ATMOSPHERIC THERMODYNAMIC FUNCTIONS This section contains the standard thermodynamic functions that can be used with a specific procedure. Go to page for the start of any specific calculations Enter Freezing temperature and freezing band TFC FB. Define CONSTANTS CPA. RA CPA KA KA KV CPV. RV CPV KV RA ε RV fcpa( M) := CPA + CPV M fra( M) := RA + RV M fka( M) := fra( M) fcpa( M) P PV. T. TF T + TFC G 9. PG TG MG. LV LF CW 9 CI 9 αl.9 αi.9 βl. βi.9 ΓL.9 ΓI.
. FUNCTIONS: Vapor pressure, mixing ratio, virtual temperature fpvl( T) exp αl βl T ΓLlnT ( ) fpvi( T) exp αi βi T ΓIlnT ( ) fpv( T) if( T > TF, fpvl( T), fpvi( T) ) fmvs( P, T) ε fmvl( P, T) ε fpv( T) P fpv( T) fpvl( T) P fpvl( T) fpv( P, M) M P M + fms( P, T, M) fmvs( P, T) ε fmv( P, T, M) if( fmvs( P, T) > M, M, fmvs( P, T) ) fu( P, T, M) := ( M) fmvs( P, T) fpv( P, T, M) fmv( P, T, M) ε P + fmv( P, T, M) fpa( P, T, M) P fpv( P, T, M) fmc( P, T, M) if( fmvs( P, T) < M, M fmvs( P, T), ) fmlb( P, T, M) ift > TF FB, fmc( P, T, M) T TF + FB, FB fml( P, T, M) if( T TF, fmc( P, T, M), fmlb( P, T, M) ) fmib( P, T, M) ift < TF, fmc( P, T, M) TF T, FB fmi( P, T, M) if( T < TF FB, fmc( P, T, M), fmib( P, T, M) )
. ENTROPY fsa( P, T) CPA ln T RA ln P T P fsv( P, T) CPV ln T T RV ln P PV + LV T fsl( T) CW ln T T fsi( T) CI ln T T LF T fsa( P, T, M) fsa( fpa( P, T, M), T) fsv( P, T, M) fmv( P, T, M) fsv( fpv( P, T, M), T) fsl( P, T, M) fml( P, T, M) fsl( T) fsi( P, T, M) fmi( P, T, M) fsi( T) fst( P, T, M) fsa( P, T, M) + fsv( P, T, M) + fsl( P, T, M) + fsi( P, T, M) fst( P, T, M) fsm( P, T, M) := ( + M) fsc( T) if( T > TF, fsl( T), fsi( T) ) faa( P, T, M) fsa( P, T) + M fsc( T) fθm( P, T, M, PC) T PC P RA+ M RV CPA+ M CPV
. ENTHALPY fha( T) CPA ( T T) fhv( T) [ CPV ( T T) + LV] fhl( T) [ CW ( T T) ] fhi( T) [ CI ( T T) LF] fha( P, T, M) fha( T) fhv( P, T, M) fmv( P, T, M) fhv( T) fhl( P, T, M) fml( P, T, M) fhl( T) fhi( P, T, M) fmi( P, T, M) fhi( T) fht( P, T, M) fha( P, T, M) + fhv( P, T, M) + fhl( P, T, M) + fhi( P, T, M) fhc( T) if( T > TF, fhl( T), fhi( T) ) fee( P, T, M) fha( T) + M fhc( T) fhw( P, T, M, W) fha( T) + M fhv( T) + if( W > TF, fhl( W), fhi( W) ) ( fmvs( P, W) M) fσ( P, T, M, Z) fht( P, T, M) + G ( + M) Z. VIRTUAL TEMPERATURE AND HEIGHT M + ε ftv( T, M) T + M ftv( P, T, M) if M < fmvs( P, T), ftv( T, M), ftv( T, fmvs( P, T) ) + fmvs( P, T) + M fα( P, P, TV, TV) G RA ln ln TV TV P P fz( P, P, TV, TV) TV TV fα( P, P, TV, TV)
