CONDENSE.DOC 2007/10/11 Constants --------- ELHE (J/kg) = latent energy of vaporization at 0°C = 2500000 ELHM (J/kg) = latent energy of melting at 0°C = 334000 ELHS (J/kg) = latent energy of sublimation at 0°C = 2834000 SHCD (J/kg°C) = specific heat capacity of dry air = 1003.5 SHCV (J/kg°C) = specific heat capacity of water vapor = 1911 SHCW (J/kg°C) = specific heat capacity of liquid water = 4185 SHCI (J/kg°C) = specific heat capacity of solid ice = 2060 RDRY (J/kg°C) = gas constant of dry air = 287 RVAP (J/kg°C) = gas constant of water vapor = 461.5 TKF (°K) = temperature of freezing = 273.16 PPVSF (Pa) = saturated partial pressure of water vapor at freezing = = 610.571 Variables --------- MG (kg/m²) = mass of gaseous air per unit area = MD + MV MD (kg/m²) = mass of dry air per unit area MV (kg/m²) = mass of water vapor per unit area ML (kg/m²) = mass of liquid water per unit area MI (kg/m²) = mass of solid ice per unit area TC (°C) = temperature TK (°K) = temperature P (Pa) = mean pressure of layer Q = specific humidity = MV / (MD+MV) RAIR (J/kg°C) = (1-Q)*RDRY + Q*RVAP = (MD*RDRY+MV*RVAP) / (MD+MV) RHOA (kg/m³) = density of air = P / RAIR*TK PPV (Pa) = partial pressure of water vapor = Q*P*RVAP/RAIR = = MV*RVAP*P / (MD*RDRY+MV*RVAP) The following quantities are independent of the Energy Reference Level ---------------------------------------------------------------------- ELHV (J/kg) = ELHE + (SHCV-SHCW)*TC = latent heat of vaporization = = enthalpy of vaporization ELHN (J/kg) = ELHM + (SHCW-SHCI)*TC = latent heat of melting = = enthalpy of melting ELHT (J/kg) = ELHS + (SHCV-SHCI)*TC = latent heat of sublimation = = ELHV + ELHN = enthalpy of sublimation Energy Reference Level X is used: sensible heat of dry air is measured from 0°K, but sensible heat of liquid water is measured from 0°C ---------------------------------------------------------------- SE (J/m²) = static energy GE (J/m²) = geopotential energy Derivation of SE-GE ------------------- Start with a parcel with the following original mass fractions: MD, MV, ML and MI; and the following original temperature: TC = TK-TKF. Condensing the water vapor to liquid at temperature TC releases [ELHE+(SHCV-SHCW)*TC]*MV; whereas melting the ice to liquid releases -[ELHM+(SHCW-SHCI)*TC]*MI. Thus the original static energy excluding geopotential energy was: SHCD*MD*TK + SHCW*(MV+ML+MI)*TC + + [ELHE+(SHCV-SHCW)*TC]*MV - [ELHM+(SHCW-SHCI)*TC]*MI = = SHCD*MD*TK + [SHCV*MV + SHCW*ML + SHCI*MI]*TC + ELHE*MV - ELHM*MI = = [SHCD*MD + SHCV*MV]*TK + [SHCW*ML + SHCI*MI]*TC + + [ELHE - SHCV*TKF]*MV - ELHM*MI Division of SE-GE into components --------------------------------- ELV (J/m²) = latent energy of vapor = [ELHE - SHCV*TKF]*MV ELI (J/m²) = latent energy of ice = - ELHM*MI EK (J/m²) = gaseous sensible heat = [SHCD*MD + SHCV*MV]*TK