ERL.TXT Energy Reference Levels 2007/10/11 Definitions ----------- ELHE (J/kg) = latent heat of vaporization at 0°C = 2500000 ELHM (J/kg) = latent heat of melting at 0°C = 334000 ELHS (J/kg) = latent heat 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 ice = 2060 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 ice per unit area TC (°C) = temperature TK (°K) = temperature TKF (°K) = temperature of freezing = 273.16 The book "Thermodynamics and Statistics" states on page 334: ELHV = latent heat of vaporization = 2502000 - 2380*TC (J/kg). In the GCM we use: ELHV = ELHE + (SHCV-SHCW)*TC = 2500000 - 2274*TC. This "latent heat of vaporization" is also consistent with Kerry Emanuel's book "Atmospheric Convection". I interpret this "latent heat of vaporization" in the following way. Given M (kg) of water at temperature TC (°C), it requires M*ELHV (J) of energy to produce M (kg) of water vapor at the same temperature TC (°C). Alternatively, given M (kg) of water vapor at temperature TC (°C), M*ELHV (J) of energy is released to the environment when the M (kg) condenses to liquid water at the same temperature TC (°C). The GCM formula for "latent heat of melting" is: ELHM + (SHCW-SHCI)*TC There are three possible Energy Reference Levels (ERL) from which to measure sensible heat or static energy minus surface geopotential energy of dry air and liquid water. The latent energy form for each ERL is derived subsequently. ERL=C: all sensible heat is measured from 0°C: EN (J/m²) = (SHCD*MD + SHCW*ML)*TC ERL=K: all sensible heat is measured from 0°K: EN (J/m²) = (SHCD*MD + SHCW*ML)*TK ERL=X: air is measured from 0°K but liquid water is measured from 0°C: EN (J/m²) = SHCD*MD*TK + SHCW*ML*TC Given a mixture of dry air and water in different phases, we can measure static energy minus surface geopotential energy (SE-GE) of the mixture for each of the different Energy Reference Levels (ERL). We do this by condensing vapor to liquid using the known "latent heat of vaporization" or by melting ice to liquid using the "latent heat of melting". The original energy (SE-GE) is a linear function of temperature. The part that does not depend on the current temperature we call latent energy, and the part that is a multiplicative factor times the temperature we call sensible heat. ERL=C: ------ Start with a parcel with the following original mass fractions: MD, MV, ML and MI; and the following original temperature: TC. 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 + SHCW*(MV+ML+MI)]*TC + [ELHE+(SHCV-SHCW)*TC]*MV - [ELHM+(SHCW-SHCI)*TC]*MI = [SHCD*MD + SHCV*MV + SHCW*ML + SHCI*MI]*TC + ELHE*MV - ELHM*MI Latent energy: ELHE*MV - ELHM*MI Sensible heat: [SHCD*MD + SHCV*MV + SHCW*ML + SHCI*MI]*TC ERL=K: ------ 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 + SHCW*(MV+ML+MI)]*TK + [ELHE+(SHCV-SHCW)*(TK-TKF)]*MV - [ELHM+(SHCW-SHCI)*(TK-TKF)]*MI = [SHCD*MD + SHCV*MV + SHCW*ML + SHCI*MI]*TK + [ELHE-(SHCV-SHCW)*TKF]*MV - [ELHM-(SHCW-SHCI)*TKF]*MI Latent energy: [ELHE-(SHCV-SHCW)*TKF]*MV - [ELHM-(SHCW-SHCI)*TKF]*MI Sensible heat: [SHCD*MD + SHCV*MV + SHCW*ML + SHCI*MI]*TK ERL=X: ------ 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 Latent energy: [ELHE - SHCV*TKF]*MV - ELHM*MI Sensible heat: [SHCD*MD + SHCV*MV]*TK + [SHCW*ML + SHCI*MI]*TC As with radiative and conductive heating, the energy required to vaporize water at the same temperature does not depend on the Energy Reference Level. Thus "latent heat of vaporization", "latent heat of melting", and "latent heat of sublimation" do not depend on ERL. ERL is only needed when moving mass in or out of a parcel, from one grid box to another, or from the ground to the atmosphere. This is because the temperature of the mass must be measured from some reference level. In the GCM, Energy Reference Level X is used -------------------------------------------- SE (J/m²) = static energy GE (J/m²) = geopotential energy SE-GE = [SHCD*MD + SHCV*MV]*TK + [SHCW*ML + SHCI*MI]*TC + + [ELHE - SHCV*TKF]*MV - ELHM*MI Division of SE-GE into components --------------------------------- ELVA (J/m²) = atmos. latent energy of vapor = [ELHE - SHCV*TKF]*MV ELIA (J/m²) = atmos. latent energy of ice = - ELHM*MI EKA (J/m²) = atmos. gaseous enthalpy = [SHCD*MD + SHCV*MV]*TK ECA (J/m²) = atmos. condensate enthalpy = [SHCW*ML + SHCI*MI]*TC H0M (J) = atmos. potential enthalpy = EKA*DXYP / P^RKAP ELIG (J/m²) = ground latent energy of ice = - ELHM*MI ECG (J/m²) = ground condensate enthalpy = [SHCW*ML + SHCI*MI]*TC Components of downward surface fluxes from atmosphere to ground --------------------------------------------------------------- SNHDT (J/m²) = sensible heat flux is independent of the choice of ERL = dry heating from atmosphere to ground = = dECG = - dEKA DEW (kg/m²) = water vapor flux from atmosphere to ground EVHDT (J/m²) = water vapor heat flux depends on choice of ERL = = energy to condense water vapor to liquid and drop to ground leaving atmospheric temperature unchanged = = ELHV*DEW + SHCW*TCG*DEW = (ELHE + SHCV*TCG)*DEW Over liquid: dECG = (ELHE + SHCV*TCG)*DEW = - dELVA - dEKA dELVA = - (ELHE-SHCV*TKF)*DEW dEKA = - SHCV*(TCG+TKF)*DEW Over ice: dELIG + dECG = (ELHE + SHCV*TCG)*DEW = - dELVA - dEKA dELIG = - ELHM*DEW dECG = (ELHS+SHCV*TCG)*DEW dELVA = - (ELHE - SHCV*TKF)*DEW dEKA = - SHCV*(TCG+TKF)*DEW