ENRGTRAN.TXT ENeRGy TRANsport 2002/11/04 Definitions ----------- GRAV (m/s²) = Earth's constant gravitational acceleration at surface (GRAV at 32000 m is 1% less than GRAV at the surface) RDRY (J/kgC) = gas constant of air SHCD (J/kgC) = specific heat capacity of dry air at constant pressure X = exponent of exner function = RDRY/SHCD M (kg/m²) = vertical coordinate = mass above P (Pa) = pressure = GRAVŽM P  (Pa) = constant global mean sea level pressure Z (m) = altitude measured from mean sea level T (K) = temperature = TPŽ(P/P )Í TP (K) = potential temperature = TŽ(P /P)Í á (m³/kg) = specific volume EPS (J/kg) = specific potential enthalpy = SHCDŽTP ES (J/kg) = specific enthalpy = SHCDŽT EIS (J/kg) = specific internal energy = (SHCD-RDRY)ŽT EPT (J/m²) = vertically integrated total potential enthalpy EP (J/m²) = vertically integrated potential enthalpy E (J/m²) = vertically integrated enthalpy EG (J/m²) = vertically integrated geopotential energy EI (J/m²) = vertically integrated internal energy MS (kg/m²) = vertical coordinate of surface MT (kg/m²) = vertical coordinate at dynamical top of Model ÄM (kg/m²) = mass transported from grid cell 1 to grid cell 2 ÄEP (J/m²) = potential enthalpy transported from grid cell 1 to 2 TPK (K) = potential temperature of transported air ZS¡ (m) = surface topography of grid cell 1 ZK¡ (m) = altitude near removal in grid cell 1 MS¡ (kg/m²) = vertical coordinate of surface in grid cell 1 MK¡ (kg/m²) = vertical coordinate where removal occurred in grid cell 1 T¡ (K) = temprature of removed air = TPKŽ(GRAVŽMK¡/P )Í ZS¢ (m) = surface topography of grid cell 2 ZK¢ (m) = altitude near insertion in grid cell 2 MS¢ (kg/m²) = vertical coordinate of surface in grid cell 2 MK¢ (kg/m²) = vertical coordinate where insertion occurred in cell 2 T¢ (K) = temprature of inserted air = TPKŽ(GRAVŽMK¢/P )Í Relationships ------------- Where GRAV is nearly constant in the Earth's lower atmosphere, choose the vertical coordinate, M, so that: P = GRAVŽM Hydrostatic assumption: dP/dZ = - GRAV/á => á dM = - dZ Equation of state for ideal gases: P = RGASŽT/á => á = RDRYŽT/GRAVŽM (M-ÄM)Í = MÍ - XŽMÍ»±ÄM + ÑŽXŽ(X-1)ŽMÍ»² - ... Vertically integrated energies ------------------------------ ”MS ”MS ”MS EG = GRAVŽZ dM = GRAV d(ZŽM) - GRAV M dZ = —MT —MT —MT ”MS = GRAVŽ(ZSŽMS - ZTŽMT) + GRAV áŽM dM = —MT ”MS = GRAVŽ(ZSŽMS - ZTŽMT) + RDRYŽT dM —MT ”MS EI = (SHCD-RDRY)ŽT dM —MT ”MS EG + EI = GRAVŽ(ZSŽMS - ZTŽMT) + SHCDŽT dM —MT ”MS EPT = GRAVŽZSŽ(MS-MT) + SHCDŽT dM = EG + EI + GRAVŽMTŽ(ZT-ZS) —MT As MT tends to 0, ZT tends to infinity more slowly, so that ZTŽMT tends to 0. In energy exchanges in the Model, the value of MT is irrelevant. Removal of mass and energy from grid cell 1 ------------------------------------------- ”MS¡ EPT¡old = GRAVŽZS¡(MS¡-MT) + SHCDŽTPŽ(GRAVŽM/P )Í dM —MT ”MK¡+ÑÄM EPT¡new = EPT¡old - GRAVŽZS¡ÄM - SHCDŽTPKŽ(GRAVŽM/P )Í dM - —MK¡-ÑÄM ”MS¡ - SHCDŽTPŽ{(GRAVŽM/P )Í - [GRAVŽ(M-ÄM)/P ]Í} dM ß —MK¡+ÑÄM ß EPT¡old - GRAVŽZS¡ÄM - SHCDŽTPKŽ(GRAVŽMK¡/P )ÍÄM - ”MS¡ - SHCDŽTPŽ(GRAV/P )Í[MÍ - (MÍ-XŽMÍ»±ÄM)] dM = —MK¡+ÑÄM = EPT¡old - GRAVŽZS¡ÄM - SHCDŽTK¡ÄM - ”MS¡ - SHCDŽTPŽ(GRAV/P )ÍXŽMÍ»±ÄM dM = —MK¡+ÑÄM = EPT¡old - GRAVŽZS¡ÄM - SHCDŽTK¡ÄM - ”MS¡ - ÄM (RGASŽT/M) dM = —MK¡+ÑÄM ”MS¡ = EPT¡old - GRAVŽZS¡ÄM - SHCDŽTK¡ÄM - ÄM GRAVŽá dM = —MK¡+ÑÄM ”MS¡ = EPT¡old - GRAVŽZS¡ÄM - SHCDŽTK¡ÄM + ÄM