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Lecture5
• AtmosphericPressure• HydrostaticBalance• GeopotentialHeight• HypsometricEquation
ATMOS5130
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IntroductiontoHydrostaticBalance
• Pressureatanypointintheatmosphereequalstheweightperunitareaabovethatpoint.• Pressure=Forceperunitarea• Weightofanobject=Force=Fg =mg• g=accelerationduetogravity(9.81ms-2)atsealevel
AckermanandKnox,2012
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![Page 4: Atmospheric Pressure - home.chpc.utah.eduhallar/Thermo/Lectures/... · 2017. 1. 22. · 850 hPa 700 hPa T v = 240 K T v Δz = 1,364 gpm = 280 K Δz = 1,591 gpm Fig. 4.3 Hypsometric](https://reader035.vdocuments.us/reader035/viewer/2022081620/6117c532987d99013c5589f5/html5/thumbnails/4.jpg)
HydrostaticEquation
![Page 5: Atmospheric Pressure - home.chpc.utah.eduhallar/Thermo/Lectures/... · 2017. 1. 22. · 850 hPa 700 hPa T v = 240 K T v Δz = 1,364 gpm = 280 K Δz = 1,591 gpm Fig. 4.3 Hypsometric](https://reader035.vdocuments.us/reader035/viewer/2022081620/6117c532987d99013c5589f5/html5/thumbnails/5.jpg)
Lecture5
• AtmosphericPressure• HydrostaticBalance• GeopotentialHeight• HypsometricEquation
ATMOS5130
![Page 6: Atmospheric Pressure - home.chpc.utah.eduhallar/Thermo/Lectures/... · 2017. 1. 22. · 850 hPa 700 hPa T v = 240 K T v Δz = 1,364 gpm = 280 K Δz = 1,591 gpm Fig. 4.3 Hypsometric](https://reader035.vdocuments.us/reader035/viewer/2022081620/6117c532987d99013c5589f5/html5/thumbnails/6.jpg)
Fig.4.1
HydrostaticApproximationfortheAtmosphere
F = 𝐹$% − 𝐹'()* − 𝐹+
𝐹 = 𝐴𝑝 𝑧 − 𝐴𝑝 𝑧 + 𝑑𝑧 − 𝜌 𝐴𝑑𝑧 𝑔
DividebytheMasstogetaccelerationofatmosphere
𝑎 = 𝐹𝑀 =
𝐹𝜌𝑉 =
𝐹𝜌𝐴𝑑𝑧 =
𝐴𝑝 𝑧 − 𝐴𝑝 𝑧 + 𝑑𝑧 − 𝜌 𝐴𝑑𝑧 𝑔𝜌𝐴𝑑𝑧
𝑎 = 7[% 9 :% 9;'9 :< '9 +]<7'9
= −> <⁄% 9;'9 :% 9
'9− 𝑔
𝑎 + 𝑔 = −> <⁄% 9;'9 :%(9)
'9
−(𝑎 + g) = > <⁄C%C9
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Fig.4.1
HydrostaticApproximationfortheAtmosphere
𝜕𝑝𝜕𝑧 = −𝜌 𝑎 + 𝑔
Inallbutthemostextrememeteorologicalconditionstheverticalaccelerationofofairparcelsismuchsmallerthangravity.
C%C9≈ −𝜌𝑔
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ApplicationtotheAtmosphere𝜕𝑝𝜕𝑧 = −𝜌𝑔 = −
𝑝𝑔𝑅'𝑇H
1𝑝𝜕𝑝𝜕𝑧 = −
𝑔𝑅'𝑇H
𝜕ln𝑝𝜕𝑧 = −
𝑔𝑅'𝑇H
Therateofchangeofthelogarithmofpressurewithheightisinverselyproportionaltotheabsolutetemperatureanddoesnotdependonpressure.
Pressuregenerallyfallsoffexponentiallywithheight!
