the following using a Skew-T Diagram: Saturation Mixing Ratio (w sw ) Mixing Ratio (w) Specific Humidity (q) Relative Humidity (r) Application: The Skew-T Diagram

19. ThermodynamicsM. D. Eastin Given: p = 1000 mb Saturation Mixing Ratio: T = 26°C T d =18°C 1. Place a large dot at the location that corresponds to (p, T) 2. Obtain value for w sw from the saturation mixing ratio line that corresponds to (p, T) T = 26°C P = 1000 mb w sw = 22 g/kg Application: The Skew-T Diagram

20. ThermodynamicsM. D. Eastin Given: p = 1000 mb Mixing Ratio: T = 26°C Specific Humidity: T d =18°C 1. Place a large dot at the location that corresponds to (p, T d ) 2. Obtain value for w from the saturation mixing ratio line that corresponds to (p, T d ) 3. Compute q using the w value → 0.0123 / (1 + 0.0123) T d = 18°C P = 1000 mb w = 12.3 g/kg q = 12.2 g/kg Application: The Skew-T Diagram

21. ThermodynamicsM. D. Eastin Given: p = 1000 mb Relative Humidity: T = 26°C T d =18°C 1. Place a large dot at the location that corresponds to (p, T d ) 2. Place a large dot at the location that corresponds to (p, T) 3. Obtain value for w and w sw from the saturation mixing ratio lines that corresponds to Td and T, respectively 4. Compute r → 0.0123 / 0.022 Application: The Skew-T Diagram T d = 18°C P = 1000 mb T = 26°C w sw = 22 g/kg r = 56% w = 12.3 g/kg

22. ThermodynamicsM. D. Eastin Our Approach: Examine the following:Isobaric processes (occurring at the surface) Processes involving ascent → Unsaturated → Saturated Learn how to compute:Parameters that are conserved during typical atmospheric processes (isobaric, adiabatic) Each parameter using standard observations and/or thermodynamic diagrams (Skew-Ts) What do we regularly observe? Total Pressure (p) Temperature (T) Dewpoint Temperature (T d ) or Relative Humidity (r) Moist Air Parameters during Processes

23. ThermodynamicsM. D. Eastin Isobaric Cooling: Dew Point Temperature (T d ) Definition: Temperature at which saturation (with respect to liquid water) is reached when an unsaturated moist air parcel is cooled at constant pressure Parcel is a closed system Mass of water vapor and dry air are constant Isobaric transformation Total pressure (p) constant Vapor pressure (e) constant Mixing ratio (w) constant Saturation vapor pressure (e sw ) and saturation mixing ratio (w sw ) change since they are both functions of the temperature Moist Air Parameters during Processes Temperature T2T2 T1T1 e sw1 Vapor pressure TdTd Temperature Cools: T 1 → T 2 e sw2 e e sw (T)

24. ThermodynamicsM. D. Eastin Isobaric Cooling: Dew Point Temperature (T d ) Such a process regularly occurs Radiational cooling near surface Often occurs at night (no solar heating) Can occur at ground level (dew) or through a layer (fog) Moist Air Parameters during Processes

25. ThermodynamicsM. D. Eastin Isobaric Cooling: Dew Point Temperature (T d ) Obtained by integrating the Clausius-Clapeyron equation between our initial [e sw = e sw (T 1 ), T = T 1 ] and final [e sw = e, T = T 2 ] states, solving for T 2, and setting T 1 = T, e/e sw = r, and T 2 = T d (see your text) Needed Information for Computation: Observed variables:T, r Computed variables:----- Physical Constants:R v, l v Units:K Moist Air Parameters during Processes

26. ThermodynamicsM. D. Eastin Given: p = 1000 mb Dew Point Temperature: T = 26°C r = 56% 1. Place a large dot at the location that corresponds to (p, T) 2. Obtain value for w sw from the saturation mixing ratio line that corresponds to (p, T) 3. Compute w using r and w sw → 0.56(0.022) 4. The T d value is the temperature at (p, w) Application: The Skew-T Diagram T d = 18°C P = 1000 mb T = 26°C w sw = 22 g/kg r = 56% w = 12.3 g/kg

27. ThermodynamicsM. D. Eastin Adiabatic Isobaric Process: Wet-Bulb Temperature (T w ) Definition: Temperature at which saturation (with respect to liquid water) is reached when an unsaturated moist air parcel is cooled by the evaporation of liquid water where: w sw is the saturation mixing ratio at T w w is the mixing ratio at T d See your text for the full derivation… Needed Information for Computation: Can not be mathematically solved for without iteration Easiest to solve for graphically on a Skew-T diagram Moist Air Parameters during Processes Important

28. ThermodynamicsM. D. Eastin Moist Air Parameters during Processes Adiabatic Isobaric Process: Wet-Bulb Temperature (T w ) Such a process regularly occurs Evaporational cooling occurs near the surface during light rain The temperature often feels colder when its raining → It is!

