Water Vapor in the Air

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 M. D. Eastin

Water Vapor in the Air • 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 M. D. Eastin

p (mb) C 221000 Fusion Liquid Vaporization Solid 1013 6.11 T Sublimation Vapor 0 100 374 T (ºC) Review of Clausius-Clapeyron Equation • 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) P (mb) Vapor esw T Liquid Sections of the P-V and P-T diagrams for which the Clausius-Clapeyron equation is derived in the following slides Liquid and Vapor V M. D. Eastin

p (mb) C 221000 Fusion Liquid Vaporization Solid 1013 6.11 T Sublimation Vapor 0 100 374 T (ºC) Review of Clausius-Clapeyron Equation • Mathematical Representation: • Application of the Carnot Cycle… • where: T = Temperature of the system • l = Latent heat for given phase change • dps = Change in system pressure at saturation • dT = Change in system temperature • Δα = Change in specific volumes between • the two phases M. D. Eastin

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

Liquid Water pw, T, ρw, mw Open sub-system Dry Air (gas) pd, T, ρd, md, Rd Closed sub-system Water Vapor e, T, ρv, mv, Rv Open sub-system Ice Water pi, T, ρi, mi Open sub-system Energy Exchange Mass Exchange Review of Systems • Our atmosphere is a heterogeneous • closed system consisting of multiple • sub-systems • We will now begin to account for the • entire system… M. D. Eastin

Moist Air Parameters • 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 (Td) • or • Relative Humidity (r) M. D. Eastin

Basic Moisture Parameters • 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 Dry Air (N2 and O2) Water Vapor (H2O) pd = Partial pressure of dry air ρd = Density of dry air T = Temperature of dry air Rd = 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) ρv = Density of water vapor (called vapor density) T = Temperature of water vapor Rv = Gas constant for water vapor ( Based on the mean molecular weights ) ( of the constituents in water vapor ) = 461 J / kg K M. D. Eastin

Basic Moisture Parameters • 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? M. D. Eastin

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

Basic Moisture Parameters • 3. Saturation Mixing Ratio (wsw): • 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 “esw” • from observations? M. D. Eastin

Basic Moisture Parameters • 3. Saturation Mixing Ratio (wsw): • How do we find “esw”? Our integrated Clausius-Clapeyron equation • Use T to find the saturation vapor pressure (esw) • where: esw has units of mb • T has units of K • Needed Information for Computation: • Observed variables: p, T • Computed variables: esw • Physical Constants: Rd, Rv, lv • Units: g/kg M. D. Eastin

Basic Moisture Parameters • 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 M. D. Eastin

Basic Moisture Parameters 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 “esw” from observations? M. D. Eastin

Basic Moisture Parameters • 5. Relative Humidity (r): • Finding “e” and “esw”: • where: e and esw have units of mb • Td and T has units of K • Needed Information for Computation: • Observed variables: Td, T • Computed variables: e, esw • Physical Constants: lv, Rv Units: % M. D. Eastin

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

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

Application: The Skew-T Diagram Given: p = 1000 mb Saturation Mixing Ratio: T = 25°C Td =16°C 1. Place a large dot at the location that corresponds to (p, T) 2. Obtain value for wsw from the saturation mixing ratio line that corresponds to (p, T) wsw = 22 g/kg T = 25°C P = 1000 mb M. D. Eastin

Application: The Skew-T Diagram Given: p = 1000 mb Mixing Ratio: T = 25°C Specific Humidity: Td =16°C 1. Place a large dot at the location that corresponds to (p, Td) 2. Obtain value for w from the saturation mixing ratio line that corresponds to (p, Td) 3. Compute q using the w value → 0.0123 / (1 + 0.0123) w = 12.3 g/kg q = 12.2 g/kg Td = 16°C P = 1000 mb M. D. Eastin

Application: The Skew-T Diagram Given: p = 1000 mb Relative Humidity: T = 25°C Td =16°C 1. Place a large dot at the location that corresponds to (p, Td) 2. Place a large dot at the location that corresponds to (p, T) 3. Obtain value for w and wsw from the saturation mixing ratio lines that corresponds to Td and T, respectively 4. Compute r → 0.0123 / 0.022 r = 56% w = 12.3 g/kg wsw = 22 g/kg Td = 16°C T = 25°C P = 1000 mb M. D. Eastin

Moist Air Parameters during Processes • 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 (Td) • or • Relative Humidity (r) M. D. Eastin

Temperature Cools: T1→ T2 esw(T) esw1 Td Vapor pressure esw2 e T2 T1 Temperature Moist Air Parameters during Processes • Isobaric Cooling: Dew Point Temperature (Td) • 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 (esw) and • saturation mixing ratio (wsw) change • since they are both functions of • the temperature M. D. Eastin

Moist Air Parameters during Processes • Isobaric Cooling: Dew Point Temperature (Td) • 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) M. D. Eastin

Moist Air Parameters during Processes Isobaric Cooling: Dew Point Temperature (Td) Obtained by integrating the Clausius-Clapeyron equation between our initial [esw = esw(T1), T = T1] and final [esw = e, T = T2] states, solving for T2, and setting T1 = T, e/esw = r, and T2 = Td (see your text) Needed Information for Computation: Observed variables: T, r Computed variables: ----- Physical Constants: Rv, lv Units: K M. D. Eastin