![Page 9: Atmospheric Pressure - home.chpc.utah.eduhallar/Thermo/Lectures/... · 2017. 1. 22. · 850 hPa 700 hPa T v = 240 K T v Δz = 1,364 gpm = 280 K Δz = 1,591 gpm Fig. 4.3 Hypsometric](https://reader035.vdocuments.us/reader035/viewer/2022081620/6117c532987d99013c5589f5/html5/thumbnails/9.jpg)
1 m 1 gpm 1 m 1 gpm 1 m 1 gpm
g > go
g = g0
g < g0
Geometric vs. Geopotential Meters
Fig.4.2
GeopotentialHeightü Earth’sgravityhasAltitudeandLatitudedependences.ü Overalltheseareminor(fewtenthsofapercentunderconditionsofinterestto
meteorologist).ü GeopotentialHeightallowsusanewreference(orcoordinate)toaccommodatethese
changesingravity.ü GeopotentialheightisaverticalcoordinatereferencedtoEarth'smeansealevel— an
adjustmenttogeometricheight(elevationabovemeansealevel)usingthevariationofgravitywithlatitudeandelevation.
ü Thusitcanbeconsidereda"gravity-adjustedheight".
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1 m 1 gpm 1 m 1 gpm 1 m 1 gpm
g > go
g = g0
g < g0
Geometric vs. Geopotential Meters
Fig.4.2
GeopotentialHeightAtanelevationofz,thegeopotentialisdefined
𝜙 𝑧 = M 𝑔 𝜙, ℎ 𝑑𝑧9
P
Where𝑔 𝜙, ℎ istheaccelerationduetogravity,𝜙 islatitudeandhisthegeometricelevation(variationinlatitudeandelevationgenerallydonotmatterformeteorologicalapplications).Geopotential=>Potentialenergyperunitmass
𝑍 = 𝜙 𝑧𝑔P
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KeyPoint
• Whenmeteorologistreferenceheight– itshouldbeunderstoodthattheyreallymeangeopotentialheight.
• ThisallowsfortheHypsometricEquation
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850 hPa
700 hPa
Tv= 240 K Tv= 280 K Δz = 1,591 gpmΔz = 1,364 gpm
Fig.4.3
HypsometricEquationGiventwopressurelevelswhatisthethicknessoftheatmosphere(Δ𝑧)
M 𝑑𝑧 =𝑅'𝑔M 𝑇H𝑑ln𝑝%S
%T
9S
9T
Δ𝑧 = 𝑅'𝑇H𝑔
M 𝑑ln𝑝 =𝑅'𝑇H𝑔
%S
%Tln
𝑝>𝑝U
Thelayerthickness(Δ𝑧)isdirectlyproportionaltomeanvirtualtemperature(𝑇H)
*Mostofthetime,Tisused…verycloseifnottoohumid.
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Inclassproblem
Whatisthe1000to500millibar thicknessofdryairthathasanaveragetemperatureof5° C?
![Page 14: Atmospheric Pressure - home.chpc.utah.eduhallar/Thermo/Lectures/... · 2017. 1. 22. · 850 hPa 700 hPa T v = 240 K T v Δz = 1,364 gpm = 280 K Δz = 1,591 gpm Fig. 4.3 Hypsometric](https://reader035.vdocuments.us/reader035/viewer/2022081620/6117c532987d99013c5589f5/html5/thumbnails/14.jpg)
Inclassproblem
Whatisthe1000to500millibar thicknessofdryairthathasanaveragetemperatureof5° C?
HINT:SincetheairisdryTv =T=278K
![Page 15: Atmospheric Pressure - home.chpc.utah.eduhallar/Thermo/Lectures/... · 2017. 1. 22. · 850 hPa 700 hPa T v = 240 K T v Δz = 1,364 gpm = 280 K Δz = 1,591 gpm Fig. 4.3 Hypsometric](https://reader035.vdocuments.us/reader035/viewer/2022081620/6117c532987d99013c5589f5/html5/thumbnails/15.jpg)
Inclassproblem
Whatisthe1000to500millibar thicknessofdryairthathasanaveragetemperatureof5° C?
HINT:SincetheairisdryTv =T=278K
Z=(278K)(1/9.8ms^-2)(287JK^-1kg^-1)LN(1000/500)=5,643geopotentialmeters
Unitscheck:Kkgms^-2mK^-1kg^-1m^-1s^2=m