Lecture 6 Atmospheric Pressure
18
Lecture 6 • Atmospheric Pressure • Pressure Profiles for Idealized Atmosphere ATMOS 5130
Transcript of Lecture 6 Atmospheric Pressure
Lecture6
• AtmosphericPressure• PressureProfilesforIdealizedAtmosphere
ATMOS5130
Goal
Allowforcalculationofpressureatanyheight
Δ𝑧 = 𝑅&𝑇(𝑔 ln
𝑝-𝑝.
HypsometricEquation
Recallfromlastlecture
• Understandhowtemperature,pressureandaltitudearerelatedintheatmosphere.
PressureProfilesforIdealizedAtmosphere• ConstantDensityAtmosphere
• Unrealistic• Betterrepresentationoftheocean• Atmospherehadthesamemassasnow,buthadaconstantdensityitwouldbeonly8.3kmdeep
/ 𝑑𝑝1(3)
15= −𝜌𝑔/ 𝑑𝑧
8
9
:1:8≈ −𝜌𝑔 HydrostaticEquation
𝑝 𝑧 = 𝑝9 − 𝜌𝑔𝑧
Pressuredecreaseslinearlywithheight
PressureProfilesforIdealizedAtmosphere• ConstantDensityAtmosphere• Pressureisdecreasingwithheight• Densityisconstant• SO,Temperaturemustdeceasewithheight
𝑝 𝑧 = 𝑝9 − 𝜌𝑔𝑧
𝑝 = 𝜌𝑅&𝑇 IDEALGASLAW
𝜕𝑝𝜕𝑧 = 𝜌𝑅&
𝜕𝑇𝜕𝑧
Differentiatewithrespecttoelevation
:=:8= − >
?@=-34.1C/km
SubstituteinHydrostaticEquation 𝜕𝑝𝜕𝑧 = −𝜌𝑔
ConstantDensityAtmosphereAutoconvective LapseRate
𝑝 𝑧 = 𝑝9 − 𝜌𝑔𝑧
:=:8= − >
?@=-34.1C/km=ΓBCDE𝑊ℎ𝑒𝑛: Γ > ΓBCDE
Densityabovegreaterthanbelow!DensityInversion =>CreatesamirageAboveaveryhotroad,alayerofwarmairwhosedensityislowerthanairabove.
Thedenserairslowslightveryslightlymorethanthelessdenselayer.So,oneseesskywheretheroadshouldbe,weofteninterpretthisasareflectioncausedbywaterontheroad
PressureProfilesforIdealizedAtmosphereConstantDensity
PressureProfilesforIdealizedAtmosphere• IsothermalAtmosphere
&1&8= −𝜌𝑔 HydrostaticEquation
Againsubstituteintheidealgaslaw,𝑑𝑝𝑑𝑧 = −𝜌𝑔 = −
𝑝𝑔𝑅&𝑇
Solve1𝑝 𝑑𝑝 = −
𝑔𝑅&𝑇
𝑑𝑧
/1𝑝
1
15𝑑𝑝 = −
𝑔𝑅&𝑇
/ 𝑑𝑧8
9
ln𝑝𝑝9
= −𝑔𝑧𝑅&𝑇
𝐻 = 𝑅&𝑇9𝑔
Where:
𝑝 𝑧 = 𝑝9exp −𝑧𝐻
ScaleHeight
𝑝 𝑧 = 𝑝9 − 𝜌𝑔𝑧
ScaleHeight(H):solveforz,whenp(z)=0,
𝐻 =𝑝9𝜌𝑔
substitutetheIdealGasLaw
𝐻 = 𝑅&𝑇9𝑔
Inotherwords:scaleheightistheincreaseinaltitudeforwhichtheatmosphericpressuredecreasesbyafactorofe-1 orabout37%ofitsoriginalvalue.
ApproximatescaleheightsforselectedSolarSystembodies
Venus:15.9kmEarth:8.3kmMars:11.1kmJupiter:27kmSaturn:59.5kmTitan:40km
Uranus:27.7kmNeptune:19.1–20.3km
Pluto:~60km
ConstantDensityAtmosphere
• ConstantLapseRateatmosphere
• NormallyΓ >0,temperaturedecreaseswithaltitude• WhenΓ >0,temperatureincreaseswithaltitude
• INVERSION
PressureProfilesforIdealizedAtmosphere
𝑇 =𝑇9 − Γ𝑧
• ConstantLapseRateatmosphere
PressureProfilesforIdealizedAtmosphere
𝑇 = 𝑇9 − Γ𝑧
𝑑𝑝𝑑𝑧 = −𝜌𝑔 = −
𝑝𝑔𝑅&(𝑇9−Γ𝑧)
TakethehydrostaticequationandcombinewithIdealGasLaw
1𝑝 𝑑𝑝 = −
𝑔𝑅&
𝑑𝑧𝑇9 − Γ𝑧
/1𝑝
1
15𝑑𝑝 = −
𝑔𝑅&
/𝑑𝑧
𝑇9 − Γ𝑧
8
9
ln𝑝𝑝9
=𝑔𝑅&Γ
ln −𝑇9 − Γ𝑧𝑇9
𝑝(𝑧) = 𝑝9𝑇9 − Γ𝑧𝑇9
>?@Q
• ConstantLapseRateatmosphere
PressureProfilesforIdealizedAtmosphere
𝑇 = 𝑇9 − Γ𝑧
𝑑𝑝𝑑𝑧 = −𝜌𝑔 = −
𝑝𝑔𝑅&(𝑇9−Γ𝑧)
TakethehydrostaticequationandcombinewithIdealGasLaw
1𝑝 𝑑𝑝 = −
𝑔𝑅&
𝑑𝑧𝑇9 − Γ𝑧
/1𝑝
1
15𝑑𝑝 = −
𝑔𝑅&
/𝑑𝑧
𝑇9 − Γ𝑧
8
9
ln𝑝𝑝9
=𝑔𝑅&Γ
ln −𝑇9 − Γ𝑧𝑇9
𝑝(𝑧) = 𝑝9𝑇9 − Γ𝑧𝑇9
>?@Q
𝑇(𝑧)
PressureProfilesforIdealizedAtmosphere• ConstantLapserateatmosphere
𝑝(𝑧) = 𝑝9𝑇(𝑧)𝑇9
>?@Q
ValidforΓ < 0Exponentoftheratioisnegative,sopressurestilldecreaseswithheight
>?@Q
isratioofautoconvective toactuallapserate
So,whathappensif𝜞 = 𝜞𝒂𝒖𝒕𝒐 ?