EC (J/m²) = condensate sensible heat = [SHCW*ML + SHCI*MI]*TC When a known amount of water vapor condenses to liquid water, what is the resultant equilibrated temperature change? ------------------------------------------------------ dM = mass of water vapor that condenses to liquid dT = equilibrated temperature change MVnew = MV - dM MLnew = ML + dM TKnew = TK + dT TCnew = TC + dT ELVnew = (ELHE-SHCV*TKF)*MVnew = ELV - (ELHE-SHCV*TKF)*dM EKnew = (SHCD*MD+SHCV*MVnew)*TKnew = = (SHCD*MD+SHCV*MVnew)*TK + (SHCD*MD+SHCV*MVnew)*dT = = EK - SHCV*dM*TK + (SHCD*MD+SHCV*MV)*dT - SHCV*dM*dT ECnew = (SHCW*MLnew+SHCI*MI)*TCnew = = (SHCW*MLnew+SHCI*MI)*TC + (SHCW*MLnew+SHCI*MI)*dT = = EC + SHCW*dM*TC + (SHCW*ML+SHCI*MI)*dT + SHCW*dM*dT ELVnew + EKnew + ECnew = ELV + EK + EC - (ELHE-SHCV*TKF)*dM - SHCV*dM*TK + (SHCD*MD+SHCV*MV)*dT - SHCV*dM*dT + + SHCW*dM*TC + (SHCW*ML+SHCI*MI)*dT + SHCW*dM*dT = 0 [SHCD*MD+SHCV*MV - SHCV*dM + SHCW*ML+SHCI*MI + SHCW*dM]*dT = = [ELHE-SHCV*TKF + SHCV*TK - SHCW*TC]*dM = = [ELHE+(SHCV-SHCW)*TC]*dM = ELHV*dM dT = ELHV*dM / [SHCD*MD + SHCV*MVnew + SHCW*MLnew + SHCI*MI] How much water vapor in a parcel must be condensed to liquid water to raise the parcel's equilibrated temperature a known amount? ----------------------------------------------------------- Same situation as section above. ELVnew + EKnew + ECnew = ELV + EK + EC - (ELHE-SHCV*TKF)*dM - SHCV*dM*TK + (SHCD*MD+SHCV*MV)*dT - SHCV*dM*dT + + SHCW*dM*TC + (SHCW*ML+SHCI*MI)*dT + SHCW*dM*dT = 0 [SHCD*MD+SHCV*MV + SHCW*ML+SHCI*MI]*dT = = [ELHE-SHCV*TKF + SHCV*TK + SHCV*dT - SHCW*TC - SHCW*dT]*dM = = [ELHE+(SHCV-SHCW)*TCnew]*dM = ELHVnew*dM dM = [SHCD*MD + SHCV*MV + SHCW*ML + SHCI*MI]*dT / ELHVnew If dT = -TC , then TCnew = 0 , ELHVnew = ELHE and dM = - [SHCD*MD + SHCV*MV + SHCW*ML + SHCI*MI]*TC / ELHE When a known amount of water vapor condenses to solid ice, what is the resultant equilibrated temperature change? ------------------------------------------------------ dM = mass of water vapor that condenses to solid ice dT = equilibrated temperature change MVnew = MV - dM MInew = MI + dM TKnew = TK + dT TCnew = TC + dT ELVnew = (ELHE-SHCV*TKF)*MVnew = ELV - (ELHE-SHCV*TKF)*dM ELInew = - ELHM*MInew = ELI - ELHM*dM EKnew = (SHCD*MD+SHCV*MVnew)*TKnew = = (SHCD*MD+SHCV*MVnew)*TK + (SHCD*MD+SHCV*MVnew)*dT = = EK - SHCV*dM*TK + (SHCD*MD+SHCV*MV)*dT - SHCV*dM*dT ECnew = (SHCW*ML+SHCI*MInew)*TCnew = = (SHCW*ML+SHCI*MInew)*TC + (SHCW*ML+SHCI*MInew)*dT = = EC + SHCI*dM*TC + (SHCW*ML+SHCI*MI)*dT + SHCI*dM*dT ELVnew + ELInew + EKnew + ECnew = ELV + ELI + EK + EC - (ELHE-SHCV*TKF)*dM - ELHM*dM - SHCV*dM*TK + (SHCD*MD+SHCV*MV)*dT - - SHCV*dM*dT + SHCI*dM*TC + (SHCW*ML+SHCI*MI)*dT + SHCI*dM*dT = 0 [SHCD*MD+SHCV*MV - SHCV*dM + SHCW*ML+SHCI*MI + SHCI*dM]*dT = = [ELHE-SHCV*TKF + ELHM + SHCV*TK - SHCI*TC]*dM = = [ELHS+(SHCV-SHCI)*TC]*dM = ELHT*dM dT = ELHT*dM / [SHCD*MD + SHCV*MVnew + SHCW*ML + SHCI*MInew] How much water vapor in a parcel must be condensed to solid ice to raise the parcel's equilibrated temperature a known amount? ----------------------------------------------------------- Same situation as section above. ELVnew + ELInew + EKnew + ECnew = ELV + ELI + EK + EC - (ELHE-SHCV*TKF)*dM - ELHM*dM - SHCV*dM*TK + (SHCD*MD+SHCV*MV)*dT - - SHCV*dM*dT + SHCI*dM*TC + (SHCW*ML+SHCI*MI)*dT + SHCI*dM*dT = 0 [SHCD*MD+SHCV*MV + SHCW*ML+SHCI*MI]*dT = = [ELHE-SHCV*TKF + ELHM + SHCV*TK + SHCV*dT - SHCI*TC - SHCI*dT]*dM = = [ELHS+(SHCV-SHCI)*TCnew]*dM = ELHTnew*dM dM = [SHCD*MD + SHCV*MV + SHCW*ML + SHCI*MI]*dT / ELHTnew If dT = -TC , then TCnew = 0 , ELHTnew = ELHS and dM = - [SHCD*MD + SHCV*MV + SHCW*ML + SHCI*MI]*TC / ELHS What happens if all condensate evaporates ----------------------------------------- dT = equilibrated temperature change MVnew = MV + ML + MI MLnew = 0 MInew = 0 TKnew = TK + dT TCnew = TC + dT ELVnew = (ELHE-SHCV*TKF)*MVnew = ELV + (ELHE-SHCV*TKF)*(ML+MI) ELInew = 0 EKnew = (SHCD*MD+SHCV*MVnew)*TKnew = = (SHCD*MD+SHCV*MVnew)*TK + (SHCD*MD+SHCV*MVnew)*dT = = EK + SHCV*(ML+MI)*TK + (SHCD*MD+SHCV*MV)*dT + SHCV*(ML+MI)*dT ECnew = 0 ELVnew + EKnew = ELV + ELI + EK + EC (ELHE-SHCV*TKF)*(ML+MI) + SHCV*(ML+MI)*TK + (SHCD*MD+SHCV*MV)*dT + + SHCV*(ML+MI)*dT = - ELHM*MI + (SHCW*ML+SHCI*MI)*TC (SHCD*MD+SHCV*MV)*dT + SHCV*(ML+MI)*dT = - (ELHE-SHCV*TKF)*(ML+MI) - - SHCV*(ML+MI)*TK - ELHM*MI + (SHCW*ML+SHCI*MI)*TC (SHCD*MD+SHCV*MVnew)*dT = = - ELHE*ML - ELHS*MI - SHCV*(ML+MI)*TC + (SHCW*ML+SHCI*MI)*TC = = - [ELHE + (SHCV-SHCW)*TC]*ML - [ELHS + (SHCV-SHCI)*TC]*MI = = - ELHV*ML - ELHT*MI dT = - (ELHV*ML + ELHT*MI) / (SHCD*MD + SHCV*MVnew) Derivation of PPVSAT(TK,ELHV), the Clausius-Clapeyron Equation See "Atmospheric Dynamics" by Craig F. Bohren and Bruce A. Albrecht ------------------------------------------------------------------- ALPHAC (m³/kg) = specific volume of condensate = 1/RHOW or 1/RHOI ALPHAV (m³/kg) = specific volume of water vapor = RVAP*TK/PPV PPVSF (Pa) = saturated partial pressure of water vapor at freezing = = 610.571 ELHV (J/kg) = enthalpy of vaporization or enthalpy of sublimation = = ELHE+(SHCV-SHCW)*TC or ELHS+(SHCV-SHCI)*TC dPPVSAT/dT = ELHV/TK*(ALPHAV-ALPHAC) ~=~ ELHV/TK*ALPHAV = = PPVSAT*ELHV/RVAP*TK² (1/PPVSAT)*dPPVSAT/dT = ELHV/RVAP*TK² Above equation will be integrated from TKF to TKend. ELHV(TK) has a weak dependence on TK. ELHV(TK) is replaced during integration with: ELH2(TKend) = [ELHV(TKF) + ELHV(TKend)]/2 = = [ELHE + ELHV(TKend)]/2 or [ELHS + ELHT(TKend)]/2 , which is constant. (1/PPVSAT)*dPPVSAT/dT = ELH2(TKend)/RVAP*TK² ln PPVSAT(TKend) - ln PPVSAT(TKF) = ELH2(TKend)*[1/TKF - 1/TKend]/RVAP PPVSAT(TK) = PPVSF * exp[ELH2(TK)*(1/TKF - 1/TK)/RVAP] Determine