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Equations Sheet MT3230 Supplemental Data 8.4 Spring, 2018 Dr. Sam Miller QUANTITIES RELATED TO ATMOSPHERIC HUMIDITY 1. VAPOR PRESSURE is the partial atmospheric pressure due to the water vapor present. The simple form of the Clausius-Clapeyron equation states: 푙푣 1 1 푒 = 푒0푒푥푝 [ ( − )] 푅푣 푇0 푇푑 where: e = vapor pressure above liquid water [Pa] e0 = reference pressure (611.12 Pa) lv = latent heat of vaporization (≅ 2.5008 x 106 J/kg) (temperature dependent) Rv = individual gas constant for water vapor (461.2 J/(kg K)) T0 = reference temperature (273.16 K) Td = in situ dew point temperature [K] To compute the saturation vapor pressure (es), substitute in situ temperature (T) for dew point (Td). 2. ABSOLUTE HUMIDITY is the density of the water vapor in a sample of air. 푒 휌푣 = 푅푣푇푑 where: 3 v = absolute humidity [kg/m ] e = in situ vapor pressure [Pa] Rv = individual gas constant for water vapor (461.2 J/(kg K)) Td = in situ dew point temperature [K] To compute the saturation absolute humidity (vs), substitute in situ temperature (T) for dew point (Td). 3. MIXING RATIO is the ratio of the mass of the vapor in a sample, to the mass of the dry air in the sample of air. 푒휀 푤 = 푝 − 푒 where: w= mixing ratio [kgvapor/kgdry air] e = in situ vapor pressure [Pa] p= in situ total pressure [Pa] = ratio of molar masses of vapor and dry air (0.622) To compute the saturation mixing ratio (ws), substitute saturation vapor pressure (es) for in situ vapor pressure (e). MT3230 Atmospheric Thermodynamics 1 4. SPECIFIC HUMIDITY is the ratio of the mass of the vapor in a sample, to the mass of the moist air in the sample of air. 푒휀 푞 = 푝 − (1 − 휀)푒 where: 푞 = specific humidity [kgvapor/kg moist air] e = in situ vapor pressure [Pa] = ratio of molar masses of vapor and dry air (0.622) p= in situ total pressure [Pa] To compute the saturation specific humidity (qs), substitute saturation vapor pressure (es) for vapor pressure (e). 5. VIRTUAL TEMPERATURE is the temperature of a parcel of dry air with the same density as one containing moist air. It can be computed by: 푤 1 + 휀 푇 = 푇 ( ) 푣 1 + 푤 or approximated by: 푇푣 ≅ 푇(1 + 0.61푤) where: Tv = virtual temperature [K] T = in situ temperature [K] w= in situ mixing ratio [kgvapor/kgdry air] = ratio of molar masses of vapor and dry air (0.622) When using a Skew-T, virtual temperature can be approximated by: 푤 푇 ≅ 푇 + 푣 6 where w = in situ mixing ratio [gvapor/kgdry air] The U.S. Standard Atmosphere approximates virtual temperature by: 푇 푇푣 ≅ 푒 1 − 0.0379 푝 where: Tv = virtual temperature [K] T = in situ temperature [K] e = in situ vapor pressure p = in situ total pressure MT3230 Atmospheric Thermodynamics 2 6. WET-BULB TEMPERATURE (usually directly measured using a sling psychrometer or similar device) is the temperature to which an air parcel (or some other object) may be cooled by forcing water to evaporate at constant pressure. 푙푣 휀 −퐵 푇푤 = 푇 − [ 퐴푒푥푝 ( ) − 푤] 푐푝 푝 푇푤 where: Tw = wet bulb temperature [K] T = in situ temperature [K] cp = specific heat of dry air at constant pressure (≅ 1003.8 J/(kg K)) = ratio of molar masses of vapor and dry air (0.622) p = in situ total pressure [Pa] 11 A = pressure constant (2.5 x 10 Pa) 3 B = temperature constant (5.4 x 10 K) w = in situ mixing ratio [kgvapor/kgdry air] 7. DEW-POINT TEMPERATURE is the temperature to which an air parcel must be cooled at constant pressure, without adding vapor (w is constant), to induce saturation. It can be computed from psychrometric data by: 