SOLVER BLOCKS ISENTROPIC EXPANSION TEMPERATURE Given fst( P, TG, M) = S ftsol( S, P, M) := Find( TG) ISENTROPIC EXPANSION PRESSURE Given fst( PG, T, M) = S fpsol( S, T, M) := Find( PG) ISENTROPIC DESSICATION TEMPERATURE Given faa( P, TG, M) = S fasol( S, P, M) := Find( TG) ISENTHALPIC DESSICATION TEMPERATURE, EQUIVALENT TEMPERATURE Given fee( P, TG, M) = H fesol( H, P, M) := Find( TG) MIXING RATIO FROM WET BULB Given fht( P, W, fmvs( P, W) ) = fhw( P, T, MG, W) fmsol( P, T, W) := Find( MG) WET BULB FROM MIXING RATIO Given fht( P, TG, fmvs( P, TG) ) = fhw( P, T, M, TG) fwsol( P, T, M) := Find( TG) LIFTING CONDENSATION LEVEL Given fu( PG, fθm( P, T, M, PG), M) = fcsol( P, T, M) := Find( PG) DEW POINT Given fpv( P, M) = fpv( TG) fdsol( P, T, M) := Find( TG)
by: L.Michaud, Nov. Fetch and display sounding data Ending α denotes property of the sounding Fetch the San Juan average sounding for Sept Z and Z Sept READ SOUNDING DATA SDAT := Pα := ZW := SDAT TCα := SDAT TDα := SDAT UW := SDAT MW := SDAT READPRN ("SANJUAN.prn" ) Sounding Pressure (kpa) SDAT Sounding Level (m) Sounding Temperature (C) Sounding Dew Point (C) Sounding Realtive Humidity (%) Sounding Mixing Ratiol (g/kg)..... 9.....9. -. -. -9. -9.9 -.9 Pα = 9 TCα = 9 -. -. TDα = 9-9. -. -. -. -.9 -. -. -9..9 - -. -. -. -9. -. -. -. -. -.
Calculate sounding properties im := last( Pα) i:=.. im im = jm := last( Pα) j:=.. jm jm = Temperature in degree Kelvin Tα := TCα + T Mixing Ratio from dew point Mα := fmvl( Pα, TDα + T) Entropy Sα := fst( Pα, Tα, Mα) Enthalpy Hα := fht( Pα, Tα, Mα) Relative Humidity from mixing ratio- Uα := fu( Pα, Tα, Mα) Virtual Temperature TVα := ftv( Tα, Mα) ρα := Pα ( RA + Mα RV) Tα ( + Mα)
Mα i 9............ = MW i 9............. = Sα i.9...9 9.. 9....9...9 9.. 9.. = Hα i 9 - -99 - - - - - - - -99-9 - = Pα i. 9..9 = Tα i.9.9 9. 9.... 9.. 9.. 9. 9. 9.... = Pα i. 9..9 = Uα i 9.... 9...9.... 9.....9.9 = UW i 9 9 9 9 = TVα i.. 9. 9... 9.. 9.. 9. 9. 9.... = ρα i...........9.....9. =
Entropy of sounding Pressure (kpa) Pα i Sα i Entropy (J/kg K)
Calculate lapse rate and heights α := α j ( ) + := fα Pα j, Pα j+, TVα j, TVα j+ Z j ( ) + := fz Pα j, Pα j+, TVα j, TVα j+ ZC := ZW ZC j := + + Z j+ ZC j ZD := ZW α ( ) = -. Z = ZC = 9 ZD = 9 -. 9 9 9 -.9 -. -. -.9 -. -. -. -. -...9... 9 9 9 9 9 99 9 9 9 9 9 9 9 9
Calculate surface pressure reduction using the TWO GUESSES METHOD. Define air properties at large radius - State P := Pα P =. T := Tα T T =. MW M := M = 9. H := fht( P, T, M) H = S := fst( P, T, M) S =. Define P pressure - State n:= P := Pα n P = ENTER SEA SURFACE TEMPERATURE SSTC :=. SST := T + SSTC ENTER TEMPERATURE AND HUMIDITY APPROACHES A := B := Calculate T and U T := SST A T = 9. T T =. U := B U = 9 Enter first guess for eyewall base pressure and calculate second guess PC := 9 PC := PC. Calculate surface air mixing ratio U MC := fmvs( PC, T) MC = Calculate air properties at State.. SC := fst( PC, T, MC) HC := fht( PC, T, MC) SC = HC =..