29. ThermodynamicsM. D. Eastin Wet-bulb Temperature (T w ): 1. Place a large dot at the location that corresponds to (p, T d ) 2. Place a large dot at the location that corresponds to (p, T) 3. Draw a line from (p, T d ) upward along a saturation mixing ratio line 4. Draw a line from (p, T) upward along a dry adiabat 5. From the intersection point of the two lines, draw another line downward along a pseudo-adiabat to the original pressure (p) 6. The T w is the resulting temperature at that pressure Application: The Skew-T Diagram T d = 6°C P = 1000 mb T = 26°C T w = 15ºC Given: p = 1000 mb T = 26ºC T d = 6ºC

30. ThermodynamicsM. D. Eastin In Class Activity Calculations: Observations from this morning at CLT:p = 1000 mb T = 8.3ºC T d = 2.8ºC Compute: w, q, w sw, r Skew-T Practice: Observations from yesterday afternoon at CLT:p = 1000 mb T = 13.5ºC r = 32% Graphically estimate: T d, T w Write your answers on a sheet of paper and turn in by the end of class…

31. ThermodynamicsM. D. Eastin Adiabatic Expansion (or Compression): Moist Potential Temperature (θ m ) Definition: Temperature an unsaturated moist air parcel would have if it were to expand or compress from (p, T) to the 1000 mb level Needed Information for Computation: Observed variables:p, T, T d (or r) Computed variables:e, w, q (also e sw if using r) Physical Constants:c p, R d, R v, l v Units:K Moist Air Parameters during Processes

32. ThermodynamicsM. D. Eastin Adiabatic Expansion (or Compression): Moist Potential Temperature (θ m ) Note: Since q << 1 in our atmosphere, the difference between the moist potential temperature ( θ m ) and the dry potential temperature ( θ ) is extremely small Therefore: The two are essentially equal: The moist potential temperature ( θ m ) is rarely used in practice Rather, the dry potential temperature ( θ ) is used Moist Air Parameters during Processes

33. ThermodynamicsM. D. Eastin Reaching Saturation by Adiabatic Ascent: An unsaturated air parcel that rises adiabatically will cool via expansion During the parcel’s ascent the following occurs: Potential temperature remains constant Moisture content (w or q) remains constant Saturation vapor pressure (e sw ) decreases Saturation mixing ratio (w sw ) decreases Relative humidity (r) increases Eventually:  Relative humidity will reach 100% and saturation occurs  Condensation must take place to maintain the equilibrium Lifting Condensation Level (LCL): Definition: Level were an ascending unsaturated moist air parcel first achieves saturation due to adiabatic cooling and condensation begins to occur Moist Air Parameters during Processes

34. ThermodynamicsM. D. Eastin Reaching Saturation by Adiabatic Ascent: Where is the typical Lifting Condensation Level (LCL)? Moist Air Parameters during Processes LCL Cloud Base Rising unsaturated parcels cool to saturation

35. ThermodynamicsM. D. Eastin Temperature at the Lifting Condensation Level (T LCL ): Definition: Temperature at which an ascending unsaturated moist air parcel first achieves saturation due to adiabatic cooling and condensation begins to occur See your text for the full derivation… Needed Information for Computation: Observed variables:T, r (or T d ) Computed variables:----- (e, e sw if using T d ) Physical Constants:----- Units:K Moist Air Parameters during Processes

36. ThermodynamicsM. D. Eastin Temperature of the Lifting Condensation Level (T LCL ): 1. Place a large dot at the location that corresponds to (p, T d ) 2. Place a large dot at the location that corresponds to (p, T) 3. Draw a line from (p, T d ) upward along a saturation mixing ratio line 4. Draw a line from (p, T) upward along a dry adiabat 5. The T LCL is found at the intersection point of the two lines 6. The corresponding pressure p LCL also defines the LCL Application: The Skew-T Diagram T d = 6°C P = 1000 mb T = 26°C T LCL = 2ºC Given: p = 1000 mb T = 26ºC T d = 6ºC P LCL = 745 mb

37. ThermodynamicsM. D. Eastin Saturated (Moist) Adiabatic Ascent:  Once saturation is achieved (at the LCL), further ascent produces additional cooling (adiabatic expansion) and condensation must occur  Cloud drops begin to form! Two Extreme Possibilities: 1. Condensation Remains  All liquid water stays with the rising air parcel  Implies no precipitation Closed system → no mass exchanged with environment Adiabatic → no heat exchanged with environment Reversible process → if the parcel descends, drops evaporate Implies no entrainment mixing Moist Air Parameters during Processes