1 푇푑 = 1 푅푣 푝푐푝 퐴 −퐵 − 푙푛 [( ) (푇푤 − 푇) + 푒푥푝 ( )] 푇0 푙푣 푒0휀푙푣 푒0 푇푤 or by: 퐵 푇 = 푑 퐴휀 푙푛 ( ) 푤푝 where: Td = in situ dew point temperature [K] T0 = reference temperature (273.16 K) Rv = individual gas constant for water vapor (461.2 J/(kg K)) e0 = reference pressure (611.12 Pa) 6 lv = latent heat of vaporization (≅ 2.5008 x 10 J/kg) (temperature dependent) p = in situ total pressure [Pa] cp = specific heat of dry air at constant pressure (≅ 1003.8 J/(kg K)) = ratio of molar masses of vapor and dry air (0.622) Tw = in situ wet bulb temperature [K] 11 A = pressure constant (2.5 x 10 Pa) 3 B = temperature constant (5.4 x 10 K) w = in situ mixing ratio [kgvapor/kgdry air] MT3230 Atmospheric Thermodynamics 3 8. EQUIVALENT TEMPERATURE is the temperature a parcel of moist air would have if all the vapor in the parcel were condensed at constant pressure. It can be computed by: 푙푣푤푠 푇푒 = 푇푒푥푝 [ ] 푐푝푇 or estimated by the linear approximation: 푙푣푤 푇푒 ≅ 푇 + 푐푝 where: Te = equivalent temperature [K] 6 lv = latent heat of vaporization (≅ 2.5008 x 10 J/kg) (temperature dependent) w = in situ mixing ratio [kgvapor/kg dry air] ws = saturation mixing ratio [kgvapor/kg dry air] cp = specific heat of dry air at constant pressure (≅ 1003.8 J/(kg K)) T = in situ temperature [K] 9. EQUIVALENT POTENTIAL TEMPERATURE is the temperature a parcel of moist air would have if all the vapor in the parcel were condensed at constant pressure and the parcel is taken dry adiabatically to 1000 hPa. 푙푣푤푠 휃푒 = 휃푒푥푝 [ ] 푐푝푇 where: e = equivalent potential temperature [K] = potential temperature [K] 6 lv = latent heat of vaporization (≅ 2.5008 x 10 J/kg) (temperature dependent) ws = saturation mixing ratio [kgvapor/kg dry air] cp = specific heat of dry air at constant pressure (≅ 1003.8 J/(kg K)) T = in situ temperature [K] 10. ISENTROPIC CONDENSATION TEMPERATURE is the temperature at which saturation is reached when a parcel of moist air is lifted dry adiabatically, with its vapor content (w) held constant, that is, the temperature of a lifted parcel at the LCL. An analytical approximation is given by: 퐵 푇푖푐 ≅ 1 퐴휀 푇 휅 푙푛 [ ( 푖 ) ] 푤푝푖 푇푖푐 where: Tic = isentropic condensation temperature [K] 11 A = pressure constant (2.5 x 10 Pa) 3 B = temperature constant (5.4 x 10 K) w = in situ mixing ratio [kgvapor/kg dry air] Ti = initial temperature [K] pi = initial pressure [Pa] = Rd/cp for dry air (≅ 0.286) MT3230 Atmospheric Thermodynamics 4 Tic LCL w M D Td Tw T pLCL p0 ILLUSTRATION OF THE RELATIONSHIP BETWEEN SEVERAL THERMODYNAMIC VARIABLES USING A SKEW-T LOG P DIAGRAM. A parcel with initial temperature T (20 °C) and dew point Td (-10 °C) originates at p0 (1000 hPa). Then an external force lifts the parcel aloft. Its temperature cools at the Dry Adiabatic Lapse Rate (D), and its dew point follows a line of constant water vapor mixing ratio (w). (Note that Td decreases as z increases, even though w remains constant.) The parcel temperature continues to fall dry adiabatically until reaching the isentropic condensation temperature (Tic; about -15 °C) at the Lifted Condensation Level (LCL; near 640 hPa), where T = Td. Visible cloud bases appear at the LCL as vapor condenses into liquid cloud droplets. If the Tic is followed moist adiabatically (M) back to the parcel’s point of origin (1000 hPa), one can compute the parcel’s wet bulb temperature (Tw; about 7 °C). (Background image courtesy of Plymouth State University meteorology program, 2012). MT3230 Atmospheric Thermodynamics 5 .
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