Calculate air properties at Point TC := ftsol( SC, P, MC) TC = HC := fht( P, TC, MC) HC = Calculate static energy at state σc := fσ P, TC, MC, ZD n σc = ( ) Calculate the work for the two guesses WC := ( HC σc) WC = Calculate the pressure P required to make the work zero 99. 99. 9 99 9 9 MINMUM EYEWALL PRESSURE (kpa) WC P := PC ( PC PC ) P = 9. WC WC Calculate base pressure reduction ( ) P := P P P =. Calculate air properties at State M := fmvs( P, T) U S := fst( P, T, M) M =. S =. H := fht( P, T, M) H = 9 Calculate air properties at State Check temperatures at equilibrium level T := ftsol( S, P, M) T = 9. T T =. TV := ftv( P, T, M) TV = 9. TV T =. TVα n = 9. TVα n T =. H := fht( P, T, M) H = 9 Calculate air properties at State T := ftsol( S, P, M) T = 9. T T =. H := fht( P, T, M) H = 9 T = 9. T T =.
Work and Heat calculations W := H H W = MAXIMUM HURRICANE WIND VELOCITY (m/s) V := W V =. Qr := H H Qi := H H Qr = Qi = ZD n = µ:= H + ZD n ( + M) G µ = 9 M_DEL := M M M_DEL =. Calculate Updraft properties Tβ := ftsol( S, Pα, M) S =. ρβ := Pα ( RA + M RV) Tβ ( + M) M =. Tβ i =.. 9. 9..... 9... 9. 9..9. ρβ i =...9.99......9....... Pα i =. 9..9
Hydrostatic Base Pressure Calculation - Air-Sea Interaction case q:= PK := Pα q PK = M =. S =. PHK := Pα q PHK := PHK + m:=.. z :=.. PHK m := PHK + ( m ) TK z := ftsol S, PHK z, M TU z := TK z T ( ) ( ) TVK z := ftv PHK z, TK z, M UK := fu( PHK, TK, M) ρk z := PHK z ( RA + M RV) TK z ( + M)
PHK 9 9 9 9 = TK 9 9.9 9... 9...... 9.... 9. 9. 9. 9. 99... = TVK 9 9. 9....... 9....9. 9. 9. 9.9 9.9.... = UK 9 9. 9.... 9...9.9 9..9 9.... 9... =
ZK := ZD n PHK = m:=.. ZK =. DKZ m := fz PHK m, PHK ( m+ ), TVK m, TVK m+ ( ) 9... ZK m+ ( ) := ZK m DKZ m.. 9. 9.. 9. PHK = ZK = TVK = 9.9 9. DKZ = 9 9. ZK =.. 9....9 G TVK := TVK + ZK CPA... 9.9 9..9.... 9... 9. 9 -.....9...9. 9.9 ρk =. PHK =......9.99. 9.9. 9 9 9 9
Pressure versus Height Elevation (km) ZK m ZD i Updraft Pressure Ambient Pressure PHK m, Pα i Pressure (kpa) Pressure versus Height - Lower End Elevation (km) ZK m ZC i 9 9 9 9 9 Updraft Pressure Ambient Pressure PHK m, Pα i Pressure (kpa)
Pressure versus Height - Top End Elevation (km) ZK m ZC i Updraft Ambient PHK m, Pα i Pressure (kpa) Density versus Height Elevation (km) ZK m ZC i ZC i...... ρk m, ρα i, ρβ i Air Density (kg/m) Updraft Density Ambient Density Updraft Density ignoring pressure effect
Temperature versus Height Elevation (km) ZK m ZC i Updraft Temperature Ambient Temperature TK m, Tα i Temperature (K) Temperature versus Pressure Pressure (kpa) Updraft Temperature Ambient Dew Point Temperature Ambient Dry Bulb Temperature (K)
Temperature versus Pressure Pressure (kpa) 9 9 9 Updraft Virtual Temperature Ambient Temperature Temperature (K) Density versus Pressure Pressure (kpa) 9.........9.. Updraft Density Ambient Density Density (kg/m)
ZPPLOT ZC := ZPPLOT ZK := ZPPLOT ρk := ZPPLOT ρα := ZPPLOT TK := ZPPLOT Tα := ZPPLOT....... 9..9....9........ 9..9.....9....9.9....9....9...........9.9.99..9.................9...9 9... 9...... 9.... 9. 9. 9. 9. 99....9.9 9. 9.... 9.. 9.. 9. 9. 9.... = WRITEPRN "ISABEL_PLOT.prn" ( ) ZPPLOT :=