38. ThermodynamicsM. D. Eastin Saturated (Moist) Adiabatic Ascent:  Once saturation is achieved (at the LCL), further ascent produces additional cooling (adiabatic expansion) and condensation must occur  Cloud drops begin to form! Two Extreme Possibilities: 2. Condensation is Removed  All condensed water falls out of rising air parcel  Parcel always consists of only dry air and water vapor  Implies heavy precipitation and no cloud drops Open system → Condensed water mass removed from system → Irreversible process Pseudo-adiabatic → No heat exchanged with environment → No dry air mass exchanged → No water vapor exchanged Implies no entrainment mixing Moist Air Parameters during Processes

39. ThermodynamicsM. D. Eastin Saturated (Moist) Adiabatic Ascent: Which one occurs in reality? Moist Air Parameters during Processes Clouds with no precipitation Shallow No loss of condensed water Experience some entrainment Ascent is almost reversible Clouds with precipitation Shallow or Deep Loss of condensed water Experience some entrainment Ascent is almost pseudo-adiabatic

40. ThermodynamicsM. D. Eastin Reversible Equivalent Potential Temperature ( θ e ): Definition: Temperature an unsaturated moist parcel would have if it: Dry adiabatically ascends to saturation (to its LCL) Moist adiabatically ascends until all water vapor was condensed and retained within the parcel Dry adiabatically descends to 1000 mb where: Needed Information for Computation: Difficult to compute for since m w is unknown Can be computed if m w is observed (e.g. by radar) or estimated Moist Air Parameters during Processes Important Cannot be determined on a Skew-T diagram

41. ThermodynamicsM. D. Eastin Pseudo-Adiabatic Equivalent Potential Temperature ( θ ep ): Definition: Temperature an unsaturated moist parcel would have if it: Dry adiabatically ascends to saturation (to its LCL) Moist adiabatically ascends until all water vapor was condensed and falls out of the parcel Dry adiabatically descends to 1000 mb Needed Information for Computation: Observed variables:p, T, T d, r Computed variables:e, w, T LCL Physical Constants:R d, R v, l v Units:K Moist Air Parameters during Processes

42. ThermodynamicsM. D. Eastin Pseudo-Adiabatic Equivalent Potential Temperature ( θ ep ): 1. Place large dots at the locations that correspond to (p, T d ) and (p, T) 2. Draw a line from (p, T d ) upward along a saturation mixing ratio line 3. Draw a line from (p, T) upward along a dry adiabat 4. From the intersection point of the two lines, draw another line upward along a pseudo-adiabat until it parallels the dry adiabats 5. From this “parallel point” (where all vapor has been condensed) draw a line downward along a dry adiabat to 1000 mb. 6. The θ ep is the resulting temperature at 1000 mb. Application: The Skew-T Diagram T d = 2°C P = 1000 mb T = 22°C θ ep = 307 K (34ºC + 273) Given: p = 1000 mb T = 22ºC T d = 2ºC

43. ThermodynamicsM. D. Eastin Saturated (Moist) Adiabatic Descent:   A descending saturated air parcel will warm (adiabatic compression)  The amount of temperature increase will depend on whether condensed water is present in the parcel Two possible scenarios; 1. Parcel does not contain condensed water The parcel immediately become unsaturated Dry adiabatic descent occurs Potential temperature ( θ ) remains constant Mixing ratio (w) remains constant Similar to the final leg of determining θ ep on the Skew-T diagram Moist Air Parameters during Processes

44. ThermodynamicsM. D. Eastin Saturated (Moist) Adiabatic Descent:   A descending saturated air parcel will warm (adiabatic compression)  The amount of temperature increase will depend on whether condensed water is present in the parcel Two possible scenarios; 2. Parcel does contain condensed water Initial descent warms air to a unsaturated state Produces an unstable state for the condensed water drops Some water drops evaporate → cools the air parcels → moistens the air parcel → brings parcel back to saturation Subsequent descent requires additional droplet evaporation in order to maintain the saturated state  Saturated descent can occur as long as condensed water is present  Once all the condensed water evaporates → dry-adiabatic descent Moist Air Parameters during Processes

45. ThermodynamicsM. D. Eastin Wet-Bulb Potential Temperature ( θ w ): Definition: Temperature a saturated moist air parcel that contains condensed water would have if it descends adiabatically to 1000 mb where: w is the mixing ratio at θ w See your text for the full derivation… Needed Information for Computation: Can not be mathematically solved for without iteration Easiest to solve for graphically on a Skew-T diagram Moist Air Parameters during Processes Important