GRAV dZ ß —MK¡+ÑÄM ß EPT¡old - GRAVŽZS¡ÄM - SHCDŽTK¡ÄM + ÄMŽGRAVŽ(ZS¡ - ZK¡) = = EPT¡old - SHCDŽTK¡ÄM - GRAVŽZK¡ÄM Insertion of mass and energy into grid cell 2 Assume that area of grid cell 1 equals area of grid cell 2 ---------------------------------------------------------- ”MS¡ EPT¢old = GRAVŽZS¡(MS¡-MT) + SHCDŽTPŽ(GRAVŽM/P )Í dM —MT ”MK¢+ÄM EPT¢new = EPT¢old + GRAVŽZS¢ÄM + SHCDŽTPKŽ(GRAVŽM/P )Í dM + —MK¢ ”MS¡ + SHCDŽTPŽ{(GRAVŽM/P )Í - [GRAVŽ(M-ÄM)/P ]Í} dM ß —MK¢ ß EPT¢old + GRAVŽZS¢ÄM + SHCDŽTPKŽ(GRAVŽMK¢/P )ÍÄM + ”MS¢ + SHCDŽTPŽ(GRAV/P )Í[MÍ - (MÍ-XŽMÍ»±ÄM)] dM = —MK¢ = EPT¢old + GRAVŽZS¢ÄM + SHCDŽTK¢ÄM + ”MS¢ + SHCDŽTPŽ(GRAV/P )ÍXŽMÍ»±ÄM dM = —MK¢ ”MS¢ = EPT¢old + GRAVŽZS¢ÄM + SHCDŽTK¢ÄM - ÄM GRAV dZ = —MK¢ = EPT¢old + GRAVŽZS¢ÄM + SHCDŽTK¢ÄM - ÄMŽGRAVŽ(ZS¢ - ZK¢) = = EPT¢old + SHCDŽTK¢ÄM + GRAVŽZK¢ÄM EPT¡new - EPT¡old + EPT¢new - EPT¢old = = SHCDŽ(TK¢-TK¡)ŽÄM + GRAVŽ(ZK¢-ZK¡)ŽÄM When transport of air from column 1 to column 2 occurs at the same altitude (ZK¡ = ZK¢), then the net change in EPT summed over the two columns is SHCDŽ(TK¢-TK¡)ŽÄM . If the transport is adiabatic, then the potential temperature of the parcel is maintained but actual temperatures may vary dependent on different presuures of the columns. When transport of air from column 1 to column 2 occurs at the same pressure (MK¡ = MK¢) and if the transport is adiabatic, then TK¡ = TK¢ and the net change in EPT, summed over the two columns, is GRAVŽ(ZK¢-ZK¡)ŽÄM . Total Potential Energy transport estimated from Potential Enthalpy Tran. Assume that potential temperature is constant inside each layer. ---------------------------------------------------------------- RRT3 = 1/Ò3 ß .57735 MAË (kg/m²) = mass of layer K PEË (Pa) = edge pressure above layer K = PE˪¡ + GRAVŽMA˪¡ PE  (Pa) = surface pressure PUPË (Pa) = PEË + Ñ(1-RRT3)ŽGRAVŽMAË = pressure inside layer K PDNË (Pa) = PEË + Ñ(1+RRT3)ŽGRAVŽMAË = pressure inside layer K TPË (K) = constant potential temperature of layer K TUPË (K) = TPË(PUPË/P )Í = temperature at pressure PUPË in layer K TDNË (K) = TPË(PDNË/P )Í = temperature at pressure PDNË in layer K TË (K) = Ñ(TDNË+TUPË) = average temperature of layer K áUPË(m³/kg) = RDRYŽTUPË/PUPË = specific volume at pressure PUPË áDNË(m³/kg) = RDRYŽTDNË/PDNË = specific volume at pressure PDNË ZE  (m) = surface topography above mean sea level ZEË (m) = edge altitude above layer K = ZEË«¡ + Ñ(áDNË+áUPË)ŽMAË ZË (m) = mass weighted altitude of layer K = = ZEË«¡ + [áDNË(1-RRT3) + áUPË(1+RRT3)]ŽMAË/4 ÄMAË (kg/m²) = air mass transported from column 1 to 2 at layer K ÄEPË (J/m²) = potential enthalpy transported from 1 to 2 at layer K ÄEPTË (J/m²) = total potential energy transported at layer K ÄEPTË = SHCDŽTËŽÄM + GRAVŽZËŽÄMË = = SHCDŽÑ(TDNË + TUPË)ŽÄM + GRAVŽZËŽÄMË = = SHCDŽTPËŽÑ(PDNËÍ + PUPËÍ)ŽÄM/P Í + GRAVŽZËŽÄMË = = ÄPEËŽÑ(PDNËÍ + PUPËÍ)/P Í + GRAVŽZËŽÄMË = When potential temperature, TP, is constant in the column, then adiabatic transport conserves total potential energy --------------------------------------------------------- ”ZK ”MK GRAVŽZK = GRAV dZ = - RDRYŽTPŽMÍ»±(GRAV/P )Í dM = —0 —M  MK = - RGASŽTPŽMÍ(GRAV/P )Í/X = M  = SHCDŽTPŽ[(GRAVŽM /P )Í - (GRAVŽMK/P )Í] = = SHCDŽTPŽ[1 - (GRAVŽMK/P )Í] = SHCDŽ(TP - TK) EPT¡new - EPT¡old + EPT¢new - EPT¢old = = SHCDŽ(TK¢-TK¡)ŽÄM + GRAVŽ(ZK¢-ZK¡)ŽÄM = = SHCDŽ(TK¢-TK¡)ŽÄM + SHCDŽ(TP-TK¢ - TP+TK¡)ŽÄM = 0