Application: The Skew-T Diagram Given: p = 1000 mb Dew Point Temperature: T = 25°C r = 56% 1. Place a large dot at the location that corresponds to (p, T) 2. Obtain value for wsw from the saturation mixing ratio line that corresponds to (p, T) 3. Compute w using randwsw → 0.56(0.022) 4. The Tdvalue is the temperature at (p, w) r = 56% w = 12.3 g/kg wsw = 22 g/kg Td = 16°C T = 25°C P = 1000 mb M. D. Eastin

Moist Air Parameters during Processes • Adiabatic Isobaric Process: Wet-Bulb Temperature (Tw) • 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: wsw is the saturation mixing ratio at Tw • w is the mixing ratio at Td • 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 Important M. D. Eastin

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

Application: The Skew-T Diagram Wet-bulbTemperature (Tw): 1. Place a large dot at the location that corresponds to (p, Td) 2. Place a large dot at the location that corresponds to (p, T) 3. Draw a line from (p, Td) 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 Tw is the resulting temperature at that pressure Given: p = 1000 mb T = 25ºC Td = 6ºC Tw = 14ºC Td = 6°C T = 25°C P = 1000 mb M. D. Eastin

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

Moist Air Parameters during Processes Adiabatic Expansion (or Compression): Moist Potential Temperature (θm) 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, Td (or r) Computed variables: e, w, q (also esw if using r) Physical Constants: cp, Rd, Rv, lv Units: K M. D. Eastin

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

Moist Air Parameters during Processes • 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 (esw) decreases • Saturation mixing ratio (wsw) 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 M. D. Eastin

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

Moist Air Parameters during Processes Temperature at the Lifting Condensation Level (TLCL): 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 Td) Computed variables: ----- (e, esw if using Td) Physical Constants: ----- Units: K M. D. Eastin

Application: The Skew-T Diagram Temperature of the Lifting Condensation Level (TLCL): 1. Place a large dot at the location that corresponds to (p, Td) 2. Place a large dot at the location that corresponds to (p, T) 3. Draw a line from (p, Td) upward along a saturation mixing ratio line 4. Draw a line from (p, T) upward along a dry adiabat 5. The TLCL is found at the intersection point of the two lines 6. The corresponding pressure pLCL also defines the LCL Given: p = 1000 mb T = 25ºC Td = 6ºC TLCL = 2ºC PLCL = 740 mb Td = 6°C T = 25°C P = 1000 mb M. D. Eastin

Moist Air Parameters during Processes • 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 M. D. Eastin

Moist Air Parameters during Processes • 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 M. D. Eastin

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

Moist Air Parameters during Processes • Reversible Equivalent Potential Temperature (θe): • 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 retainedwithin the parcel • Dry adiabatically descends to 1000 mb • where: • Needed Information for Computation: • Difficult to compute for since mw is unknown • Can be computed if mw is observed • (e.g. by radar) or estimated Important Cannot be determined on a Skew-T diagram M. D. Eastin

Moist Air Parameters during Processes • Pseudo-Adiabatic Equivalent Potential Temperature (θe): • 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, Td, r • Computed variables: e, w, TLCL • Physical Constants: Rd, Rv, lv • Units: K M. D. Eastin

Application: The Skew-T Diagram Pseudo-Adiabatic Equivalent Potential Temperature (θep): 1. Place large dots at the locations that correspond to (p, Td) and (p, T) 2. Draw a line from (p, Td) 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 θep is the resulting temperature at 1000 mb. Given: p = 1000 mb T = 25ºC Td = 6ºC θep = 307 K (34ºC + 273) T = 20°C Td = 0°C P = 1000 mb M. D. Eastin

Moist Air Parameters during Processes • 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 θep on the Skew-T diagram M. D. Eastin

Moist Air Parameters during Processes • 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 M. D. Eastin

Moist Air Parameters during Processes • Wet-Bulb Potential Temperature (θw): • 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 θw • 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 Important M. D. Eastin

Application: The Skew-T Diagram Wet-bulb Potential Temperature (θw): 1. Place a large dot at the location that corresponds to (p, Td) 2. Place a large dot at the location that corresponds to (p, T) 3. Draw a line from (p, Td) 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 θw is the resulting temperature at 1000 mb Given: p = 700 mb T = 8ºC Td = -11ºC Td = -11°C T = 8°C P = 700 mb θw = 287 K (14ºC + 273) P = 1000 mb M. D. Eastin

Additional Parameters 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… M. D. Eastin

Additional Parameters Virtual Temperature (Tv): 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, Td (or r) Computed variables: e, w, q Physical Constants: Rd, Rv, lv Units: K Cannot be determined on a Skew-T diagram M. D. Eastin

Additional Parameters Virtual Potential Temperature (θv) Definition: Temperature a moist air parcel would have if it were to expand or compress from (p, Tv) 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 θm 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, Td (or r) Computed variables: e, w, q Physical Constants: cp, Rd, Rv, lv Units: K Cannot be determined on a Skew-T diagram M. 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 M. D. Eastin

Water Vapor in the Air