Lecture5

• AtmosphericPressure• HydrostaticBalance• GeopotentialHeight• HypsometricEquation

ATMOS5130

IntroductiontoHydrostaticBalance

• Pressureatanypointintheatmosphereequalstheweightperunitareaabovethatpoint.• Pressure=Forceperunitarea• Weightofanobject=Force=Fg =mg• g=accelerationduetogravity(9.81ms-2)atsealevel

AckermanandKnox,2012

HydrostaticEquation

Lecture5

• AtmosphericPressure• HydrostaticBalance• GeopotentialHeight• HypsometricEquation

ATMOS5130

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

Fig.4.1

HydrostaticApproximationfortheAtmosphere

𝜕𝑝𝜕𝑧 = −𝜌 𝑎 + 𝑔

Inallbutthemostextrememeteorologicalconditionstheverticalaccelerationofofairparcelsismuchsmallerthangravity.

C%C9≈ −𝜌𝑔

ApplicationtotheAtmosphere𝜕𝑝𝜕𝑧 = −𝜌𝑔 = −

𝑝𝑔𝑅'𝑇H

1𝑝𝜕𝑝𝜕𝑧 = −

𝑔𝑅'𝑇H

𝜕ln𝑝𝜕𝑧 = −

𝑔𝑅'𝑇H

Therateofchangeofthelogarithmofpressurewithheightisinverselyproportionaltotheabsolutetemperatureanddoesnotdependonpressure.

Pressuregenerallyfallsoffexponentiallywithheight!

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".

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

KeyPoint

• Whenmeteorologistreferenceheight– itshouldbeunderstoodthattheyreallymeangeopotentialheight.

• ThisallowsfortheHypsometricEquation

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.

Inclassproblem

Whatisthe1000to500millibar thicknessofdryairthathasanaveragetemperatureof5° C?

Inclassproblem

Whatisthe1000to500millibar thicknessofdryairthathasanaveragetemperatureof5° C?

HINT:SincetheairisdryTv =T=278K

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