46. ThermodynamicsM. D. Eastin Wet-bulb Potential Temperature ( θ w ): 1. Place a large dot at the location that corresponds to (p, T d ) 2. Place a large dot at the location that corresponds to (p, T) 3. Draw a line from (p, T d ) upward along a saturation mixing ratio line 4. Draw a line from (p, T) upward along a dry adiabat 5. From the intersection point of the two lines, draw another line downward along a pseudo-adiabat to 1000 mb 6. The θ w is the resulting temperature at 1000 mb Application: The Skew-T Diagram T d = -11°C P = 1000 mb T = 9°C Given: p = 700 mb T = 9ºC T d = -11ºC P = 700 mb θ w = 288 K (15ºC + 273)

47. ThermodynamicsM. D. Eastin Equation of State for Moist Air: Obtained by combining the Equations of State for both dry air and water vapor with the mixing ratio and specific humidity (see your text) where: Advantage: Defines total density (combinations of dry air and water vapor) Used to more easily define the total density gradients that determine atmospheric stability (or parcel buoyancy) Will use more in next chapter… Additional Parameters

48. ThermodynamicsM. D. Eastin Virtual Temperature (T v ): Definition: The temperature a moist air parcel would have if the parcel contained no water vapor (i.e. vapor was replaced by dry air) See your text for the full derivation… Advantage: Simple way to account for variable moisture in an air parcel Will use more in next chapter… Needed Information for Computation: Observed variables:p, T, T d (or r) Computed variables:e, w, q Physical Constants:R d, R v, l v Units:K Additional Parameters Cannot be determined on a Skew-T diagram

49. ThermodynamicsM. D. Eastin Virtual Potential Temperature (θ v ) Definition: Temperature a moist air parcel would have if it were to expand or compress from (p, T v ) to the 1000 mb level, and the parcel contained no water vapor (i.e. vapor was replaced by dry air) Advantage: Similar to θ and θ m but accounts for variable moisture in a parcel Used to define atmospheric stability Will use more in next chapter… Needed Information for Computation: Observed variables:p, T, T d (or r) Computed variables:e, w, q Physical Constants:c p, R d, R v, l v Units:K Additional Parameters Cannot be determined on a Skew-T diagram

50. ThermodynamicsM. D. Eastin Summary: Relationship of Parameters Lots of Temperatures! Each temperature defines the state of an air parcel at a single location Differences result from → Whether moisture is included → Type of process involved Lots of Potential Temperatures! Each potential temperature defines the state of an air parcel at 1000 mb Differences result from → Whether moisture is included → Type of process involved

51. ThermodynamicsM. D. Eastin Can be used to estimate (or simplify the computation of): Mixing ratio (w) Saturation mixing ratio (w sw ) Relative humidity (r) Specific humidity (q) Potential temperature ( θ ) Wet-bulb temperature (T w ) Note: All parameter symbols are color-coded with their locations Summary: The Skew-T Diagram T d, w P = 1000 mb T, w sw Given: p = 800 mb T = 9.5ºC T d = -8.0ºC P = 800 mb θwθw T LCL θ ep θ TwTw P LCL Temperature at the LCL (T LCL ) Pressure at the LCL (P LCL ) Wet-bulb potential temperature ( θ w ) Pseudo-adiabatic equivalent potential temperature ( θ ep )

52. ThermodynamicsM. D. Eastin Review: Review of the Clausius-Clapeyron Equation Review of our Atmosphere as a System Basic parameters that describe moist air Definitions Application: Use of Skew-T Diagrams Parameters that describe atmospheric processes for moist air Isobaric Cooling Adiabatic – Isobaric processes Adiabatic expansion (or compression) Unsaturated Saturated Application: Use of Skew-T Diagrams Additional useful parameters Summary Water Vapor in the Air

53. ThermodynamicsM. D. Eastin References Petty, G. W., 2008: A First Course in Atmospheric Thermodynamics, Sundog Publishing, 336 pp. Tsonis, A. A., 2007: An Introduction to Atmospheric Thermodynamics, Cambridge Press, 197 pp. Wallace, J. M., and P. V. Hobbs, 1977: Atmospheric Science: An Introductory Survey, Academic Press, New York, 467 pp. Also (from course website): NWSTC Skew-T Log-P Diagram and Sounding Analysis, National Weather Service, 2000 The Use of the Skew-T Log-P Diagram in Analysis and Forecasting, Air Weather Service, 1990