TKnew and MVnew so that PPVnew = PPVSAT(TKnew,ELH2new). Use linear iterations to solve for dT and dM. ------------------------------------------------ dT = increase in parcel's temperature for each iteration dM = reduction in parcel's water vapor mass for each iteration ELH2 = ELHE+(SHCV-SHCW)*TC/2 or ELHS+(SHCV-SHCI)*TC/2 PPV = RVAP*MV*P / (RDRY*MD + RVAP*MV) PPVSAT = PPVSF * exp[ELH2*(1/TKF-1/TK)/RVAP] dPPVSAT/dT = PPVSAT*ELH2 / RVAP*TK² PPVSATnew ~=~ PPVSAT + dT * dPPVSAT/dT PPVnew ~=~ PPV*(1 - dM/MV) dM ~=~ dT*(SHCD*MD + SHCV*MV + SHCW*ML + SHCI*MI) / ELHV PPVnew ~=~ PPVSATnew PPV*(1 - dM/MV) ~=~ PPVSAT + dT*dPPVSAT/dT PPV - PPVSAT ~=~ dT*dPPVSAT/dT + + PPV*dT*(SHCD*MD + SHCV*MV + SHCW*ML + SHCI*MI) / ELHV*MV ELHV*MV*(PPV - PPVSAT) ~=~ ~=~ dT*[ELHV*MV*dPPVSAT/dT + PPV*(SHCD*MD + SHCV*MV + SHCW*ML + SHCI*MI)] dT = ELHV*MV*(PPV - PPVSAT) / [ELHV*MV*dPPVSAT/dT + PPV*(SHCD*MD + SHCV*MV + SHCW*ML + SHCI*MI)] Calculate exact change dM , associated with change dT TKnew = TK + dT ELHVnew = ELHV + (SHCV-SHCW)*dT or ELHV + (SHCV-SHCI)*dT dM = dT*(SHCD*MD + SHCV*MV + SHCW*ML + SHCI*MI) / ELHVnew Conservation of energy when condensate falls from layer 2 to layer 1 -------------------------------------------------------------------- MC (kg/m²) = mass of condensate H0ML (J) = gaseous potential enthalpy of layer L H0ML*PL^RKAP (J) = gaseous enthalpy of layer L P1new (Pa) = mean pressure of layer 1 after falling = P1-.5*GRAV*MC P2new (Pa) = mean pressure of layer 2 after falling = P2-.5*GRAV*MC Change in thermal energy (J) by air (which is negative): H0M1*P1new^RKAP + H0M2*P2new^RKAP - H0M1*P1^RKAP - H0M2*P2^RKAP This is equal to the loss of geopotential energy of the condensate, and should be added back as thermal energy to either the condensate or the air. When water vapor condenses to saturation and the resultant temperature is 0°C, determine the condensate mixture ML and MI ----------------------------------------------------- MC = ML + MI = amount of condensate PPVSF = PPV = (MV-MC)*RVAP*P / [MD*RDRY + (MV-MC)*RVAP] PPVSF*[MD*RDRY + (MV-MC)*RVAP] = (MV-MC)*RVAP*P PPVSF*MD*RDRY = (MV-MC)*RVAP*(P-PPVSF) MC = MV - RDRY*MD*PPVSF / RVAP*(P-PPVSF) dT = - TC = TKF - TK TKnew = TKF TCnew = 0 MVnew = MV - MC ELVnew = (ELHE-SHCV*TKF)*MVnew = ELV - (ELHE-SHCV*TKF)*MC ELInew = - ELHM*MI EKnew = (SHCD*MD+SHCV*MVnew)*TKnew = = (SHCD*MD+SHCV*MVnew)*TK + (SHCD*MD+SHCV*MVnew)*dT = = EK - SHCV*MC*TK + (SHCD*MD+SHCV*MV)*dT - SHCV*MC*dT ECnew = 0 = EC ELVnew + ELInew + EKnew + ECnew = ELV + ELI + EK + EC - (ELHE-SHCV*TKF)*MC - ELHM*MI - SHCV*MC*TK + (SHCD*MD+SHCV*MV)*dT - - SHCV*MC*dT = 0 - ELHE*MC + SHCV*TKF*MC - ELHM*MI - SHCV*MC*TK + + (SHCD*MD+SHCV*MV)*(TKF-TK) - SHCV*MC*(TKF-TK) = 0 ELHM*MI = - ELHE*MC - (SHCD*MD+SHCV*MV)*(TK-TKF) MI = - [ELHE*MC + (SHCD*MD+SHCV*MV)*(TK-TKF)] / ELHM ML = MC - MI