PressureProfilesforIdealizedAtmosphere• ConstantLapserateatmosphere
𝑝(𝑧) = 𝑝9𝑇(𝑧)𝑇9
>?@Q
ValidforΓ < 0Exponentoftheratioisnegative,sopressurestilldecreaseswithheight
>?@Q
isratioofautoconvective toactuallapserate
So,whathappensif𝜞 = 𝜞𝒂𝒖𝒕𝒐 ?
𝑝(𝑧) = 𝑝9=(8)=5
- Pressureislinearfunctionofz
0 200 400 600 800 1000 1200
0
5
10
15
20
Γ=0
Γ=6.5 K/km
Γ=Γauto
p [hPa]
z [km]
Pressure vs. Altitude
0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4
0
5
10
15
20
Γ=0
Γ=6.5 K/km
Γ=Γauto
ρ (kg/m3)
z [km]
Density vs. Altitude
0 100 200 300
0
5
10
15
20
Γ=0
Γ=6.5 K
/km
Γ=Γauto
T [K]
z [km]
Temperature vs. Altitude
Fig.4.4
Γ =0;IsothermalΓ =6.5K/km;ConstantLapseRateΓ =Γauto;Autoconvective;ConstantDensity=g/Rd
0 200 400 600 800 1000 1200
0
5
10
15
20
Γ=0
Γ=6.5 K/km
Γ=Γauto
p [hPa]
z [km]
Pressure vs. Altitude
0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4
0
5
10
15
20
Γ=0
Γ=6.5 K/km
Γ=Γauto
ρ (kg/m3)
z [km]
Density vs. Altitude
0 100 200 300
0
5
10
15
20
Γ=0
Γ=6.5 K
/km
Γ=Γauto
T [K]
z [km]
Temperature vs. Altitude
Fig.4.4
Γ =0;IsothermalΓ =6.5K/km;ConstantLapseRateΓ =Γauto;Autoconvective ConstantDensity
𝑝 𝑧 = 𝑝9exp −𝑧𝐻
𝑝(𝑧) = 𝑝9𝑇(𝑧)𝑇9
>?@Q
𝑝 𝑧 = 𝑝9 − 𝜌𝑔𝑧Autoconvective
ConstantLapseRate
Isothermal
0 200 400 600 800 1000 1200
0
5
10
15
20
Γ=0
Γ=6.5 K/km
Γ=Γauto
p [hPa]
z [km]
Pressure vs. Altitude
0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4
0
5
10
15
20
Γ=0
Γ=6.5 K/km
Γ=Γauto
ρ (kg/m3)
z [km]
Density vs. Altitude
0 100 200 300
0
5
10
15
20
Γ=0
Γ=6.5 K
/km
Γ=Γauto
T [K]
z [km]
Temperature vs. Altitude
Fig.4.4
Γ =0;IsothermalΓ =6.5K/km;ConstantLapseRateΓ =Γauto;Autoconvective ConstantDensity
PiecewiseLinearTemperatureProfile
• UltimateApproximationismodelbasedonseriesofstackedlayers• Eachlayerexhibitsadifferentconstantlapserate• Generallyspeaking,ifalayeristhinenough,thenalltheabovemethodsgiveyouaboutthesameresult.
𝑝YZ- = 𝑝Y𝑇YZ-𝑇Y
>?@Q[
-40 -30 -20 -10 0 10 20 30 40
Temperature [!
C]-50-60-80-90 -70
10
20
30
50
70
100
150
200
250
300
400
500
700
850925
1000
Three Layers
Pre
ssu
re [h
Pa
]
-40 -30 -20 -10 0 10 20 30 40
Temperature [!
C]-50-60-80-90 -70
10
20
30
50
70
100
150
200
250
300
400
500
700
850925
1000
24 Layers
Pre
ssu
re [h
Pa
]
Piecewise Linear Profiles
Fig.4.5
